**Meet the editor**

Dr. Orhan Yalçın was born in Niğde / Cappadocia, Turkey. He received the Ph. D. degree in Physics from Gebze Institute of Technology, Kocaeli, Turkey. He studied inorganic spin-Peierls systems. Dr. Yalçın worked for many years in Gaziosmanpaşa University, Gebze Institute of Technology, Erciyes University and Bozok University. Since January 2010 he is an academic partner

of the Niğde University. His current research interests include ferromagnetic resonance, microwave technology, magnetic nanostructures (nanowires, nanoparticles and magnetic nano O-ring), and spin-Peierls systems. Dr. Yalçın has published many papers in journals listed in the Science Citation Index, and several invited chapters in books released by international publishers.

Contents

**Preface VII** 

Chi-Kuen Lo

Chapter 1 **Ferromagnetic Resonance 1**  Orhan Yalçın

Chapter 2 **Instrumentation for Ferromagnetic Resonance Spectrometer 47** 

Chapter 3 **Detection of Magnetic Transitions** 

H. Montiel and G. Alvarez

R. Singh and S. Saipriya

Chapter 7 **Microwave Absorption** 

and Raúl Valenzuela

**by Means of Ferromagnetic Resonance and Microwave Absorption Techniques 63** 

Chapter 4 **FMR Measurements of Magnetic Nanostructures 93**  Manish Sharma, Sachin Pathak and Monika Sharma

Chapter 5 **FMR Studies of [SnO2/Cu-Zn Ferrite] Multilayers 111** 

Chapter 6 **Dynamic and Rotatable Exchange Anisotropy in Fe/KNiF3/FeF2 Trilayers 147**  Stefania Widuch, Robert L. Stamps, Danuta Skrzypek and Zbigniew Celinski

> **in Nanostructured Spinel Ferrites 169**  Gabriela Vázquez-Victorio, Ulises Acevedo-Salas

Chapter 8 **Unusual Temperature Dependence of Zero-Field** 

**on Al-Substituted ε-Fe2O3 195** 

Shu-Fang Fu and Xuan-Zhang Wang

**Photonic Crystals 211** 

**Ferromagnetic Resonance in Millimeter Wave Region** 

Marie Yoshikiyo, Asuka Namai and Shin-ichi Ohkoshi

Chapter 9 **Optical Properties of Antiferromagnetic/Ion-Crystalic** 

## Contents

### **Preface XI**


Preface

The book "Ferromagnetic Resonance - Theory and Applications" highlights recent advances at the interface between the science and technology of nanostructures (bilayer-multilayers, nanowires, spinel type nanoparticles, photonic crystal, etc.). The electromagnetic resonance techniques have become a central field of modern scientific and technological activities. The modern technological applications of ferromagnetic resonance are in spintronics, electronics, space navigation, remote-control equipment, radio engineering, electronic computers, maritime, electrical engineering, instrument-

The resonance arises when the energy levels of a quantized system of electronic or nuclear moments are Zeeman split by a uniform magnetic field and the system absorbs energy from an oscillating magnetic field at sharply defined frequencies corresponding to the transitions between the energy levels. Classically, the resonance event occurs when a transverse *ac* field is applied at the Larmor frequency. The resonance behavior is usually called electromagnetic resonance and nuclear magnetic resonance. Main types of resonance phenomenon can be listed as electron paramagnetic/spin resonance, nuclear magnetic resonance, nuclear quadrupole

This book begins with a brief overview of the historical development of the ferromagnetic resonance. In the scope of this book, types of resonance phenomenon and their applications were systematically examined in terms of the technical evolution of the magnetic smart nanostructures. The ferromagnetic resonance signal occurs at high field values while electron paramagnetic resonance signal occurs at low magnetic field values as the applied field decrease from high to low field region. Obviously, one of the most employed techniques to characterize the magnetic materials is the magnetic resonance, also well-known as the ferromagnetic resonance at temperatures below Curie temperature and the electron paramagnetic resonance at

This is a fundamental book for students, researchers and materials scientists. The chapters in this book are clear, to the point, and appealing for reader. Over the last years the interest in smart magnetic nanostructures and their applications in various electronic devices, effective optoelectronic devices, biosensors, photo detectors, solar

making and geophysical methods of prospecting.

resonance, and ferromagnetic resonance.

temperatures above Curie temperature.

## Preface

The book "Ferromagnetic Resonance - Theory and Applications" highlights recent advances at the interface between the science and technology of nanostructures (bilayer-multilayers, nanowires, spinel type nanoparticles, photonic crystal, etc.). The electromagnetic resonance techniques have become a central field of modern scientific and technological activities. The modern technological applications of ferromagnetic resonance are in spintronics, electronics, space navigation, remote-control equipment, radio engineering, electronic computers, maritime, electrical engineering, instrumentmaking and geophysical methods of prospecting.

The resonance arises when the energy levels of a quantized system of electronic or nuclear moments are Zeeman split by a uniform magnetic field and the system absorbs energy from an oscillating magnetic field at sharply defined frequencies corresponding to the transitions between the energy levels. Classically, the resonance event occurs when a transverse *ac* field is applied at the Larmor frequency. The resonance behavior is usually called electromagnetic resonance and nuclear magnetic resonance. Main types of resonance phenomenon can be listed as electron paramagnetic/spin resonance, nuclear magnetic resonance, nuclear quadrupole resonance, and ferromagnetic resonance.

This book begins with a brief overview of the historical development of the ferromagnetic resonance. In the scope of this book, types of resonance phenomenon and their applications were systematically examined in terms of the technical evolution of the magnetic smart nanostructures. The ferromagnetic resonance signal occurs at high field values while electron paramagnetic resonance signal occurs at low magnetic field values as the applied field decrease from high to low field region. Obviously, one of the most employed techniques to characterize the magnetic materials is the magnetic resonance, also well-known as the ferromagnetic resonance at temperatures below Curie temperature and the electron paramagnetic resonance at temperatures above Curie temperature.

This is a fundamental book for students, researchers and materials scientists. The chapters in this book are clear, to the point, and appealing for reader. Over the last years the interest in smart magnetic nanostructures and their applications in various electronic devices, effective optoelectronic devices, biosensors, photo detectors, solar cells, nanodevices and plasmonic structures have been increasing tremendously. This is caused by the unique properties of smart magnetic nanostructures and the outstanding performance of nanoscale devices. Especially, microwave absorption properties of the nano scale materials are important of course by themselves in telecommunications device applications. The ferromagnetic resonance technique is advantageous because it does not cause damage to nano materials. On the other hand, the ferromagnetic resonance is a powerful technique for studying the spin structure, magnetic properties in bulk samples, thin films, anisotropies, the damping constant, *g*factor, spin relaxation, electromagnetic wave absorbers, multilayers, nanoparticles, exchange-spring magnets, medical-agricultural and military applications.

There are nine chapters in the book "Ferromagnetic Resonance - Theory and Applications" and each chapter is about a different point of ferromagnetic resonance. Yalçn presented a review of the historical development of the ferromagnetic resonance theory and their applications for some nanostructures in Chapter 1. The instrumentation for ferromagnetic resonance spectrometer was shown in Chapter 2 by Lo. The detection of magnetic transitions by means of ferromagnetic resonance and microwave absorption techniques are reviewed by Montiel and Alvarez in Chapter 3. In Chapter 4, the ferromagnetic and multilayer nanowires are presented by Manish. The ferromagnetic resonance studies of (SnO2/Cu-Zn ferrite) multilayers are indicated by Singh and Saipriya in Chapter 5. The dynamic and rotatable exchange anisotropy in Fe/KNiF3/FeF2 trilayers is displayed in Chapter 6 by Widuch, Stamps, Skrzypek and Celinski. The microwave absorption in nanostructured spinel ferrites is presented by Vázquez-Victorio, Acevedo-Salas and Valenzuela in Chapter 7. In Chapter 8, Yoshikiyo, Namai and Ohkoshi studied unusual temperature dependence of zero-field ferromagnetic resonance in millimeter wave region on Al-substituted ε-Fe2O3. The optical properties of antiferromagnetic/ion-crystalic photonic crystals are discussed by Fuand Wang in Chapter 9.

This book is of interest to both fundamental magnetic resonance research, such as the one conducted in nanoscience, nanobiotechnology, nanotechnology and medicine etc., and also to practicing scientists, students, researchers, in applied material sciences and engineers for spintronics.

I would like to thank all the authors for their efforts in preparing their chapters. The scientists and technicians have a duty to share their knowledge about their scientific studies. I am very grateful to Ms. Marina Jozipovic for her assistance, guidance and support. I would like to invite magnetic resonance, nanobiotechnology and nanotechnology scientists to read and share the knowledge and contents of this book.

> **Dr. Orhan Yalçn**  Faculty of Arts and Sciences, Physics Department, Nigde University Nigde, Turkey

### **Chapter 1**

## **Ferromagnetic Resonance**

Orhan Yalçın

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/56134

### **1. Introduction**

Ferromagnetism is used to characterize magnetic behavior of a material, such as the strong attraction to a permanent magnet. The origin of this strong magnetism is the presence of a spontaneous magnetization which is produced by a parallel alignment of spins. Instead of a parallel alignment of all the spins, there can be an anti-parallel alignment of unequal spins. This results in a spontaneous magnetization which is called ferrimagnetism.

The resonance arises when the energy levels of a quantized system of electronic or nuclear moments are Zeeman split by a uniform magnetic field and the system absorbs energy from an oscillating magnetic field at sharply defined frequencies corresponding to the transitions between the levels. Classically, the resonance event occurs when a transverse *ac* field is applied at the Larmor frequency.

The resonance behaviour usually called magnetic resonance (MR) and nuclear magnetic resonance (NMR). Main types of resonance phenomenon can be listed as nuclear magnetic resonance (NMR), nuclear quadrupole resonance (NQR), electron paramagnetic/spin resonance (EPR, ESR), spin wave resonance (SWR), ferromagnetic resonance (FMR), antiferromagnetic resonance (AFMR) and conductor electron spin resonance (CESR). The resonant may be an isolated ionic spin as in electron paramagnetic resonance (EPR) or a nuclear magnetic resonance (NMR). Also, resonance effects are associated with the spin waves and the domain walls. The resonance methods are important for investigating the structure and magnetic properties of solids and other materials. These methods are used for imaging and other applications.

The following information can be accessed with the help of such resonance experiments. (i) Electrical structure of point defects by looking at the absorption in a thin structure. (ii) The line width with the movement of spin or surroundings isn't changed. (iii) The distribution of the magnetic field in solid by looking at the of the resonance line position (chemical shift and etc.). (iv) Collective spin excitations.

© 2013 Yalçın, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The atoms of ferromagnetic coupling originate from the spins of *d*-electrons. The size of *μ* permanent atomic dipoles create spontaneously magnetized. According to the shape of dipoles materials can be ferromagnetic, antiferromagnetic, diamagnetic, paramagnetic and etc.

Ferromagnetic Resonance 3

properties, it also allows one to study the fundamental excitations and technological applications of a magnetic system (Schmool, 1998; Voges, 1998; Zianni, 1998; Grünberg, 2000, 2001; Vlasko-Vlasov, 2001; Zhai, 2003; Akta*ş*, 2004; Birkhäuser Verlag, 2007; Seib, 2009). The various thickness, disk array, half-metallic ferromagnetic electrodes, magnon scattering and other of some properties of samples have been studied using the FMR tehniques (Mazur, 1982; da Silva, 1993; Chikazumi, 1997; Song, 2003; Mills, 2003; Rameev, 2003(a), 2003(b), 2004(a), 2004(b); An, 2004; Ramprasad, 2004; Xu, 2004; Wojtowicz, 2005; Zakeri, 2007; Tsai, 2009; Chen, 2009). The magnetic properties of single-crystalline (Kambe, 2005; Brustolon, 2009), polycrystalline (Singh, 2006; Fan, 2010), alloy films (Sihues, 2007), temperature dependence and similar qualities have been studied electromagnetic spectroscopy techniques (Özdemir, 1998; Birlikseven, 1999(a), 1999(b); Fermin, 1999; Rameev, 2000; Aktaş, 2001; Budak, 2003; Khaibullin, 2004). The magnetic resonance techniques (EPR, FMR) have been applied to the iron oxides, permalloy nanostructure (Kuanr, 2005), clustered, thermocouple connected to the ferromagnet, thin permalloy layer and et al. (Guimarães, 1998; Spoddig, 2005; Can, 2012; Rousseau, 2012; Valenzuela, 2012; Bakker, 2012; Maciá, 2012; Dreher, 2012; Kind, 2012; Li, 2012; Estévez, 2012; Sun, 2012(a), 2012(b), 2012(c); Richard, 2012). Magneto-optic (Paz, 2012), dipolar energy contributions (Bose, 2012), nanocrystalline (Maklakov, 2012; Raita, 2012), La0.7Sr0.3MnO3 films (Golosovsky, 2012), La0.67Ba0.33Mn1-**y**A**y**O3, A - Fe, Cr (Osthöver, 1998), voltage-controlled magnetic anisotropy (VCMA) and spin transfer torque (Zhu, 2012) and the typical properties of the inertial resonance are investigated (Olive, 2012). The exchange bias (Backes, 2012), Q cavities for magnetic material (Beguhn, 2012), MgO/CoFeB/Ta structure (Chen, 2012), the interfacial origin of the giant magnetoresistive effect (GMR) phenomenon (Prieto, 2012), selfdemagnetization field (Hinata, 2012), Fe3O4/InAs(100) hybrid spintronic structures (Huang, 2012), granular films (Kakazei, 1999, 2001; Sarmiento, 2007; Krone, 2011; Kobayashi, 2012), nano-sized powdered barium (BaFe12O19) and strontium (Sr Fe12O19) hexaferrites (Korolev, 2012), Ni0.7Mn0.3-x CoxFe2O4 ferrites (NiMnCo: x = 0.00, 0.04, 0.06, and 0.10) (Lee, 2012), thin films (Demokritov, 1996,1997; Nakai, 2002; Lindner, 2004; Aswal, 2005; Jalali-Roudsar, 2005; Cochran, 2006; Mizukami, 2007; Seemann, 2010), Ni2MnGa films (Huang, 2004), magnetic/electronic order of films (Shames, 2012), Fe1-xGd(Tb)x films (Sun, 2012), in ε-Al0.06Fe1.94O3 (Yoshikiyo, 2012). 10 nm thick Fe/GaAs(110) film (Römer, 2012), triangular shaped permalloy rings (Ding, 2012) and Co2-Y hexagonal ferrite single rod (Bai, 2012) structures and properties have been studied by FMR tecniques (Spaldin, 2010). Biological applications (Berliner, 1981; Wallis, 2005; Gatteschi, 2006; Kopp, 2006; Fischer, 2008; Mastrogiacomo, 2010), giant magneto-impedance (Valenzuela, 2007; Park, 2007), dynamics of feromagnets (Vilasi, 2001; Rusek, 2004; Limmer, 2006; Sellmyer, 2006; Spinu, 2006; Azzerboni, 2006; Krivoruchko, 2012), magneto-optic kerr effect (Suzuki, 1997; Neudecker, 2006), Heusler alloy (HA) films (Kudryavtsev, 2007), ferrites (Kohmoto, 2007), spin polarized electrons (Rahman, 2008) and quantum mechanics (Weil, 2007) have been studied by FMR technique in generally (Hillebrands, 2002, 2003, 2006). In additional, electric and magnetic properties of pure, Cu2+ ions doped hydrogels have been studied by ESR

techniques (Coşkun, 2012).

Ferromagnetic resonance (FMR) technique was initially applied to ferromagnetic materials, all magnetic materials and unpaired electron systems. Basically, it is analogous to the electron paramagnetic resonance (EPR). The EPR technique gives better results at unpaired electron systems. The FMR technique depends on the geometry of the sample at hand. The demagnetization field is observed where the sample geometry is active. The resonance area of the sample depends on the properties of material. The FMR technique is advantageous because it does not cause damage to materials. Also, it allows a three dimensional analysis of samples. The FMR occurs at high field values while EPR occurs at low magnetic field values. Also, line-width of ferromagnetic materials is large according to paramagnetic materials. Exchange interaction energy between unpaired electron spins that contribute to the ferromagnetism causes the line narrowing. So, ferromagnetic resonance lines appear sharper than expected.

The FMR studies have been increased since the EPR was discovered in 1945 (Zavosky, 1945; Kittel, 1946, 1947, 1949, 1953, 1958; Kip, 1949; Bloembergen, 1950, 1954; Crittenden, 1953; Van Vleck, 1950; Herring, 1950; Anderson, 1953; Damon, 1953; Young, 1953; Ament, 1955; Ruderman, 1954; Reich, 1955; Kasuya, 1956; White, 1956; Macdonald, 1956; Mercereau, 1956; Walker, 1957; Yosida, 1957; Tannenwald, 1957; Jarrett, 1958; Rado, 1958; Brown, 1962; Frait, 1965; Sparks, 1969). The beginnings of theoretical and experimental studies of spectroscopic investigations of basic sciences are used such as physics, chemistry, especially nanosciences and nanostructures (Rodbell, 1964; Kooi, 1964; Bhagat, 1967, 1974; Sparks, 1970(a), 1970(b), 1970(c), 1970(d); Rachford, 1981; Dillon, 1981; Schultz, 1983; Artman, 1957, 1979; Ramesh, 1988(a), 1988(b); Fraitova, 1983(a), 1983(b), 1984; Teale, 1986; Speriosu, 1987; Vounyuk, 1991; Roy, 1992; Puszkarski, 1992; Weiss, 1955). The FMR technique can provide information on the magnetization, magnetic anisotropy, dynamic exchange/dipolar energies and relaxation times, as well as the damping in the magnetization dynamics (Wigen, 1962, 1984, 1998; De Wames, 1970; Wolfram, 1971; Yu, 1975; Frait, 1985, 1998; Rook, 1991; Bland, 1994; Patton, 1995, 1996; Skomski, 2008; Coey, 2009). This spectroscopic method/FMR have been used to magnetic properties (Celinski, 1991; Farle, 1998, 2000; Fermin, 1999; Buschow, 2004; Heinrich, 2005(a), 2005(b)), films (Özdemir, 1996, 1997), monolayers (Zakeri, 2006), ultrathin and multilayers films (Layadi, 1990(a), 1990(b), 2002, 2004; Wigen, 1993; Zhang, 1994(a), 1994(b); Farle, 2000; Platow, 1998; Anisimov, 1999; Yldz, 2004; Heinrich, 2005(a); Lacheisserie, 2005; de Cos, 2006; Liua, 2012; Schäfer, 2012), the angular, the frequency (Celinski, 1997; Farle, 1998), the temperature dependence (Platow, 1998), interlayer exchange coupling (Frait, 1965, 1998; Parkin, 1990, 1991(a), 1991(b), 1994; Schreiber, 1996; Rook, 1991; Wigen, 1993; Layadi, 1990(a); Heinrich, 2005; Paul, 2005), Brillouin light scattering (BLS) (Grünberg, 1982; Cochran, 1995; Hillebrands, 2000) and sample inhomogeneities (Artman, 1957, 1979; Damon, 1963; McMichael, 1990; Arias, 1999; Wigen, 1998; Chappert, 1986; Gnatzig, 1987; Fermin, 1999) of samples. Besides using FMR to characterize magnetic properties, it also allows one to study the fundamental excitations and technological applications of a magnetic system (Schmool, 1998; Voges, 1998; Zianni, 1998; Grünberg, 2000, 2001; Vlasko-Vlasov, 2001; Zhai, 2003; Akta*ş*, 2004; Birkhäuser Verlag, 2007; Seib, 2009). The various thickness, disk array, half-metallic ferromagnetic electrodes, magnon scattering and other of some properties of samples have been studied using the FMR tehniques (Mazur, 1982; da Silva, 1993; Chikazumi, 1997; Song, 2003; Mills, 2003; Rameev, 2003(a), 2003(b), 2004(a), 2004(b); An, 2004; Ramprasad, 2004; Xu, 2004; Wojtowicz, 2005; Zakeri, 2007; Tsai, 2009; Chen, 2009). The magnetic properties of single-crystalline (Kambe, 2005; Brustolon, 2009), polycrystalline (Singh, 2006; Fan, 2010), alloy films (Sihues, 2007), temperature dependence and similar qualities have been studied electromagnetic spectroscopy techniques (Özdemir, 1998; Birlikseven, 1999(a), 1999(b); Fermin, 1999; Rameev, 2000; Aktaş, 2001; Budak, 2003; Khaibullin, 2004). The magnetic resonance techniques (EPR, FMR) have been applied to the iron oxides, permalloy nanostructure (Kuanr, 2005), clustered, thermocouple connected to the ferromagnet, thin permalloy layer and et al. (Guimarães, 1998; Spoddig, 2005; Can, 2012; Rousseau, 2012; Valenzuela, 2012; Bakker, 2012; Maciá, 2012; Dreher, 2012; Kind, 2012; Li, 2012; Estévez, 2012; Sun, 2012(a), 2012(b), 2012(c); Richard, 2012). Magneto-optic (Paz, 2012), dipolar energy contributions (Bose, 2012), nanocrystalline (Maklakov, 2012; Raita, 2012), La0.7Sr0.3MnO3 films (Golosovsky, 2012), La0.67Ba0.33Mn1-**y**A**y**O3, A - Fe, Cr (Osthöver, 1998), voltage-controlled magnetic anisotropy (VCMA) and spin transfer torque (Zhu, 2012) and the typical properties of the inertial resonance are investigated (Olive, 2012). The exchange bias (Backes, 2012), Q cavities for magnetic material (Beguhn, 2012), MgO/CoFeB/Ta structure (Chen, 2012), the interfacial origin of the giant magnetoresistive effect (GMR) phenomenon (Prieto, 2012), selfdemagnetization field (Hinata, 2012), Fe3O4/InAs(100) hybrid spintronic structures (Huang, 2012), granular films (Kakazei, 1999, 2001; Sarmiento, 2007; Krone, 2011; Kobayashi, 2012), nano-sized powdered barium (BaFe12O19) and strontium (Sr Fe12O19) hexaferrites (Korolev, 2012), Ni0.7Mn0.3-x CoxFe2O4 ferrites (NiMnCo: x = 0.00, 0.04, 0.06, and 0.10) (Lee, 2012), thin films (Demokritov, 1996,1997; Nakai, 2002; Lindner, 2004; Aswal, 2005; Jalali-Roudsar, 2005; Cochran, 2006; Mizukami, 2007; Seemann, 2010), Ni2MnGa films (Huang, 2004), magnetic/electronic order of films (Shames, 2012), Fe1-xGd(Tb)x films (Sun, 2012), in ε-Al0.06Fe1.94O3 (Yoshikiyo, 2012). 10 nm thick Fe/GaAs(110) film (Römer, 2012), triangular shaped permalloy rings (Ding, 2012) and Co2-Y hexagonal ferrite single rod (Bai, 2012) structures and properties have been studied by FMR tecniques (Spaldin, 2010). Biological applications (Berliner, 1981; Wallis, 2005; Gatteschi, 2006; Kopp, 2006; Fischer, 2008; Mastrogiacomo, 2010), giant magneto-impedance (Valenzuela, 2007; Park, 2007), dynamics of feromagnets (Vilasi, 2001; Rusek, 2004; Limmer, 2006; Sellmyer, 2006; Spinu, 2006; Azzerboni, 2006; Krivoruchko, 2012), magneto-optic kerr effect (Suzuki, 1997; Neudecker, 2006), Heusler alloy (HA) films (Kudryavtsev, 2007), ferrites (Kohmoto, 2007), spin polarized electrons (Rahman, 2008) and quantum mechanics (Weil, 2007) have been studied by FMR technique in generally (Hillebrands, 2002, 2003, 2006). In additional, electric and magnetic properties of pure, Cu2+ ions doped hydrogels have been studied by ESR techniques (Coşkun, 2012).

2 Ferromagnetic Resonance – Theory and Applications

etc.

sharper than expected.

The atoms of ferromagnetic coupling originate from the spins of *d*-electrons. The size of *μ* permanent atomic dipoles create spontaneously magnetized. According to the shape of dipoles materials can be ferromagnetic, antiferromagnetic, diamagnetic, paramagnetic and

Ferromagnetic resonance (FMR) technique was initially applied to ferromagnetic materials, all magnetic materials and unpaired electron systems. Basically, it is analogous to the electron paramagnetic resonance (EPR). The EPR technique gives better results at unpaired electron systems. The FMR technique depends on the geometry of the sample at hand. The demagnetization field is observed where the sample geometry is active. The resonance area of the sample depends on the properties of material. The FMR technique is advantageous because it does not cause damage to materials. Also, it allows a three dimensional analysis of samples. The FMR occurs at high field values while EPR occurs at low magnetic field values. Also, line-width of ferromagnetic materials is large according to paramagnetic materials. Exchange interaction energy between unpaired electron spins that contribute to the ferromagnetism causes the line narrowing. So, ferromagnetic resonance lines appear

The FMR studies have been increased since the EPR was discovered in 1945 (Zavosky, 1945; Kittel, 1946, 1947, 1949, 1953, 1958; Kip, 1949; Bloembergen, 1950, 1954; Crittenden, 1953; Van Vleck, 1950; Herring, 1950; Anderson, 1953; Damon, 1953; Young, 1953; Ament, 1955; Ruderman, 1954; Reich, 1955; Kasuya, 1956; White, 1956; Macdonald, 1956; Mercereau, 1956; Walker, 1957; Yosida, 1957; Tannenwald, 1957; Jarrett, 1958; Rado, 1958; Brown, 1962; Frait, 1965; Sparks, 1969). The beginnings of theoretical and experimental studies of spectroscopic investigations of basic sciences are used such as physics, chemistry, especially nanosciences and nanostructures (Rodbell, 1964; Kooi, 1964; Bhagat, 1967, 1974; Sparks, 1970(a), 1970(b), 1970(c), 1970(d); Rachford, 1981; Dillon, 1981; Schultz, 1983; Artman, 1957, 1979; Ramesh, 1988(a), 1988(b); Fraitova, 1983(a), 1983(b), 1984; Teale, 1986; Speriosu, 1987; Vounyuk, 1991; Roy, 1992; Puszkarski, 1992; Weiss, 1955). The FMR technique can provide information on the magnetization, magnetic anisotropy, dynamic exchange/dipolar energies and relaxation times, as well as the damping in the magnetization dynamics (Wigen, 1962, 1984, 1998; De Wames, 1970; Wolfram, 1971; Yu, 1975; Frait, 1985, 1998; Rook, 1991; Bland, 1994; Patton, 1995, 1996; Skomski, 2008; Coey, 2009). This spectroscopic method/FMR have been used to magnetic properties (Celinski, 1991; Farle, 1998, 2000; Fermin, 1999; Buschow, 2004; Heinrich, 2005(a), 2005(b)), films (Özdemir, 1996, 1997), monolayers (Zakeri, 2006), ultrathin and multilayers films (Layadi, 1990(a), 1990(b), 2002, 2004; Wigen, 1993; Zhang, 1994(a), 1994(b); Farle, 2000; Platow, 1998; Anisimov, 1999; Yldz, 2004; Heinrich, 2005(a); Lacheisserie, 2005; de Cos, 2006; Liua, 2012; Schäfer, 2012), the angular, the frequency (Celinski, 1997; Farle, 1998), the temperature dependence (Platow, 1998), interlayer exchange coupling (Frait, 1965, 1998; Parkin, 1990, 1991(a), 1991(b), 1994; Schreiber, 1996; Rook, 1991; Wigen, 1993; Layadi, 1990(a); Heinrich, 2005; Paul, 2005), Brillouin light scattering (BLS) (Grünberg, 1982; Cochran, 1995; Hillebrands, 2000) and sample inhomogeneities (Artman, 1957, 1979; Damon, 1963; McMichael, 1990; Arias, 1999; Wigen, 1998; Chappert, 1986; Gnatzig, 1987; Fermin, 1999) of samples. Besides using FMR to characterize magnetic

The FMR measurements were performed in single crystals of silicon- iron, nickel-iron, nickel and hcp cobalt (Frait, 1965), thin films (Knorr, 1959; Davis, 1965; Hsia, 1981; Krebs, 1982; Maksymowich, 1983, 1985, 1992; Platow, 1998; Durusoy, 2000; Baek, 2002; Kuanr, 2004), CoCr magnetic thin films (Cofield, 1987), NiFe/FeMn thin films (Layadi, 1988), single-crystal Fe/Cr/Fe(100) sandwiches (Krebs, 1989), polycrystalline single films (Hathaway, 1981; Rezende, 1993) and ultrathin multilayers of the system Au/Fe/Au/Pd/Fe (001) prepared on GaAs(001) (Woltersdorf, 2004). The FMR techniques have been succesfully applied peak-topeak linewidth (Yeh, 2009; Sun, 2012), superconducting and ferromagnetic coupled structures (Richard, 2012) and thin Co films of 50 nm thick (Maklakov, 2012). The garnet materials (Ramesh, 1988 (a), 1988 (b)), polar magneto-optic kerr effect and brillouin light scattering measurements (Riedling, 1999), giant-magnetoresistive (GMR) multilayers (Grünberg, 1991; Borchers, 1998) and insulated multilayer film (de Cos, 2006; Lacheisserie, 2005) are the most intensely studied systems.

Ferromagnetic Resonance 5

FMR spectra originated from the iron/soft layers as shown in the exchange spring magnets in Fig.9. Finally, superparamagnetic/single-domain nanoparticles and their resonance are

Magnetic materials are classified as paramagnetic, ferromagnetic, ferrimagnetic, antiferromagnetic and diamagnetic to their electronic order. Magnetic orders are divided in two groups as (i) paramagnetic, ferromagnetic, ferrimagnetic, antiferromagnetic and (ii) diamagnetic. The magnetic moments in diamagnetic materials are opposite to each other as well as the moments associated with the orbiting electrons so that a zero magnetic moment

 is produced on macroscopic scale. In the paramagnetic materials, each atom possesses a small magnetic moment. The orientation of magnetic moment of each atom is random, the net magnetic moment of a large sample (macroscopic scale) of dipole and the magnetization

Nanoscience, nanotechnology and nanomaterials have become a central field of scientific and technical activity. Over the last years the interest in magnetic nanostructures and their applications in various electronic devices, effective opto-electronic devices, bio-sensors, photo-detectors, solar cells, nanodevices and plasmonic structures have been increasing tremendously. This is caused by the unique properties of magnetic nanostructures and the outstanding performance of nanoscale devices. Dimension in the range of one to hundred nanometers, is called the nano regime. In recent years, nanorods, nanoparticles, quantum dots, nanocrystals etc. are in a class of nanostructures (Yalçn, 2012; Kartopu & Yalçn, 2010; Aktaş, 2006) studied extensively. As the dimensions of nano materials decrease down to the nanometer scale, the surface of nanostructures starts to exhibit new and interesting

The magnetization of a matter is derived by electrons moving around the nucleus of an atom. Total magnetic moment occurs when the electrons such as a disc returns around its axis consist of spin angular momentum and returns around the nucleus consist orbital angular momentum. The most of matters which have unpaired electrons have a little magnetic moment. This natural angular momentum consists of the result of charged particle return around its own axis and is called spin of the particle. The origin of spin is not known exactly, although electron is point particle the movement of an electron in an external magnetic field is similar to the movement of the disc. In other words, the origin of the spin is quantum field theoretical considerations and comes from the representations of the Poincare algebra for the elementary particles. The magnetism related to spin angular momentum, orbital angular momentum and spin-orbit interactions angular momentum. The movement of the electron around the nucleus can be considered as a current loop while electron spin is considered very small current loop which generate magnetic field. Here, orbital angular

described in detail.

( ) 

**2. Magnetic order** 

vector are zero when there is no applied field.

properties mainly due to quantum size effects.

**3. Origin of magnetic moment** 

The technique of FMR can be applied to nano-systems (Poole, 2003; Parvatheeswara, 2006; Mills, 2006; Schmool, 2007; Vargas, 2007; Seemann, 2009; Wang, 2011; Patel, 2012; De Biasi, 2013). The FMR measurement on a square array of permalloy nanodots have been comparion a numerical simulation based on the eigenvalues of the linearized Landau-Lifshitz equation (Rivkin, 2007). The dynamic fluctuations of the nanoparticles and their anisotropic behaviour have been recorded with FMR signal (Owens, 2009). Ferromagnetic resonance (FMR) modes for Fe70Co30 magnetic nanodots of 100 nm in diameter in a monodomain state are studied under different in-plane and out-of-plane magnetic fields (Miyake, 2012). The FMR techniques have been accomplished applied to magnetic microwires and nanowire arrays (Adeyeye, 1997; Wegrowe, 1999, 2000; García-Miquel, 2001; Jung, 2002; Arias, 2003; Raposo, 2011; Boulle, 2011; Kraus, 2012; Klein, 2012). In additional, FMR measurements have been performed for nanocomposite samples of varying particles packing fractions with demagnetization field (Song, 2012). The ferromagnetic resonance of magnetic fluids were theoretically investigated on thermal and particles size distribution effects (Marin, 2006). The FMR applied to nanoparticles, superparamagnetic particles and catalyst particles (de Biasi, 2006; Vargas, 2007; Duraia, 2009).

In the scope of this chapter, we firstly give a detailed account of both magnetic order and their origin. The origin of magnetic orders are explained and the equations are obtained using Fig.1 which shows rotating one electron on the table plane. Then, the dynamic equation of motion for magnetization was derived. We mentioned MR and damping terms which have consisted three terms as the Bloch-Bloembergen, the Landau-Lifshitz and the Gilbert form. We indicated electron EPR/ESR and their historical development. The information of spin Hamiltonian and *g*-tensor is given. The dispersion relations of monolayer, trilayers, five-layers and multilayer/*n*-layers have regularly been calculated for ferromagnetic exchange-couple systems (Grünberg, 1992; Nagamine, 2005, Schmool, 1998). The theoretical FMR spectra were obtained by using the dynamic equation of motion for magnetization with the Bloch-Bloembergen type damping term. The exchange-spring (hard/soft) system which is the best of the sample for multilayer structure has been explained by using the FMR technique and equilibrium condition of energy of system. The FMR spectra originated from the iron/soft layers as shown in the exchange spring magnets in Fig.9. Finally, superparamagnetic/single-domain nanoparticles and their resonance are described in detail.

### **2. Magnetic order**

4 Ferromagnetic Resonance – Theory and Applications

2005) are the most intensely studied systems.

catalyst particles (de Biasi, 2006; Vargas, 2007; Duraia, 2009).

The FMR measurements were performed in single crystals of silicon- iron, nickel-iron, nickel and hcp cobalt (Frait, 1965), thin films (Knorr, 1959; Davis, 1965; Hsia, 1981; Krebs, 1982; Maksymowich, 1983, 1985, 1992; Platow, 1998; Durusoy, 2000; Baek, 2002; Kuanr, 2004), CoCr magnetic thin films (Cofield, 1987), NiFe/FeMn thin films (Layadi, 1988), single-crystal Fe/Cr/Fe(100) sandwiches (Krebs, 1989), polycrystalline single films (Hathaway, 1981; Rezende, 1993) and ultrathin multilayers of the system Au/Fe/Au/Pd/Fe (001) prepared on GaAs(001) (Woltersdorf, 2004). The FMR techniques have been succesfully applied peak-topeak linewidth (Yeh, 2009; Sun, 2012), superconducting and ferromagnetic coupled structures (Richard, 2012) and thin Co films of 50 nm thick (Maklakov, 2012). The garnet materials (Ramesh, 1988 (a), 1988 (b)), polar magneto-optic kerr effect and brillouin light scattering measurements (Riedling, 1999), giant-magnetoresistive (GMR) multilayers (Grünberg, 1991; Borchers, 1998) and insulated multilayer film (de Cos, 2006; Lacheisserie,

The technique of FMR can be applied to nano-systems (Poole, 2003; Parvatheeswara, 2006; Mills, 2006; Schmool, 2007; Vargas, 2007; Seemann, 2009; Wang, 2011; Patel, 2012; De Biasi, 2013). The FMR measurement on a square array of permalloy nanodots have been comparion a numerical simulation based on the eigenvalues of the linearized Landau-Lifshitz equation (Rivkin, 2007). The dynamic fluctuations of the nanoparticles and their anisotropic behaviour have been recorded with FMR signal (Owens, 2009). Ferromagnetic resonance (FMR) modes for Fe70Co30 magnetic nanodots of 100 nm in diameter in a monodomain state are studied under different in-plane and out-of-plane magnetic fields (Miyake, 2012). The FMR techniques have been accomplished applied to magnetic microwires and nanowire arrays (Adeyeye, 1997; Wegrowe, 1999, 2000; García-Miquel, 2001; Jung, 2002; Arias, 2003; Raposo, 2011; Boulle, 2011; Kraus, 2012; Klein, 2012). In additional, FMR measurements have been performed for nanocomposite samples of varying particles packing fractions with demagnetization field (Song, 2012). The ferromagnetic resonance of magnetic fluids were theoretically investigated on thermal and particles size distribution effects (Marin, 2006). The FMR applied to nanoparticles, superparamagnetic particles and

In the scope of this chapter, we firstly give a detailed account of both magnetic order and their origin. The origin of magnetic orders are explained and the equations are obtained using Fig.1 which shows rotating one electron on the table plane. Then, the dynamic equation of motion for magnetization was derived. We mentioned MR and damping terms which have consisted three terms as the Bloch-Bloembergen, the Landau-Lifshitz and the Gilbert form. We indicated electron EPR/ESR and their historical development. The information of spin Hamiltonian and *g*-tensor is given. The dispersion relations of monolayer, trilayers, five-layers and multilayer/*n*-layers have regularly been calculated for ferromagnetic exchange-couple systems (Grünberg, 1992; Nagamine, 2005, Schmool, 1998). The theoretical FMR spectra were obtained by using the dynamic equation of motion for magnetization with the Bloch-Bloembergen type damping term. The exchange-spring (hard/soft) system which is the best of the sample for multilayer structure has been explained by using the FMR technique and equilibrium condition of energy of system. The Magnetic materials are classified as paramagnetic, ferromagnetic, ferrimagnetic, antiferromagnetic and diamagnetic to their electronic order. Magnetic orders are divided in two groups as (i) paramagnetic, ferromagnetic, ferrimagnetic, antiferromagnetic and (ii) diamagnetic. The magnetic moments in diamagnetic materials are opposite to each other as well as the moments associated with the orbiting electrons so that a zero magnetic moment ( ) is produced on macroscopic scale. In the paramagnetic materials, each atom possesses a small magnetic moment. The orientation of magnetic moment of each atom is random, the net magnetic moment of a large sample (macroscopic scale) of dipole and the magnetization vector are zero when there is no applied field.

Nanoscience, nanotechnology and nanomaterials have become a central field of scientific and technical activity. Over the last years the interest in magnetic nanostructures and their applications in various electronic devices, effective opto-electronic devices, bio-sensors, photo-detectors, solar cells, nanodevices and plasmonic structures have been increasing tremendously. This is caused by the unique properties of magnetic nanostructures and the outstanding performance of nanoscale devices. Dimension in the range of one to hundred nanometers, is called the nano regime. In recent years, nanorods, nanoparticles, quantum dots, nanocrystals etc. are in a class of nanostructures (Yalçn, 2012; Kartopu & Yalçn, 2010; Aktaş, 2006) studied extensively. As the dimensions of nano materials decrease down to the nanometer scale, the surface of nanostructures starts to exhibit new and interesting properties mainly due to quantum size effects.

### **3. Origin of magnetic moment**

The magnetization of a matter is derived by electrons moving around the nucleus of an atom. Total magnetic moment occurs when the electrons such as a disc returns around its axis consist of spin angular momentum and returns around the nucleus consist orbital angular momentum. The most of matters which have unpaired electrons have a little magnetic moment. This natural angular momentum consists of the result of charged particle return around its own axis and is called spin of the particle. The origin of spin is not known exactly, although electron is point particle the movement of an electron in an external magnetic field is similar to the movement of the disc. In other words, the origin of the spin is quantum field theoretical considerations and comes from the representations of the Poincare algebra for the elementary particles. The magnetism related to spin angular momentum, orbital angular momentum and spin-orbit interactions angular momentum. The movement of the electron around the nucleus can be considered as a current loop while electron spin is considered very small current loop which generate magnetic field. Here, orbital angular

momentum was obtained by the result of an electron current loop around the nucleus. Thus, both it is exceeded the difficulty of understanding the magnetic moment and the magnetic moment for an electron orbiting around the nucleus is used easily. The result of orbitalangular momentum ( *L* ) adapted for spin-angular momentum ( *S* ) (Cullity, 1990).

One electron is rotating from left to right on the table plane as shown in Fig.1. The rotating electron creates a current (*i*) on the circle with radius of *r*.

**Figure 1.** Schematic representation of the precession of a single electron on the table plane.

The magnetic moment of a single electron is defined as below

$$
\vec{\mu} = \mathbf{i} \cdot \vec{\mathbf{A}}.\tag{1}
$$

Ferromagnetic Resonance 7

, the magnetic field

. The equation of

(5)

*H*

/ 2 /2 *H* , is called the Larmor

*H* in literature.

(6)

 

 

when the angle

*H dt* . The magnetic moment vector make precession movement about *H*

 

Magnetic Resonance (MR) is a research branch which examines magnetic properties of matters. The magnetic properties of atom originate from electrons and nucleus. So, it is studied in two groups such as electron paramagnetic resonance (EPR)/electron spin resonance (ESR) and nuclear magnetic resonance (NMR). At ESR and NMR all of them are the sample is placed in a strong static magnetic field and subjected to an orthogonally amplitude-frequency. While EPR uses a radiation of microwave frequency in general, NMR is observed at low radio frequency range. The energy absorption occurs when radio

in the external

). The motion of

of magnetic moment

in two dimensional motions on the plane and

) of amount

<sup>1</sup> . *d dL dt dt* 

in one dimensional motion. This motion corresponds to Newton's dynamic

<sup>1</sup> . *<sup>d</sup> <sup>H</sup> dt* 

and external magnetic field does not change. Therefore, in time ( *dt* ), the tip of the vector

This expression is called the equation of motion for magnetic moment (

. This frequency,

frequency. In general this Larmor frequency is used this form

) forms a cone related to *<sup>H</sup>*

) on the magnetic moment (

*dL dt*

equations. When an electron is placed in an applied magnetic field *H*

**Figure 2.** Schematic representation of precession of a single magnetic moment

This equation is related to

motion for magnetic moment (

magnetic field around the z-axis.

 *H* / 2

**4. Magnetic resonance** 

magnetic moment (

moves an angle ( )

at a frequency of

will produce a torque (

*F dP dt*

) is found by equating the torque as below

Where, *A* is the circle area. The magnetic moment is written as follows by using the current ( *e it* ) , one cycle ( 2*r vt* ) and angular momentum ( *<sup>e</sup> L m vr* ) definition.

$$
\vec{\mu} = -\frac{e}{2m\_e}\vec{L}\tag{2}
$$

Where / 2 *<sup>e</sup> e m* and *L* is the gyromagnetic (magneto-mechanical or magneto-gyric) ratio and the orbital-angular momentum, respectively. Therefore, the magnetic moment is obtained from Eq. (2) as below

$$
\vec{\mu} = -\gamma \,\vec{L}.\tag{3}
$$

The following expression is obtained when derivative of Eq.(3)

$$d\vec{\mu} + \chi \, d\vec{L} + \vec{L} \, d\chi = 0.\tag{4}$$

For our purpose, we only need to know that is a constant and *d* 0 . From this results, *d dL* <sup>0</sup> . The derivative of time of this equation, the equation of motion for magnetic moments of an electron is found as below

Ferromagnetic Resonance 7

$$\frac{1}{\gamma} \frac{d\vec{\mu}}{dt} = \frac{d\vec{L}}{dt} = \vec{\tau}. \tag{5}$$

This equation is related to *dL dt* in two dimensional motions on the plane and *F dP dt* in one dimensional motion. This motion corresponds to Newton's dynamic equations. When an electron is placed in an applied magnetic field *H* , the magnetic field will produce a torque ( ) on the magnetic moment ( ) of amount *H* . The equation of motion for magnetic moment ( ) is found by equating the torque as below

**Figure 2.** Schematic representation of precession of a single magnetic moment in the external magnetic field around the z-axis.

$$\frac{1}{\gamma} \frac{d\vec{\mu}}{dt} = \vec{\mu} \times \vec{H}.\tag{6}$$

This expression is called the equation of motion for magnetic moment ( ). The motion of magnetic moment ( ) forms a cone related to *<sup>H</sup>* when the angle of magnetic moment and external magnetic field does not change. Therefore, in time ( *dt* ), the tip of the vector moves an angle ( ) *H dt* . The magnetic moment vector make precession movement about *H* at a frequency of *H* / 2 . This frequency, / 2 /2 *H* , is called the Larmor frequency. In general this Larmor frequency is used this form *H* in literature.

#### **4. Magnetic resonance**

6 Ferromagnetic Resonance – Theory and Applications

electron creates a current (*i*) on the circle with radius of *r*.

angular momentum ( *L*

Where, *A* 

*d dL* 

( *e it* ) , one cycle ( 2

Where / 2 *<sup>e</sup>* 

*e m* and *L*

obtained from Eq. (2) as below

momentum was obtained by the result of an electron current loop around the nucleus. Thus, both it is exceeded the difficulty of understanding the magnetic moment and the magnetic moment for an electron orbiting around the nucleus is used easily. The result of orbital-

One electron is rotating from left to right on the table plane as shown in Fig.1. The rotating

(1)

(2)

(3)

(4)

0 . From this results,

is

) (Cullity, 1990).

) adapted for spin-angular momentum ( *S*

**Figure 1.** Schematic representation of the precession of a single electron on the table plane.

*i A*

and the orbital-angular momentum, respectively. Therefore, the magnetic moment

 *L*.

is the circle area. The magnetic moment is written as follows by using the current

2 *<sup>e</sup> <sup>e</sup> <sup>L</sup> m*

 

<sup>0</sup> . The derivative of time of this equation, the equation of motion for magnetic

*r vt* ) and angular momentum ( *<sup>e</sup> L m vr* ) definition.

is the gyromagnetic (magneto-mechanical or magneto-gyric) ratio

is a constant and *d*

The magnetic moment of a single electron is defined as below

The following expression is obtained when derivative of Eq.(3)

0. *d dL Ld*

For our purpose, we only need to know that

moments of an electron is found as below

.

Magnetic Resonance (MR) is a research branch which examines magnetic properties of matters. The magnetic properties of atom originate from electrons and nucleus. So, it is studied in two groups such as electron paramagnetic resonance (EPR)/electron spin resonance (ESR) and nuclear magnetic resonance (NMR). At ESR and NMR all of them are the sample is placed in a strong static magnetic field and subjected to an orthogonally amplitude-frequency. While EPR uses a radiation of microwave frequency in general, NMR is observed at low radio frequency range. The energy absorption occurs when radio

frequency is equal with energy difference between electrons two levels. But, the transition must obey the selection rules. The splitting between the energy levels occurs when total angular moment of electron is different from zero. On the other hand, the splitting of energy levels has not been observed in the filled orbit. The precession motion of a paramagnetic sample in magnetic field is seen schematically in Fig. 2. If microwave field with frequency at perpendicular is applied to the static field, it comes out power absorption when precession (<sup>0</sup> ) is same with -frequency. The power increases when these frequencies come near to each other and it occurs maximum occurs at point when they are equal. This behaviour is called magnetic resonance (MR).

The magnetic materials contain a large number of atomic magnetic moment in generally. Net atomic magnetic moment can be calculated by *M N* . Where, *N* is the number of atomic magnetic moment in materials.

$$\frac{1}{\gamma} \frac{d\vec{M}}{dt} = \vec{M} \times \vec{H}\_{\text{eff}} \tag{7}$$

Ferromagnetic Resonance 9

is due completely

 *M* . In

towards *<sup>H</sup>*

, we must require that

1950) type does not. Landau and Lifshitz observed that the ferromagnetic exchange forces between spins are much greater than the Zeeman forces between the spins and the magnetic fields in their formulation of the damping term. Therefore, the exchange will conserve the

this small damping limit, the Landau-Lifshitz and the Gilbert forms are equivalence so that whether one uses one or the other is simply a matter of convenience or familiarity. However, Callen has obtained a dynamic equation by quantizing the spin waves into magnons and treating the problem quantum-mechanically (Callen, 1958). Subsequently, Fletcher, Le Craw, and Spencer have reproduced the same equation using energy consideration (Fletcher, 1960). In their reproduction, they found the mean the rate of energy transfer between the uniform precession, the spin waves (Grünberg, 1979, 1980) and the

Stern and Gerlach (Gerlach, 1922) proved that the electron-magnetic moment of an atom in an external magnetic field originates only in certain directions in the experiment in 1922. Uhlenbek and Goudsmit found that the connection between the magnetic moment and spin angular momentum of electron (Uhlenbek, 1925), Rabi and Breit found the transition between the energy levels in oscillating magnetic field (Rabi, 1938). This also proved to be observed in the event of the first magnetic resonance. The EPR technique is said to be important of Stern-Gerlach experiment. Zavoisky observed the first peak in the electron paramagnetic resonance for CuCI22H2O sample and recorded (Zavoisky, 1945). The most of EPR experiments were made by scientists in the United Kingdom and the United States. Important people mentioned in the experimental EPR studies; Abragam, Bleaney and Van Vleck. The historical developments of MR have been summarized by Ramsey (Ramsey, 1985). NMR experiments had been done by Purcell et al. (Purcell, 1946). Today it has been used as a tool for clinical medicine. MRI was considered as a basic tool of CT scan in 1970s. The behaviors of spin system under the external magnetic field with the gradient of spin system are known NMR tomography. This technique is used too much for medicine, clinics, diagnostic and therapeutic purposes. General structure of the EPR spectrometer consist four basic parts in general. (i) Source system (generally used in the microwave 1-100 GHz), (ii) cavity-grid system, (iii) Magnet system and (iv) detector and modulation system. EPR/ESR is subject of the MR. An atom which has free electron when it is put in magnetic field the electron's energy levels separate (Yalçn, 2003, 2007(a), 2007(b)). This separation originates from the interaction of the electrons magnetic moment with external magnetic field. Energy

> <sup>ˆ</sup> <sup>ˆ</sup> *H g HS <sup>B</sup>*

It is called Zeeman Effect. If the applied magnetic field oriented z-axis energy levels are;

(8)

(9)

and *<sup>H</sup>*

. In this formulation, since the approach of *<sup>M</sup>*

to the relatively weak interaction between *M*

**5. Electron paramagnetic resonance** 

separating has been calculated by the following Hamiltonian.

. *MsB s E g HM*

magnitude of *M*

lattice.

This precession movement continue indefinitely would take forever when there is no damping force. The damping term may be introduced in different ways. Indeed, since the details of the damping mechanism in a ferromagnet have not been completely resolved, different mathematical forms for the damping have been suggested. The three most common damping terms used to augment the right-hand side of Eq. (7) are as follows:

$$\text{(i) The Bloch-Blockgergen form: } -\frac{\vec{M}\_{\theta,\phi}}{T\_2} - \frac{\vec{M}\_z - M\_0}{T\_1}$$

$$\text{(iii) The Landau-Liffishtz form: } \frac{-\vec{\lambda}}{\left|\vec{M}\right|^2} \vec{M} \times \vec{M} \times \vec{H}$$

$$\text{(iii) The Gült form: } \frac{\alpha}{\left| \vec{M} \right|} \vec{M} \times \frac{d\vec{M}}{dt}$$

Bloch-Bloembergen type damping does not converse *M* so it is equivalent to the type of Landau-Lifshitz and the Gilbert only when is small and for small excursion of *M* . For large excursion of *M* , the magnitude of *M* is certainly not protected, as the damping torque is in the direction of the magnetization component in this formularization. Hence, the observation of *M* in the switching experiments in thin films should be provide a sensitive test on the appropriate form of the damping term for ferromagnetism since *M* which is conserved during switching. This would suggest that the form of the Bloch-Bloembergen damping term would not be applicable for this type of experiment. The Gilbert type (Gilbert, 1955) is essentially a modification of the original form which is proposed firstly by Landau and Lifshitz (Landau & Lifshitz, 1935). It is very important to note that the Landau-Lifshitz and Gilbert type of damping conserve while the Bloch-Bloembergen (Bloembergen, 1950) type does not. Landau and Lifshitz observed that the ferromagnetic exchange forces between spins are much greater than the Zeeman forces between the spins and the magnetic fields in their formulation of the damping term. Therefore, the exchange will conserve the magnitude of *M* . In this formulation, since the approach of *<sup>M</sup>* towards *<sup>H</sup>* is due completely to the relatively weak interaction between *M* and *<sup>H</sup>* , we must require that *M* . In this small damping limit, the Landau-Lifshitz and the Gilbert forms are equivalence so that whether one uses one or the other is simply a matter of convenience or familiarity. However, Callen has obtained a dynamic equation by quantizing the spin waves into magnons and treating the problem quantum-mechanically (Callen, 1958). Subsequently, Fletcher, Le Craw, and Spencer have reproduced the same equation using energy consideration (Fletcher, 1960). In their reproduction, they found the mean the rate of energy transfer between the uniform precession, the spin waves (Grünberg, 1979, 1980) and the lattice.

### **5. Electron paramagnetic resonance**

8 Ferromagnetic Resonance – Theory and Applications

<sup>0</sup> ) is same with

behaviour is called magnetic resonance (MR).

(i) The Bloch-Bloembergen form: , <sup>0</sup>

(ii) The Landau-Lifshitz form: 2 *M M H*

 

Landau-Lifshitz and the Gilbert only when

large excursion of *M* , the magnitude of *M*

(iii) The Gilbert form: *dM <sup>M</sup>*

atomic magnetic moment in materials.

Net atomic magnetic moment can be calculated by *M N*

precession (

frequency is equal with energy difference between electrons two levels. But, the transition must obey the selection rules. The splitting between the energy levels occurs when total angular moment of electron is different from zero. On the other hand, the splitting of energy levels has not been observed in the filled orbit. The precession motion of a paramagnetic sample in magnetic field is seen schematically in Fig. 2. If microwave field with

frequency at perpendicular is applied to the static field, it comes out power absorption when

come near to each other and it occurs maximum occurs at point when they are equal. This

The magnetic materials contain a large number of atomic magnetic moment in generally.

*dM <sup>M</sup> <sup>H</sup>*

This precession movement continue indefinitely would take forever when there is no damping force. The damping term may be introduced in different ways. Indeed, since the details of the damping mechanism in a ferromagnet have not been completely resolved, different mathematical forms for the damping have been suggested. The three most common damping terms used to augment the right-hand side of Eq. (7) are as follows:

*dt*

2 1 *M Mz M T T*

Bloch-Bloembergen type damping does not converse *M* so it is equivalent to the type of

torque is in the direction of the magnetization component in this formularization. Hence, the observation of *M* in the switching experiments in thin films should be provide a sensitive

conserved during switching. This would suggest that the form of the Bloch-Bloembergen damping term would not be applicable for this type of experiment. The Gilbert type (Gilbert, 1955) is essentially a modification of the original form which is proposed firstly by Landau and Lifshitz (Landau & Lifshitz, 1935). It is very important to note that the Landau-Lifshitz and Gilbert type of damping conserve while the Bloch-Bloembergen (Bloembergen,

test on the appropriate form of the damping term for ferromagnetism since *M*

 

*M* 

*M dt*

*eff*

1


(7)

is small and for small excursion of *M*

is certainly not protected, as the damping

. Where, *N* is the number of


. For

which is

Stern and Gerlach (Gerlach, 1922) proved that the electron-magnetic moment of an atom in an external magnetic field originates only in certain directions in the experiment in 1922. Uhlenbek and Goudsmit found that the connection between the magnetic moment and spin angular momentum of electron (Uhlenbek, 1925), Rabi and Breit found the transition between the energy levels in oscillating magnetic field (Rabi, 1938). This also proved to be observed in the event of the first magnetic resonance. The EPR technique is said to be important of Stern-Gerlach experiment. Zavoisky observed the first peak in the electron paramagnetic resonance for CuCI22H2O sample and recorded (Zavoisky, 1945). The most of EPR experiments were made by scientists in the United Kingdom and the United States. Important people mentioned in the experimental EPR studies; Abragam, Bleaney and Van Vleck. The historical developments of MR have been summarized by Ramsey (Ramsey, 1985). NMR experiments had been done by Purcell et al. (Purcell, 1946). Today it has been used as a tool for clinical medicine. MRI was considered as a basic tool of CT scan in 1970s. The behaviors of spin system under the external magnetic field with the gradient of spin system are known NMR tomography. This technique is used too much for medicine, clinics, diagnostic and therapeutic purposes. General structure of the EPR spectrometer consist four basic parts in general. (i) Source system (generally used in the microwave 1-100 GHz), (ii) cavity-grid system, (iii) Magnet system and (iv) detector and modulation system. EPR/ESR is subject of the MR. An atom which has free electron when it is put in magnetic field the electron's energy levels separate (Yalçn, 2003, 2007(a), 2007(b)). This separation originates from the interaction of the electrons magnetic moment with external magnetic field. Energy separating has been calculated by the following Hamiltonian.

$$
\hat{H} = \lg \mu\_{\mathcal{B}} \,\vec{H} \cdot \hat{\mathcal{S}} \tag{8}
$$

It is called Zeeman Effect. If the applied magnetic field oriented z-axis energy levels are;

$$E\_{\rm Ms} = \mathcal{g} \,\,\mu\_{\mathcal{B}} \, H \cdot \mathcal{M}\_s. \tag{9}$$

Here, *g* is the *g*-value (or Landé *g*-value) (for free electron 2.0023193 *<sup>e</sup> g* and proton 2.7896 *Ng* ), *<sup>B</sup>* is Bohr magneton ( <sup>24</sup> 4 9.2740 10 / *B e eh m J T* ) and *Ms* is the number of magnetic spin quantum. If the orbital angular momentum of electron is large of zero ( 0 *L* ) *g* -value for free atoms is following

$$\mathbf{g} = \mathbf{1} + \frac{S(S+1) - L(L+1) + f(J+1)}{2J(J+1)}.\tag{10}$$

Ferromagnetic Resonance 11

versus magnetic field. At the

which have crystal lattice

If this equation is provided the system absorbs energy from applied electromagnetic wave (see Fig. 3). It is called resonance effect. Material absorbs energy in two different ways from applied electromagnetic wave by according to the Eq.(12). Firstly, in Eq.(12), frequency of electromagnetic wave doesn't change while the external magnetic field changes. Secondly,

same time it is said the absorption curve. The magnetic field derivative beneath of this

<sup>ˆ</sup> ˆˆ ˆ ˆ ˆˆ ˆ *H H gS SDS SAI Hg I I PI B N <sup>N</sup>*

Zeeman effect, second one is thin layer effects, the third one is the effect of between electronic spin and nucleus-spin of ion and it is known that thin layer effects. The fourth term is the effect of nucleus with the magnetic field. The last one is quadrupole effect of

 

operators of electronic and nucleus, respectively. In this equation, first term is

*g* factor is equal free electron's *g* -factor. But, *g* -factor in the exited energy levels separated from the *g* -factor of free electron. The hamiltonian for an ion which is in the magnetic field

ˆˆ ˆ ˆ ˆ ( ) *H H L gS L S B e*

In this equation, first term is Zeeman effects, second one is spin-orbit interaction. The first order energy of ion which shows *J M*, and it is excepted not degenerate is seen at below.

ˆ ˆ , , , ( ), *J eB z <sup>z</sup> BZ z E JMg HS JM JM H S JM*

We can write the hamiltonian equation which uses energy equations. There is two terms in the hamiltonian equations. The first term is the independent temperature coefficient for paramagnetic for paramagnetic, the last terms are only for spin variables. If the angular

 

> 


(13)

). Here, *pp* and 2 1 *T* are linewidth, *Hres* is

is the ratio of total angular momentum to

(15)

is zero,

. For the base energy level if *L*

(14)

its opposite can be provided.

 

**5.1. Spin Hamiltonian** 

resonance field,

ˆ *S* and *I* 

**5.2.** *g*

 **tensor** 

is following (Weil, 1994).

In this Fig.(3) it has been seen that magnetic susceptibility 2

is resonance frequency.

The spin Hamiltonian is total electronic spins and nucleon spin *I*

nucleus. It can be added different terms in Eq.13 (Slichter, 1963).

 *g J* . *<sup>J</sup>*

figure is FMR absorption spectrum ( <sup>2</sup> *d dH*

under the static magnetic field following;

The total magnetic moment of ion is *<sup>B</sup>*

moment of ion occurs because of spin, *g*

Planck constant. Landé factor *g* is depend on *SLJ* , ,

The anisotropy of the *g*-factor is described by taking into account the spin–orbit interaction combined (Yalçn, 2004(c)). The total magnetic moment can be written at below;

$$
\mu\_{eff} = \lg \mu\_B \sqrt{I(J+1)}.
$$

The values of orbital angular momentum of unpaired electrons for most of the radicals and radical ions are zero or nearly zero. Hence, the number of total electron angular momentum *J* equals only the number of spin quantum *S* . So, these values are nearly 2. For free electron ( 1/2 *Ms* ) and for this electron;

$$
\Delta E = E\_{\text{+1/2}} - E\_{\text{-1/2}} = \text{g } \mu\_{\text{B}}H.\tag{11}
$$

When the electromagnetic radiation which frequency is applied to such an electron system;

$$
\hbar\nu = \mathcal{g}\,\mu\_{\mathcal{B}}\,\mathcal{H}.\tag{12}
$$

**Figure 3.** The energy levels and resonance of free electron at zero field and increasing applied magnetic field. In this figure, while the value of magnetic field increases, the separating between energy levels increase. Arbitrary units used in vertical axes for 2 and 2 *d dH* .

If this equation is provided the system absorbs energy from applied electromagnetic wave (see Fig. 3). It is called resonance effect. Material absorbs energy in two different ways from applied electromagnetic wave by according to the Eq.(12). Firstly, in Eq.(12), frequency of electromagnetic wave doesn't change while the external magnetic field changes. Secondly, its opposite can be provided.

In this Fig.(3) it has been seen that magnetic susceptibility 2 versus magnetic field. At the same time it is said the absorption curve. The magnetic field derivative beneath of this figure is FMR absorption spectrum ( <sup>2</sup> *d dH* ). Here, *pp* and 2 1 *T* are linewidth, *Hres* is resonance field, is resonance frequency.

### **5.1. Spin Hamiltonian**

10 Ferromagnetic Resonance – Theory and Applications

zero ( 0 *L* ) *g* -value for free atoms is following

( 1/2 *Ms* ) and for this electron;

When the electromagnetic radiation which frequency

. *<sup>B</sup> h gH*

increase. Arbitrary units used in vertical axes for 2

2.7896 *Ng* ), *<sup>B</sup>*

Here, *g* is the *g*-value (or Landé *g*-value) (for free electron 2.0023193 *<sup>e</sup> g* and proton

number of magnetic spin quantum. If the orbital angular momentum of electron is large of

( 1) ( 1) ( 1) 1 . 2 ( 1)

The anisotropy of the *g*-factor is described by taking into account the spin–orbit interaction

( 1) *eff B*

The values of orbital angular momentum of unpaired electrons for most of the radicals and radical ions are zero or nearly zero. Hence, the number of total electron angular momentum *J* equals only the number of spin quantum *S* . So, these values are nearly 2. For free electron

1/2 1/2 . *<sup>B</sup> EE E g H*

**Figure 3.** The energy levels and resonance of free electron at zero field and increasing applied magnetic field. In this figure, while the value of magnetic field increases, the separating between energy levels

> and 2 *d dH* .

  (11)

is applied to such an electron system;

(12)

 *g JJ*

 

*eh m J T* ) and *Ms* is the

(10)

 is Bohr magneton ( <sup>24</sup> 4 9.2740 10 / *B e* 

*SS LL J J <sup>g</sup> J J*

combined (Yalçn, 2004(c)). The total magnetic moment can be written at below;

The spin Hamiltonian is total electronic spins and nucleon spin *I* which have crystal lattice under the static magnetic field following;

$$
\hat{H} = \mu\_B \vec{H} \cdot \vec{\mathcal{S}} + \hat{\mathbf{S}} \cdot \vec{\mathcal{D}} \cdot \hat{\mathbf{S}} + \vec{\mathcal{S}} \cdot \vec{A} \cdot \hat{\mathbf{I}} - \mu\_N \cdot \vec{H} \cdot \vec{\mathcal{g}}\_N \cdot \hat{\mathbf{I}} + \hat{\mathbf{I}} \cdot \vec{\mathcal{P}} \cdot \hat{\mathbf{I}} \tag{13}
$$

ˆ *S* and *I* operators of electronic and nucleus, respectively. In this equation, first term is Zeeman effect, second one is thin layer effects, the third one is the effect of between electronic spin and nucleus-spin of ion and it is known that thin layer effects. The fourth term is the effect of nucleus with the magnetic field. The last one is quadrupole effect of nucleus. It can be added different terms in Eq.13 (Slichter, 1963).

#### **5.2.** *g*  **tensor**

The total magnetic moment of ion is *<sup>B</sup> g J* . *<sup>J</sup>* is the ratio of total angular momentum to Planck constant. Landé factor *g* is depend on *SLJ* , , . For the base energy level if *L* is zero, *g* factor is equal free electron's *g* -factor. But, *g* -factor in the exited energy levels separated from the *g* -factor of free electron. The hamiltonian for an ion which is in the magnetic field is following (Weil, 1994).

$$
\hat{H} = \mu\_{\text{B}} \vec{H} \cdot (\hat{L} + \mathcal{g}\_{e} \hat{\mathbf{S}}) + \mathcal{X} \hat{L} \cdot \hat{\mathbf{S}} \tag{14}
$$

In this equation, first term is Zeeman effects, second one is spin-orbit interaction. The first order energy of ion which shows *J M*, and it is excepted not degenerate is seen at below.

$$E\_f = \left\langle I, M \Big| \varrho\_e \mu\_B \, H\_z \hat{S}\_z \Big| J, M \right\rangle + \left\langle I, M \Big| (\mu\_B \, H\_Z + \mathcal{L} \hat{S}\_z) \Big| J, M \right\rangle \tag{15}$$

We can write the hamiltonian equation which uses energy equations. There is two terms in the hamiltonian equations. The first term is the independent temperature coefficient for paramagnetic for paramagnetic, the last terms are only for spin variables. If the angular moment of ion occurs because of spin, *g* -tensor is to be isotropic.

### **6. Ferromagnetic resonance**

The most important parameters for ferromagnet can be deduced by the ferromagnetic resonance method. FMR absorption curves may be obtained from Eq.(12) by chancing frequency or magnetic field. FMR signal can be detected by the external magnetic field and frequency such as EPR signal. The field derivative FMR absorption spectra are greater than in EPR as a generally. The linear dependence of frequency of resonance field may be calculated from 1 GHz to 100 GHz range in frequency spectra (L-, S-, C-, X-, K-, Q-, V-, E-, W-, F-, and D-band). The resonance frequency, relaxation, linewidth, Landé *g*-factor (spectroscopic *g*-factor), the coercive force, the anisotropy field, shape of the specimen, symmetry axes of the crystal and temperature characterized FMR spectra. The broadening of the FMR absorption line depend on the line width (so called 2 1 *T* on the Bloch-Bloembergen type damping form). The nonuniform modes are seen in the EPR signal. The nonlinear effects for FMR are shown by the relationship between the uniform precessions of magnetic moments. The paramagnetic excitation of unstable oscillation of the phonons displays magneto-elastic interaction in ferromagnetic systems. This behaviour so called magnetostriction. The FMR studies have led to the development of many micro-wave devices. These phenomenon are microwave tubes, circulators, oscillators, amplifiers, parametric frequency converters, and limiters. The resonance absorption curve of electromagnetic waves at centimeter scale by ferromagnet was first observed by Arkad'ev in 1913 (Arkad'ev, 1913) .

Ferromagnetic Resonance 13

(16)

is the effective

are obtained from

*HH H*

 

are the angles for magnetization and

is the effective field for a single magnetic films. The

(17)

(18)

for (x,y,z)

can be easily calculated using the Eq. (16).

sin sin cos( ) cos cos

 

246

sin sin sin ...

Where, Z a d ex *E EEE* , , , are Zeeman, magnetocrystalline anisotropy, demagnetization and

applied magnetic field vector in the spherical coordinates, respectively. Magnetic anisotropy energy arises from either the interaction of electron spin magnetic moments with the lattice via spin-orbit coupling. On the other hand, anisotropy energy induced due to local atomic ordering. The *θ* in anisotropy energy is the angle between magnetization orientation and local easy axis of the magnetic anisotropy. *K*u1 and *K*u2 are energy density constants. The demagnetization field is proportional to the magnetic free pole density. The exchange energy for thin magnetic film may be neglected in generally. Because associated energies is small. But, this exchange energy are not neglected for multilayer structures. This energy occurs between the magnetic layers, so that this energy called interlayer exchange energy.

uniaxial anisotropy term and *Ku* takes into account some additional second-order uniaxial

 

Neglecting the damping term one can write the equation of motion for the magnetization

<sup>1</sup> . *eff dM M H*

internal field due to the anisotropy energy. The dynamic equation of motion for

<sup>0</sup> <sup>1</sup> . *iz eff dM M M M H dt T*

Here, <sup>221</sup> *T TTT* (, ,) represents both transverse (for *Mx* and *My* components) and the longitudinal (for *Mz* components) relaxation times of the magnetization. That is, 1 *T* is the

projections of the magnetization. In the spherical coordinates the Bloch-Bloembergen

is the effective magnetic field that includes the applied magnetic field and the

*dt*

for the magnetization vector *M*

 and ( , ) *H H* 

...

Z a d ex

2 sin

*E E EEE*

a u1 u2 u3 2 2

 

*EK K K*

 

This expression is seen at the end of this subject in details. 2 *K MK eff <sup>U</sup>*

magnetization with the Bloch-Bloembergen type damping term is given as Eq.(18).

spin-lattice relaxation time, 2 *<sup>T</sup>* is the spin-spin relaxation time, and (0,0,1) *iz*

 

Z

*T*

d

ferromagnetic exchange energy. (,)

anisotropy and 2 2 *H M KM eff S u s* 

static equilibrium conditions. *E EE E* , , and

equilibrium values of polar angles

equation can be written as below;

vector *M*

Here the *Heff*

as

*ex i j*

*E M E JS S*

 

*E MH*

The sample geometry, relative orientation of the equilibrium magnetization *M* , the applied *dc* magnetic field *H* and experimental coordinate systems are shown in Fig.4.

**Figure 4.** Sample geometries and relative orientations of equilibrium magnetization *M* and the *dc*  components of external magnetic field, *H* for thin films.

The ferromagnetic resonance data analyzed using the free energy expansion similar to that employed

$$\begin{aligned} E\_{\rm T} &= E\_{\rm Z} + E\_{\rm a} + E\_{\rm d} + E\_{\rm ex} + \dots \\ E\_{\rm Z} &= -M \cdot H \Big( \sin \theta \sin \theta\_{\rm H} \cos(\phi - \phi\_{\rm H}) + \cos \theta \cos \theta\_{\rm H} \Big) \\ E\_{\rm a} &= K\_{\rm u1} \sin^{2} \theta + K\_{\rm u2} \sin^{4} \theta + K\_{\rm u3} \sin^{6} \theta \dots \\ E\_{\rm d} &= -2\pi \, M^{2} \sin^{2} \theta \\ E\_{\rm ex} &= -JS\_{\rm i} S\_{\rm j} \end{aligned} \tag{16}$$

Where, Z a d ex *E EEE* , , , are Zeeman, magnetocrystalline anisotropy, demagnetization and ferromagnetic exchange energy. (,) and ( , ) *H H* are the angles for magnetization and applied magnetic field vector in the spherical coordinates, respectively. Magnetic anisotropy energy arises from either the interaction of electron spin magnetic moments with the lattice via spin-orbit coupling. On the other hand, anisotropy energy induced due to local atomic ordering. The *θ* in anisotropy energy is the angle between magnetization orientation and local easy axis of the magnetic anisotropy. *K*u1 and *K*u2 are energy density constants. The demagnetization field is proportional to the magnetic free pole density. The exchange energy for thin magnetic film may be neglected in generally. Because associated energies is small. But, this exchange energy are not neglected for multilayer structures. This energy occurs between the magnetic layers, so that this energy called interlayer exchange energy. This expression is seen at the end of this subject in details. 2 *K MK eff <sup>U</sup>* is the effective uniaxial anisotropy term and *Ku* takes into account some additional second-order uniaxial anisotropy and 2 2 *H M KM eff S u s* is the effective field for a single magnetic films. The equilibrium values of polar angles for the magnetization vector *M* are obtained from static equilibrium conditions. *E EE E* , , and can be easily calculated using the Eq. (16). Neglecting the damping term one can write the equation of motion for the magnetization vector *M* as

12 Ferromagnetic Resonance – Theory and Applications

The most important parameters for ferromagnet can be deduced by the ferromagnetic resonance method. FMR absorption curves may be obtained from Eq.(12) by chancing frequency or magnetic field. FMR signal can be detected by the external magnetic field and frequency such as EPR signal. The field derivative FMR absorption spectra are greater than in EPR as a generally. The linear dependence of frequency of resonance field may be calculated from 1 GHz to 100 GHz range in frequency spectra (L-, S-, C-, X-, K-, Q-, V-, E-, W-, F-, and D-band). The resonance frequency, relaxation, linewidth, Landé *g*-factor (spectroscopic *g*-factor), the coercive force, the anisotropy field, shape of the specimen, symmetry axes of the crystal and temperature characterized FMR spectra. The broadening of the FMR absorption line depend on the line width (so called 2 1 *T* on the Bloch-Bloembergen type damping form). The nonuniform modes are seen in the EPR signal. The nonlinear effects for FMR are shown by the relationship between the uniform precessions of magnetic moments. The paramagnetic excitation of unstable oscillation of the phonons displays magneto-elastic interaction in ferromagnetic systems. This behaviour so called magnetostriction. The FMR studies have led to the development of many micro-wave devices. These phenomenon are microwave tubes, circulators, oscillators, amplifiers, parametric frequency converters, and limiters. The resonance absorption curve of electromagnetic waves at centimeter scale by ferromagnet was first observed by Arkad'ev in

The sample geometry, relative orientation of the equilibrium magnetization *M*

**Figure 4.** Sample geometries and relative orientations of equilibrium magnetization *M*

for thin films.

The ferromagnetic resonance data analyzed using the free energy expansion similar to that

and experimental coordinate systems are shown in Fig.4.

, the applied

and the *dc* 

**6. Ferromagnetic resonance** 

1913 (Arkad'ev, 1913) .

components of external magnetic field, *H*

employed

*dc* magnetic field *H*

$$\frac{1}{\gamma} \frac{d\vec{M}}{dt} = \vec{M} \times \vec{H}\_{\text{eff}}.\tag{17}$$

Here the *Heff* is the effective magnetic field that includes the applied magnetic field and the internal field due to the anisotropy energy. The dynamic equation of motion for magnetization with the Bloch-Bloembergen type damping term is given as Eq.(18).

$$\frac{1}{\chi} \frac{d\vec{M}}{dt} = \vec{M} \times \vec{H}\_{\text{eff}} - \frac{\vec{M} - \delta\_{\text{iz}} M\_0}{T}. \tag{18}$$

Here, <sup>221</sup> *T TTT* (, ,) represents both transverse (for *Mx* and *My* components) and the longitudinal (for *Mz* components) relaxation times of the magnetization. That is, 1 *T* is the spin-lattice relaxation time, 2 *<sup>T</sup>* is the spin-spin relaxation time, and (0,0,1) *iz* for (x,y,z) projections of the magnetization. In the spherical coordinates the Bloch-Bloembergen equation can be written as below;

**Figure 5.** Damped precession of a magnetic moment *M* toward the effective magnetic field *Heff* according to the Bloch-Bloembergen type equation (Aktaş, 1993, 1994; Yalçn, 2008(a)).

$$\frac{1}{\gamma} \frac{d\vec{M}}{dt} = \frac{\vec{M}}{|\vec{M}|} \times \vec{\nabla}E - \frac{\vec{M}\_{\theta,\phi}}{\gamma T\_2} - \frac{\vec{M}\_z - M\_0}{\gamma T\_1}.\tag{19}$$

Ferromagnetic Resonance 15

(23b)

. Here, restoring force

arises when the coordinate system is

(24)

(25)

and the magnetic susceptibility

2 2

 

2 2

 

*T*

2

(26)

is given

*E* . The

(23a)

2

2

is given by the second derivative of the kinetic part in the energy with

2 2 1 <sup>1</sup> . <sup>2</sup> *P h* 

is the microwave frequency, 1 *h* is the amplitude of the magnetic field component

0

2 2

 

*M T*

0

 

<sup>2</sup> <sup>4</sup>

*<sup>E</sup> M i*

is the imaginary part of the high-frequency susceptibility. The field derivative FMR

*m h* 

The theoretical absorption curves are obtained by using the imaginary part of the high frequency magnetic susceptibility as a function of applied field (Öner, 1997; Min, 2006;

> 1 2 <sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>2</sup> 0

The dispersion relation can be derived by substituting Eq.(16) into Eq. (23) (Aktaş, 1997;

*s*

Yalçn, 2004(a), 2004(b), 2008(a); Güner, 2006; Kharmouche, 2007; Stashkevich, 2009)

   sin

 

> 

 *<sup>i</sup>* 

*E* . The restoring constant in this chapter corresponds to

2

 sin sin <sup>0</sup>

2

 2 2

*g H* is the Larmour frequency of the magnetization in the external *dc*

*EE E M T* 

is the second derivative of the potential part in the energy of system *xx*

1 1

1

*T M M m E E m*

*s s*

> 

2 2

sin

*i M T M*

*E E*

*s s*

effective magnetic field. This dispersion relation can be related as the angular momentum

. The *E*

4

 

*s*

*i*

 

(Sparks, 1964; Morrish, 1965; Vittoria, 1993; Gurevich, 1996; Chikazumi, 1997)

not parallel to the symmetry and last term originated from relaxation term in Eq.(23)

The power absorption from radio frequency (*rf*) field in a unit volume of sample is given by

2

1

analogue to be linear momentum oscillator described

*i pp* 

*i*

Here *<sup>B</sup>* 

inverse mass <sup>1</sup>

constant

*E*

where

and 2 

as

Cullity, 2009)

 

*i* 

respect to linear momentum <sup>1</sup>

. The inverse mass is proportional to *E*

absorption spectrum is proportional to 2 *d dH*

Where, the torque is obtained from the energy density through the expression

$$
\vec{\nabla}E = -\left(\frac{\partial E}{\partial \theta}\right)\hat{e}\_{\phi} + \frac{1}{\sin \theta} \left(\frac{\partial E}{\partial \phi}\right)\hat{e}\_{\theta}.\tag{20}
$$

For a small deviation from the equilibrium orientation, the magnetization vector *M* can be approximated by

$$
\vec{M} = M\_s \hat{\mathbf{e}}\_r + m\_\theta \hat{\mathbf{e}}\_\theta + m\_\phi \hat{\mathbf{e}}\_\phi. \tag{21}
$$

Where the dynamic transverse components are assumed to be sufficiently small and can be given as

$$\begin{aligned} m\_{\vartheta}(z,t) &= m\_{\vartheta}^{0} \exp i(\cot \pm kz) \\ m\_{\phi}(z,t) &= m\_{\phi}^{0} \exp i(\cot \pm kz) \end{aligned} \tag{22}$$

Dispersion relation for films can be derived by using these solutions (Eq.(22)) in Eqs. (19) and (20). On the other hand, the eigen frequency of thin films mode is determined by the static effective field and can be derived directly from the total free energy for magnetic system/ferromagnet. It is given by the second derivatives of the total energy with respect to the and (Smit, 1955; Artman, 1957; Wigen, 1984, 1988, 1992; Baseglia, 1988; Layadi, 1990; Farle, 1998). The matrices form for *m m* and is calculated using the Eq.(19) with Eq.(20, 21, 22).

#### Ferromagnetic Resonance 15

$$
\begin{pmatrix}
i\frac{\partial\boldsymbol{\theta}}{\partial\boldsymbol{\gamma}} + \frac{1}{\boldsymbol{\gamma}T\_{2}} + \frac{E\_{\theta\phi}}{M\_{s}\sin\theta} & \frac{E\_{\phi\phi}}{M\_{s}\sin^{2}\theta} \\
\end{pmatrix}
\cdot
\begin{pmatrix}
m\_{\theta} \\
m\_{\phi}
\end{pmatrix} = 0
\tag{23a}
$$

$$\left(\frac{\alpha\nu}{\mathcal{I}}\right)^2 = \frac{1}{M^2 \sin^2\theta} \left(E\_{\theta\theta}E\_{\phi\phi} - E\_{\theta\phi}^2\right) + \left(\frac{1}{\mathcal{I}\,T\_2}\right)^2\tag{23b}$$

Here *<sup>B</sup> g H* is the Larmour frequency of the magnetization in the external *dc* effective magnetic field. This dispersion relation can be related as the angular momentum analogue to be linear momentum oscillator described *<sup>i</sup>* . Here, restoring force constant is the second derivative of the potential part in the energy of system *xx E* . The inverse mass <sup>1</sup> *i* is given by the second derivative of the kinetic part in the energy with respect to linear momentum <sup>1</sup> *i pp E* . The restoring constant in this chapter corresponds to *E* . The inverse mass is proportional to *E* . The *E* arises when the coordinate system is not parallel to the symmetry and last term originated from relaxation term in Eq.(23) (Sparks, 1964; Morrish, 1965; Vittoria, 1993; Gurevich, 1996; Chikazumi, 1997)

14 Ferromagnetic Resonance – Theory and Applications

**Figure 5.** Damped precession of a magnetic moment *M*

approximated by

given as

the and

22).

according to the Bloch-Bloembergen type equation (Aktaş, 1993, 1994; Yalçn, 2008(a)).

Where, the torque is obtained from the energy density through the expression

For a small deviation from the equilibrium orientation, the magnetization vector *M*

<sup>1</sup> . *<sup>M</sup> M M <sup>z</sup> dM M <sup>E</sup> dt M T T*

> *E E Ee e*

ˆˆˆ . *M Me me me s r* 

Where the dynamic transverse components are assumed to be sufficiently small and can be

0 0 ( , ) exp ( ) ( , ) exp ( ) *m z t m i t kz m z t m i t kz*

 

 

 

Dispersion relation for films can be derived by using these solutions (Eq.(22)) in Eqs. (19) and (20). On the other hand, the eigen frequency of thin films mode is determined by the static effective field and can be derived directly from the total free energy for magnetic system/ferromagnet. It is given by the second derivatives of the total energy with respect to

Farle, 1998). The matrices form for *m m* and

 

<sup>1</sup> ˆ ˆ . sin

(Smit, 1955; Artman, 1957; Wigen, 1984, 1988, 1992; Baseglia, 1988; Layadi, 1990;

 

toward the effective magnetic field *Heff*

, 0 2 1

(19)

(20)

(21)

(22)

is calculated using the Eq.(19) with Eq.(20, 21,

can be

The power absorption from radio frequency (*rf*) field in a unit volume of sample is given by

$$P = \frac{1}{2} \alpha \chi \chi\_2 h\_1^2. \tag{24}$$

where is the microwave frequency, 1 *h* is the amplitude of the magnetic field component and 2 is the imaginary part of the high-frequency susceptibility. The field derivative FMR absorption spectrum is proportional to 2 *d dH* and the magnetic susceptibility is given as

$$\mathcal{X} = 4\pi \left( \frac{m\_{\phi}}{h\_{\phi}} \right)\_{\phi = 0} \tag{25}$$

The theoretical absorption curves are obtained by using the imaginary part of the high frequency magnetic susceptibility as a function of applied field (Öner, 1997; Min, 2006; Cullity, 2009)

$$\chi = \chi\_1 + i\chi\_2 = \frac{4\pi M\_s \left(\frac{E\_{\theta\theta}}{M\_s}\right) \left(\frac{\alpha\_0}{\mathcal{I}}\right)^2 - \left(\frac{\alpha}{\mathcal{I}}\right)^2 + i\left(\frac{2\alpha\nu}{\mathcal{I}^2 T\_2}\right)\right)}{\left(\left(\frac{\alpha\_0}{\mathcal{I}}\right)^2 - \left(\frac{\alpha}{\mathcal{I}}\right)^2\right)^2 + \left(\frac{2\alpha\nu}{\mathcal{I}^2 T\_2}\right)^2} \tag{26}$$

The dispersion relation can be derived by substituting Eq.(16) into Eq. (23) (Aktaş, 1997; Yalçn, 2004(a), 2004(b), 2008(a); Güner, 2006; Kharmouche, 2007; Stashkevich, 2009)

$$\left(\frac{\alpha}{\gamma}\right)^2 = \left[H\cos(\theta-\theta\_H) + H\_{\text{eff}}\cos(2\theta)\right] \cdot \left[H\cos(\theta-\theta\_H) + H\_{\text{eff}}\cos^2\theta\right] + \left(\frac{1}{\gamma T\_2}\right)^2\tag{27}$$

Ferromagnetic Resonance 17

(28)

is the azimuth

The experimental data were analyzed by using magnetic energy density for a system consisting of *n* magnetic layers with saturation magnetization *Ms* and layer thickness *ti*. The magnetic energy density for the nanoscale multilayer structures the energy per unit surface

   

 

 

 

sin sin cos( ) cos cos

2 2 4 4

*is i H i H i H*

 

sin cos sin cos

*i ai i i i bi i i*

, , 1 1

*t K t K*

is the polar angle of the magnetization *Ms* to the z-axis and *<sup>i</sup>*

1998; Schmool, 1998; Lindner, 2003; Sklyuyev, 2009; Topkaya, 2010; Erkovan, 2011).

This type exchange-coupling system is located in an external magnetic field, the magnetic moment in each layer. The suitable theoretical expression may be derived in order to deduce magnetic parameter for *ac* susceptibility. The equation of precession motion for

> , 21 <sup>1</sup> . 1 <sup>1</sup> , *<sup>i</sup>*

  *T*

 

*M*

*dM M M M M E dt t M T*

*i is*

*th* layer in the spherical coordinates with the Bloch-Bloembergen type

(29)

*i i zi is*

, , ,

*n*

*i*

 

*n n*

*i i*

, 1 1 1 1

sin sin cos( ) cos cos

1 2 , 1 1 1 1

*ii i i i i i i*

angle to the x-axis in the film plane. The first term is the Zeeman energy. The second and third terms correspond to first and second order magnetocrystalline energy with respectively. These energies due to the demagnetization field and any induced perpendicular anisotropy energy. On the other hand, these energies qualitatively have the same angular dependence with respect to the film normal. The second order magnetocrystalline energy term can be neglected for most of the ferromagnetic systems. The last two terms corresponds to bilinear and biquadratic interactions of ferromagnetic layers through nonmagnetic spacer via conduction energies. *Ai i*, <sup>1</sup> and *i i*, <sup>1</sup> *B* are bilinear and biquadratic coupling constants, respectively. The bilinear exchange interaction can be written from Eq.16. *Ai i*, <sup>1</sup> can be either negative and positive depending on antiferromagnetic and ferromagnetic interactions, respectively. The antiparallel/perpendicular and parallel alignments of magnetization of nearest neighboring layers are energetically favorable for a negative/positive value of *i i*, <sup>1</sup> *B* . Biquadratic interaction for spin systems have been analysed for Ising system in detail (Chen, 1973; Erdem, 2001). The biquadratic term is smaller than the bilinear interaction term. Therefore, it can be neglected for most of the ferromagnetic systems. The indirect exchange energy depends on spacer thickness and even shows oscillatory behavior with spacer thickness (Ruderman, 1954; Yosida, 1957; Parkin, 1990, 1991(a), 19901(b), 1994). The current literature on single ultrathin films and multilayers is given in below at table (Layadi, 1990(a), 1990(b); Wigen, 1992; Zhang, 1994(a), 1994 (b); Goryunov, 1995; Ando, 1997; Farle; 1998; Platow,

 

*ii i i i i i i*

 

sin sin cos( ) cos cos

area can be written as below

Where, *<sup>i</sup>* 

magnetization of the *i*

relaxation term can be written as

1

*tM H*

*i*

*n*

1

 

*A*

*B*

*n*

*E*

*i n*

*i*

1

1

here, 0 2 is the circular frequency of the EPR spectrometer. Fitting Eq.(27) with experimental results of the FMR measurement at different out-of-plane-angle ( ) *<sup>H</sup>* , the values for the effective magnetization can be obtained.

Figure 6 uses of both experimental and theoretical coordinate systems for the nanowire sample geometry. Equilibrium magnetization *M* and *dc*-magnetic field *<sup>H</sup>* are shown in this figure and also the geometric factor and hexagonal nanowire array presentation of nanowire are displayed. The ferromagnetic resonance theory has been developed for thin films applied to nanowires with the help of the following Fig.6. The effective uniaxial anisotropy term for nanowire arrays films <sup>2</sup> 1 3 *K M PK eff <sup>U</sup>* is written in this manner for arrayed nanowires. The first term in the *Keff* is due to the magnetostatic energy of perpendicularly-arrayed NWs (Dubowik, 1996; Encinas-Oropesa, 2001; Demand, 2002; Yalçn, 2004(a); Kartopu, 2009, 2010, 2011(a)) and constant with the symmetry axis along wire direction. The second term in the *Keff* is packing factor for a perfectly ordered hcp NW

arrays. The packing factor is defined as <sup>2</sup> *P dr* ( 2 3)( ) . The packing factor (*P*) of nanowires increases, nanowire diameter increases, the preferential orientation of the easy direction of magnetization changes from the parallel to the perpendicular direction to the wire axis (Kartopu, 2011(a)). As further, the effective uniaxial anisotropy (*Keff*) for a perfectly ordered hcp NWs should decrease linearly with increasing packing factor. 2 13 2 *H M P KM eff S u s* , which is the effective anisotropy field derived from the total magnetic anisotropy energy of NWs Eq. (16). The values for total magnetization have been obtained by fitting *Heff* with experimental results of FMR measurements at different angles ( *<sup>H</sup>* ) of external field *H* . The experimental spectra are proportional to the derivative of the absorbed power with respect to the applied field which is also proportional to the imaginary part of the magnetic susceptibility.

**Figure 6.** (a) Schematic representation of the cobalt nanowires and the relative orientation of the equilibrium magnetization *M* and the dc component of the external magnetic field *H*, for the FMR experiments and their theoretical calculations. (b) Hexagonal NW array exhibiting a total of seven wires and the dashed lines bottom of the seven wires indicate the six fold symmetry. (c) Sample parameters used in the packing factors *P* calculation.

The experimental data were analyzed by using magnetic energy density for a system consisting of *n* magnetic layers with saturation magnetization *Ms* and layer thickness *ti*. The magnetic energy density for the nanoscale multilayer structures the energy per unit surface area can be written as below

16 Ferromagnetic Resonance – Theory and Applications

 

values for the effective magnetization can be obtained.

anisotropy term for nanowire arrays films <sup>2</sup> 1 3 *K M PK eff <sup>U</sup>*

arrays. The packing factor is defined as <sup>2</sup> *P dr* ( 2 3)( )

2 13 2 *H M P KM eff S u s*

) of external field *H*

used in the packing factors *P* calculation.

imaginary part of the magnetic susceptibility.

angles ( *<sup>H</sup>*

sample geometry. Equilibrium magnetization *M*

here, 0 2 

2 2

 

experimental results of the FMR measurement at different out-of-plane-angle ( ) *<sup>H</sup>*

Figure 6 uses of both experimental and theoretical coordinate systems for the nanowire

this figure and also the geometric factor and hexagonal nanowire array presentation of nanowire are displayed. The ferromagnetic resonance theory has been developed for thin films applied to nanowires with the help of the following Fig.6. The effective uniaxial

for arrayed nanowires. The first term in the *Keff* is due to the magnetostatic energy of perpendicularly-arrayed NWs (Dubowik, 1996; Encinas-Oropesa, 2001; Demand, 2002; Yalçn, 2004(a); Kartopu, 2009, 2010, 2011(a)) and constant with the symmetry axis along wire direction. The second term in the *Keff* is packing factor for a perfectly ordered hcp NW

nanowires increases, nanowire diameter increases, the preferential orientation of the easy direction of magnetization changes from the parallel to the perpendicular direction to the wire axis (Kartopu, 2011(a)). As further, the effective uniaxial anisotropy (*Keff*) for a perfectly ordered hcp NWs should decrease linearly with increasing packing factor.

total magnetic anisotropy energy of NWs Eq. (16). The values for total magnetization have been obtained by fitting *Heff* with experimental results of FMR measurements at different

of the absorbed power with respect to the applied field which is also proportional to the

**Figure 6.** (a) Schematic representation of the cobalt nanowires and the relative orientation of the equilibrium magnetization *M* and the dc component of the external magnetic field *H*, for the FMR experiments and their theoretical calculations. (b) Hexagonal NW array exhibiting a total of seven wires and the dashed lines bottom of the seven wires indicate the six fold symmetry. (c) Sample parameters

, which is the effective anisotropy field derived from the

. The experimental spectra are proportional to the derivative

<sup>1</sup> cos( ) cos(2 ) cos( ) cos *H HH H H eff H eff <sup>T</sup>*

 

is the circular frequency of the EPR spectrometer. Fitting Eq.(27) with

2

and *dc*-magnetic field *<sup>H</sup>*

 

is written in this manner

. The packing factor (*P*) of

2

(27)

, the

are shown in

 

$$E = -\begin{pmatrix} \sum\_{i=1}^{n} t\_i M\_s \left( \sin \theta\_i \sin \theta\_H \cos(\phi\_i - \phi\_H) + \cos \theta\_i \cos \theta\_H \right) + \\ \sum\_{i=1}^{n} t\_i K\_{a,i} \sin^2 \phi\_i \cos^2 \theta\_i + \sum\_{i=1}^{n} t\_i K\_{b,i} \sin^4 \phi\_i \cos^4 \theta\_i + \\ \sum\_{i=1}^{n-1} A\_{i,i\mp 1} \left( \sin \theta\_i \sin \theta\_{i\mp 1} \cos(\phi\_{i\mp 1} - \phi\_i) + \cos \theta\_i \cos \theta\_{i\mp 1} \right) + \\ \sum\_{i=1}^{n-1} B\_{i,i\mp 1} \left( \sin \theta\_i \sin \theta\_{i\mp 1} \cos(\phi\_{i\mp 1} - \phi\_i) + \cos \theta\_i \cos \theta\_{i\mp 1} \right)^2 \end{pmatrix} \tag{28}$$

Where, *<sup>i</sup>* is the polar angle of the magnetization *Ms* to the z-axis and *<sup>i</sup>* is the azimuth angle to the x-axis in the film plane. The first term is the Zeeman energy. The second and third terms correspond to first and second order magnetocrystalline energy with respectively. These energies due to the demagnetization field and any induced perpendicular anisotropy energy. On the other hand, these energies qualitatively have the same angular dependence with respect to the film normal. The second order magnetocrystalline energy term can be neglected for most of the ferromagnetic systems. The last two terms corresponds to bilinear and biquadratic interactions of ferromagnetic layers through nonmagnetic spacer via conduction energies. *Ai i*, <sup>1</sup> and *i i*, <sup>1</sup> *B* are bilinear and biquadratic coupling constants, respectively. The bilinear exchange interaction can be written from Eq.16. *Ai i*, <sup>1</sup> can be either negative and positive depending on antiferromagnetic and ferromagnetic interactions, respectively. The antiparallel/perpendicular and parallel alignments of magnetization of nearest neighboring layers are energetically favorable for a negative/positive value of *i i*, <sup>1</sup> *B* . Biquadratic interaction for spin systems have been analysed for Ising system in detail (Chen, 1973; Erdem, 2001). The biquadratic term is smaller than the bilinear interaction term. Therefore, it can be neglected for most of the ferromagnetic systems. The indirect exchange energy depends on spacer thickness and even shows oscillatory behavior with spacer thickness (Ruderman, 1954; Yosida, 1957; Parkin, 1990, 1991(a), 19901(b), 1994). The current literature on single ultrathin films and multilayers is given in below at table (Layadi, 1990(a), 1990(b); Wigen, 1992; Zhang, 1994(a), 1994 (b); Goryunov, 1995; Ando, 1997; Farle; 1998; Platow, 1998; Schmool, 1998; Lindner, 2003; Sklyuyev, 2009; Topkaya, 2010; Erkovan, 2011).

This type exchange-coupling system is located in an external magnetic field, the magnetic moment in each layer. The suitable theoretical expression may be derived in order to deduce magnetic parameter for *ac* susceptibility. The equation of precession motion for magnetization of the *i th* layer in the spherical coordinates with the Bloch-Bloembergen type relaxation term can be written as

$$\frac{1}{\gamma} \frac{d\vec{M}}{dt} = \frac{1}{t\_i} \frac{\vec{M}}{M\_{i,s}} \times \vec{\nabla}\_{M\_i} E - \frac{\vec{M}\_{\theta i, \phi i}}{\gamma T\_2} - \frac{\vec{M}\_{z,i} - M\_{i,s}}{\gamma T\_1}. \quad \left(i = 1, \ n\right) \tag{29}$$

The matrices form for , 1 ,1 , , ,1 ,1 , , , , and *m m mm m m i i ii i i* of each magnetic layers calculated using the Eq.(29) with Eq.(20, 21,28).

Ferromagnetic Resonance 19

trilayers (one nonmagnetic and two magnetic)

4 2

 

 

 

five-layers (three magnetic and two nonmagnetic layer)

642

 

multilayer/n-layers (n magnetic and n-1 nonmagnetic layer)

2 1 0

*C C*

 

0

...

and the dc component of the

2 2 2 (2 2)/2

 

 

*n n C <sup>n</sup>*

 

**Figure 7.** Schematic representation of the (a) one layer, (b) three layer, (c) five layer and (d) n magnetic

for the FMR experiments and their theoretical calculations.

layer and their relative orientation of the equilibrium magnetization *M*

external magnetic field *H*

<sup>210</sup> *CCC* 0

1 0 *C C* 0

1 1 1 1 2 1 1 2 2 3 2 3 1 1 1 1 1 2 1 2 1 3 1 3 1 2 1 2 <sup>2</sup> 1 1 1 1 1 1 1 12 1 1 1 13 1 11 1 12 1 33 2 22 sin sin sin sin sin sin sin sin sin sin sin sin sin *s s s s s s s s ss ss s s EE E E E E t M t M tM tM tM tM E EE E E E tM tM tM tM tM tM E E tM tM* 2 2 2 2 3 2 3 3 1 2 2 1 1 2 2 2 1 3 2 2 1 3 1 3 2 3 2 3 <sup>2</sup> 12 2 2 2 2 2 3 2 23 2 21 2 22 3 33 3 3 3 13 3 2 3 sin sin sin sin sin sin sin sin sin sin sin sin sin *s s s s ss s s ss ss s E E E E t M t M tM tM EEE E E E tM tM tM tM tM tM EEEE tM tM tM t* 1 1 2 2 3 3 3 3 3 3 1 3 3 1 2 3 3 2 3 3 3 3 <sup>2</sup> 23 3 3 3 3 3 33 3 32 3 33 sin sin sin sin sin sin sin *s s s ss s s s s m m m m m E E m M t M t M EEEEE E tM tM tM tM tM tM* 0 (30)

1, 2, 3 number in this Eq.(30) corresponds to *ii i* 1, and 1 , respectively. Here 2 <sup>1</sup> *<sup>i</sup> T* . Then dispersion relation for ferromagnetic exchange-coupled *n*-layers has been calculated using the (2nx2n) matrix on the left-hand side of Eq. (30) in below in detail.

$$\left(\frac{\rho}{\gamma}\right)^{2n} + \mathbb{C}\_{\left(2n-2\right)/2} \left(\frac{\rho}{\gamma}\right)^{2n-2} + \dots + \mathbb{C}\_1 \left(\frac{\rho}{\gamma}\right)^2 + \mathbb{C}\_0 = 0\tag{31}$$

Here, *n* is the number of ferromagnetic layer. *C0*, *C1,*... etc. are constant related to ,, , , *i s ii ii ii tME E E* , sin *<sup>i</sup>* and <sup>2</sup> sin *<sup>i</sup>* . The dispersion relations for monolayer, trilayers, five-layers obtained from the Eq.(31). For tri-layers detail information are seen in ref. (Zhang, 1994(a); Schmool, 1998; Lindner, 2003). It is given that the dispersion relation for monolayer, trilayers, five-layers and multilayers/n-layers in Fig. 7.

1 2 1 2

sin sin

2

*T*

,, , , *i s ii ii ii tME E E* , sin *<sup>i</sup>* 

*s s*

*E E tM tM*

2 22

 

 

 

> 

 

> 

<sup>1</sup> *<sup>i</sup>*

 

calculated using the Eq.(29) with Eq.(20, 21,28).

1 3 1 3 2 3 2 3

*EEEE tM tM tM t*

 

 

> 

 

3 3 3 13 3 2 3

sin sin sin sin

 

 

*ss s*

 

 

The matrices form for , 1 ,1 , , ,1 ,1 , , , , and *m m mm m m* 

 

 

> 

> >

 

 

> 

 

*n n*

and <sup>2</sup> sin *<sup>i</sup>*

monolayer, trilayers, five-layers and multilayers/n-layers in Fig. 7.

 

 

 

1 1 1 1 2 1 1 2 2 3 2 3

<sup>2</sup> 1 1 1 1 1 1 1 12 1 1 1 13

sin sin sin sin sin sin sin sin

 

1 1 1 1 1 2 1 2 1 3 1 3

*s s ss ss*

1 2 2 1 1 2 2 2 1 3 2 2

*EEE E E E tM tM tM tM tM tM*

*ss s s ss*

2 21 2 22 3 33

1 3 3 1 2 3 3 2 3 3 3 3

*EEEEE E tM tM tM tM tM tM*

*ss s s s s*

3 33 3 32 3 33

*s s s s s s*

*EE E E E E t M t M tM tM tM tM E EE E E E tM tM tM tM tM tM*

1 11 1 12 1 33

 

sin sin sin

<sup>2</sup> 12 2 2 2 2 2 3 2 23

*s s s s*

*E E E E t M t M tM tM*

sin sin sin sin sin sin

sin sin sin

sin sin sin

 

 

> 

 

 

> 

1, 2, 3 number in this Eq.(30) corresponds to *ii i* 1, and 1 , respectively. Here

been calculated using the (2nx2n) matrix on the left-hand side of Eq. (30) in below in detail.

Here, *n* is the number of ferromagnetic layer. *C0*, *C1,*... etc. are constant related to

five-layers obtained from the Eq.(31). For tri-layers detail information are seen in ref. (Zhang, 1994(a); Schmool, 1998; Lindner, 2003). It is given that the dispersion relation for

2 2 2 2

 

 

 *i i ii i i* of each magnetic layers

> 

> >

 

> 

3 3 3 3 3

   

 

*E E m*

 

 

0

(30)

 

*m m m m m*

 

 

> 

 

> 

sin sin sin sin

 

 

*s s s*

(31)

. The dispersion relations for monolayer, trilayers,

monolayer

<sup>0</sup> *C* 0

2

 

*M t M t M*

. Then dispersion relation for ferromagnetic exchange-coupled *n*-layers has

(2 2)/2 <sup>1</sup> <sup>0</sup> ... 0

*C C <sup>n</sup> C*

2 2 2 2 3 2 3 3

<sup>2</sup> 23 3 3 3 3

 

**Figure 7.** Schematic representation of the (a) one layer, (b) three layer, (c) five layer and (d) n magnetic layer and their relative orientation of the equilibrium magnetization *M* and the dc component of the external magnetic field *H* for the FMR experiments and their theoretical calculations.

### **6. Example: Exchange spring (hard/soft) behaviour**

The Bloch wall, Néel line and magnetization vortex are well known properties for magnetic domain in magnetic systems. The multilayer structures are ordered layer by layer. The best of the sample for multilayer structure are exchange-spring systems. The equilibrium magnetic properties of nano-structured exchange-spring magnets may be studied in detail for some selected magnetic systems. The exchange systems are oriented from the exchange coupling between ferromagnetic and antiferromagnetic films or between two ferromagnetic films. This type structure has been extensively studied since the phenomenon was discovered (Meiklejohn, 1956, 1957). Kneller and Hawing have been used firstly the "exchange-spring" expression (Kneller, 1991). Spring magnet films consist of hard and soft layers that are coupled at the interfaces due to strong exchange coupling between relatively soft and hard layers. The soft magnet provides a high magnetic saturation, whereas the magnetically hard material provides a high coercive field. Skomski and Coey explored the theory of exchanged coupled films and predicted that a huge energy about three times of commercially available permanent magnets (120 MGOe) can be induced (Skomski, 1993; Coey, 1997). The magnetic reversal proceeds via a twisting of the magnetization only in the soft layer after saturating hard layers, if a reverse magnetic field that is higher than exchange field is applied. The spins are sufficiently closed to the interface are pinned by the hard layer, while those in deep region of soft layer rotate up to some extent to follow the applied field (Szlaferek, 2004). To be more specific, the angle of the rotation depends on the distance to the hard layer. That is the angle of rotating in a spiral spin structure similar to that of a Bloch domain wall. If the applied field is removed, the soft spins rotate back into alignment with the hard layer.

Ferromagnetic Resonance 21

spring magnets SmCo(hard)/Fe(soft). For theoretical analysis, the exchange-spring magnet SmCo/Fe is divided into subatomic multi-layers (*d*=2 Å), and the spins in each layer are characterized by the average magnetization *Mi*, and the uniaxial anisotropy constant *Ki*,

Sublayers are coupled by an exchange constant *Ai i*, <sup>1</sup> (Astalos, 1998; Fullerton, 1998, 1999; Jiang, 1999, 2002, 2005; Grimsditch, 1999; Scholz, 2000; Hellwig, 2000; Pollmann, 2001;

plane (where the external field is always perpendicular to the film normal) easy axis of the hard layer (Yldz, 2004(a), 2004(b)). The FMR spectra for exchange-spring magnet of SmCo/Fe have been analyzed using the Eqs. (26, 27, and 33) in Fig.9. Sm-Co (200 Å)/Fe (200 Å and 100 Å) bilayers have been grown on epitaxial 200 Å Cr(211) buffer layer on single crystal MgO(110) substrates by magnetron sputtering technique (Wüchner, 1997). To prevent oxidation Sm-Co/Fe film was coated with a 100 Å thick Cr layer. The FMR spectra for exchange-spring magnets of 200 Å and 100 Å Fe samples for different angles of the

There are three peaks that are one of them corresponds to the bulk mode and the remaining to the surface modes for 200 Å Fe sample. For more information about the FMR studies exchange spring magnets look at the ref. (Yildiz, 2004(a), 2004(b) ). Exchange-spring coupled magnets are promising systems for applications in perpendicular magnetic data recordingstorage devices and permanent magnet (Schrefl, 1993(a), 1993(b),1998, 2002; Mibu, 1997,

is the angle formed by the magnetization of the *i th* plane with the in-

**Figure 8.** Schematic illustration of phases of exchange spring magnets.

applied magnetic field in the film plane are presented in Fig.9.

(Fig.8).

Dumesnil, 2002). *<sup>i</sup>*

1998).

The general expression of the free energy for exchange interaction spring materials at film ( , 1 / 2 *i i* and / 2 *<sup>H</sup>* ) plane in spherical coordinate system as below.

$$\begin{split} E &= -\sum\_{i=1}^{N} \vec{H} \cdot \vec{M}\_{i} - \sum\_{i=1}^{N} K\_{a,i} \cos^{2} \phi\_{i} - \sum\_{i=1}^{N} K\_{b,i} \cos^{4} \phi\_{i} \\ &- \sum\_{i=1}^{N-1} \frac{A\_{i,i\mp 1}}{t^{2}} \cos \left(\phi\_{i\mp 1} - \phi\_{i}\right) - \sum\_{i=1}^{N-1} \frac{B\_{i,i\mp 1}}{t^{2}} \cos \left(\phi\_{i\mp 1} - \phi\_{i}\right)^{2} \end{split} \tag{32}$$

The expression is obtained as following using ' *i i* for magnetization's equilibrium orientations of each layer at a state of equilibrium under the external magnetic field.

$$\begin{aligned} \tan \phi\_i &= \\ \frac{HM\_i t^2 \sin \phi\_{\rm II} + A\_{i,i\mp 1} \sin \phi\_{i\mp 1} + 2B\_{i,i\mp 1} \sin \phi\_{i\mp 1} \cos(\phi\_i - \phi\_{i\mp 1})}{HM\_i t^2 \cos \phi\_{\rm II} + 2t^2 K\_{a,i} \cos \phi\_i + 4t^2 K\_{b,i} \cos^3 \phi\_i + A\_{i,i\mp 1} \cos \phi\_{i\mp 1} + 2B\_{i,i\mp 1} \cos \phi\_{i\mp 1} \cos(\phi\_i - \phi\_{i\mp 1})} \end{aligned} \tag{33}$$

In this example, second-order anisotropy term ( , 0 *Kb i* ) and biquadratic interaction constant ( , 1 0 *i i B* ) considered and the result obtained show as following as adapted with spring magnets SmCo(hard)/Fe(soft). For theoretical analysis, the exchange-spring magnet SmCo/Fe is divided into subatomic multi-layers (*d*=2 Å), and the spins in each layer are characterized by the average magnetization *Mi*, and the uniaxial anisotropy constant *Ki*, (Fig.8).

**Figure 8.** Schematic illustration of phases of exchange spring magnets.

20 Ferromagnetic Resonance – Theory and Applications

alignment with the hard layer.

 and / 2 *<sup>H</sup>* 

( , 1 / 2 *i i* 

 

'

*i*

tan

**6. Example: Exchange spring (hard/soft) behaviour** 

The Bloch wall, Néel line and magnetization vortex are well known properties for magnetic domain in magnetic systems. The multilayer structures are ordered layer by layer. The best of the sample for multilayer structure are exchange-spring systems. The equilibrium magnetic properties of nano-structured exchange-spring magnets may be studied in detail for some selected magnetic systems. The exchange systems are oriented from the exchange coupling between ferromagnetic and antiferromagnetic films or between two ferromagnetic films. This type structure has been extensively studied since the phenomenon was discovered (Meiklejohn, 1956, 1957). Kneller and Hawing have been used firstly the "exchange-spring" expression (Kneller, 1991). Spring magnet films consist of hard and soft layers that are coupled at the interfaces due to strong exchange coupling between relatively soft and hard layers. The soft magnet provides a high magnetic saturation, whereas the magnetically hard material provides a high coercive field. Skomski and Coey explored the theory of exchanged coupled films and predicted that a huge energy about three times of commercially available permanent magnets (120 MGOe) can be induced (Skomski, 1993; Coey, 1997). The magnetic reversal proceeds via a twisting of the magnetization only in the soft layer after saturating hard layers, if a reverse magnetic field that is higher than exchange field is applied. The spins are sufficiently closed to the interface are pinned by the hard layer, while those in deep region of soft layer rotate up to some extent to follow the applied field (Szlaferek, 2004). To be more specific, the angle of the rotation depends on the distance to the hard layer. That is the angle of rotating in a spiral spin structure similar to that of a Bloch domain wall. If the applied field is removed, the soft spins rotate back into

The general expression of the free energy for exchange interaction spring materials at film

, , 11 1

*A B t t*

orientations of each layer at a state of equilibrium under the external magnetic field.

 

> 

*NN N*

*ii i N N i i i i*

 

*E HM K K*

The expression is obtained as following using '

2

2 2 23

 

*i i*

*HM t A B HM t t K t K A B*

1 1

 

) plane in spherical coordinate system as below.

*i i i i*

*i i* 

,1 1 ,1 1 1

sin sin 2 sin cos( ) cos 2 cos 4 cos cos 2 cos cos( )

 

*i H ii i ii i i i i H ai i bi i ii i ii i i i*

In this example, second-order anisotropy term ( , 0 *Kb i* ) and biquadratic interaction constant ( , 1 0 *i i B* ) considered and the result obtained show as following as adapted with

, , , 1 1, 11 1

 

   

> 

> >

(32)

for magnetization's equilibrium

 

  (33)

2 4

1 1 , 1 , 1 <sup>2</sup> 2 2 1 1

cos cos

*i ai i bi i*

cos cos

Sublayers are coupled by an exchange constant *Ai i*, <sup>1</sup> (Astalos, 1998; Fullerton, 1998, 1999; Jiang, 1999, 2002, 2005; Grimsditch, 1999; Scholz, 2000; Hellwig, 2000; Pollmann, 2001; Dumesnil, 2002). *<sup>i</sup>* is the angle formed by the magnetization of the *i th* plane with the inplane (where the external field is always perpendicular to the film normal) easy axis of the hard layer (Yldz, 2004(a), 2004(b)). The FMR spectra for exchange-spring magnet of SmCo/Fe have been analyzed using the Eqs. (26, 27, and 33) in Fig.9. Sm-Co (200 Å)/Fe (200 Å and 100 Å) bilayers have been grown on epitaxial 200 Å Cr(211) buffer layer on single crystal MgO(110) substrates by magnetron sputtering technique (Wüchner, 1997). To prevent oxidation Sm-Co/Fe film was coated with a 100 Å thick Cr layer. The FMR spectra for exchange-spring magnets of 200 Å and 100 Å Fe samples for different angles of the applied magnetic field in the film plane are presented in Fig.9.

There are three peaks that are one of them corresponds to the bulk mode and the remaining to the surface modes for 200 Å Fe sample. For more information about the FMR studies exchange spring magnets look at the ref. (Yildiz, 2004(a), 2004(b) ). Exchange-spring coupled magnets are promising systems for applications in perpendicular magnetic data recordingstorage devices and permanent magnet (Schrefl, 1993(a), 1993(b),1998, 2002; Mibu, 1997, 1998).

Ferromagnetic Resonance 23

. This equation for SPR system so called modified Bloch for

fine particle magnets. The SPR microwave absorption is proportional to the imaginary part of the dynamic susceptibility. The line shape and resonance field for superparamagnet is obtained. The temperature evolution for the SPR line-width for nanoparticles can be calculated by ( ) *H T L x* . In this expression *<sup>T</sup>* is a saturation line-width at a temperature T, *Lx x x* ( ) coth( ) (1 / ) is the Langevin function with / *eff B x MVH k T* , *V* is the particle volume. The superparamagnetic (Chastellain, 2004; Dormer, 2005; Hamoudeh, 2007), coreshell nanoparticles and nanocrystalline nanoparticles (Woods, 2001; Wiekhorst, 2003; Tartaj, 2004) have been performed for possible biological applications (Sun,2005; Zhang, 2008). In additional, superparamagnetic nanoparticles have been used for hydrogels, memory effects

The EPR, FMR and SPR signals have been observed in Fig.10. The EPR signal has reached approaching peak level about 3000 G as seeing at Fig.10. It's symmetric and line width are narrower than resonance field, in generally. If EPR samples show crystallization, resonance field value starts to change. The EPR signal can be observed at lower temperature about 3000 G and the signal can show crystalline property. The signal is observed in two different areas at FMR spectra as the magnetic field is parallel and perpendicular to the film. The FMR spectra are observed at low field when the magnetic field is parallel to the film, in generally. On the other hand, the FMR spectra are observed at highest field when the magnetic field is perpendicular to the film. For other conditions FMR signals are observed between these two conditions for thin films. FMR spectra can be seen a wide range of field

**Figure 10.** (**a**) The EPR/ESR experimental signal for La0.7Ca0.3MnO3 samples at room temperature (see, Kartopu, 2011 (b)). (**b**) Theoretical FMR spectra calculated from Eq. (26) with Eq.(27) at parallel (

dot line in online) and experimental FMR spectra for Ni NWs (*P*= 29,6; *L*=0,8 μm, *τ*=13 ) (see for detail, Kartopu, 2011 (a)). (**d**) The theoretical SPR signal for superparamagnet by Eq.(34) at room temperature.

0 ;~7000 G) position of OPG case. (**c**) The theoretical (red-

Here, 2 1 *<sup>H</sup>*

**8. Result and discussions** 

so as to the thin films are full.

o

90 ;~ 2000 G) and perpendicular ( <sup>o</sup>

*T* , ( ) *Hr* 

and electronic devices (Raikher, 2003; Sasaki, 2005; Heim, 2007).

**Figure 9.** FMR spectra for SmCo(200 Å)/Fe(200 Å) (black line) and SmCo(200 Å)/Fe(100 Å) (blue line) samples. These FMR spectra originated from the iron/soft layers.

### **7. Superparamagnetic resonance**

Magnetic nanoparticles have been steadily interested in science and nanotechnology. As the dimensions of magnetic nanoparticles decrease to the nanometer scale, these nanoparticles start to exhibit new and interesting physical properties mainly due to quantum size effects (Yalçn, 2004(a), 2008(b), 2012). A single domain particle is commonly referred to as superparamagnetic (Held, 2001; Diaz, 2002; Fonseca, 2002). The superparamagnetic/singledomain nanoparticles are important for non surgical interfere of human body. Even the intrinsic physical characteristics of nanoparticles are observed to change drastically compared to their macroscopic counterparts. Stoner-Wohlfarth (Stoner, 1948) and Heisenberg model (Heisenberg, 1928) to describe the fine structure were firstly used in detail. A simple (Bakuzis, 2004) and the first atomic-scale models of the ferrimagnetic and heterogeneous systems in which the exchange energy plays a central role in determining the magnetization of the NPs, were studied (Kodama, 1996, 1999; Kodama & Berkowitz, 1999). Superparamagnetic resonance (SPR) studies of fine magnetic nanoparticles is calculated a correlation between the line-width and the resonance field for superparamagnetic structures (Berger, 1997, 1998, 2000(a), 2000(b), 2001; Kliava, 1999). The correlation of the line-width and the resonance field is calculated from Bloch-Bloembergen equation of motion for magnetization. The SPR spectra, line width and resonance field may be analyzed by using the Eq.(34) in below. The equation of motion for magnetization with Bloch-Bloembergen type relaxation term for FMR adapted for superparamagnetic structures from Eqs.(18) and (19) in below.

$$\chi\_2(H) = \frac{1}{\pi} \frac{\Lambda\_H \left(\Lambda\_H^2 + H^2 + H\_r^2\right)}{\left(\Lambda\_H^2 + \left(H - H\_r\right)^2\right) \cdot \left(\Lambda\_H^2 + \left(H + H\_r\right)^2\right)}\tag{34}$$

Here, 2 1 *<sup>H</sup> T* , ( ) *Hr* . This equation for SPR system so called modified Bloch for fine particle magnets. The SPR microwave absorption is proportional to the imaginary part of the dynamic susceptibility. The line shape and resonance field for superparamagnet is obtained. The temperature evolution for the SPR line-width for nanoparticles can be calculated by ( ) *H T L x* . In this expression *<sup>T</sup>* is a saturation line-width at a temperature T, *Lx x x* ( ) coth( ) (1 / ) is the Langevin function with / *eff B x MVH k T* , *V* is the particle volume. The superparamagnetic (Chastellain, 2004; Dormer, 2005; Hamoudeh, 2007), coreshell nanoparticles and nanocrystalline nanoparticles (Woods, 2001; Wiekhorst, 2003; Tartaj, 2004) have been performed for possible biological applications (Sun,2005; Zhang, 2008). In additional, superparamagnetic nanoparticles have been used for hydrogels, memory effects and electronic devices (Raikher, 2003; Sasaki, 2005; Heim, 2007).

### **8. Result and discussions**

22 Ferromagnetic Resonance – Theory and Applications

**Figure 9.** FMR spectra for SmCo(200 Å)/Fe(200 Å) (black line) and SmCo(200 Å)/Fe(100 Å) (blue line)

Magnetic nanoparticles have been steadily interested in science and nanotechnology. As the dimensions of magnetic nanoparticles decrease to the nanometer scale, these nanoparticles start to exhibit new and interesting physical properties mainly due to quantum size effects (Yalçn, 2004(a), 2008(b), 2012). A single domain particle is commonly referred to as superparamagnetic (Held, 2001; Diaz, 2002; Fonseca, 2002). The superparamagnetic/singledomain nanoparticles are important for non surgical interfere of human body. Even the intrinsic physical characteristics of nanoparticles are observed to change drastically compared to their macroscopic counterparts. Stoner-Wohlfarth (Stoner, 1948) and Heisenberg model (Heisenberg, 1928) to describe the fine structure were firstly used in detail. A simple (Bakuzis, 2004) and the first atomic-scale models of the ferrimagnetic and heterogeneous systems in which the exchange energy plays a central role in determining the magnetization of the NPs, were studied (Kodama, 1996, 1999; Kodama & Berkowitz, 1999). Superparamagnetic resonance (SPR) studies of fine magnetic nanoparticles is calculated a correlation between the line-width and the resonance field for superparamagnetic structures (Berger, 1997, 1998, 2000(a), 2000(b), 2001; Kliava, 1999). The correlation of the line-width and the resonance field is calculated from Bloch-Bloembergen equation of motion for magnetization. The SPR spectra, line width and resonance field may be analyzed by using the Eq.(34) in below. The equation of motion for magnetization with Bloch-Bloembergen type relaxation term for FMR adapted for superparamagnetic structures from Eqs.(18) and

2 2 22 2 <sup>1</sup> ( ) ( )

2 22

*H H*

(34)

() () *HH r H rH r*

*HH HH*

samples. These FMR spectra originated from the iron/soft layers.

*H*

**7. Superparamagnetic resonance** 

(19) in below.

The EPR, FMR and SPR signals have been observed in Fig.10. The EPR signal has reached approaching peak level about 3000 G as seeing at Fig.10. It's symmetric and line width are narrower than resonance field, in generally. If EPR samples show crystallization, resonance field value starts to change. The EPR signal can be observed at lower temperature about 3000 G and the signal can show crystalline property. The signal is observed in two different areas at FMR spectra as the magnetic field is parallel and perpendicular to the film. The FMR spectra are observed at low field when the magnetic field is parallel to the film, in generally. On the other hand, the FMR spectra are observed at highest field when the magnetic field is perpendicular to the film. For other conditions FMR signals are observed between these two conditions for thin films. FMR spectra can be seen a wide range of field so as to the thin films are full.

**Figure 10.** (**a**) The EPR/ESR experimental signal for La0.7Ca0.3MnO3 samples at room temperature (see, Kartopu, 2011 (b)). (**b**) Theoretical FMR spectra calculated from Eq. (26) with Eq.(27) at parallel (

o 90 ;~ 2000 G) and perpendicular ( <sup>o</sup> 0 ;~7000 G) position of OPG case. (**c**) The theoretical (reddot line in online) and experimental FMR spectra for Ni NWs (*P*= 29,6; *L*=0,8 μm, *τ*=13 ) (see for detail, Kartopu, 2011 (a)). (**d**) The theoretical SPR signal for superparamagnet by Eq.(34) at room temperature. FMR spectra are similar to thin films at nanowire samples. In case of occupancy rate is that as the theoretical *P* 33% for Nickel (Ni) it behaviors like thin film. But, in case of occupancy rate is that as the theoretical *P* 33% it behaviors different from thin film. This situation is clearly visible from 2 13 2 *H M P KM eff <sup>S</sup> u s* . If the occupancy rate is *P* 33% sample's signals show the opposite behavior according to thin film FMR signals. Look at for more information (Kartopu, 2011 (a)). This is perceived as changes the direction of the easy axis. The changes of easy axes depend on magnetization (Terry, 1917) and porosity (Kartopu, 2011 (a)) for magnetic materials/transition elements. The SPR signal is similar to EPR signal. SPR peak may show symmetrical properties both at room temperature and low temperatures. The SPR signal is in the form of Lorentzian and Gaussian line shapes at all temperature range. Specially prepared nanoparticles SPR peak exhibit shift in symmetry. The line width of SPR peak expands at low temperature.

Ferromagnetic Resonance 25

Ament, W. S.; Rado, G. T. (1955). Electromagnetic effects of spin wave resonance in

Ando, Y.; Koizumi, H.; Miyazaki, T. (1997). Exchange coupling energy determined by ferromagnetic resonance in 80 Ni-Fe/Cu multilayer films. *J. Magn. Magn. Mater*. Vol.

Anisimov, A. N.; Farle, M.; Poulopoulos, P.; Platow, W.; Baberschke, K.; Isberg, P.; Wäppling, R.; Niklasson, A. M. N.; Eriksson, O. (1999). Orbital magnetism and magnetic anisotropy probed with ferromagnetic resonance. *Phys. Rev. Lett*. Vol. 82, pp. 2390. An, S. Y.; Krivosik, P.; Kraemer, M. A.; Olson, H. M.; Nazarov, A. V.; Patton, C. E. (2004). High power ferromagnetic resonance and spin wave instability processes in permalloy

Anderson, P. W. (1953). Exchange narrowing in paramagnetic resonance. *Rev. Mod. Phy.* Vol.

Arias, R.; Mills, D. L. (1999). Extrinsic contributions to the ferromagnetic resonance response

Arias, R.; Mills, D. L. (2003). Theory of collective spin waves and microwave response of

Arkad'ev, V.K. (1913). The Reflection of Electric Waves from a Wire, *Sov. Phys.-JETP*, Vol.

Artman, J. O. (1957). Ferromagnetic resonance in metal single crystals. *Phys. Rev.* Vol. 105,

Artman, J. O. (1979). Domain mode FMR in materials with *K1* and *Ku*. *J. Appl. Phys*. Vol. 50,

Astalos, R. J.; Camley, R. E. (1998). Magnetic permeability for exchange-spring magnets:

Aswal, D.K.; Singh, A.; Kadam, R.M.; Bhide, M.K.; Page, A.G.; Bhattacharya, S.; Gupta, S.K.; Yakhmi, J.V.; Sahni, V.C. (2005). Ferromagnetic resonance studies of nanocrystalline

Azzerboni, B.; Asti, G.; Pareti, L.; Ghidini, M. (2006). Magnetic nanostructures in modern technology, spintronics, magnetic MEMS and recording. Proceedings of the NATO advanced study institute on magnetic nanostructures for micro-electromechanical systems and spintronic applications catona, *Published by Springer.* Italy. ISBN 978-1-

Backes, D.; Bedau, D., Liu, H., Langer, J.; Kent, A.D. (2012). Characterization of interlayer interactions in magnetic random access memory layer stacks using ferromagnetic

Baek, J.S.; Min, S.G.; Yu, S.C.; Lim, W.Y. (2002). Ferromagnetic resonance of Fe–Sm–O thin

Bai, Y .; Xu, F.; Qiao, L. (2012). The twice ferromagnetic resonance in hexagonal ferrite single

Bakker, F.L.; Flipse, J.; Slachter, A.; Wagenaar, D.; van Wees, B.J. (2012). Thermoelectric detection of ferromagnetic resonance of a nanoscale ferromagnet. *Phys. Rev. Lett.* Vol.

ferromagnetic metals. *Phys. Rev*. Vol. 97, pp. 1558.

thin films. *J. Appl. Phys.* Vol. 96, pp. 1572.

in ultrathin films, *Phys. Rev. B.* Vol. 60, pp. 7395.

ferromagnetic nanowire arrays. *Phys. Rev. B.* Vol. 67, pp. 094423.

application to Fe/Sm-Co. *Phys. Rev. B.* Vol. 58, pp. 8646.

La0.6Pb0.4 MnO3 thin films. *Mater. Lett.* Vol. 59, pp. 728.

resonance. *J. Appl. Phys*. Vol. 111, pp. 07C721.

rod and paired rods. *Phys. Lett. A*. Vol.376, pp. 563.

films. *J. App. Phys*. Vol. 93, pp. 7604.

166, pp. 75.

25, pp. 269.

pp. 74.

pp. 2024.

4020-6337-4.

108, pp. 167602.

45A, issue 45, pp. 312.

### **Author details**

Orhan Yalçn *Niğde University, Niğde, Turkey* 

### **Acknowledgement**

I would like to thank **Muhittin Öztürk** and **Songül Özüm** of Niğde University for valuable discussions an the critical reading of the chapter. This study was supported by Research found (Grant No. FEB2012/12) of Niğde University.

### **9. References**


Ament, W. S.; Rado, G. T. (1955). Electromagnetic effects of spin wave resonance in ferromagnetic metals. *Phys. Rev*. Vol. 97, pp. 1558.

24 Ferromagnetic Resonance – Theory and Applications

**Author details** 

*Niğde University, Niğde, Turkey* 

found (Grant No. FEB2012/12) of Niğde University.

in NiMn alloys. *Solid State Commun*.Vol. 87, pp. 1067.

in materials science, Vol. 94, ISBN 978-3-540-49334-1.

**Acknowledgement** 

**9. References** 

307, pp. 250.

pp. 298.

pp. 3319.

Orhan Yalçn

FMR spectra are similar to thin films at nanowire samples. In case of occupancy rate is that as the theoretical *P* 33% for Nickel (Ni) it behaviors like thin film. But, in case of occupancy rate is that as the theoretical *P* 33% it behaviors different from thin film. This

*P* 33% sample's signals show the opposite behavior according to thin film FMR signals. Look at for more information (Kartopu, 2011 (a)). This is perceived as changes the direction of the easy axis. The changes of easy axes depend on magnetization (Terry, 1917) and porosity (Kartopu, 2011 (a)) for magnetic materials/transition elements. The SPR signal is similar to EPR signal. SPR peak may show symmetrical properties both at room temperature and low temperatures. The SPR signal is in the form of Lorentzian and Gaussian line shapes at all temperature range. Specially prepared nanoparticles SPR peak exhibit shift in

I would like to thank **Muhittin Öztürk** and **Songül Özüm** of Niğde University for valuable discussions an the critical reading of the chapter. This study was supported by Research

Adeyeye, A.O.; Bland, J.A.C.; Daboo, C.; Hasko, D.G. (1997). Magnetostatic interactions and magnetization reversal in ferromagnetic wires. *Phys. Rev. B*. Vol. 56, pp. 3265. Aktaş, B. (1993). Clear evidence for field induced unidirectional exchange surface anisotropy

Aktaş, B.; Özdemir, M. (1994). Simulated spin wave resonance absorption curves for ferromagnetic thin films and application to NiMn films. *Physica B*. Vol. 119, pp. 125. Aktaş, B. (1997). FMR properties of epitaxial Fe3O4 films on MgO(100). *Thin Solid Films*. Vol.

Aktaş, B.; Özdemir, M.; Yilgin, R.; Öner, Y.; Sato, T.; Ando, T. (2001). Thickness and temperature dependence of magnetic anisotropies of Ni77 Mn23 films. *Physica B.* Vol. 305,

Akta*ş*, B.; Yildiz, F.; Rameev, B.; Khaibullin, R.; Tagirov, L.; Özdemir, M. (2004). Giant room temperature ferromagnetism in rutile TiO2 implanted by Co. *Phys. stat. sol. (c).* Vol. 12,

Aktaş, B.; Tagirov, L. & Mikailov, F. (October, 2006). *Magnetic Nanostructures,* Springer Series

. If the occupancy rate is

situation is clearly visible from 2 13 2 *H M P KM eff <sup>S</sup> u s*

symmetry. The line width of SPR peak expands at low temperature.


Bakuzis, A.F.; Morais, P.C. (2004). Magnetic nanoparticle systems: an Ising model approximation. *J. Magn. Magn. Mater*. Vol. 272-276, pp. e1161.

Ferromagnetic Resonance 27

Boulle, O.; Malinowski, G.; Kläui, M. (2011). Current-induced domain wall motion in nanoscale ferromagnetic elements. *Materials Science and Engineering* R. Vol. 72, pp. 159. Brown, F.M. (1962).Magnetostatic principles in ferromagnetism. *North Holland Publishing* 

Brustolon, M.; Giamello, E. (2009). Electron paramagnetic resonance. *John Wiley & Sons, Inc.*

Budak, S.; Yildiz, F.; Özdemir, M.; Aktaş, B. (2003). Electron spin resonance studies on single

Buschow, K.H.J.; de Boer, F.R. (2004). Physics of magnetism and magnetic materials. *Kluwer* 

Callen, H.B. (1958). A ferromagnetic dynamical equation. *J. Phys. Chem. Solids*. Vol. 4, pp.

Can, M.M.; Coşkun, M.; Frat, T. (2012). A comparative study of nanosized iron oxide particles; magnetite (Fe3O4), maghemite (γ-Fe2O3) and hematite (α-Fe2O3), using

Celinski, Z.; Urquhart, B.; Heinrich, B. (1997). Using ferromagmetic resonance to measure the magnetic moments of ultrathin films, *J. Magn. Magn. Mater.* Vol. 166, pp. 6. Celinski, Z.; Heinrich, B. (1991). Ferromagnetic resonance line width of Fe ultrathin films

Chappert, C.; Le Dang, K.; Beauvillain, P.; Hurdequint, H.; Renard, D. (1986). Ferromagnetic resonance studies of very thin cobalt films on a gold substrate. *Phys. Rev. B.* Vol. 34, pp.

Chastellain, M.; Petri, A.; Hofmann, H. (2004). Particle size investigations of a multi-step synthesis of PVA coated superparamagnetic nanoparticles. *J. Colloid. İnterf. Sci.* Vol. 278,

Chen, Y.S.; Cheng, C.W.; Chern, G.; Wu, W.F.; Lin, J.G. (2012). Ferromagnetic resonance probed annealing effects on magnetic anisotropy of perpendicular CoFeB/MgO bilayer.

Chen, S.H.; Chang, C.R.; Xiao, J.Q.; Nikolić, K.B. (2009). Spin and charge pumping in magnetic tunnel junctions with precessing magnetization: A nonequilibrium green

Chen, H.H.; Levy, P.M. (1973). Dipole and quadrupole phase transitions in spin-1 models.

Chikazumi, S. (1997). Physics of ferromagnetism. *Oxford University Press.* ISBN 0-19-851776-

Cochran, J.F. (1995). Light scattering from ultrathin magnetic layers and bilayers in magnetic ultrathin films. Heinrich, B.; Bland, J.A.C. (Eds.) *Springer, Berlin, Heidelberg* Vol. II, pp. 222. B. Hillebrands: Brillouin light scattering in magnetic superlattices. ibid pp. 258. Cochran, J.F.; Kambersky, V. (2006). Ferromagnetic resonance in very thin films. *J. Magn.* 

Coey, J.M.D. (2009). Magnetism and magnetic materials. *Cambridge University Press*. ISBN-13

Coey, J.M.D. (1997). Permanent magnetism. *Solid State Comm*. Vol. 102, pp. 101.

crystalline Fe3O4 films. *J. Magn. Magn. Mater.* Vol. 258–259, pp. 423.

ferromagnetic resonance. *J. Alloy. Compd.* Vol. 542, pp. 241.

grown on a bcc substrate. *J. Appl. Phys*. Vol. 70, pp.5936.

function approach. *Phys. Rev. B.* Vol. 79, pp. 054424.

*Company*.

256.

3192.

pp. 353.

9.

*J. Appl. Phys*. Vol. 111, pp.07C101.

*Phys. Rev. B. Vol.* 7, 4267.

*Magn. Mater.* Vol. 302, pp. 348.

978052181614-4.

ISBN 978-0-470-25882.

*Academic Publishers*. ISBN: 0-306-47421-2.


Boulle, O.; Malinowski, G.; Kläui, M. (2011). Current-induced domain wall motion in nanoscale ferromagnetic elements. *Materials Science and Engineering* R. Vol. 72, pp. 159.

26 Ferromagnetic Resonance – Theory and Applications

*Appl. Phys*. Vol. 111, pp. 07A503.

*Condens. Matter.* Vol. 12, pp. 9347.

*Magn. Mater*. Vol. 234, pp. 535.

*Magn. Mater.* Vol. 192, pp. 258.

transition metals. *Phys. Rev*. *B*. Vol. 10, pp. 179.

978-3-7643-8999-4.

540-57407-7.

*Rev*. Vol. 78, pp. 572.

*Phys. Rev*. Vol. 93, pp. 72.

*Rev. Lett.* Vol. 82, pp. 2796.

*Cond-mat. Mes-hall.* pp. 1.

Bakuzis, A.F.; Morais, P.C. (2004). Magnetic nanoparticle systems: an Ising model

Baseglia, L.; Warden, M.; Waldner, F.; Hutton, S.L.; Drumheller, J.E.; He, Y.Q.; Wigen, P.E.; Maryško M. (1988). Derivation of the resonance frequency from the free energy of

Beguhn, S.; Zhou, Z.; Rand, S.; Yang, X.; Lou, J.; Sun, N.X. (2012). A new highly sensitive broadband ferromagnetic resonance measurement system with lock-in detection. *J.* 

Berger, R.; Bissey, J.; Kliava, J. (2000(a)). Lineshapes in magnetic resonance spectra. *J. Phys.:* 

Berger, R.; Kliava, J.; Bissey, J.C. (2000(b)). Magnetic resonance of superparamagnetic iron-

Berger, R.; Bissey, J.C.; Kliava, J.; Soulard, B. (1997). Superparamagnetic resonance of ferric

Berger, R.; Bissey, J. C.; Kliava, J.; Daubric, H.; Estournѐs, C. (2001). Temperature dependence of superparamagnetic resonance of iron oxide nanoparticles. *J. Magn.* 

Berger, R.; Kliava, J.; Bissey, J. C.; Baїettoz, V. (1998). Superparamagnetic resonance of annealed iron containing borate glass. *J. Phys.: Condens. Matter*. Vol. 10, pp. 8559. Birkhäuser Verlag, A.G. (2007). Spin glasses statics and dynamics. *Basel, Boston,* Berlin. ISBN

Birlikseven, C.; Topacli, C.; Durusoy, H.Z.; Tagirov, L.R.; Koymen, A.R.; Aktaş, B. (1999 (a)). Magnetoresistance, magnetization and FMR study of Fe/Ag/Co multilayer film. *J. Magn.* 

Birlikseven, C.; Topacli, C.; Durusoy, H.Z.; Tagirov, L.R.; Koymen, A.R.; Aktaş, B. (1999 (b)). Layer-sensitive magnetization, magnetoresistance and ferromagnetic resonance (FMR)

Bhagat, S.M.; Anderson, J.R.; Wu, N. (1967). Influence of the anomalous skin effect on the

Bhagat, S.M.; Lubitz, P. (1974). Temperature variation of ferromagnetic relaxation in the 3d

Bland, J.A.C.; Heinrich, B. (1994). Ultrathin magnetic structures I: An introduction to the electronic magnetic and structural properties. *Springer-Verlag Berlin Heidelberg*. ISBN 3-

Bloembergen, B. (1950). On the ferromagnetic resonance in nickel and supermalloy. *Phys.* 

Bloembergen, N.; Wang, S. (1954). Relaxation effects in para- and ferromagnetic resonace.

Borchers, J.A.; Dura, J.A.; Unguris, J.; Tulchinsky, D.; Kelley, M.H.; Majkrzak, C.F. (1998). Observation of antiparallel magnetic order in weakly coupled CoyCu multilayers. *Phys.* 

Bose, T.; Trimper, S. (2012). Nonlocal feedback in ferromagnetic resonance. *Arxiv: 1204-5342.* 

study of NiFe/Ag/CoNi trilayer film. *J. Magn. Magn. Mater*. Vol. 202, pp. 342.

ferromagnetic resonance linewidth in iron. *Phys. Rev*. Vol. 155, pp. 510.

containing nanoparticles in annealed glass. *J. Appl. Phys*. Vol. 87, pp. 7389.

ions in devitrifield borate glass. *J. Magn. Magn. Mater*. Vol. 167, pp. 129.

approximation. *J. Magn. Magn. Mater*. Vol. 272-276, pp. e1161.

ferromagnets. Phys. Rev. B. Vol. 38, pp. 2237.


Cofield, M.L.; Glocker, D.; Gau, J.S. (1987). Spin-wave resonance in CoCr magnetic thin films. *J. Appl. Phys.* Vol. 61, pp. 3810.

Ferromagnetic Resonance 29

Dormer, K.; Seeney, C.; Lewelling, K.; Lian, G.; Gibson, D.; Johnson, M. (2005). Epithelial internalization of superparamagnetic nanoparticles and response to external magnetic

Dreher, L.; Weiler, M.; Pernpeintner, M.; Huebl, H.; Gross, R.; Brandt, M.S.; Goennenwein, S.T.B. (2012). Surface acoustic wave-driven ferromagnetic resonance in nickel thin films:

Dubowik, J. (1996). Shape anisotropy of magnetic heterostructures. *Phys. Rev. B.* Vol. 54,

Dumesnil, K.; Dufour, C.; Mangin, Ph.; Rogalev, A. (2002). Magnetic springs in exchangecoupled DyFe2/YFe2 superlattices: An element-selective x-ray magnetic circular

Duraia, E.M.; Abdullin, Kh.A. (2009). Ferromagnetic resonance of cobalt nanoparticles used as a catalyst for the carbon nanotubes synthesis. *J. Magn. Magn. Mater.* Vol. 321, pp. 69. Durusoy, H.Z.; Aktaş, B.; Yilgin, R.; Terada, N.; Ichikawa, M.; Kaneda, T.; Tagirov, L.R. (2000). New technique for measuring the microwave penetration depth in high- Tc

Encinas-Oropesa, A.; Demand, M.; Piraux, L.; Huynen, I.; Ebels, U. (2001). Dipolar interactions in arrays of nickel nanowires studied by ferromagnetic resonance. *Phys.* 

Erdem, R.; Keskin, M. (2001). Dynamics of a spin-1 Ising system in the neighborhood of

Erkovan, M.; Öztürk, S.T.; Topkaya, R.; Özdemir, M.; Aktaş, B.; Öztürk, O. (2011). Ferromagnetic resonance investigation of Py/Cr multilayer system*. J. Appl. Phys*. Vol.

Estévez, D.C.; Betancourt, I.; Montiel, H. (2012). Magnetization dynamics and ferromagnetic resonance behavior of melt spun FeBSiGe amorphous alloys. *J. Appl. Phys*. Vol. 112, pp.

Fan, W.J.; Qiu, X.P.; Shi, Z.; Zhou, S.M.; Cheng, Z.H. (2010). Correlation between isotropic ferromagnetic resonance field shift and rotatable anisotropy in polycrystalline

Farle, M. (1998). Ferromagnetic resonance of ultrathin metallic layers. *Rep. Prog. Phys*. Vol.

Farle, M.; Lindner, J.; Baberschke, K. (2000). Ferromagnetic resonance of Ni(111) on Re(0001*).* 

Fermin, J.R.; Azevedo, A.; Aguiar, F.M.; Li, B.; Rezende, S.M. (1999). Ferromagnetic resonance linewidth and anisotropy dispersion in thin Fe films. *J. Appl. Phys.* Vol. 85,

Fischer, H.; Mastrogiacomo, G.; Löffler, J.F.; Warthmann, R.J.; Weidler, P.G.; Gehring, A.U. (2008). Ferromagnetic resonance and magnetic characteristics of intact magnetosome chains in Magnetospirillum gryphiswaldense. *Earth and Planetary Science Letters*. Vol.

Fletcher, R.C.; Le Craw, R.C.; Spencer, E.G. (1960). Electron spin relaxation in ferromagnetic

Theory and experiment. *Arxiv:Cond-mat. Mes-hall.* Vol. 1208, pp. 1.

dichroism study. *Phys. Rev. B.* Vol. 65, pp. 094401.

superconducting thin films. *Phys. B*. Vol. 284-288, pp. 953.

equilibrium states, *Phys. Rev. E*. Vol. 64, pp. 026102.

NiFe/FeMn bilayers. *Thin Solid Films.* Vol. 518, pp. 2175.

*J. Magn. Magn. Mater.* Vol. 212, pp. 301.

insulators. *Phys. Rev.* Vol. 117, pp. 955.

field. *Biomaterials.* Vol. 26, pp. 2061.

*Rev. B*. Vol. 63, pp. 104415.

110, pp. 023908.

053923.

61, pp. 755.

pp. 7316.

270, pp. 200.

pp.1088.


Dormer, K.; Seeney, C.; Lewelling, K.; Lian, G.; Gibson, D.; Johnson, M. (2005). Epithelial internalization of superparamagnetic nanoparticles and response to external magnetic field. *Biomaterials.* Vol. 26, pp. 2061.

28 Ferromagnetic Resonance – Theory and Applications

*Phys*. Vol. 25, pp. 310.

199.

25, pp. 239.

*Magnetism.* Vol. I

Vol. 36, pp. 3520.

*Appl. Phys.* Vol. 41, pp. 87.

*Magn. Mater*.Vol. 249, pp. 228.

pp. 21.

films. *J. Appl. Phys.* Vol. 61, pp. 3810.

Cofield, M.L.; Glocker, D.; Gau, J.S. (1987). Spin-wave resonance in CoCr magnetic thin

Coşkun, R.; Okutan, M.; Yalçn, O.; Kösemen, A. (2012). Electric and magnetic properties of

Crittenden, E.C.Jr; Hoffman, R.W. (1953). Thin films of ferromagnetic materials. *Rev. Mod.* 

Cullity, B.D.; Graham, C.D. (1990). Introduction to Magnetic Materials. *Wiley.* New York. pp.

Damon, R.W. (1953). Relaxation effect in the ferromagnetic resonance. *Rev. Mod. Phys*. Vol.

Damon, R.W. (1963). Ferromagneticre sonance at high power in Rado, G.T.; Suhl, H. (Eds.).

da Silva, E.C.; Meckenstock, R.; von Geisau, O.; Kordecki, R.; Pelzl, J.; Wolf, J.A.; Grünberg, P. (1993). Ferromagnetic resonance investigations of anisotropy fields of Fe(001)

Davis, J.A. (1965). Effect of surface pinning on the magnetization of thin films. *J. Appl. Phys.*

Demokritov, S.; Rücker, U.; Grünberg, P. (1996). Enhancement of the Curie temperature of epitaxial EuS(100) films caused by growth dislocations. *J. Magn. Magn. Mater.* Vol. 163,

Demokritov, S.; Rücker, U.; Arons, R.R.; Grünberg, P. (1997). Antiferromagnetic interlayer

De Wames, R.E.; Wolfram, T. (1970). Dipole-exchange spinwaves in ferromagnetic films*. J.* 

Demand, M.; Encinas-Oropesa, A.; Kenane, S.; Ebels, U.; Huynen, I.; Pirax, L. (2002). Ferromagnetic resonance studies of nickel and permalloy nanowire arrays. *J. Magn.* 

De Biasi, E.; Lima, Jr. E.; Ramos, C.A.; Butera, A.; Zysler, R.D. (2013). Effect of thermal fluctuations in FMR experiments in uniaxial magnetic nanoparticles: Blocked vs.

De Biasi, R.S.; Gondim, E.C. (2006). Use of ferromagnetic resonance to determine the size distribution of γ-Fe2O3 nanoparticles. Solid State Commun. Vol. 138, pp. 271. De Cos, D.; Arribas, A.G.; Barandiaran, J.M. (2006). Ferromagnetic resonance in gigahertz magneto-impedance of multilayer systems. *J. Magn. Magn. Mater.* Vol. 304, pp. 218. Diaz, L.L.; Torres, L.; Moro, E. (2002). Transition from ferromagnetism to superparamagnetism on the nanosecond time scale. *Phys. Rev. B.* Vol. 65, pp. 224406. Dillon, J.F.Jr.; Gyorgy, E.M.; Rupp, L.W.Jr.; Yafet, Y.; Testardi,L.R. (1981). Ferromagnetic resonance in compositionally modulated CuNi films. *J. Appl. Phys.* Vol. 52, pp. 2256. Ding, J.; Kostylev, M.; Adeyeye, A.O. (2012). Broadband ferromagnetic resonance spectroscopy of permalloy triangular nanorings. *J. Appl. Phys*. Vol. 100, pp. 062401.

coupling in epitaxial Fe/EuS (100) bilayers. J. Appl. Phys. Vol. 81, pp. 5348.

superparamagnetic regimes. *J. Magn. Magn. Mater.* Vol. 326, pp. 138.

hydrogels doped with Cu ions. Acta. Phys. Pol. A*.* Vol. 122, pp. 683.

Cullity, B.D.; Graham, C.D. (2009). *Introduction to magnetic materials* 2nd Edition.

epitaxial layers. *J. Magn. Magn. Mater.* Vol. 121, pp. 528.


Fonseca, F.C.; Goya, G.F.; Jardim, R.F.; Muccillo, R.; Carreño, N.L.V.; Longo, E.; Leite, E.R. (2002). Superparamagnetism and magnetic properties of Ni nanoparticles embedded in SiO2. *Phys. Rev. B*. Vol. 66, pp.104406.

Ferromagnetic Resonance 31

Grünberg, P. (1980). Brillouin scattering from spin waves in thin ferromagnetic films. *J.* 

Grünberg, P.; Mayr, C.M.; Vach, W. (1982). Determination of magnetic parameters by means of brillouin scattering. Examples: Fe, Ni, Ni0.8Fe0.2 . *J. Magn. Magn. Mater.* Vol. 28, pp.

Grünberg, P.; Barnas, J.; Saurenbach, F.; Fuβ, J.A.; Wolf, A.; Vohl, M. (1991). Layered magnetic structures: antiferromagnetic type interlayer coupling and magnetoresistance

Grünberg, P.; Demokritov, S.; Fuss, A.; Schreiber, R.; Wolf, J.A.; Purcell, S.T. (1992). Interlayer exchange, magnetotransport and magnetic domains in Fe/Cr layered

Grünberg, P. (2000). Layered magnetic structures in research and application. *Acta mater.* Vol. 48, pp. 239. Grünberg, P. (2001). Layered magnetic structures: history, facts and

Guimarães, A.P. (2009). Principles of nanomagnetism. *Springer-Verlag Berlin Heidelberg*. ISBN 978-3-642-01481-9. Guimarães, A.P. (1998). Magnetism and magnetc resonance in

Gurevich, A.G.; Melkov, G.A. (1996). *Magnetization Oscillations and Waves* (CRC, Boca Raton). Güner, S.; Yalçn, O.; Kazan, S.; Yldz, F.; Şahingöz, R. (2006). FMR studies of bilayer Co90Fe10/Ni81Fe19, Ni81Fe19/Co90Fe10 and monolayer Ni81Fe19 thin films. *Phys. Stat. Solid (a)*.

Hamoudeh, M.; Al Faraj, A.; Canet-Soulas, E.; Bessueille, F.; Leonard, D.; Fessi, H. (2007). Elaboration of PLLA-based superparamagnetic nanoparticles: Characterization, magnetic behaviour study and in vitro elaxivity evaluation. *Int. J. Pharmaceut.* Vol. 338,

Hathaway, K.; Cullen, J. (1981). Magnetoelastic softening of moduli and determination of

Heim, E.; Harling, S.; Pöhlig, K.; Ludwig, F.; Menzel, H.; Schilling, M. (2007). Fluxgate magnetorelaxometry of superparamagnetic nanoparticles for hydrogel characterization.

Heinrich, B. (2005(a)). Ferromagnetic resonance in ultrathin film structures. In magnetic ultrathin films. Heinrich, B.; Bland, J.A.C. (Eds.) *Springer, Berlin, Heidelberg*. Vol. II, pp.

Heinrich, B.; Bland, J.A.C. (2005(b)). Ultrathin magnetic structures IV, applications of

spring films studied by resonant soft-x-ray magneto-optical Kerr effect. *Phys. Rev. B.* Vol. 62,

magnetic anisotropy in RE-TM compounds. *J. Appl. Phys.* Vol. 52, pp. 2282.

nanomagnetism. *Springer Berlin Heidelberg* New York . ISBN 3-540-21954-4. Heisenberg, W. (1928). Theory of ferromagnetism. *ZeitschriftfürPhysik*. Vol. 49, pp. 619. Held, G.A.; Grinstein, G.; Doyle, H.; Sun, S.; Murray, C.B. (2001). Competing interactions in dispersions of superparamagnetic nanoparticles. *Phys. Rev. B*. Vol. 64, pp. 012408. Hellwig, O.; Kortrigh, J.B.; Takano, K.; Fullerton, E.E. (2000). Switching behavior of Fe-Pt/Ni-

due to antiparallel alignment. *J. Magn. Magn. Mater.* Vol. 93, pp. 58.

structures. *J. Magn. Magn. Mater.* Vol. 104-107, pp. 1734.

figures. *J. Magn. Magn. Mater.* Vol. 226-230, pp. 1688.

solids. *A Wiley-Interscience Publication*. Canada.

*J. Magn. Magn. Mater*. Vol. 311, pp. 150.

Vol. 203, pp. 1539.

pp. 248.

195.

Fe exchange-

pp. 11694.

*Magn. Magn. Mater.* Vol. 15–18, pp. 766.

319.


Grünberg, P. (1980). Brillouin scattering from spin waves in thin ferromagnetic films. *J. Magn. Magn. Mater.* Vol. 15–18, pp. 766.

30 Ferromagnetic Resonance – Theory and Applications

*Phys. Rev. V*ol.139, pp.A1173.

*Phys. Rev*. Vol. 139, pp. 1173.

*Phys. Stat. Sol.* Vol. 120, pp. 341.

*Phys. Stat. Sol.* Vol. 120, pp. 659.

*Stat. Sol.* Vol. 124, pp. 587.

*Physik*. Vol. 9, pp. 353.

No.11.

SiO2. *Phys. Rev. B*. Vol. 66, pp.104406.

Fonseca, F.C.; Goya, G.F.; Jardim, R.F.; Muccillo, R.; Carreño, N.L.V.; Longo, E.; Leite, E.R. (2002). Superparamagnetism and magnetic properties of Ni nanoparticles embedded in

Frait, Z.; Macfaden, H. (1965). Ferromagnetic resonance in metals frequency dependence.

Frait, Z.; Fraitova, D.; Zarubova, N. (1985). Observation of FMR surface spin wave modes in

Frait, Z.; Fraitova, D. (1998). Low energy spinwave excitation in highly conducting thin films and surfaces, in *Frontiers in Magnetism of Reduced Dimension Systems.* Nato ASI Series,

Fraitova, D. (1983(a)). An analytical theory of FMR in bulk metals, I. Dispersion relations.

Fraitova, D. (1983(b)). An analytical theory of FMR in bulk metals, II. Penetration depths.

Fraitova, D. (1984). An analytical theory of FMR in bulk metals, III. Surface impedance. *Phys.* 

Fullerton, E.E.; Jiang, J.S.; Bader, S.D. (1999). Hard/soft magnetic heterostructures: model

Fullerton, E.E.; Jiang, J.S.; Grimsditch, M.; Sowers, C.H.; Bader, S.D. (1998). Exchange-spring behavior in epitaxial hard/soft magnetic bilayers. *Phys. Rev. B.* Vol. 58, pp.12193. Gatteschi, D.; Sessoli, R.; Villain, J. (2006). Molecular nanomagnets. *Oxford University Press*. García-Miquel, H.; García, J.M.; García-Beneytez, J.M.; Vázquez, M. (2001). Surface magnetic anisotropy in glass-coated amorphous microwires as determined from ferromagnetic

Gerlach, W.; Stern, O. (1922). Dasmagnetische moment dessilber atoms. *Zeitschrift für* 

Gilbert, T.A. (1955). Armour research foundation rept. *Armour Research Foundation*. Chicago.

Gnatzig, K.;. Dötsch, H.; Ye, M.; Brockmeyer, A. (1987). Ferrimagnetic resonance in garnet films at large precession angles*. J. Appl. Phys.* Vol. 62, pp. 4839. Golosovsky, M.; Monod, P.; Muduli, P.K.; Budhani, R.C. (2012). Low-field microwave absorption in epitaxial La0.7Sr0.3MnO3 films resulting from the angle-tuned ferromagnetic resonance in the

Goryunov, Yu. V.; Garifyanov, N.N.; Khaliullin, G.G.; Garifullin, I.A.; Tagirov, L.R.; Schreiber, F.; Mühge, Th.; Zabel, H. (1995). Magnetican isotropies of sputtered Fe films

Grimsditch, M.; Camley, R.; Fullerton, E.E.; Jiang, S.; Bader, S.D.; Sowers, H. (1999). Exchange-spring systems: Coupling of hard and soft ferromagnets as measured by magnetization and Brillouin light scattering (invited). *J. Appl. Phys.* Vol. 85, pp. 5901. Grünberg, P.; Schwarz, B.; Vach, W.; Zinn, W.; Dabkowski, D. (1979). Light scattering from

Bar'yakhtar, V.G.; Wigen, P.E.; Lesnik, N.A. (Eds.) (Kluwer, Dordrecht) pp. 121. Frait, Z.; Macfaden, H. (1965). Ferromagnetic resonance in metals frequency dependence.

bulk amorphous ferromagnets. *Phys. Stat. Sol.* (b). Vol. 128, pp. 219.

exchange-spring magnets. *J. Magn. Magn. Mater.* Vol. 200, pp. 392.

resonance measurements. *J. Magn. Magn. Mater*. Vol. 231, pp. 38.

multidomain state. *Arxiv: 1206-3041. Cont-mat. mtrl-sci.* pp. 1.

spin waves in bubble films. *J. Magn. Magn. Mater.* Vol. 13, pp. 181.

on MgO substrates. *Phys. Rev. B*. Vol. 52, pp. 13450.

	- Herring, C.; Kittel, C. (1950). On the theory of spin waves in ferromagnetic media. *Phys. Rev*. Vol. 81, pp. 869.

Ferromagnetic Resonance 33

Kambe, T.; Kajiyoshi, K.; Oshima, K.; Tamura, M.; Kinoshita, M. (2005). Ferromagnetic resonance in β-p-NPNN at radio-frequency region. *Polyhedron.* Vol. 24, pp. 2468. Kartopu, G.; Yalçn, O.; Kazan, S.; Aktaş, B. (2009). Preparation and FMR analysis of Co

Kartopu, G.; Yalçn, O. (2010). Electrodeposited nanowires and their applications. edited by N. Lupu. *INTECH*. available from: http://sciyo.com/articles/show/title/fabrication-andapplications-of-metalnanowire-arrays-electrodeposited-in-ordered-porous-templates. Kartopu, G.; Yalçn, O.; Choy, K.-L.; Topkaya, R.; Kazan, S.; Aktaş, B. (2011 (a)). Size effects and origin of easy-axis in nickel nanowire arrays. *J. Appl. Phys.* Vol. 109, pp. 033909. Kartopu, G; Yalçn, O; Demiray, A.S. (2011 (b)). Magnetic and transport properties of chemical solution deposited (100)-textured La0.7Sr0.3MnO3 and La0.7Ca0.3MnO3

Kasuya, T. (1956). A theory of metallic ferro- and antiferromagnetism on Zener's model.

Kind, J.; van Raden, U.J.; García-Rubio, I.; Gehring, A.U. (2012). Rock magnetic techniques complemented by ferromagnetic resonance spectroscopy to analyse a sediment record.

Kip, A.F.; Arnold, R.D. (1949). Ferromagnetic resonance at microwave frequencies in iron

Kittel, C. (1947). On the theory of ferromagnetic resonance absorption. *Phys. Rev.* Vol. 73, pp.

Kittel, C. (1946).Theory of the structure of ferromagnetic domains in films and small

Kittel, C. (1949). On the gyromagnetic ratio and spectroscopic splitting factor of

Kittel, C.; Abrahams, E. (1953). Relaxation processes in ferromagnetism. *Rev. Mod. Phys.* Vol.

Kip, A.F.; Arnold, R.D. (1949). Ferromagnetic Resonance at Microwave Frequencies in Iron

Kittel, C. (1958). Interaction of spin waves and ultrasonic waves in ferromagnetic crystals.

Kharmouche, A.; Ben Youssef, J.; Layadi, A.; Chérif, S.M. (2007). Ferromagnetic resonance in evaporated Co/Si(100) and Co/glass thin films. *J. App. Phys*. Vol. 101, pp. 13910. Khaibullin, R.I.; Tagirov, L.R.; Rameev, B.Z.; Ibragimov, S.Z.; Yldz, F.; Aktaş, B. (2004). High curie-temperature ferromagnetism in cobalt-implanted single-crystalline rutile. *J.* 

Klein, P.; Varga, R.; Infante, G.; Vázquez, M. (2012). Ferromagnetic resonance study of FeCoMoB microwires during devitrification process. *J. Appl. Phys*. Vol. 111, pp. 053920. Kliava, J.; Berger, R. (1999). Size and shape distribution of magnetic nanoparticles in disordered systems: computer simulations of superparamagnetic resonance spectra. *J.* 

Kneller, E.F.; Hawing, R. (1991). The exchange-spring magnet: a new material principle for

nanowires in alumina templates. *J. Magn. Magn. Mater*. Vol. 321, pp. 1142.

nanocrystalline thin films. *Phys. Scr.* Vol. 83, pp. 015701.

*Prog. Theor. Phys.* Vol. 16, pp. 45.

*Geophys. J. Int.* Vol*.* 191, pp. 51.

155.

25, pp. 233.

single crystal. *Phys. Rev*. Vol. 75, pp. 1556.

Single Crystal. *Phys. Rev*. Vol. 75, pp. 1556.

*Phys.: Condens. Matter.* Vol. 16, pp. 1.

*Magn. Magn. Mater*. Vol. 205, pp. 328.

permanent magnets. *IEEE Trans. Magn*. Vol. 27, pp. 3588.

*Phys. Rev*. Vol. 110, pp. 836.

ferromagnetic substances. *Phys. Rev.* Vol. 76, pp. 743.

Particles. *Phys. Rev.* Vol. 70, pp. 965.


Kambe, T.; Kajiyoshi, K.; Oshima, K.; Tamura, M.; Kinoshita, M. (2005). Ferromagnetic resonance in β-p-NPNN at radio-frequency region. *Polyhedron.* Vol. 24, pp. 2468.

32 Ferromagnetic Resonance – Theory and Applications

*Springer-Verlag Berlin Heidelberg*.

Güntherodt, G. (Eds.). *Springer, Berlin, Heidelberg.*

resonance. *J. Appl. Phys*. Vol. 111, pp. 07C108.

*J. Magn. Magn. Mater.* Vol. 288, pp. 15.

*Springer-Verlag Berlin Heidelberg*. ISBN-10 3-540-20108-4.

Vol. 81, pp. 869.

07B722.

pp. 2031.

pp. 1223.

pp. 3229.

10K311.

Herring, C.; Kittel, C. (1950). On the theory of spin waves in ferromagnetic media. *Phys. Rev*.

Hillebrands, B. (2000). Light scattering in solids VII. Topics. *Appl. Phys.* Vol. 75, M. Cardona,

Hillebrands, B.; Thiaville, A. (2006). Spin dynamics in confined magnetic structures III.

Hillebrands, B.; Ounadjela, K. (2002). Spin dynamics in confined magnetic structures I.

Hillebrands, B.; Ounadjela, K. (2003). Spin dynamics in confined magnetic structures II. *Springer-Verlag Berlin Heidelberg*. Hinata, S.; Saito, S.; Takahashi, M. (2012). Ferromagnetic resonance analysis of internal effective field of classified grains by switching field for granular perpendicular recording media. *J. Appl. Phys*.. Vol. 111, pp.

Hsia, L.C.; Wigen, P.E. (1981). Enhancement of uniaxial anisotropy constant by introducing

Huang, Z.C.; Hu, X.F.; Xu, Y.X.; Zhai, Y.; Xu, Y.B.; Wu, J.; Zhai, H. R. (2012). Magnetic properties of ultrathin single crystal Fe3O4 film on InAs(100) by ferromagnetic

Huang, M.D.; Lee, N.N.; Hyun, Y.H.; Dubowik, J.; Lee, Y.P. (2004). Ferromagnetic resonance study of magnetic-shape-memory Ni2MnGa films. *J. Magn. Magn. Mater*. Vol. 272–276,

Jalali-Roudsar, A.A.; Denysenkov, V.P.; Khartsev, S.I. (2005). Determination of magnetic anisotropy constants for magnetic garnet epitaxial films using ferromagnetic resonance.

Jarrett, H.S.; Waring, R.K. (1958). Ferrimagnetic resonance in NiMnO3. *Phys. Rev.* Vol. 111,

Jiang, J.S.; Fullerton, E.E.; Sowers, C.H.; Inomata, A.; Bader, S.D.; Shapiro, A.J.; Shull, R.D.; Gornakov, V. S.; Nikitenko, V.I. (1999). Spring magnet films. *IEEE Trans. Magn.* Vol. 35,

Jiang, J.S.; Pearson, J.E.; Liu, Z.Y.; Kabius, B.; Trasobares, S.; Miller, D.J.; Bader, S.D. (2005). A new approach for improving exchange-spring magnets. *J. Appl. Phys.* Vol. 97, pp.

Jiang, J.S.; Bader, S.D.; Kaper, H.; Leaf, G.K.; Shull, R.D.; Shapiro, A.J.; Gornakov, V.S.; Nikitenko, V.I.; Platt, C.L.; Berkowitz, A.E.; David, S.; Fullerton, E.E. (2002). Rotational

Kakazei, G.N.; Kravets, A.F.; Lesnik, N.A.; de Azevedo, M.M.P.; Pogorelov, Y. G.; Sousa, J.B. (1999). Ferromagnetic resonance in granular thin films. *J. Appl. Phys.* Vol. 85, pp. 5654. Kakazei, G.N.; Pogorelov, Yu.G.; Sousa, J.B.; Golub, V.O.; Lesnik, N.A.; Cardoso, S.; Freitas, P.P. (2001). FMR in CoFe(*t*)/Al2O3 multilayers: from continuous to discontinuous

hysteresis of exchange-spring Magnets*. J. Phys. D: Appl. Phys.* Vol. 35, pp. 2339. Jung, S.; Watkins, B.; DeLong, L.; Ketterson, J.B.; Chandrasekhar, V. (2002). Ferromagnetic

resonance in periodic particle arrays. *Phys. Rev. B.* Vol. 66, pp. 132401.

regime. *J. Magn. Magn. Mater.* Vol. 226-230, pp. 1828.

oxygen vacancies in Ca-doped YIG. *J. Appl. Phys.* Vol. 52, pp. 1261.


Kobayashi, T.; Ishida, N.; Sekiguchi, K.; Nozaki, Y. (2012). Ferromagnetic resonance properties of granular Co-Cr-Pt films measured by micro-fabricated coplanar waveguides. *J. Appl. Phys*. Vol. 111, pp. 07B919.

Ferromagnetic Resonance 35

Landau, E.; Lifshitz, E. (1935). On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. *Physik Z. Sowjetunnion.* Vol. 8, pp. 153. Layadi, A.; Lee, J.M.; Artman, J.O. (1988). Spin-wave FMR in annealed NiFe/FeMn thin films*. J. Appl. Phys.*

Layadi, A. (2002). Exchange anisotropy: A ferromagnetic resonance study. *Phys. Rev. B.* Vol.

Layadi, A. (2004). Theoretical study of resonance modes of coupled thin films in the rigid

Layadi, A.; Artman, J.O. (1990(a)). Ferromagnetic resonance in a coupled two-layer System.

Layadi A.; Artman, J.O. (1990(b)). Study of antiferromagnetic coupling by ferromagnetic

Lee, J.; Hong, Y.K.; Lee, W.; Abo, G.S.; Park, J.; Syslo, R.; Seong, W.M.; Park, S.H.; Ahn, W.K. (2012). High ferromagnetic resonance and thermal stability spinel Ni0.7Mn0.3-x CoxFe2O4

Limmer, W.; Glunk, M.; Daeubler, J.; Hummel, T.; Schoch, W.; Bihler, C.; Huebl, H.; Brandt, M.S.; Goennenwein, S.T.B.; Sauer, R. (2006). Magnetic anisotropy in (Ga, Mn)As on GaAs(1 1 3)As studied by magnetotransport and ferromagnetic resonance.

Lindner, J.; Baberschke, K. (2003). In situ ferromagnetic resonance: an ultimate tool to investigate the coupling in ultrathin magnetic films. *J. Phys.: Condens. Matter*. Vol. 15,

Lindner, J.; Tolinski, T.; Lenz, K.; Kosubek, E.; Wende, H.; Baberschke, K.; Ney, A.; Hesjedal, T.; Pampuch, C.; Koch, R.; Däweritz, L.; Ploog, K.H. (2004). Magnetic anisotropy of MnAs-films on GaAs(0 0 1) studied with ferromagnetic resonance. *J. Magn. Magn.* 

Li, N.; Schäfer, S.; Datta, R.; Mewes, T.; Klein, T. M.; Gupta, A. (2012). Microstructural and ferromagnetic resonance properties of epitaxial nickel ferrite films grown by chemical

Liua, B.; Yang, Y.; Tang, D.; Zhang, B.; Lu, M.; Lu, H. (2012). The contributions of intrinsic damping and two magnon scattering on the ferromagnetic resonance linewidth in

Macdonald, J.R. (1956). Spin exchange effects in ferromagnetic resonance. *Phys. Rev.* Vol.

Maciá, F.; Warnicke, P.; Bedau, D.; Im, M.Y.; Fischer, P.; Arena, D.A.; Kent, A.D. (2012). Perpendicular magnetic anisotropy in ultrathin Co9Ni multilayer films studied with ferromagnetic resonance and magnetic x-ray microspectroscopy. *J. Magn. Magn. Mater.*

Maklakov, S.S.; Maklakov, S.A.; Ryzhikov, I.A.; Rozanov, K.N.; Osipov, A.V. (2012). Thin Co films with tunable ferromagnetic resonance frequency. *J. Magn. Magn. Mater*. Vol. 324,

Maksymowich, L.J.; Sendorek, D. (1983). Surface modes in magnetic thin amorphous films

ferrite for ultra high frequency devices. *J. Appl. Phys*. Vol. 111, pp. 07A516.

Vol. 63, pp. 3808.

66, pp. 184423.

pp. R193.

103, pp. 280.

pp. 2108.

Vol. 324, pp. 3629.

layer model. *Phys. Rev. B.* Vol. 69, pp. 144431.

resonance (FMR).*J. Magn. Magn. Mater.* Vol. 92, pp. 143.

vapor deposition. *Appl. Phys. Lett.* Vol. 101, pp. 132409.

of GdCoMo alloys. *J. Magn. Magn. Mater.* Vol. 37, pp. 177.

[Fe65Co35/SiO2]n multilayer films. J. Alloy. compd*.* Vol. 524, pp. 69.

*J. Magn. Magn. Mater.* Vol. 92, pp. 143.

*Microelectron. J.* Vol. 37, pp. 1490.

*Mater.* Vol. 277, pp. 159.


Landau, E.; Lifshitz, E. (1935). On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. *Physik Z. Sowjetunnion.* Vol. 8, pp. 153. Layadi, A.; Lee, J.M.; Artman, J.O. (1988). Spin-wave FMR in annealed NiFe/FeMn thin films*. J. Appl. Phys.* Vol. 63, pp. 3808.

34 Ferromagnetic Resonance – Theory and Applications

07E113.

276.

*Magn. Mater*. Vol. 310, pp. 2271.

*Springer science + Business Media, Inc*. Boston.

waveguides. *J. Appl. Phys*. Vol. 111, pp. 07B919.

nanoparticles. *Phys. Rev. B. Vol.* 59, pp. 6321.

NiFe2*O*4 nanoparticles. *Phys. Rev. Lett.* Vol. 77, pp. 394.

inhomogeneity model. *J. Appl. Phys.* Vol. 35, pp. 791.

spectroscopy. *Earth Planet Sc. Lett.* Vol. 247, pp. 10.

*Magn. Magn. Mater.* Vol. 310, pp. 2561.

iron films. *Phys. Rev.* Vol. 113, pp. 1039.

Kobayashi, T.; Ishida, N.; Sekiguchi, K.; Nozaki, Y. (2012). Ferromagnetic resonance properties of granular Co-Cr-Pt films measured by micro-fabricated coplanar

Kodama, R.H.; Berkowitz, A.E.; McNiff Jr. E.J.; Foner S. (1996). Surface spin disorder in

Kooi, C.F.; Wigen, P.E.; Shanabarger, M.R.; Kerrigan J.V. (1964). Spin-wave resonance in magnetic films on the basis of surface –spin –pinning model and the volume

Kopp, R.E.; Weiss, B.P.; Maloof, A.C.; Vali, H.; Nash, C.Z.; Kirschvink, J.L. (2006). Chains, clumps, and strings: magneto fossil taphonomy with ferromagnetic resonance

Korolev, K.A.; McCloy, J.S.; Afsar, M.N. (2012). Ferromagnetic resonance of micro- and nano-sized hexagonal ferrite powders at millimeter waves. *J. Appl. Phys*. Vol. 111, pp.

Kohmoto, O. (2007). Ferromagnetic resonance equation of hexagonal ferrite in c-plane. *J.* 

Knorr, T.G.; Hoffman, R.W. (1959). Dependence of geometric magnetic anisotropy in thin

Kraus, L.; Frait, Z.; Ababei, G.; Chayka, O.; Chiriac, H. (2012). Ferromagnetic resonance in

Krebs, J.J.; Rachford, F.J.; Lubitz, P.; Prinz, G.A. (1982). Ferromagnetic resonance studies of

Krebs, J.J.; Lubitz, P.; Chaiken, A.; Prinz, G.A. (1989). Magnetic resonance determination of the antiferromagnetic coupling of Fe layers through Cr. *Phys. Rev. Lett*. Vol. 63, pp. 1645. Krivoruchko, V.N.; Marchenko, A.I. (2012). Spatial confinement of ferromagnetic resonances

Krone, P.; Albrecht, M.; Schrefl, T. (2011). Micromagnetic simulation of ferromagnetic resonance of perpendicular granular media: Influence of the intergranular exchange on the Landau–Lifshitz–Gilbert damping constant. *J. Magn. Magn. Mater.* Vol. 323, pp. 432. Kuanr, B.K.; Camley, R.E.; Celinski, Z. (2004). Relaxation in epitaxial Fe films measured by

Kuanr, B.K.; Camley, R.E.; Celinski, Z. (2005). Extrinsic contribution to Gilbert damping in sputtered NiFe films by ferromagnetic resonance. *J. Magn. Magn. Mater*. Vol. 286, pp.

Kudryavtsev, Y.V.; Oksenenko, V.A.; Kulagin, V.A.; Dubowik, J.; Lee, Y.P. (2007). Ferromagnetic resonance in Co2MnGa films with various structural ordering. *J. Magn.* 

Lacheisserie, E.T.; Gignoux, D.; Schlenker, M. (2005). Magnetism, materials and aplications.

submicron amorphous wires. *J. Appl. Phys*. Vol. 111, pp. 053924.

very thin epitaxial single crystals of iron*. J. Appl. Phys.* Vol. 53, pp. 8058.

in honeycomb antidot lattices. *J. Magn. Magn. Mater.* Vol. 324, pp. 3087.

ferromagnetic resonance. *J. Appl. Phys.* Vol. 93, pp. 6610.

Kodama, R.H. (1999). Magnetic Nanoparticles. *J. Magn. Magn. Mater.* Vol. 200, pp. 359. Kodama, R.H.; Berkowitz, A.E. (1999). Atomic-scale magnetic modeling of oxide


Maksymowicz, L.J.; Sendorek, D.; Żuberek, R. (1985). Surface anisotropy energy of thin amorphous magnetic films of (Gd1-xCox)1-yMoy alloys; experimental results. *J. Magn. Magn. Mater.* Vol. 46, pp. 295. Maksymowicz, L.J.; Jankowski, H. (1992). FMR experiment in multilayer structure of FeBSi/Pd. *J. Magn. Magn. Mater.* Vol. 109, pp. 341.

Ferromagnetic Resonance 37

Neudecker, I.; Woltersdorf, G.; Heinrich, B.; Okuno, T.; Gubbiotti, G.; Back, C.H. (2006). Comparison of frequency, field, and time domain ferromagnetic resonance methods. *J.* 

Olive, E.; Lansac, Y.; Wegrowe, J.E. (2012). Beyond ferromagnetic resonance: The inertial

Osthöver, C.; Grünberg, P.; Arons, R.R. (1998). Magnetic properties of doped La0.67Ba0.33Mn1-

Owens, F.J. (2009). Ferromagnetic resonance observation of a phase transition in magnetic

Öner, Y.; Özdemir, M.; Aktaş, B.; Topacli, C.; Haris, E.A.; Senoussi, S. (1997). The role of Pt impurities on both bulk and surface anisotropies in amorphous NiMn films. *J. Magn.* 

Özdemir, M.; Aktaş, B.; Öner, Y.; Sato, T.; Ando, T. (1996). A spin- wave resonance study on

Özdemir, M.; Aktaş, B.; Öner, Y.; Sato, T.; Ando, T. (1997). Anomalous anisotropy of re-

Özdemir, M.; Öner, Y.; Aktaş, B. (1998). Evidence of superparamagnetic behaviour in an

Patel, R.; Owens, F.J. (2012). Ferromagnetic resonance and magnetic force microscopy evidence for above room temperature ferromagnetism in Mn doped Si made by a solid

Patton, C.E.; Hurben, M.J. (1995). Theory of magnetostatic waves for in-plane magnetized isotropic films*. J. Magn. Magn. Mater*. Vol. 139, pp. 263. Patton, C.E.; Hurben, M.J. (1996). Theory of magnetostatic waves for in-plane magnetized anisotropic films. *J. Magn.* 

Park, D.G.; Kim, C.G.; Kim, W.W.; Hong, J.H. (2007). Study of GMI-valve characteristics in the Co-based amorphous ribbon by ferromagnetic resonance. *J. Magn. Magn. Mater.* Vol.

Parkin, S.S.P.; More, N.; Roche, K.P. (1990). Oscillations in exchange coupling and magneto resistance in metallic superlattice structures: Co/Ru, Co/Cr, and Fe/Cr. *Phys. Rev. Lett.*

Parkin, S.S.P.; Bhadra, R.; Roche, K.P. (1991(a)). Oscillatory magnetic exchange coupling

Parkin, S.S.P. (1991(b)). Systematic variation of the strength and oscillation period of indirect magnetic exchange coupling through the 3*d*, 4*d*, and 5*d* transition metals. *Phys. Rev. Lett.*

Parkin, S.S.P.; Farrow, R.F.C.; Marks, R.F.; Cebollada, A.; Harp, G.R.; Savoy, R.J. (1994). Oscillations of interlayer exchange coupling and giant magnetoresistance in (111)

Parvatheeswara, R.B.; Caltun, O.; Dumitru, I.; Spinu, L. (2006). Ferromagnetic resonance parameters of ball-milled Ni–Zn ferrite nanoparticles. *J. Magn. Magn. Mater*. Vol. 304,

amorphous Ni62Mn38 film by ESR measurements. *Phys. B.* Vol. 252, pp. 138.

field-aligned Fe2O3 nanoparticles. *J. Magn. Magn. Mater.* Vol. 321, pp. 2386.

reentrant Mn77Mn23 thin films. *J. Magn. Magn. Mater.* Vol. 164, pp. 53.

entrant Ni77Mn23 film. *J. Phys.: Condens. Matter.* Vol. 9, pp. 6433.

state sintering process. *Solid State Commun*. Vol. 152, pp. 603.

through thin copper layers. *Phys. Rev. Lett.* Vol. 66, pp. 2152.

oriented permalloy/Au multilayers. *Phys. Rev. Lett.* Vol. 72, pp. 3718.

regime of the magnetization. *Appl. Phys. Lett.* Vol. 100, pp. 192407.

yAyO3, A - Fe, Cr. *J. Magn. Magn. Mater.* Vol. 177-181, pp. 854.

*Magn. Magn. Mater.* Vol. 307, pp. 148.

*Magn. Mater*. Vol. 170, pp. 129.

*Magn. Mater*. Vol. 163, pp. 39.

310, pp. 2295.

Vol. 64, pp. 2304.

Vol. 67, pp. 3598.

pp. 752.


Neudecker, I.; Woltersdorf, G.; Heinrich, B.; Okuno, T.; Gubbiotti, G.; Back, C.H. (2006). Comparison of frequency, field, and time domain ferromagnetic resonance methods. *J. Magn. Magn. Mater.* Vol. 307, pp. 148.

36 Ferromagnetic Resonance – Theory and Applications

*Magn. Mater.* Vol. 322, pp. 661.

*Phys. Rev*. Vol. 104, pp. 63.

*Phys.* Vol. 99, pp. 08Q510.

*Mater*. Vol. 310, pp. 2248.

Vol. 48, pp. 1782.

58, pp. 6442.

014419.

Maksymowicz, L.J.; Sendorek, D.; Żuberek, R. (1985). Surface anisotropy energy of thin amorphous magnetic films of (Gd1-xCox)1-yMoy alloys; experimental results. *J. Magn. Magn. Mater.* Vol. 46, pp. 295. Maksymowicz, L.J.; Jankowski, H. (1992). FMR experiment in multilayer structure of FeBSi/Pd. *J. Magn. Magn. Mater.* Vol. 109, pp. 341. Marin, C.N. (2006). Thermal and particle size distribution effects on the ferromagnetic

Mastrogiacomo, G.; Fischer H.; Garcia-Rubio, I.; Gehring, A.U. (2010). Ferromagnetic resonance spectroscopic response of magnetite chains in a biological matrix. *J. Magn.* 

Mazur, P.; Mills, D.L. (1982). Inelastic scattering of neutrons by surface spin waves on ferromagnets. *Phys. Rev. B.* Vol. 26, pp. 5175. McMichael, R.D.; Wigen, P.E. (1990). High power FMR without a degenerate spin wave manifold. *Phys. Rev. Lett.* Vol. 64, pp. 64. Meiklejohn, W.H.; Bean, C.P. (1956). New magnetic anisotropy. *Phys. Rev.* Vol. 102, pp. 1413. Meiklejohn, W.H.; Bean, C.P. (1957). New magnetic anisotropy. *Phys. Rev.* Vol. 105, pp. 904. Mercereau, J.E.; Feynman, R.P. (1956). Physical conditions for ferromagnetic resonance.

Mibu, K.; Nagahama, T.; Ono, T.; Shinjo, T. (1997). Magnetoresistance of quasi-Bloch-wall induced in NiFe/CoSm exchange. *J. Magn. Magn. Mater.* Vol. 177-181, pp. 1267. Mibu, K.; Nagahama, T.; Shinjo, T.; Ono, T. (1998). Magnetoresistance of Bloch-wall-type magnetic structures induced in NiFe/CoSm exchange-spring bilayers. *Phys. Rev.* B. Vol.

Mills, D.L.; Bland, J.A.C. ( 2006). Nanomagnetism ultrathin films, multilayers and nanostructures. *Elsevier B.V.* Mills, D.L. (2003). Ferromagnetic resonance relaxation in ultrathin metal films: The role of the conduction electrons. *Phys. Rev. B.* Vol. 68, pp.

Min, J.H.; Cho, J.U.; Kim, Y.K.; Wu, J.H.; Ko, Y.D.; Chung, J.S. (2006). Substrate effects on microstructure and magnetic properties of electrodeposited Co nanowire arrays. *J. Appl.* 

Miyake, K.; Noh, S.M.; Kaneko, T.; Imamura, H.; Sahashi, M. (2012). Study on highfrequency 3–D magnetization precession modes of circular magnetic nano-dots using coplanar wave guide vector network analyzer ferromagnetic resonance. *IEEE T. Magn.*

Mizukami, S.; Nagashima, S.; Yakata, S.; Ando, Y.; Miyazaki, T. (2007). Enhancement of DC voltage generated in ferromagnetic resonance for magnetic thin film. *J. Magn. Magn.* 

Nagamine, L.C.C.M.; Geshev, J.; Menegotto, T.; Fernandes, A.A.R.; Biondo, A.; Saitovitch, E.B. (2005). Ferromagnetic resonance and magnetization studies in exchange-coupled

Nakai, T.; Yamaguchi, M.; Kikuchi, H.; Iizuka, H.; Arai, K.I. (2002). Remarkable improvement of sensitivity for high-frequency carrier-type magnetic field sensor with

Morrish, A. H. (1965).The physical principles of magnetism *Wiley*, New York.

NiFe/Cu/NiFe structures. *J. Magn. Magn. Mater.* Vol. 288, pp. 205.

ferromagnetic resonance. *J. Magn. Magn. Mater.* Vol. 242–245, pp. 1142.

resonance in magnetic fluids. *J. Magn. Magn. Mater*. Vol. 300, pp. 397.


Paul, A.; Bürgler, D.E.; Grünberg, P. (2005). Enhanced exchange bias in ferromagnet/ antiferromagnet multilayers. *J. Magn. Magn. Mater.* Vol. 286, pp. 216.

Ferromagnetic Resonance 39

Rameev, B.Z.; Yilgin, R.; Aktaş, B.; Gupta, A.; Tagirov, L.R. (2003(b)). FMR studies of CrO

Rameev, B.Z.; Gupta, A.; Anguelouch, A.; Xiao, G.; Yldz, F.; Tagirov, L.R.; Aktaş, B. (2004 (b)). Probing magnetic anisotropies in half-metallic CrO2 epitaxial films by FMR. *J.* 

Rameev, B.; Okay, C.; Yldz, F.; Khaibullin, R.I.; Popok, V.N.; Aktas, B. (2004 (a)). Ferromagnetic resonance investigations of cobalt-implanted polyimides. *J. Magn. Magn.* 

Ramesh, M.; Wigen, P.E. (1988 (a)). Ferromagnetic resonance of parallel stripe domains– domain wall system. *J. Magn. Magn. Mater.* Vol. 74, pp. 123. Ramesh, M.; Ren, E.W.; Artman, J.O.; Kryder, M.H. (1988 (b)). Domain mode ferromagnetic resonance studies in

Ramprasad, R.; Zurcher P.; Petras, M.; Miller, M.; Renaud, P. (2004). Magnetic properties of metallic ferromagnetic nanoparticle composites. *J. Appl. Phys.* Vol. 96, pp. 519.

Raposo, V.; Zazo, M.; Iñiguez, J. (2011). Comparison of ferromagnetic resonance between

Reich, K.H. (1955). Ferromagnetic resonance absorption in a nickel single crystal at low

Rezende, S.M.; Moura, J.A.S.; de Aguiar, F.M. (1993). Ferromagnetic resonance in Ag

Riedling, S.; Knorr, N.; Mathieu, C.; Jorzick, J.; Demokritov, S.O.; Hillebrands, B.; Schreiber, R.; Grünberg, P. (1999). Magnetic ordering and anisotropies of atomically layered

Richard, C.; Houzet, M.; Meyer, J.S. (2012). Andreev current induced by ferromagnetic

Rivkin, K.; Xu, W., De Long, L.E.; Metlushko, V.V.; Ilic, B.; Ketterson, J.B. (2007). Analysis of ferromagnetic resonance response of square arrays of permalloy nanodots. *J. Magn.* 

Rodbell, D.S. (1964). Ferromagnetic resonance absorption linewidth of nickel metal.

Rook, K.; Artman, J.O. (1991). Spin wave resonance in FeAlN films. *IEEE Trans. Magn.* Vol. 27, pp. 5450. Roy, W.V.; Boeck, J.D.; Borghs, G. (1992). Optimization of the magnetic field of perpendicular ferromagnetic thin films for device applications. *Appl. Phys. Lett.*

Rousseau, O.; Viret, M. (2012). Interaction between ferromagnetic resonance and spin

Römer, F.M.; Möller, M.; Wagner, K.; Gathmann, L.; Narkowicz, R.; Zähres, H.; Salles, B.R.; Torelli, P.; Meckenstock, R.; Lindner, J.; Farle, M. (2012). In situ multifrequency ferromagnetic resonance and x-ray magnetic circular dichroism investigations on

Evidence for Landau- Lifshitz damping. *Phys. Rev*. *Lett.* Vol. 13, pp. 471.

currents in nanostructures. *Phys. Rev. B.* Vol. 85, pp. 144413.

Fe/GaAs(110): Enhanced g-factor. *J. Appl. Phys*. Vol. 100, pp. 092402.

bismuth-substituted magnetic garnet films. *J. Appl. Phys.* Vol. 64, pp. 5483.

amorphous wires and microwires. *J. Magn. Magn. Mater*. Vol. 323, pp. 1170.

Ramsey, N.F. (1985). Molecular beams. *Oxford University Press.* New York.

Fe/Au(001) multilayers. *J. Magn. Magn. Mater.* Vol.198-199, pp. 348.

epitaxial thin films. *Microelectron. Eng.* Vol. 69, pp. 336.

*Magn. Magn. Mater*. Vol. 272–276, pp. 1167.

temperature. *Phys. Rev*. Vol. 101, pp. 1647.

coupled Ni films. *J. Appl. Phys.* Vol. 73, pp. 6341.

resonance. *Appl. Phys. Lett*. Vol. 109, pp. 057002.

*Magn. Mater.* Vol. 309, pp. 317.

Vol. 61, pp. 3056.

*Mater.* Vol. 278, pp. 164.


*Magn. Mater.* Vol. 300, pp. 382.

7165.

07C105.

0-471-07935-9.

Paul, A.; Bürgler, D.E.; Grünberg, P. (2005). Enhanced exchange bias in ferromagnet/

Paz, E.; Cebollada, F.; Palomares, F.J.; González, J.M.; Martins, J.S.; Santos, N.M., Sobolev, N.A. (2012). Ferromagnetic resonance and magnetooptic study of submicron epitaxial

Pires, M.J.M.; Mansanares, A.M., da Silva, E.C.; Schmidt, J.E.; Meckenstock, R.; Pelzl, J. (2006). Ferromagnetic resonance studies in granular CoCu codeposited films. *J. Magn.* 

Platow, W.; Anisimov, A.N.; Dunifer, G.L.; Farle, M.; Baberschke, K. (1998). Correlations between ferromagnetic-resonance linewidths and sample quality in the study of

Pollmann, J.; Srajer, G.; Hakel, D.; Lang, J.C.; Maser, J.; Jiang, J.S.; Bader, S.D. (2001). Magnetic imaging of a buried SmCo layer in a spring magnet. *J. Appl. Phys.* Vol. 89, pp.

Poole, C.P.; Owens J.F. (2003). Introduction to nanotechnology. *John Wiley* & *Sons, Inc.* ISBN:

Prieto, A.G.; Fdez-Gubieda, M.L.; Lezama, L.; Orue, I. (2012). Study of surface effects on CoCu nanogranular alloys by ferromagnetic resonance. *J. Appl. Phys*. Vol. 111, pp.

Purcell, E.M.; Torrey, H.C.; Pound, R.V. (1946). Resonance absorption by nuclear magnetic

Puszkarski, H. (1992). Spectrum of interface coupling-affected spin-wave modes in

Rabi, I.I.; Zacharias, J.R.; Millman, S.; Kusch, P. (1938). A new method of measuring nuclear

Rachford, F.J.; Vittoria, C. (1981). Ferromagnetic anti-resonance in non-saturated magnetic

Rado, G.T. (1958). Effect of electronic mean free path on spin-wave resonance in ferromagnetic metals. *J. Appl. Phys.* Vol. 29, pp. 330. Raikher, Yu.L.; Stepanov, V.I. (2003). Nonlinear dynamic response of superparamagnetic nanoparticles. *Microelectron.* 

Raita, O.; Popa, A.; Stan, M.; Suciu, R.C.; Biris, A.; Giurgiu, L.M. (2012). Effect of Fe concentration in ZnO powders on ferromagnetic resonance spectra. *Appl. Magn. Reson.* 

Rahman, F. (2008). Nanostructures in electronics and photonics. *Pan Stanford Publishing Pte.* 

Rameev, B.Z.; Aktaş, B.; Khaibullin, R.I.; Zhikharev, V.A.; Osin, Yu.N.; Khaibullin, I.B. (2000). Magnetic properties of iron-and cobalt-implanted silicone polymers. *Vacuum.*

Rameev, B.Z.; Yldz, F.; Aktaş, B.; Okay, C.; Khaibullin, R.I.; Zheglov, E.P.; Pivin, J.C.; Tagirov, L.R. (2003(a)). I on synthesis and FMR studies of iron and cobalt nanoparticles

antiferromagnet multilayers. *J. Magn. Magn. Mater.* Vol. 286, pp. 216.

Fe(001) stripes. *J. Appl. Phys*. Vol. 111, pp. 123917.

metallic ultrathin films. *Phys. Rev. B.* Vol. 58, pp. 5611.

moments in a solid. Phys. Rev. Vol. 69, pp. 37.

magnetic moment, *Phys. Rev.* Vol. 53, pp. 318.

in polyimides. *Microelectron. Eng.* Vol. 69, pp. 330.

metals. *J. Appl. Phys.* Vol. 52, pp. 2253.

*Eng*. Vol. 69, pp. 317.

*Ltd.* Singapore, 596224.

Vol. 42, pp. 499.

Vol. 58, pp. 551.

ferromagnetic bilayer films. *Phys. stat. sol*. Vol. 171, pp. 205.


Ruderman, M.A.; Kittel, C. (1954). Indirect exchange coupling of nuclear magnetic moments by conduction electrons. *Phys. Rev.* Vol. 96, pp. 99. Rusek, P. (2004). Spin dynamics of ferromagnetic spin glass. *J. Magn. Magn. Mater*. Vol. 272–276, pp. 1332.

Ferromagnetic Resonance 41

Singh, A.; Chowdhury, P.; Padma, N.; Aswal, D.K.; Kadam, R.M.; Babu, Y.; Kumar, M.L.J.; Viswanadham, C.S.; Goswami, G.L.; Gupta, S.K.; Yakhmi, J.V. (2006). Magnetotransport and ferromagnetic resonance studies of polycrystalline La0.6Pb0.4NO3 thin

Shames, A.I.; Rozenberg, E.; Sominski, E.; Gedanken, A. (2012). Nanometer size effects on magnetic order in La1-x CaxMnO3 (x = 50.5 and 0.6) manganites, probed by

Sklyuyev, A.; Ciureanu, M.; Akyel, C.; Ciureanu, P.; Yelon, A. (2009). Microwave studies of magnetic anisotropy of Co nanowire arrays. *J. App. Phys*. Vol. 105, pp. 023914. Skomski, R.; Coey, J.M.D. (1993). Giant energy product in nanostructured two-phase

Skomski, R. (2008) .Simple models of magnetism. *Oxford University Press*. ISBN 978–0–19–

Slichter, C.P. (1963). Principle of Magnetic Resonance. *Harper & Row,* New York. Smit, J.; Beljers, H. G. (1955). Ferromagnetic resonance absorption in BaFe12O19, a high

Spaldin, N.A. (2010). Magnetic materials: Fundamentals and applications. *Cambridge* 

Sparks, M. (1969). Theory of surface spin pinning in ferromagnetic resonance. *Phys. Rev. Lett.*

Sparks, M. (1970(a)). Ferromagnetic resonance in thin films. III. Theory of mode Intensities.

Sparks, M. (1970(b)). Ferromagnetic resonance in thin films I and II. *Phys. Rev. B.* Vol. 1, pp. 3831. Sparks, M. (1970(c)). Ferromagnetic resonance in thin films. I. Theory of normal-

Sparks, M. (1970(d)). Ferromagnetic resonance in thin films. I. Theory of normal-mode

Speriosu, V.S.; Parkin, S.S.P. (1987). Standing spin waves in FeMn/NiFe/FeMn exchange bias

Spinu, L.; Dumitru, I.; Stancu, A.; Cimpoesu, D. (2006). Transverse susceptibility as the lowfrequency limit of ferromagnetic resonance. *J. Magn. Magn. Mater.* Vol. 296, pp. 1. Spoddig, D.; Meckenstock, R.; Bucher, J.P.; Pelzl, J. (2005). Studies of ferromagnetic resonance line width during electrochemical deposition of Co films on Au(1 1 1). *J.* 

Song, Y.Y.; Kalarickal, S.; Patton, C.E. (2003). Optimized pulsed laser deposited barium ferrite thin films with narrow ferromagnetic resonance linewidths. *J. Appl. Phys.* Vol. 94,

Song, H.; Mulley, S.; Coussens, N.; Dhagat, P.; Jander, A.; Yokochi, A. (2012). Effect of packing fraction on ferromagnetic resonance in NiFe2O4 nanocomposites. *J. Appl. Phys*.

Stashkevich, A.A.; Roussigné, Y.; Djemia, P.; Chérif, S.M.; Evans, P.R.; Murphy, A.P.; Hendren, W.R.; Atkinson, R.; Pollard, R.J.; Zayats, A.V.; Chaboussant, G.; Ott, F. (2009).

Sparks, M. (1964). Ferromagnetic Relaxation Theory*. McGraw-Hill,* New York.

films. *Solid State Commun.* Vol. 137, pp. 456.

magnets. *Phys. Rev. B.* Vol. 48, pp. 15812.

anisotropy crystal. *Philips Res. Rep*. Vol. 10, pp. 113.

*Universty Press.* ISBN 13 978 0 521 88669 7.

mode frequencies. *Phys. Rev*. *B.* Vol. 1, pp. 3831.

structures. *IEEE Trans. magn.* Vol. 23, pp. 2999.

intensities. *Phys. Rev*. *B*. Vol. 1, pp. 3869.

*Magn. Magn. Mater.* Vol. 286, pp. 286.

857075–2.

Vol. 22, pp. 1111.

pp. 5103.

Vol. 111, pp. 07E348.

*Phys. Rev. B.* Vol. 1, pp. 3869.

ferromagnetic resonance. *J. Appl. Phys*. Vol. 111, pp. 07D701.


Singh, A.; Chowdhury, P.; Padma, N.; Aswal, D.K.; Kadam, R.M.; Babu, Y.; Kumar, M.L.J.; Viswanadham, C.S.; Goswami, G.L.; Gupta, S.K.; Yakhmi, J.V. (2006). Magnetotransport and ferromagnetic resonance studies of polycrystalline La0.6Pb0.4NO3 thin films. *Solid State Commun.* Vol. 137, pp. 456.

40 Ferromagnetic Resonance – Theory and Applications

10679.

*Mater.* Vol. 127, pp. 273.

322, pp. 2979.

*Science+Business Media, Inc.* 

Ruderman, M.A.; Kittel, C. (1954). Indirect exchange coupling of nuclear magnetic moments by conduction electrons. *Phys. Rev.* Vol. 96, pp. 99. Rusek, P. (2004). Spin dynamics of

Sarmiento, G.; Fdez-Gubieda, M.L.; Siruguri, V.; Lezama, L.; Orue, I. (2007). Ferromagnetic resonance study of Fe50Ag50 granular film. *J. Magn. Magn. Mater.* Vol. 316, pp. 59. Sasaki, M.; Jönsson, P.E.; Takayama, H.; Mamiya, H. (2005). Aging and memory effects in

Schäfer, S.; Pachauri, N.; Mewes, C.K.A.; Mewes, T.; Kaiser, C.; Leng, Q.; Pakala, M. (2012). Frequency-selective control of ferromagnetic resonance linewidth in magnetic

Schmool, D.S.; Barandiarán, J.M. (1998). Ferromagnetic resonance and spin wave resonance in multiphase materials: theoretical considerations. *J. Phys.: Condens. Matter.* Vol. 10, pp.

Schmool, D.S.; Schmalzl, M. (2007). Ferromagnetic resonance in magnetic nanoparticle

Scholz, W.; Suess, D.; Schrefl, T.; Fidler, J. (2000). Micromagnetic simulation of structure-

Schreiber, F.; Frait, Z. (1996). Spinwave resonance in high conductivity films: The Fe–Co alloy system. *Phys. Rev. B.* Vol. 54, pp. 6473. Schrefl, T.; Kronmüller, H.; Fidler, J. (1993(a)). Exchange hardening in nano-structured permanent magnets*. J. Magn. Magn.* 

Schrefl, T.; Schmidts, H.F.; Fidler, J.; Kronmüller, H. (1993(b)). The role of exchange and dipolar coupling at grain boundaries in hard magnetic materials. *J. Magn. Magn. Mater.* Vol. 124, pp. 251. Schrefl, T.; Fidler, J. (1998). Modelling of exchange-spring permanent

Schrefl, T.; Forster, H.; Fidler, J.; Dittrich, R.; Suess, D.; Scholz, W. (2002). Magnetic hardening of exchange spring multilayers. Proof XVII Rare Earth Magnets Workshop,

Schultz, S.; Gullikson, E.M. (1983). Measurement of static magnetization using electron spin

Seemann, K.; Leiste, H.; Klever, C. (2009). On the relation between the effective ferromagnetic resonance linewidth Δfeff and damping parameter αeff in ferromagnetic

Seemann, K.; Leiste, H.; Klever, Ch. (2010). Determination of intrinsic FMR line broadening in ferromagnetic (Fe44Co56)77Hf12N11 nanocomposite films. *J. Magn. Magn. Mater.* Vol.

Seib, J.; Steiauf, D.; Fähnle, M. (2009).Linewidth of ferromagnetic resonance for systems with

Sellmyer, D.; Skomski, R. (2006). Advanced magnetic nanostructures. *Springer* 

Sihues, M.D.; Durante-Rincón, C.A.; Fermin, J.R. (2007). A ferromagnetic resonance study of

Fe–Co–Hf–N nanocomposite films. *J. Magn. Magn. Mater.* Vol. 321, pp. 3149.

property relations in hard and soft magnets. *Comp. Mater. Sci*. Vol. 18, pp. 1.

ferromagnetic spin glass. *J. Magn. Magn. Mater*. Vol. 272–276, pp. 1332.

superparamagnets and superspin glasses. *Rev. B.* Vol. 71, pp. 104405.

multilayers. *J. Appl. Phys*. Vol. 100, pp. 032402.

assemblies. *J. Non-Cryst. Solids.* Vol. 353, pp. 738.

magnets. *J. Magn. Magn. Mater.* Vol. 177-181, pp. 970.

anisotropic damping. *Phys. Rev. B.* Vol. 79, pp. 092418.

NiFe alloy thin films. *J. Magn. Magn. Mater*. Vol. 316, pp. 462.

resonance*. Rev. Sci. Insrum.* Vol. 54, pp. 1383.

University of Delaware, ed: Hadjipanayis, G.; Bonder M.J. pp. 1006.


Spin-wave modes in Ni nanorod arrays studied by Brillouin light scattering. *Phys. Rev. B.* Vol. 80, pp. 144406.

Ferromagnetic Resonance 43

Valenzuela, R.; Herbst, F.; Ammar, S. (2012). Ferromagnetic resonance in Ni–Zn ferrite nanoparticles in different aggregation states. *J. Magn. Magn. Mater.* Vol. 324, pp. 3398. Van Vleck, J.H. (1950). Concerning the theory of ferromagnetic resonance absorption. *Phys.* 

Vargas, J.M.; Zysler, R.D.; Butera, A. (2007). Order–disorder transformation in FePt nanoparticles studied by ferromagnetic resonance. *Appl. Surf. Sci.* Vol. 254, pp. 274.

Vittoria, C. (1993). Microwave Properties in Magnetic Films. *World Scientific,* Singapore. pp.

Vlasko-Vlasov, K.; Welp, U.; Jiang, J.S.; Miller, D.J.; Crabtree, G.W.; Bader, S.D. (2001). Field induced biquadratic exchange in hard/soft ferromagnetic bilayers. *Phys. Rev. Lett.* Vol.

Voges, F.; de Gronckel, H.; Osthöver, C.; Schreiber, R.; Grünberg, P. (1998). Spin valves with

Vounyuk, B.P.; Guslienko, K.Y.; Kozlov, V.I.; Lesnik, N.A.; Mitsek, A.I. (1991). Effect on the interaction of layers on a ferromagnetic resonance in two layer feromagnetic films*. Sov.* 

Walker, L.R. (1957). Magnetostatic modes in ferromagnetic resonance. *Phys. Rev.* Vol. 105,

Wallis, T.M.; Moreland, J.; Riddle, B.; Kabos, P. (2005). Microwave power imaging with ferromagnetic calorimeter probes on bimaterial cantilevers. *J. Magn. Magn. Mater.* Vol.

Wang, X.; Deng, L.J.; Xie, J.L.; Li, D. (2011). Observations of ferromagnetic resonance modes on FeCo-based nanocrystalline alloys. *J. Magn. Magn. Mater.* Vol. 323, pp. 635. Weil, J.A.; Bolton, J.R.; Wertz, J.E. (1994). Electron Paramagnetic Resonance: elementary Theory and Practical Application. *John Wiley-Sons*. New York. Weil, J.A.; Bolton, J.R. (2007). Electron paramagnetic resonance. *John Wiley & Sons, Inc.* Hoboken, New Jersey.

Weiss, M.T.; Anderson, P.W. (1955). Ferromagnetic resonance in ferroxdure. *Phys. Rev*. Vol.

Wegrowe, J.E.; Kelly, D.; Franck, A.; Gilbert, S.E.; Ansermet, J.Ph. (1999). Magnetoresistance

Wegrowe, J.E.; Comment, A.; Jaccard, Y.; Ansermet, J.Ph. (2000). Spin-dependent scattering

White, R.L.; Solt, I.H. (1956). Multiple ferromagnetic resonance in ferrite spheres. *Phys. Rev.*

Wiekhorst, F.; Shevchenko, E.; Weller, H.; Kötzler, J. (2003). Anisotropic superparamagnetism of mono dispersive cobalt-platinum nanocrystals. *Phys. Rev. B.*

Wigen P.E.; Zhang, Z. (1992). Ferromagnetic resonance in coupled magnetic multilayer systems. *Braz. J. Phys*. Vol. 22, pp. 267. Wigen, P.E.; Kooi, C.F.; Shanaberger, M.R.; Rosing, T.R. (1962). Dynamic pinning in thin-film spin-wave resonance. *Phys. Rev. Lett.*

of ferromagnetic nanowires. *Phys. Rev. Lett*. Vol. 82, pp. 3681.

of a domain wall of controlled size. *Phys. Rev. B*. Vol. 61, pp. 12216.

Vilasi, D. (2001). Hamiltonian dynamics. *World Scientific*. ISBN 981-02-3308-6.

CoO as an exchange bias layer. *J. Magn. Magn. Mater.* Vol. 190, pp. 183.

*Rev.* Vol. 78, pp. 266.

87.

86, pp. 4386.

pp. 390.

286, pp. 320.

98, pp. 925.

Vol. 104, pp. 56.

Vol. 67, pp. 224416.

ISBN 978-0471-75496-1.

*Phys. Solid State.* Vol. 33, pp. 250.


alloys. *Phil. Trans. R. Soc. A*. Vol. 240, pp. 599.

impurities. *J. Appl. Phys*. Vol. 111, pp. 07A328.

particles. *Phys. Rev. B*. Vol. 69, pp. 094401.

*Magn. Mater.* Vol. 165, pp. 134.

*Phys. Rev*. Vol. 105, pp. 377.

*Appl. Phys.* Vol. 108, pp. 023910.

*Phys. Rev. B.* Vol. 80, pp. 014423.

Naturwissenschaften. Vol. 13, Issue (47), pp. 953.

amorphous ferromagnets. *J. Non-Cryst. Solids*. Vol. 353, pp. 768.

1312.

*B.* Vol. 80, pp. 144406.

Spin-wave modes in Ni nanorod arrays studied by Brillouin light scattering. *Phys. Rev.* 

Stoner, E.C.; Wohlfarth, E. P. (1948). A mechanism of magnetic hysteresis in heterogeneous

Sun, Y.; Duan, L.; Guo, Z.; DuanMu, Y.; Ma, M.; Xu, L.; Zhang, Y.; Gu, N. (2005). An improved way to prepare superparamagnetic magnetite-silica core-shell nanoparticles

Sun, Y.; Song, Y.Y.; Chang, H.; Kabatek, M.; Jantz, M.; Schneider, W.; Wu, M.; Schultheiss, H.; Hoffmann, A. (2012(a)). Growth and ferromagnetic resonance properties of

Sun, L.; Wang, Y.; Yang, M.; Huang, Z.; Zhai, Y.; Xu, Y.; Du, J.; Zhai, H. (2012(c)). Ferromagnetic resonance studies of Fe thin films with dilute heavy rare-earth

Suzuki, Y.; Katayama, T.; Takanashi, K.; Schreiber, R.; Gtinberg, P.; Tanaka, K. (1997) . The magneto-optical effect of Cr( 001) wedged ultrathin films grown on Fe( 001). *J. Magn.* 

Szlaferek, A. (2004). Model exchange-spring nanocomposite. *Status Solidi B,* Vol. 241, pp.

Tannenwald, P.E.; Seawey, M.H. (1957). Ferromagnetic resonance in thin films of permalloy.

Tartaj, P.; González-Carreño, T.; Bomati-Miguel, O.; Serna, C. J. (2004). Magnetic behavior of superparamagnetic Fe nanocrystals confined inside submicron-sized spherical silica

Teale, R.W.; Pelegrini, F. (1986). Magnetic surface anisotropy and ferromagnetic resonance

Terry, E.M. (1917). The magnetic properties of iron, nickel and cobalt above the curie point,

Topkaya, R.; Erkovan, M.; Öztürk, A.; Öztürk, O.; Aktaş, B.; Özdemir, M. (2010). Ferromagnetic resonance studies of exchange coupled ultrathin Py/Cr/Py trilayers. *J.* 

Tsai, C.C.; Choi, J.; Cho, S.; Lee, S.J.; Sarma, B.K.; Thompson, C.; Chernyashevskyy, O.; Nevirkovets, I.; Metlushko, V.; Rivkin, K.; Ketterson, J.B. (2009). Vortex phase boundaries from ferromagnetic resonance measurements in a patterned disc array.

Uhlenbeck, G.E.; Goudsmit,S. (1925). Ersetzung der hypothese vom unmechanischen zwang durch eine forderung bezüglich des inneren verhaltens jedes einzelnen elektrons. Die

Valenzuela, R.; Zamorano, R.; Alvarez, G.; Gutiérrez, M.P.; Montiel, H. (2007). Magnetoimpedance, ferromagnetic resonance, and low field microwave absorption in

nanometer-thick yttrium iron garnet films. *Appl. Phys. Lett.* Vol. 101, pp. 152405. Sun, Y.; Song Y.Y.; Wu, M. (2012(b)). Growth and ferromagnetic resonance of yttrium iron

for possible biological application. *J. Magn. Magn. Mater*. Vol. 285, pp. 65.

garnet thin films on metals. *Appl. Phys. Lett.* Vol. 101, pp. 082405.

in the single crystal GdAl2. *J. Phys. F:Met. Phys.* Vol. 16, pp. 621.

and Keeson's quantum theory of magnetism. *Phys. Rev.* Vol. 9, pp. 394.


Vol. 9, pp. 206. Wigen, P.E. (1984). Microwave properties of magnetic garnet thin films. *Thin Solid Films.* Vol. 114, pp. 135.

Ferromagnetic Resonance 45

Yalçn, O.; Erdem, R.; Övünç, S. (2008(b)). Spin-1 model of noninteracting nanoparticles. *Acta Phys. Pol. A.* Vol. 114, pp. 835. Yalçn, O.; Erdem,R.; Demir, Z. (2012). Magnetic properties and size effects of spin-1/2 and spin-1 models of core-surface nanoparticles in different type lattices, smart nanoparticles technology, Abbass Hashim (Ed.), ISBN: 978-

properties-and-size-effects-of-spin-1-2-and-spin-1-models-of-core-shell-nanoparticles-

Yeh, Y.C.; Jin, J.D.; Li, C.M.; Lue, J.T. (2009). The electric and magnetic properties of Co and Fe films percept from the coexistence of ferromagnetic and microstrip resonance or a T-

Yldz, F.; Yalçn, O.; Özdemir, M.; Aktaş, B.; Köseoğlu, Y.; Jiang, J.S. (2004(a)). Magnetic properties of SmCo/Fe exchange spring magnets. *J. Magn. Magn. Mater.* Vol. 272–276,

Yldz, F.; Yalçn,O.; Aktaş, B.; Özdemir, M.; Jiang, J.S. (2004(b)). Ferromagnetic resonance studies on Sm-Co/Fe thin films. *MSMW'04 Symposium Proceedings. Kharkov, Ukraine*. Yldz, F.; Kazan, S.; Aktas, B.; Tarapov, S.; Samofalov, V.; Ravlik, A. (2004(c)). Magnetic anisotropy studies on FeNiCo/Ta/FeNiCo three layers film by layer sensitive

Yosida, K. (1957). Magnetic properties of Cu-Mn alloys. *Phys. Rev.* Vol. 106, pp. 893. Yoshikiyo, M.; Namai, A.; Nakajima, M.; Suemoto, T.; Ohkoshi, S. (2012). Anomalous behavior of high-frequency zero-field ferromagnetic resonance in aluminum-

Young, J.A.; Uehling, E.A. (1953). The tensor formulation of ferromagnetic resonance. *Phys.* 

Yu, J.T.; Turk, R.A.; Wigen, P.E. (1975). Exchange dominated surface spinwaves in yttrium-

Zakeri, K.; Kebe, T.; Lindner, J.; Farle, M. (2006). Magnetic anisotropy of Fe/GaAs(001) ultrathin films investigated by in situ ferromagnetic resonance. *J. Magn. Magn. Mater*.

Zakeri, Kh.; Lindner, J.; Barsukov, I.; Meckenstock, R.; Farle, M.; von Hörsten, U.; Wende, H.; Keune, W. (2007). Spin dynamics in ferromagnets: Gilbert damping and two-

Zavoisky, E. (1945). Spin-magnetic resonance in paramagnetics. *J. Phys*. *USSR.* Vol. 9, pp.

Zianni, X.; Trohidou, K.N. (1998). Monte carlo simulations the coercive behaviour of oxide

Zhai, Y.; Shi, L.; Zhang, W.; Xu, Y.X.; Lu, M.; Zhai, H.R.; Tang, W.X.; Jin, X.F.; Xu, Y.B.; Bland, J.A.C. (2003). Evolution of magnetic anisotropy in epitaxial Fe films by

Zhang, Z.; Zhou, L.; Wigen, P.E.; Ounadjela, K. (1994 (a)). Angular dependence of ferromagnetic resonance in exchange coupled Co/Ru/Co trilayer structures, *Phys. Rev. B.* Vol. 50, pp. 6094. Zhang, Z.; Zhou, L.; Wigen, P.E.; Ounadjela, K. (1994 (b)). Using

coated ferromagnetic particles*. J. Phys.: Condens. Matter.* Vol. 10, pp. 7475.

ferromagnetic resonance technique. *Phys. stat. sol.(c).* Vol. 12, pp. 3694.

substituted ε-Fe2O3. *J. Appl. Phys*. Vol. 111, pp. 07A726.

iron-garnet films. *Phys. Rev. B.* Vol. 11, pp. 420.

magnon scattering. *Phys. Rev. B.* Vol. 76, pp. 104416.

ferromagnetic resonance. *J. Appl. Phys.* Vol. 93, pp. 7622.

953-51-0500-8, InTech, DOI: 10.5772/34706. Available from:

type microstrip. *Measurement.* Vol. 42, pp. 290.

in-di.

pp. 1941.

*Rev*. Vol. 93, pp. 544.

Vol. 299, pp. L1.

211.

http://www.intechopen.com/books/smart-nanoparticles-technology/magnetic-


Yalçn, O.; Erdem, R.; Övünç, S. (2008(b)). Spin-1 model of noninteracting nanoparticles. *Acta Phys. Pol. A.* Vol. 114, pp. 835. Yalçn, O.; Erdem,R.; Demir, Z. (2012). Magnetic properties and size effects of spin-1/2 and spin-1 models of core-surface nanoparticles in different type lattices, smart nanoparticles technology, Abbass Hashim (Ed.), ISBN: 978- 953-51-0500-8, InTech, DOI: 10.5772/34706. Available from:

44 Ferromagnetic Resonance – Theory and Applications

*Thin Solid Films.* Vol. 114, pp. 135.

3125.

pp. 7007-7009.

Vol. 12, pp. 3698.

Kocaeli, Turkey.

Vol. 8, pp. 841.

Vol. 272-276, pp. 1684.

Chemistry. Vol. 143, pp. 347.

*J. Magn. Magn. Mater.* Vol. 258–259, pp. 137.

CuGeO3. *Spectrochim. Acta Part A.* Vol. 66, pp. 307.

CuGeO3 *Spectrochim. Acta Part A.* Vol. 68, pp. 1320.

Vol. 9, pp. 206. Wigen, P.E. (1984). Microwave properties of magnetic garnet thin films.

Wigen, P.E.; Zhang, Z.; Zhou, L.; Ye, M.; Cowen, J.A. (1993). The dispersion relation in

Wigen, P.E. (1998). Routes to chaos in ferromagnetic resonance and the return trip: Controlling and synchronizing Chaos, in Bar'yakhtar, V.G.; Wigen, P.E.; Lesnik, N.A. (Eds.). Frontiers in magnetism of reduced dimension systems Nato ASI series (Kluwer, Dordrecht) pp. 29. Wojtowicz, T. (2005). Ferromagnetic resonance study of the free-hole contribution to magnetization and magnetic anisotropy in modulation-doped Ga1-

Wolfram, T.; De Wames, R.E. (1971). Magneto-exchange branches and spin-wave resonance in conducting and insulating films: Perpendicular resonance. *Phys. Rev. B.* Vol. 4, pp.

Woltersdorf, G.; Heinrich, B.; Woltersdorf, J.; Scholz, R. (2004). Spin dynamics in ultrathin film structures with a network of misfit dislocations. *Journal of Applied Physics*. Vol. 95,

Woods, S.I.; Kirtley, J.R.; Sun, S.; Koch, R.H. (2001). Direct investigation of superparamagnetism in Co nanoparticle films. *Phys. Rev. Lett.* Vol. 87, pp. 137205. Wüchner, S.; Toussaint, J.C.; Voiron, J. (1997). Magnetic properties of exchange-coupled trilayers of amorphous rare-earth-cobalt alloys. J. *Phys. Rev. B.* Vol. 55, pp. 11576. Xu, Y.; Zhang, D.; Zhai, Y.; Chen, J.; Long, J.G.; Sang, H.; You, B.; Du, J.; Hu, A.; Lu, M.; Zhai, H.R. (2004). FMR study on magnetic thin and ultrathin Ni-Fe films. *Phys. Stat. Sol. (c).*

Yalçn, O.; Yldz, F.; Özdemir, M.; Aktaş, B.; Köseoğlu, Y.; Bal, M.; Touminen, M.T. (2004(a)). Ferromagnetic resonance studies of Co nanowire arrays. *J. Magn. Magn. Mater*.

Yalçn, O.; Yldz, F.; Özdemir, M.; Rameev, B.; Bal, M.; Tuominen, M.T. (2004(b)). FMR Studies of Co Nanowire Arrays, Nanostructures Magnetic Materials and Their Applications. *Kluwer Academic Publisher.* Nato Science Series. Mathematics, Physics and

Yalçn, O. (2004(c)). PhD Thesis, investigation of phase transition in inorganic spin-Peierls CuGeO3 systems by ESR techgnique. Gebze institute of technology, 2004 Gebze,

Yalçn, O.; Aktaş, B. (2003). The Effects of Zn2+ doping on Spin-Peierls transition in CuGeO3

Yalçn, O.; Yldz, F.; Aktaş, B. (2007(a)). Spin-op and spin-Peierls transition in doped

Yalçn, O. (2007(b)). Comparison effects of different doping on spin-Peierls transition in

Yalçn, O.; Kazan, S.; Şahingöz, R.; Yildiz, F.; Yerli, Y.; Aktaş, B. (2008(a)). Thickness dependence of magnetic properties of Co90Fe10 nanoscale thin films. *J. Nanosci. Nanotech*.

antiparallel coupled ferromagnetic films*. J. Appl. Phys.* Vol. 73, pp. 6338.

xMnxAs/Ga1-yAlyAs: Be. *Phys. Rev. B.* Vol. 71, pp. 035307.

 http://www.intechopen.com/books/smart-nanoparticles-technology/magneticproperties-and-size-effects-of-spin-1-2-and-spin-1-models-of-core-shell-nanoparticles-

	- Yldz, F.; Yalçn, O.; Özdemir, M.; Aktaş, B.; Köseoğlu, Y.; Jiang, J.S. (2004(a)). Magnetic properties of SmCo/Fe exchange spring magnets. *J. Magn. Magn. Mater.* Vol. 272–276, pp. 1941.
	- Yldz, F.; Yalçn,O.; Aktaş, B.; Özdemir, M.; Jiang, J.S. (2004(b)). Ferromagnetic resonance studies on Sm-Co/Fe thin films. *MSMW'04 Symposium Proceedings. Kharkov, Ukraine*.
	- Yldz, F.; Kazan, S.; Aktas, B.; Tarapov, S.; Samofalov, V.; Ravlik, A. (2004(c)). Magnetic anisotropy studies on FeNiCo/Ta/FeNiCo three layers film by layer sensitive ferromagnetic resonance technique. *Phys. stat. sol.(c).* Vol. 12, pp. 3694.
	- Yosida, K. (1957). Magnetic properties of Cu-Mn alloys. *Phys. Rev.* Vol. 106, pp. 893. Yoshikiyo, M.; Namai, A.; Nakajima, M.; Suemoto, T.; Ohkoshi, S. (2012). Anomalous behavior of high-frequency zero-field ferromagnetic resonance in aluminumsubstituted ε-Fe2O3. *J. Appl. Phys*. Vol. 111, pp. 07A726.
	- Young, J.A.; Uehling, E.A. (1953). The tensor formulation of ferromagnetic resonance. *Phys. Rev*. Vol. 93, pp. 544.
	- Yu, J.T.; Turk, R.A.; Wigen, P.E. (1975). Exchange dominated surface spinwaves in yttriumiron-garnet films. *Phys. Rev. B.* Vol. 11, pp. 420.
	- Zakeri, K.; Kebe, T.; Lindner, J.; Farle, M. (2006). Magnetic anisotropy of Fe/GaAs(001) ultrathin films investigated by in situ ferromagnetic resonance. *J. Magn. Magn. Mater*. Vol. 299, pp. L1.
	- Zakeri, Kh.; Lindner, J.; Barsukov, I.; Meckenstock, R.; Farle, M.; von Hörsten, U.; Wende, H.; Keune, W. (2007). Spin dynamics in ferromagnets: Gilbert damping and twomagnon scattering. *Phys. Rev. B.* Vol. 76, pp. 104416.
	- Zavoisky, E. (1945). Spin-magnetic resonance in paramagnetics. *J. Phys*. *USSR.* Vol. 9, pp. 211.
	- Zianni, X.; Trohidou, K.N. (1998). Monte carlo simulations the coercive behaviour of oxide coated ferromagnetic particles*. J. Phys.: Condens. Matter.* Vol. 10, pp. 7475.
	- Zhai, Y.; Shi, L.; Zhang, W.; Xu, Y.X.; Lu, M.; Zhai, H.R.; Tang, W.X.; Jin, X.F.; Xu, Y.B.; Bland, J.A.C. (2003). Evolution of magnetic anisotropy in epitaxial Fe films by ferromagnetic resonance. *J. Appl. Phys.* Vol. 93, pp. 7622.
	- Zhang, Z.; Zhou, L.; Wigen, P.E.; Ounadjela, K. (1994 (a)). Angular dependence of ferromagnetic resonance in exchange coupled Co/Ru/Co trilayer structures, *Phys. Rev. B.* Vol. 50, pp. 6094. Zhang, Z.; Zhou, L.; Wigen, P.E.; Ounadjela, K. (1994 (b)). Using

ferromagnetic resonance as a sensitive method to study the temperature dependence of interlayer exchange coupling. *Phys. Rev. Lett.* Vol. 73, pp. 336.

**Chapter 2** 

© 2013 Lo, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

**Instrumentation for Ferromagnetic Resonance** 

Even FMR is an antique technique, it is still regarded as a powerful probe for one of the modern sciences, the spintronics. Since materials used for spintronics are either ferromagnetic or spin correlated, and FMR is not only employed to study their magneto static behaviors, for instances, anisotropies [1,2], exchange coupling [3,4,5,6], but also the spin dynamics; such as the damping constant [7,8,9], g factor [8,9], spin relaxation [9], etc. In this chapter a brief description about the key components and techniques of FMR will be given. For those who have already owned a commercial FMR spectrometer could find very helpful and detail information of their system from the instruction and operation manuals. The purpose of this text is for the one who want to understand a little more detail about commercial system, and for researchers who want to build their own spectrometers based

FMR spectrometer is a tool to record electromagnetic (EM) wave absorbed by sample of interest under the influence of external DC or Quasi DC magnetic field. Simply speaking, the spectrometer should consist of at least an EM wave excitation source, detector, and transmission line which bridges sample and EM source. The precession frequency of ferromagnetics lies at the regime of microwave (-wave) ranged from 0.1 to about 100 GHz, therefore, FMR absorption occurs at -wave range. The generation, detection and transmission at such this high frequency are not as simple as those for DC or low frequency electronics. According to transmission line and network theories [10,11], impedance of 50 between transmission line and load has to be matched for optimization of energy transfer. FMR spectrometer also has a resonator and an electro magnet which produces magnetic field to vary the sample's magnetization during the measurement. Sample which is mounted inside the cavity absorbs energy from the -wave source. The detector electronics records the changes on either the reflectance or transmittance of the -wave while magnetic

on vector network analyzer (VNA) would gain useful information as well.

**Spectrometer** 

http://dx.doi.org/10.5772/56069

**1. Introduction** 

Additional information is available at the end of the chapter

Chi-Kuen Lo


## **Instrumentation for Ferromagnetic Resonance Spectrometer**

Chi-Kuen Lo

46 Ferromagnetic Resonance – Theory and Applications

Vol. 322, pp. 485.

*hall.* pp. 1.

ferromagnetic resonance as a sensitive method to study the temperature dependence of

Zhang, B.; Cheng, J.; Gonga, X.; Dong, X.; Liu, X.; Ma, G.; Chang, J. (2008). Facile fabrication of multi-colors high fluorescent/superparamagnetic nanoparticles. *J. Colloid Interf. Sci.*

Zhu, J.; Katine, J.A.; Rowlands, G.E.; Chen, Y.J.; Duan, Z.; Alzate, J.G.; Upadhyaya, P.; Langer, J.; Amiri, P.K.; Wang, K.L.; Krivorotov, I.N. (2012). Voltage-induced ferromagnetic resonance in magnetic tunnel junctions. *ArXiv: 1205-2835: Cond-mat. Mes-*

interlayer exchange coupling. *Phys. Rev. Lett.* Vol. 73, pp. 336.

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/56069

### **1. Introduction**

Even FMR is an antique technique, it is still regarded as a powerful probe for one of the modern sciences, the spintronics. Since materials used for spintronics are either ferromagnetic or spin correlated, and FMR is not only employed to study their magneto static behaviors, for instances, anisotropies [1,2], exchange coupling [3,4,5,6], but also the spin dynamics; such as the damping constant [7,8,9], g factor [8,9], spin relaxation [9], etc. In this chapter a brief description about the key components and techniques of FMR will be given. For those who have already owned a commercial FMR spectrometer could find very helpful and detail information of their system from the instruction and operation manuals. The purpose of this text is for the one who want to understand a little more detail about commercial system, and for researchers who want to build their own spectrometers based on vector network analyzer (VNA) would gain useful information as well.

FMR spectrometer is a tool to record electromagnetic (EM) wave absorbed by sample of interest under the influence of external DC or Quasi DC magnetic field. Simply speaking, the spectrometer should consist of at least an EM wave excitation source, detector, and transmission line which bridges sample and EM source. The precession frequency of ferromagnetics lies at the regime of microwave (-wave) ranged from 0.1 to about 100 GHz, therefore, FMR absorption occurs at -wave range. The generation, detection and transmission at such this high frequency are not as simple as those for DC or low frequency electronics. According to transmission line and network theories [10,11], impedance of 50 between transmission line and load has to be matched for optimization of energy transfer. FMR spectrometer also has a resonator and an electro magnet which produces magnetic field to vary the sample's magnetization during the measurement. Sample which is mounted inside the cavity absorbs energy from the -wave source. The detector electronics records the changes on either the reflectance or transmittance of the -wave while magnetic

© 2013 Lo, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

field is being swept [12]. Most commercial spectrometers, for examples, ELEXSYS-II E580, Bruker Co [13], and JES-FA100, JOEL Ltd. [14], character FMR by measuring the reflectance at fixed band frequency. -wave is generated by a Klystron, which goes to a metallic cavity via a wave guide. Signal reflected back from the cavity through the same waveguide to a detector, the Schottky barrier diode. Noted that a circulator is employed, such that the wave does go to the detector and not return back to the generator. The integration of the source, detector, circulator, protected electronics, etc. in a single box, is named "Microwave Bridge". The basic configuration of most FMR spectrometer is shown in Fig.1. FMR absorption spectrum is obtained by comparing the incident and reflected signals. However, the signal to noise ratio (SNR) given by this kind of reflectometer is still not large enough to recognize the spectrum, and lock-in technique is needed to enhance SNR to acceptable value, for example, bigger than 5.

Instrumentation for Ferromagnetic Resonance Spectrometer 49

= �� (1)

(3)

�� = ���� = �� (2)

(4)

(5)

A Metallic resonator (cavity) is a space enclosed by metallic walls which sustain electromagnetic standing wave or electromagnetic oscillation. The cavity has to be coupled with external circuit which provides excitation energy. On the other hand, the excited cavity supplies energy to the sample of interest (loading) through coupling [10,11]. The basic structure of a cavity set is sketched as in Fig.2(a), which consists of a wave guide, a coupler with iris control, and a metallic cavity. The working principle of coupled cavity can be understood by using LRC equivalent circuit which couples to a transmission line [10,11] as shown in Fig.2(b) and 2(c). The coupling structure is represented by an ideal transformer with transforming ratio 1:n. The parallel RLC circuit could be transformed to Ts from AB plane via the transformer, such that C'=n2C, R'=R/n2, and L'=L/n2. The resonance frequency, fres, and Q-factor will not be affected by

> √���� <sup>=</sup> � � � �����

�� = ����� �

A coupling coefficient, , which defines the relation between energy dissipation (Ed) in a cavity and energy dissipation of external circuit (Ee), tells if the cavity is coupled with

> �� ��� �� �����

U in eq.(3) stands for the voltage applied to the cavity, and 1:n is the transforming ratio for tuning to critical couple state (ZC = �� = 1) before doing FMR measurement. The characterization impedance, ZC, is always designed to be 50 (Z0) for matching impedance as mentioned previously. Signal will be distorted and weakened if working at either undercoupling (<1) or over-coupling states (>1). The external Q factor (Qe) of a cavity, coupled to external circuit, states the energy dissipation in the external circuit, and also has the

> β = �� ��

In eq.(4), Q0 is the intrinsic Q factor of cavity. Practically, the loaded Q-factor, QL, is used in

Metallic cavities used for FMR measurement have to be TE mode, such that the cavity center which is also the sample location has maximum excitation magnetic field. Furthermore, this kind of cavity has very high Q-factor and plays key role in FMR detection. The signal

� ��� ; or � �� <sup>=</sup> � �� <sup>+</sup> � ��

<sup>=</sup> � ���� <sup>=</sup> �� ��

the transformation [10] as pointed out by eq.(1) and eq.(2), respectively:

�� � <sup>=</sup> �

β ≡ �� �� =

�� = ��

output, *VS*, from the cavity can be expressed as [15]:

�� � = �� � ��

**2. Key components of FMR** 

**2.1. Metallic cavity** 

external circuit.

experiments:

following relationship with :

The operational frequency of the source, detector and cavity cannot be varied broadly, therefore, it is necessary to change the microwave bridge and cavity while working with different band frequency. Dart a glance at a FMR spectrometer, the key parts are the microwave bridge, cavity, gauss meter, electromagnet, and lock-in amplifier for signal process. Surely, automation and data acquisition are also crucial.

**Figure 1.** Basic configuration of FMR spectrometer. The microwave bridge mainly consists of the microwave generator, circulator, and detector.

### **2. Key components of FMR**

#### **2.1. Metallic cavity**

48 Ferromagnetic Resonance – Theory and Applications

value, for example, bigger than 5.

field is being swept [12]. Most commercial spectrometers, for examples, ELEXSYS-II E580, Bruker Co [13], and JES-FA100, JOEL Ltd. [14], character FMR by measuring the reflectance at fixed band frequency. -wave is generated by a Klystron, which goes to a metallic cavity via a wave guide. Signal reflected back from the cavity through the same waveguide to a detector, the Schottky barrier diode. Noted that a circulator is employed, such that the wave does go to the detector and not return back to the generator. The integration of the source, detector, circulator, protected electronics, etc. in a single box, is named "Microwave Bridge". The basic configuration of most FMR spectrometer is shown in Fig.1. FMR absorption spectrum is obtained by comparing the incident and reflected signals. However, the signal to noise ratio (SNR) given by this kind of reflectometer is still not large enough to recognize the spectrum, and lock-in technique is needed to enhance SNR to acceptable

The operational frequency of the source, detector and cavity cannot be varied broadly, therefore, it is necessary to change the microwave bridge and cavity while working with different band frequency. Dart a glance at a FMR spectrometer, the key parts are the microwave bridge, cavity, gauss meter, electromagnet, and lock-in amplifier for signal

**Figure 1.** Basic configuration of FMR spectrometer. The microwave bridge mainly consists of the

microwave generator, circulator, and detector.

process. Surely, automation and data acquisition are also crucial.

A Metallic resonator (cavity) is a space enclosed by metallic walls which sustain electromagnetic standing wave or electromagnetic oscillation. The cavity has to be coupled with external circuit which provides excitation energy. On the other hand, the excited cavity supplies energy to the sample of interest (loading) through coupling [10,11]. The basic structure of a cavity set is sketched as in Fig.2(a), which consists of a wave guide, a coupler with iris control, and a metallic cavity. The working principle of coupled cavity can be understood by using LRC equivalent circuit which couples to a transmission line [10,11] as shown in Fig.2(b) and 2(c). The coupling structure is represented by an ideal transformer with transforming ratio 1:n. The parallel RLC circuit could be transformed to Ts from AB plane via the transformer, such that C'=n2C, R'=R/n2, and L'=L/n2. The resonance frequency, fres, and Q-factor will not be affected by the transformation [10] as pointed out by eq.(1) and eq.(2), respectively:

$$
\omega\_0' = \frac{1}{\sqrt{L'C'}} = \frac{1}{\sqrt{\frac{L}{n^2}n^2C}} = \omega\_0 \tag{1}
$$

$$Q\_0' = \omega\_0' \mathcal{C}' R' = \omega\_0 n^2 \mathcal{C} \frac{R}{n^2} = \omega\_0 \mathcal{C} R = Q\_0 \tag{2}$$

A coupling coefficient, , which defines the relation between energy dissipation (Ed) in a cavity and energy dissipation of external circuit (Ee), tells if the cavity is coupled with external circuit.

$$\beta \equiv \frac{E\_d}{E\_d} = \frac{\frac{U^2}{2Z\_C}}{\frac{U^2}{2R/n^2}} = \frac{R}{Z\_C n^2} = \frac{R'}{Z\_C} \tag{3}$$

U in eq.(3) stands for the voltage applied to the cavity, and 1:n is the transforming ratio for tuning to critical couple state (ZC = �� = 1) before doing FMR measurement. The characterization impedance, ZC, is always designed to be 50 (Z0) for matching impedance as mentioned previously. Signal will be distorted and weakened if working at either undercoupling (<1) or over-coupling states (>1). The external Q factor (Qe) of a cavity, coupled to external circuit, states the energy dissipation in the external circuit, and also has the following relationship with :

$$
\beta = \frac{\mathbf{Q}\_o}{\mathbf{Q}\_a} \tag{4}
$$

In eq.(4), Q0 is the intrinsic Q factor of cavity. Practically, the loaded Q-factor, QL, is used in experiments:

$$Q\_L = Q\_0 \frac{\beta}{1+\beta} ; \text{or } \frac{1}{Q\_L} = \frac{1}{Q\_0} + \frac{1}{Q\_0} \tag{5}$$

Metallic cavities used for FMR measurement have to be TE mode, such that the cavity center which is also the sample location has maximum excitation magnetic field. Furthermore, this kind of cavity has very high Q-factor and plays key role in FMR detection. The signal output, *VS*, from the cavity can be expressed as [15]:

$$W\_S = \chi^\* \eta Q\_L \sqrt{P Z\_0} \tag{6}$$

Instrumentation for Ferromagnetic Resonance Spectrometer 51

�� (7)

The Q-factor of commercial or home-made cavities can be determined easily by using a

Q = ����

*fres* is the resonant frequency of a cavity, and *f*, is the full width at half maximum (FWHM) of the resonant peak. Eq.(7) can be simply found by a built-in function of VNA which measures directly the reflected power of the cavity in dB at *fres*. Besides with dB unit, FWHM (f ) is also obtained at half power point, i.e. the power level at -3dB. As the values of *fres*

Q will be decreased once a sample is inserted. This is because the inserted sample and its holder absorb energy and change the coupling conditions resulting in Q reduction and resonance frequency shift. However, these deviations can be amended a little bit by adjusting the iris which controls the effective impedance by varying the aperture size between the cavity and wave guide, and by adjusting the position of metallic cap. This is equivalent to change the transformer ratio, that is, the equivalent inductance and capacitance. The iris control is a plastic screw (low -wave absorption material) which has a gold coated metal cap on one end, and the effective impedance of the whole (wave guide + iris + cavity) depends on the size of the aperture and location of the metal cap. The function of the iris acts as a device to tune the L and C, such that the cavity is critical couple again. As field is swept, the variations of sample's magnetization break the coupling condition so the cavity is deviated from critically coupled state. Consequently, wave is reflected back to the

For an ideal metallic cavity (infinite conductivity and perfectly smooth inside wall surface), there will be a unique resonant peak with infinite large Q-factor, However, this is not the case in practice. Since the cavity itself has finite conductivity, and further, the inside wall is not perfectly flat, these cause the existence of many peaks with very low Q values as extracted by a VNA shown in Fig.3(a). Fig.3(b) shows the simulation of a copper made cavity with a very rough inside wall. As the roughness is reduced, the spectrum is clearer as shown in Fig. 3(c). This could be due to the -wave which is multi-scattered by lumps on the wall surface, and these lumps could also enhance power dissipation further. In consequence, there exists many low Q peaks. Even there are many peaks other than the eigen frequency, these peaks do not response to the change of magnetization, and hence useless for FMR characterization. The manufacture of metallic cavity with Q-factor of higher than 3,000 is laboring. This is because the inside wall has to be mirror polished and coated with a few m

Fig.3(a) and Fig.3(d) are the frequency response of a Bruker X-band cavity (TE102, ER 4104OR) exhibiting in larger and smaller frequency windows, respectively. The unloaded Qfactor of this cavity is ~9,000 as claimed by the manufacturer, however, our measurement tells that the Q factor is more than 14,000. This is because frequency resolution of the VNA is

very high. As the resolution is decreased, Q is found to be decreased as well.

VNA, which can also be expressed as:

�-wave bridge, and FMR signal is resulted.

thick Au layer to reduce the imperfection.

and FWHM are known, Q is determined as shown in Fig.3(d).

", , *P* and *Z*0 in eq.(6) stand for imaginary part of magnetic susceptibility of the sample, filling factor, microwave power, and characteristic impedance, respectively. The Q-factor of a cavity directly relates to the detecting sensitivity, which depends very much on the design and manufacturing technique. A good metallic cavity normally has unloaded Q-factor of more than 5,000 which cannot be changed after being manufactured, and therefore is regarded as a constant in eq.(6). Z0 of 50 is fixed for matching the network impedance. We cannot play too much on " and as well, since the former is the intrinsic behaviour of the sample of interest to be tested, and the last parameter is the volume ratio of the sample and cavity. For thin film and multilayer samples, is very small, hence this term cannot contribute too much to *VS*. The source power, P, at the first glance, is the only possible adjustable parameter for sensitivity. Due to the occurrence of saturation, higher power may not be that helpful in increasing the FMR signal. Large excitation power drives magnetization precession in non-linear region which complicates the spectrum analysis. Besides, signal will be reduced and broadened while operating at saturation region. In order to determine the line shapes and widths precisely, this region should be avoided, hence low power is a good choice. The determination of saturation power is not difficult, since the signal intensity grows as the square root of power as indicated by eq.(6). Checking the signal intensity to see if eq.(6) is still validated as the power is increased. It is noted that the maximum excitation power of a common VNA is just a few mWs, and *VS* is not that clear to follow P.

**Figure 2.** (a) The layout of a resonant cavity, (b) The equivalent LRC circuit. The iris control is regarded as a transformer, and (c) The equivalent circuit transformed to the Ts plane [10].

The Q-factor of commercial or home-made cavities can be determined easily by using a VNA, which can also be expressed as:

50 Ferromagnetic Resonance – Theory and Applications

mWs, and *VS* is not that clear to follow P.

�� = "������ (6)

", , *P* and *Z*0 in eq.(6) stand for imaginary part of magnetic susceptibility of the sample, filling factor, microwave power, and characteristic impedance, respectively. The Q-factor of a cavity directly relates to the detecting sensitivity, which depends very much on the design and manufacturing technique. A good metallic cavity normally has unloaded Q-factor of more than 5,000 which cannot be changed after being manufactured, and therefore is regarded as a constant in eq.(6). Z0 of 50 is fixed for matching the network impedance. We cannot play too much on " and as well, since the former is the intrinsic behaviour of the sample of interest to be tested, and the last parameter is the volume ratio of the sample and cavity. For thin film and multilayer samples, is very small, hence this term cannot contribute too much to *VS*. The source power, P, at the first glance, is the only possible adjustable parameter for sensitivity. Due to the occurrence of saturation, higher power may not be that helpful in increasing the FMR signal. Large excitation power drives magnetization precession in non-linear region which complicates the spectrum analysis. Besides, signal will be reduced and broadened while operating at saturation region. In order to determine the line shapes and widths precisely, this region should be avoided, hence low power is a good choice. The determination of saturation power is not difficult, since the signal intensity grows as the square root of power as indicated by eq.(6). Checking the signal intensity to see if eq.(6) is still validated as the power is increased. It is noted that the maximum excitation power of a common VNA is just a few

**Figure 2.** (a) The layout of a resonant cavity, (b) The equivalent LRC circuit. The iris control is regarded

as a transformer, and (c) The equivalent circuit transformed to the Ts plane [10].

$$\mathbf{Q} = \frac{f\_{\rm res}}{\Delta f} \tag{7}$$

*fres* is the resonant frequency of a cavity, and *f*, is the full width at half maximum (FWHM) of the resonant peak. Eq.(7) can be simply found by a built-in function of VNA which measures directly the reflected power of the cavity in dB at *fres*. Besides with dB unit, FWHM (f ) is also obtained at half power point, i.e. the power level at -3dB. As the values of *fres* and FWHM are known, Q is determined as shown in Fig.3(d).

Q will be decreased once a sample is inserted. This is because the inserted sample and its holder absorb energy and change the coupling conditions resulting in Q reduction and resonance frequency shift. However, these deviations can be amended a little bit by adjusting the iris which controls the effective impedance by varying the aperture size between the cavity and wave guide, and by adjusting the position of metallic cap. This is equivalent to change the transformer ratio, that is, the equivalent inductance and capacitance. The iris control is a plastic screw (low -wave absorption material) which has a gold coated metal cap on one end, and the effective impedance of the whole (wave guide + iris + cavity) depends on the size of the aperture and location of the metal cap. The function of the iris acts as a device to tune the L and C, such that the cavity is critical couple again. As field is swept, the variations of sample's magnetization break the coupling condition so the cavity is deviated from critically coupled state. Consequently, wave is reflected back to the �-wave bridge, and FMR signal is resulted.

For an ideal metallic cavity (infinite conductivity and perfectly smooth inside wall surface), there will be a unique resonant peak with infinite large Q-factor, However, this is not the case in practice. Since the cavity itself has finite conductivity, and further, the inside wall is not perfectly flat, these cause the existence of many peaks with very low Q values as extracted by a VNA shown in Fig.3(a). Fig.3(b) shows the simulation of a copper made cavity with a very rough inside wall. As the roughness is reduced, the spectrum is clearer as shown in Fig. 3(c). This could be due to the -wave which is multi-scattered by lumps on the wall surface, and these lumps could also enhance power dissipation further. In consequence, there exists many low Q peaks. Even there are many peaks other than the eigen frequency, these peaks do not response to the change of magnetization, and hence useless for FMR characterization. The manufacture of metallic cavity with Q-factor of higher than 3,000 is laboring. This is because the inside wall has to be mirror polished and coated with a few m thick Au layer to reduce the imperfection.

Fig.3(a) and Fig.3(d) are the frequency response of a Bruker X-band cavity (TE102, ER 4104OR) exhibiting in larger and smaller frequency windows, respectively. The unloaded Qfactor of this cavity is ~9,000 as claimed by the manufacturer, however, our measurement tells that the Q factor is more than 14,000. This is because frequency resolution of the VNA is very high. As the resolution is decreased, Q is found to be decreased as well.

Instrumentation for Ferromagnetic Resonance Spectrometer 53

mounted at the top of a plastic screw through the shorted plate, such that its position can be adjusted for maximum signal. However, SWC does not have an iris to tune the impedance, hence critical coupling may not be easy to obtain. Although the Q is somewhat lower (less than 2,000), SWC is easy to make from commercial waveguide tube. Furthermore, different band frequency cavity can be built in the same manner, and this is particular convenient to

**Figure 4.** A shorted wave guide with possibility of sample manipulation. Magnetic field distribution

Microstrip (MS) and co-planar waveguide (CPW) which are indeed transmission lines, are commonly used to extract FMR at broad frequency range [16,17,18,19]. Sample is mounted on top of their signal lines, and -wave is conducted into MS (CPW) via Port 1 of the VNA, and the differential change in either reflected or transmitted signal is analyzed. In this operation scheme, frequency is swept at fixed field, and once the FMR conditions are fulfilled -wave absorption occurs. Fig.5 and Fig.6 are the sketches of MS and CPW. Since h << , -wave propagates in these two lines is regarded as quasi-TEM mode. In order to match the impedance of 50, the dimensions of these two planar transmission lines are

study FMR at different band frequency with network analyzer.

originated from the -wave is also sketched [16].

In case of MS, we have the followings [11]:

�� =

� � � � � �� �����

The effective dielectric constant, eff has approximately the form:

�� ��� � <sup>+</sup> �

������ �

��� <sup>=</sup> ���� �

����

������������� ���

<sup>+</sup> ���� �

The dimension of the strip line and characteristic impedance, ZC, can be worked out as

� ���������

�

��� ; for w/d � 1 

 ; for w/d ≥ 1

������/� (8)

(9)

restricted.

below:

**2.3. Microstrip and co-planar waveguide** 

**Figure 3.** (a) the frequency response of a X-band cavity. Simulation result of a Cu cavity with (b) a very rough inside wall surface, and (c) a smooth inside wall surface. (d) VNA measurement of an X-band cavity with Q higher than 87, 000. Note that there are many side peaks but not appears in this frequency range

### **2.2. Shorted waveguide cavity**

The fabrication of metallic cavity with very high Q-factor is not easy as mentioned above. If the lossy of the sample is not very big, cavity with Q of a few hundred to thousand may be good enough to recognize FMR. If so, a shorted waveguide cavity (SWC) could be an alternative. This kind of cavity is just a section of waveguide sealed by a metal plate at one end [16] as show in Fig.4. Since metallic wave guide are commercially standard with wide range of frequency from L to W band, it is worth to obtain FMR information at different bands with this cheap, simple and effective method. If the length of the waveguide tube is equal to n/2, standing wave can be formed inside. In that n and represents for integer and wave length of the -wave, respectively. In order to excite FMR signal, sample has to be placed at the n/4 position away from the shorted plate at which maximum magnetic field is located. In Fig.4, n=0 and n=1 are the suitable locations, however, n =0 gives advantages of easier sample manipulation, and lesser interference from irrelevant insertions. Sample is mounted at the top of a plastic screw through the shorted plate, such that its position can be adjusted for maximum signal. However, SWC does not have an iris to tune the impedance, hence critical coupling may not be easy to obtain. Although the Q is somewhat lower (less than 2,000), SWC is easy to make from commercial waveguide tube. Furthermore, different band frequency cavity can be built in the same manner, and this is particular convenient to study FMR at different band frequency with network analyzer.

**Figure 4.** A shorted wave guide with possibility of sample manipulation. Magnetic field distribution originated from the -wave is also sketched [16].

### **2.3. Microstrip and co-planar waveguide**

52 Ferromagnetic Resonance – Theory and Applications

range

**2.2. Shorted waveguide cavity** 

(a) (b)

(c) (d) **Figure 3.** (a) the frequency response of a X-band cavity. Simulation result of a Cu cavity with (b) a very rough inside wall surface, and (c) a smooth inside wall surface. (d) VNA measurement of an X-band cavity with Q higher than 87, 000. Note that there are many side peaks but not appears in this frequency

The fabrication of metallic cavity with very high Q-factor is not easy as mentioned above. If the lossy of the sample is not very big, cavity with Q of a few hundred to thousand may be good enough to recognize FMR. If so, a shorted waveguide cavity (SWC) could be an alternative. This kind of cavity is just a section of waveguide sealed by a metal plate at one end [16] as show in Fig.4. Since metallic wave guide are commercially standard with wide range of frequency from L to W band, it is worth to obtain FMR information at different bands with this cheap, simple and effective method. If the length of the waveguide tube is equal to n/2, standing wave can be formed inside. In that n and represents for integer and wave length of the -wave, respectively. In order to excite FMR signal, sample has to be placed at the n/4 position away from the shorted plate at which maximum magnetic field is located. In Fig.4, n=0 and n=1 are the suitable locations, however, n =0 gives advantages of easier sample manipulation, and lesser interference from irrelevant insertions. Sample is Microstrip (MS) and co-planar waveguide (CPW) which are indeed transmission lines, are commonly used to extract FMR at broad frequency range [16,17,18,19]. Sample is mounted on top of their signal lines, and -wave is conducted into MS (CPW) via Port 1 of the VNA, and the differential change in either reflected or transmitted signal is analyzed. In this operation scheme, frequency is swept at fixed field, and once the FMR conditions are fulfilled -wave absorption occurs. Fig.5 and Fig.6 are the sketches of MS and CPW. Since h << , -wave propagates in these two lines is regarded as quasi-TEM mode. In order to match the impedance of 50, the dimensions of these two planar transmission lines are restricted.

In case of MS, we have the followings [11]:

The effective dielectric constant, eff has approximately the form:

$$
\varepsilon\_{eff} = \frac{\varepsilon\_r + 1}{2} + \frac{\varepsilon\_r - 1}{2} \frac{1}{\sqrt{1 + 12 \text{h/w}}} \tag{8}
$$

The dimension of the strip line and characteristic impedance, ZC, can be worked out as below:

$$Z\_{\mathcal{C}} = \begin{cases} \frac{60}{\sqrt{\varepsilon\_{eff}}} \ln\left(\frac{8h}{w} + \frac{w}{4h}\right) & \text{; for w/d} \le 1 \\\\ \frac{120\pi}{\sqrt{\varepsilon\_{eff} \left[\frac{w}{d} + 1.393 + 0.667 \ln\left(\frac{w}{d} + 1.444\right)\right]}} & \text{; for w/d} \ge 1 \end{cases} \tag{9}$$

$$
\varepsilon\_{eff} = \frac{\varepsilon\_r + 1}{2} \tag{10}
$$

$$Z\_{C} = \begin{cases} \frac{\eta\_{o}}{\pi \sqrt{\varepsilon\_{eff}}} \ln \left( 2 \sqrt{\frac{b}{w}} \right) \,\Omega & \quad \text{; } 0 < \text{w/b} < 0.173 \\\\ \frac{\pi \,\eta\_{o}}{4 \sqrt{\varepsilon\_{eff}}} \left[ \ln \left( 2 \frac{1 + \sqrt{w/b}}{1 - \sqrt{w/b}} \right) \right]^{-1} \,\Omega & \quad \text{; } 0.173 < \frac{\text{w}}{\text{b}} < 1 \end{cases} \tag{11}$$

$$\mathbf{V} = \mathbf{S} + \mathbf{n} \tag{12}$$

$$\mathbf{V}(t) = \mathbf{S}\cos(\omega t) + n\tag{13}$$

$$V\_{ref} = V\_a \cos(\omega t + \phi) \tag{14}$$

$$\begin{array}{l} \text{V(t)}V\_{ref} = \text{SV}\_a \cos(\omega t \text{ )} \cos(\omega t + \phi) + nV\_a \cos(\omega t + \phi) = \\ \frac{\text{SV}\_a}{2} [\cos(\phi) + \cos(2\omega t + \phi)] + nV\_a \cos(\omega t + \phi) \end{array} \tag{15}$$

The second and third terms can be eliminated if these two parts are passed to a low pass filter with cutoff frequency setting at /2 or lower. Finally, we have:

$$\mathbf{V} \ll \frac{\text{SV}\_a}{2} \cos(\phi) \tag{16}$$

Instrumentation for Ferromagnetic Resonance Spectrometer 57

Signal strength is linearly proportional to the MA while MA is small. Simply speaking, MA should be small enough for sampling in linear region, but needs to be large enough for gaining good sensitivity. However, too large the MA results in signal distortion as shown in Fig.8(a). This can be understood from eq.(20) that high order term cannot be ignored for large MA which causes distortion of derivative signal. The choice of MF is critical as well that small MF cannot get rid of 1/f noise completely, and elongates the acquisition time. Large MA and MF also cause passage effect that the rate of "passage" through the absorption line is faster than the relaxation rates, which results in distorted spectrum, or even inversion of signals upon reversal of the field scan direction. Therefore, it is important to check the FMR signal with either a standard or a well-known sample with different lockin conditions for the best settings. In order to obtain correct line shape spectrum, the rule of thumb is to set the MA about 1/10 of the FWHM and increases to about 1/3 if necessary. TC also needs to coordinate with MF. According to Nyquist sampling theorem, the sampling rate should be at least twice the highest frequency contained in the signal [21]. For example, if the MF is 100 kHz, the sampling frequency would be at least 200 kHz at which TC is about 5S. If this restriction does not fulfill, sampling points cannot not be captured immediately. Consequently, information loss and line shape distortion are always resulted. The best way to avoid this is to set the scan time for a FMR signal 10 times longer than the time constant.

(a) (b) **Figure 8.** The influence of modulation amplitude (a) and modulation frequency (b) on signal. Distortion

VNA is indeed an instrument developed for characterization of electrical devices (device under test, DUT) by sending an electromagnetic wave. As an analogue of that in optics, the

will be resulted if MA and MF are not appropriate [15].

**4. Vector analyzer based FMR spectrometer** 

**4.1. Basics of vector network analyzer (VNA)** 

That is, the signal is amplified by Va, and has a maximum while is "phase-locked" to zero, hence the name of lock-in amplifier.

#### **3.2. Feld modulation lock-in technique and derivative spectrum**

It is easier to determine Hres, and Hpp by differentiating the original signal. To do so, field modulation lock-in detection is employed and described below.

There is a small field, Ha, with modulation frequency of superposing on top of external DC magnetic field, H0:

$$\mathbf{H}(t) = \mathbf{H}\_0 + \mathbf{H}\_\mathbf{a} \cos(\omega t) \tag{17}$$

Assuming we have FMR signal, VFMR, whose Taylor expansion at H0 is given below:

$$\mathbf{V\_{FMR}}\{\mathbf{H}\} = \mathbf{V\_{FMR}}\{\mathbf{H}\_0\} + \frac{\mathbf{dV\_{FMR}}}{\mathbf{d}\mathbf{H}}\Big|\_{\mathbf{H}=\mathbf{H}\_0} \mathbf{H}\_\mathbf{a} \cos(\omega t) + \cdots \tag{18}$$

Meanwhile, we also have another signal, the reference, which has the same frequency as that of the modulation field, but with a phase angle, :

$$V\_{ref} = \cos(\omega t + \phi) \tag{19}$$

The product of eq.(18) and (19) gives eq.(20) below:

$$\rm V\_{FMR} \{ H \} \times V\_{ref} = \rm V\_{FMR} \left( H\_0 \right) \cos \{ \omega t + \phi \right) \frac{dV\_{FMR}}{dH} \Big|\_{H = H\_0} + H\_\mathbf{a} \cos \{ \omega t \} \cos \{ \omega t + \phi \} + \cdots = \rm V\_{FMR} \left( H\_0 \right) \cos \{ \omega t \} + \frac{1}{2} \frac{dV\_{FMR}}{dH} \H\_\mathbf{a} \cos \{ 2\omega t + \phi \} + \cdots \tag{20}$$
 
$$\rm V\_{FMR} \left( H\_0 \right) \cos \{ \omega t + \phi \} + \frac{1}{2} \frac{dV\_{FMR}}{dH} \H\_\mathbf{a} \cos \{ \phi \} + \frac{1}{2} \frac{dV\_{FMR}}{dH} \H\_\mathbf{a} \cos \{ 2\omega t + \phi \} + \cdots \tag{20}$$

The first and third terms in eq.(20) can again be removed by using a low pass filter with cutoff frequency setting at /2 or even lower. The second term is time independent, proportional to derivative of the input signal and magnify by Ha, which has a maximum if the phase, , is locked to 0. Another advantage of use this method is that 1/f noise and drift problem can be excluded by setting the sampling window at high frequency.

The way to turn the DC or quasi DC signal from to AC is an important subject. The simple and normal way to do this is to insert two coils which sandwich the cavity. These coils are driven by a power amplify at certain high frequency, such that an AC magnetic field of a few mT at a frequency up to 200 kHz can be produced. Therefore, the output signal consists of the modulation and DC components. All these together with the reference are sent to the lock-in amplify for SNR enhancement and derivative spectrum as mentioned previously. Modulation amplitude (MA), modulation frequency (MF), and time constant (TC) which is the reciprocal of the low pass filter's cutoff frequency, have large influence on the spectra. Signal strength is linearly proportional to the MA while MA is small. Simply speaking, MA should be small enough for sampling in linear region, but needs to be large enough for gaining good sensitivity. However, too large the MA results in signal distortion as shown in Fig.8(a). This can be understood from eq.(20) that high order term cannot be ignored for large MA which causes distortion of derivative signal. The choice of MF is critical as well that small MF cannot get rid of 1/f noise completely, and elongates the acquisition time. Large MA and MF also cause passage effect that the rate of "passage" through the absorption line is faster than the relaxation rates, which results in distorted spectrum, or even inversion of signals upon reversal of the field scan direction. Therefore, it is important to check the FMR signal with either a standard or a well-known sample with different lockin conditions for the best settings. In order to obtain correct line shape spectrum, the rule of thumb is to set the MA about 1/10 of the FWHM and increases to about 1/3 if necessary. TC also needs to coordinate with MF. According to Nyquist sampling theorem, the sampling rate should be at least twice the highest frequency contained in the signal [21]. For example, if the MF is 100 kHz, the sampling frequency would be at least 200 kHz at which TC is about 5S. If this restriction does not fulfill, sampling points cannot not be captured immediately. Consequently, information loss and line shape distortion are always resulted. The best way to avoid this is to set the scan time for a FMR signal 10 times longer than the time constant.

**Figure 8.** The influence of modulation amplitude (a) and modulation frequency (b) on signal. Distortion will be resulted if MA and MF are not appropriate [15].

### **4. Vector analyzer based FMR spectrometer**

### **4.1. Basics of vector network analyzer (VNA)**

56 Ferromagnetic Resonance – Theory and Applications

hence the name of lock-in amplifier.

VFMR(H) = VFMR (H0)+

of the modulation field, but with a phase angle, :

The product of eq.(18) and (19) gives eq.(20) below:

VFMR(H) � ���� = VFMR (H0) cos(�� + )

� �VFMR

�� Hacos() <sup>+</sup> �

problem can be excluded by setting the sampling window at high frequency.

DC magnetic field, H0:

VFMR (H0) cos(�� + ) <sup>+</sup> �

The second and third terms can be eliminated if these two parts are passed to a low pass

That is, the signal is amplified by Va, and has a maximum while is "phase-locked" to zero,

It is easier to determine Hres, and Hpp by differentiating the original signal. To do so, field

There is a small field, Ha, with modulation frequency of superposing on top of external

H(�) = H� + H� cos(��) (17)

�VFMR �� � ����

Meanwhile, we also have another signal, the reference, which has the same frequency as that

� �VFMR

The first and third terms in eq.(20) can again be removed by using a low pass filter with cutoff frequency setting at /2 or even lower. The second term is time independent, proportional to derivative of the input signal and magnify by Ha, which has a maximum if the phase, , is locked to 0. Another advantage of use this method is that 1/f noise and drift

The way to turn the DC or quasi DC signal from to AC is an important subject. The simple and normal way to do this is to insert two coils which sandwich the cavity. These coils are driven by a power amplify at certain high frequency, such that an AC magnetic field of a few mT at a frequency up to 200 kHz can be produced. Therefore, the output signal consists of the modulation and DC components. All these together with the reference are sent to the lock-in amplify for SNR enhancement and derivative spectrum as mentioned previously. Modulation amplitude (MA), modulation frequency (MF), and time constant (TC) which is the reciprocal of the low pass filter's cutoff frequency, have large influence on the spectra.

�VFMR �� � ����

Assuming we have FMR signal, VFMR, whose Taylor expansion at H0 is given below:

� cos() (16)

H� cos(��) + ⋯ (18)

+Hacos(ωt) cos(�� + )+⋯=

�� Hacos(��� + ) + ⋯ (20)

���� = cos(�� + ) (19)

V ∝ <sup>S</sup>��

filter with cutoff frequency setting at /2 or lower. Finally, we have:

**3.2. Feld modulation lock-in technique and derivative spectrum** 

modulation lock-in detection is employed and described below.

VNA is indeed an instrument developed for characterization of electrical devices (device under test, DUT) by sending an electromagnetic wave. As an analogue of that in optics, the

incident wave (either optical or micro wave) will be reflected or/and transmit after interacting with the DUT. By examining the reflectance and transmittance, that is, the ratios of the powers of reflected, transmitted to that of the incident waves, scattering parameters of the DUT can be found as depicted in Fig.9. VNA cannot only find out the reflectance and transmittance, but also the impedance, phase lag, insertion loss, return loss, voltage standing ratio, etc., can be worked out. However for FMR experiment, only the first two functions are employed. VNA has at least two ports, and each port can produce and measure -wave. The scattering parameter Sij stands for scattering power ratio of incident wave produced by port i, and measured by port j. That is, for a two-port VNA, S11 and S22 characterize wave reflectance, while S12 and S21 determine the transmittance. If FMR is determined by reflected spectrum, one port is enough by analyzing either the S11 or S22. For transmittance spectrum, two ports are required for either the S12, or S21 determination. VNA has capability of microwave generation and detection, and furthermore, nowadays model has frequency ranged from about 0.1 to 100 GHz, or even higher at excitation power of tens dBm, Thus, it can serve as a microwave bridge to extract FMR parameters at a very broad frequency band. VNA measures not only the scattering amplitudes, but also their correlated phases. This is in contrast to its counterpart, the scalar network analyzer (SNA) which cannot tell phase information. FMR signal is extracted from power absorption which contains no phase information, and SNA should be good enough for the purpose. However, this kind of machine has been obsoleted for many years.

Instrumentation for Ferromagnetic Resonance Spectrometer 59

the characterization of nano scale sample is thus difficult without using lock-in. The

Metallic cavity usually has rather higher Q-factor and could be simply used with a VNA for field swept FMR detection. Taking the advantages of high Q cavity and VNA, C.K. Lo, *et. al*. [23] demonstrated FMR measurement without employing field modulation lock-in technique. They employed a TE102 cavity with unload Q of ~9,000 at X-band, and FMR signal was recorded through the measurement of reflectance, and therefore one VNA port for S11 was used. This powerful method extracts signal easily and directly as shown in Fig.10(a). One could if necessary, differentiate the original data for derivative spectrum which is used to determine the peak position and line wide as those in commercial FMR spectrometer. Further, this combination has quite good sensitivity that 1.6 nm CoFeB can be detected with

(a) (b) **Figure 10.** (a) FMR of 5 nm Py exacted at P = 10 dBm. The dotted line is the derivative of experimental

A new built spectrometer should be characterized before properly used and well known samples, such as permalloy, Fe, Co, etc., ferromagnetics are normally employed due to their H and peak positions can be found widely in literatures. Also, these samples are easy to prepare with different thicknesses, and results should be comparable to those in literatures

There are many build-in useful functions with nowadays VNA, and only some of them are used, for instances, the traces of valley (dip) position, Q factor, band width, average, etc. Dip frequency and Q-factor will be varied as the sample's magnetization state is changed by external field. The changes of these quantities were also recorded simultaneously for a Fe/Ag multilayer as shown in Fig.11 in which (a) is the FMR absorption, and (b), the post derivative of (a). The shifts of resonant frequency and Q-factor are shown in (c) and (d), respectively. Despite the variation of center frequency is just a few hundreds MHz, it allows us to determine ferromagnetic parameters precisely. The change of Q-factor is upside-down to that of the absorption. This is because any absorption of microwave inside the cavity will

application lock-in with VNA will be discussed in next section.

**4.2. Field swept VNA-FMR** 

SNR better than 5 as shown in Fig.10(b).

data. (b) Signal extracted from a 1.6nm CoFeB

reported.

reduce the Q value.

**Figure 9.** The fundamentals of microwave network analysis are analogue to that of the optics. The lower part of the panel indicates the definitions of scattering parameters [24].

VNA is commonly used with MS and CPW for FMR determination with frequency swept at fixed field. Since signal output from VNA cannot be plugged into lock-in amplifier directly, the characterization of nano scale sample is thus difficult without using lock-in. The application lock-in with VNA will be discussed in next section.

### **4.2. Field swept VNA-FMR**

58 Ferromagnetic Resonance – Theory and Applications

machine has been obsoleted for many years.

incident wave (either optical or micro wave) will be reflected or/and transmit after interacting with the DUT. By examining the reflectance and transmittance, that is, the ratios of the powers of reflected, transmitted to that of the incident waves, scattering parameters of the DUT can be found as depicted in Fig.9. VNA cannot only find out the reflectance and transmittance, but also the impedance, phase lag, insertion loss, return loss, voltage standing ratio, etc., can be worked out. However for FMR experiment, only the first two functions are employed. VNA has at least two ports, and each port can produce and measure -wave. The scattering parameter Sij stands for scattering power ratio of incident wave produced by port i, and measured by port j. That is, for a two-port VNA, S11 and S22 characterize wave reflectance, while S12 and S21 determine the transmittance. If FMR is determined by reflected spectrum, one port is enough by analyzing either the S11 or S22. For transmittance spectrum, two ports are required for either the S12, or S21 determination. VNA has capability of microwave generation and detection, and furthermore, nowadays model has frequency ranged from about 0.1 to 100 GHz, or even higher at excitation power of tens dBm, Thus, it can serve as a microwave bridge to extract FMR parameters at a very broad frequency band. VNA measures not only the scattering amplitudes, but also their correlated phases. This is in contrast to its counterpart, the scalar network analyzer (SNA) which cannot tell phase information. FMR signal is extracted from power absorption which contains no phase information, and SNA should be good enough for the purpose. However, this kind of

**Figure 9.** The fundamentals of microwave network analysis are analogue to that of the optics. The

VNA is commonly used with MS and CPW for FMR determination with frequency swept at fixed field. Since signal output from VNA cannot be plugged into lock-in amplifier directly,

lower part of the panel indicates the definitions of scattering parameters [24].

Metallic cavity usually has rather higher Q-factor and could be simply used with a VNA for field swept FMR detection. Taking the advantages of high Q cavity and VNA, C.K. Lo, *et. al*. [23] demonstrated FMR measurement without employing field modulation lock-in technique. They employed a TE102 cavity with unload Q of ~9,000 at X-band, and FMR signal was recorded through the measurement of reflectance, and therefore one VNA port for S11 was used. This powerful method extracts signal easily and directly as shown in Fig.10(a). One could if necessary, differentiate the original data for derivative spectrum which is used to determine the peak position and line wide as those in commercial FMR spectrometer. Further, this combination has quite good sensitivity that 1.6 nm CoFeB can be detected with SNR better than 5 as shown in Fig.10(b).

**Figure 10.** (a) FMR of 5 nm Py exacted at P = 10 dBm. The dotted line is the derivative of experimental data. (b) Signal extracted from a 1.6nm CoFeB

A new built spectrometer should be characterized before properly used and well known samples, such as permalloy, Fe, Co, etc., ferromagnetics are normally employed due to their H and peak positions can be found widely in literatures. Also, these samples are easy to prepare with different thicknesses, and results should be comparable to those in literatures reported.

There are many build-in useful functions with nowadays VNA, and only some of them are used, for instances, the traces of valley (dip) position, Q factor, band width, average, etc. Dip frequency and Q-factor will be varied as the sample's magnetization state is changed by external field. The changes of these quantities were also recorded simultaneously for a Fe/Ag multilayer as shown in Fig.11 in which (a) is the FMR absorption, and (b), the post derivative of (a). The shifts of resonant frequency and Q-factor are shown in (c) and (d), respectively. Despite the variation of center frequency is just a few hundreds MHz, it allows us to determine ferromagnetic parameters precisely. The change of Q-factor is upside-down to that of the absorption. This is because any absorption of microwave inside the cavity will reduce the Q value.

Instrumentation for Ferromagnetic Resonance Spectrometer 61

The advantages of the VNA-FMR with high Q cavity are: (1) the sensitivity which is comparable to that of commercials, but a lot cheaper, (2) multi frequencies can be done in

**Figure 13.** (a) data taken at Q 50,000, and (b) at ~110,000. Clearly, distorted signals are obtained.

line and VNA, otherwise damage is resulted.

*Department of Physics, National Taiwan Normal University, Taipei,* 

**Author details** 

Chi-Kuen Lo

*Taiwan* 

the same manner by just changing to another band cavity only within the frequency range of the VNA. Nevertheless, some points needed to be well aware. Firstly, VNA has built-in sweep average function which increases the SNR for every sweeping. The penalty of using high average count is the very long acquisition time. For example, a commercial FMR spectrometer takes about a few seconds to sweep a spectrum over 2 kOe field at 1 Oe resolution with time constant of a 1mS. However, the settled time for a VNA with average of 1 is found to be about 0.1S a point. Therefore, it takes at least 3 min to obtain the spectrum with the same conditions. Average of 10, and sometime 50 are often used for better SNR, then the acquisition time is very long which could overheat the electromagnet. Secondary, VNA cannot work with lock-in amplifier directly, and a circulator is also needed to force the reflected wave to a new port. Before the signal is conducted to the lock-in amplifier, a microwave transducer (Schottky diode, for example) is required. This is because the output from VNA is power which has to be transduced into voltage before plugging into the lock-in amplifier. Besides, the output of VNA has already been averaged over a range of frequencies. Thus the Schottky detector cannot recognize this tiny average change. This problem can be amended by narrowing the starting and ending frequencies of the VNA. The two frequencies indeed can be set to the same value as the dip frequency. While working with lock-in, the VNA-FMR acquires data as fast as the commercial one, since the SNR enhancement is processed by the lock-in and not by the VNA. Further, MS-, and CPW- FMR can also work together with lock-in method in this manner. If one wants to study high power VNA-FMR, a high power microwave amplifier could be used. However, if the VNA is still used as the analyzer, a suitable attenuator has to be inserted between the returned

**Figure 11.** The changes of FMR (a), resonance frequency (c), and Q-factor are recorded. (b) is the derivative spectrum taken from (a) mathematically.

The excitation power of VNA is only tens dBm, and it is unlikely to drive the sample into non-linear regime. Signal intensity is proportional to P as indicated by eq. (6), however, SNR is found to be not much different if P is bigger than -20 dBm as demonstrated in Fig.12.

**Figure 12.** FMR signal of 10nm Py was recorded as function of excitation power. Note that these spectra have the same scale, but different offset for clear comparison.

As mentioned previously that VNA has very high frequency resolution, apparently, Q could be tuned as high as 140,000 and more by simply adjusting the iris. However, distorted spectrum is resulted at very high Q, and two peaks appear while Q is adjusted close to maximum Q as seen in Fig.13(a) and (b), respectively. It is also found that the SNR does not varied significantly if Q lies between 2,000 to 13,000 (the unloaded Q is about 9,000 as claimed). Surely, this finding is just for referral, and would depend very much on cavity used.

The advantages of the VNA-FMR with high Q cavity are: (1) the sensitivity which is comparable to that of commercials, but a lot cheaper, (2) multi frequencies can be done in

**Figure 13.** (a) data taken at Q 50,000, and (b) at ~110,000. Clearly, distorted signals are obtained.

the same manner by just changing to another band cavity only within the frequency range of the VNA. Nevertheless, some points needed to be well aware. Firstly, VNA has built-in sweep average function which increases the SNR for every sweeping. The penalty of using high average count is the very long acquisition time. For example, a commercial FMR spectrometer takes about a few seconds to sweep a spectrum over 2 kOe field at 1 Oe resolution with time constant of a 1mS. However, the settled time for a VNA with average of 1 is found to be about 0.1S a point. Therefore, it takes at least 3 min to obtain the spectrum with the same conditions. Average of 10, and sometime 50 are often used for better SNR, then the acquisition time is very long which could overheat the electromagnet. Secondary, VNA cannot work with lock-in amplifier directly, and a circulator is also needed to force the reflected wave to a new port. Before the signal is conducted to the lock-in amplifier, a microwave transducer (Schottky diode, for example) is required. This is because the output from VNA is power which has to be transduced into voltage before plugging into the lock-in amplifier. Besides, the output of VNA has already been averaged over a range of frequencies. Thus the Schottky detector cannot recognize this tiny average change. This problem can be amended by narrowing the starting and ending frequencies of the VNA. The two frequencies indeed can be set to the same value as the dip frequency. While working with lock-in, the VNA-FMR acquires data as fast as the commercial one, since the SNR enhancement is processed by the lock-in and not by the VNA. Further, MS-, and CPW- FMR can also work together with lock-in method in this manner. If one wants to study high power VNA-FMR, a high power microwave amplifier could be used. However, if the VNA is still used as the analyzer, a suitable attenuator has to be inserted between the returned line and VNA, otherwise damage is resulted.

### **Author details**

60 Ferromagnetic Resonance – Theory and Applications

derivative spectrum taken from (a) mathematically.

have the same scale, but different offset for clear comparison.

**Figure 11.** The changes of FMR (a), resonance frequency (c), and Q-factor are recorded. (b) is the

The excitation power of VNA is only tens dBm, and it is unlikely to drive the sample into non-linear regime. Signal intensity is proportional to P as indicated by eq. (6), however, SNR is found to be not much different if P is bigger than -20 dBm as demonstrated in Fig.12.

**Figure 12.** FMR signal of 10nm Py was recorded as function of excitation power. Note that these spectra

As mentioned previously that VNA has very high frequency resolution, apparently, Q could be tuned as high as 140,000 and more by simply adjusting the iris. However, distorted spectrum is resulted at very high Q, and two peaks appear while Q is adjusted close to maximum Q as seen in Fig.13(a) and (b), respectively. It is also found that the SNR does not varied significantly if Q lies between 2,000 to 13,000 (the unloaded Q is about 9,000 as claimed).

Surely, this finding is just for referral, and would depend very much on cavity used.

Chi-Kuen Lo *Department of Physics, National Taiwan Normal University, Taipei, Taiwan* 

#### **5. References**

[1] M. D´az de Sihues, C. A. Durante-Rincon, and J. R. Fermin, J. Magn. Magn. Mater. 316, e462 (2007).

**Chapter 3** 

© 2013 Montiel and Alvarez, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**Detection of Magnetic Transitions by Means of** 

Due to nature of the magnetic materials, the experimental techniques to study their physical properties are generally sophisticated and expensive; and several techniques are used to obtain reliable information on the magnetic properties of these materials. One of the most employed techniques to characterize the magnetic materials is the electron magnetic resonance (EMR), also well-known as the ferromagnetic resonance (FMR) at temperatures below Curie temperature (Tc) and the electron paramagnetic resonance at temperatures above Tc. EMR is a powerful technique for studying the spin structure and magnetic properties in bulk samples, thin films and nanoparticles, being mainly characterized by means of two parameters: the resonant field (Hres) and the linewidth (HPP); these parameters reveal vital information on magnetic nature of the materials (Montiel et al., 2004, 2006; Alvarez et al., 2008, 2010). It is also necessary to mention that EMR is one of the most commonly used techniques to research the dependence of the magnetic anisotropy with respect to orientation of the sample (Montiel et al., 2007, 2008; G. Alvarez et al., 2008) and the temperature (Montiel et al., 2004; Alvarez et al., 2010); this technique is also applied to study magnetic relaxation in solid materials through their linewidth, and that it is due to

In particular, EMR technique is employed with success to determine the onset of magnetic transitions (Okamura, 1951; Okamura et al., 1951; Healy, 1952; Montiel et al., 2004; Alvarez et al., 2006, 2010), such as the Néel transition (from a paramagnetic phase to antiferromagnetic ordering) and the Curie transition (from a ferromagnetic or ferrimagnetic order to paramagnetic phase); through changes in the spectral parameters. EMR around critical temperatures have been object of an active research, e.g. Okamura et al. (1951) have

and reproduction in any medium, provided the original work is properly cited.

**Ferromagnetic Resonance and Microwave** 

**Absorption Techniques** 

Additional information is available at the end of the chapter

conduction mechanisms and intrinsic relaxation.

H. Montiel and G. Alvarez

http://dx.doi.org/10.5772/55962

**1. Introduction** 


## **Detection of Magnetic Transitions by Means of Ferromagnetic Resonance and Microwave Absorption Techniques**

H. Montiel and G. Alvarez

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/55962

### **1. Introduction**

62 Ferromagnetic Resonance – Theory and Applications

Phys. Rev. Lett. 90(18), 187601 (2003).

and E. C. Passamani, J. Appl. Phys. 99, 08C108 (2006).

Trans. Magn., VOL. 44, NO. 11, NOVEMBER 2008

K. Ong, C. P. Neo, John Wileys & Sons Ltd. 2004

[13] See http://www.bruker-biospin.com/epr-products.html.

SpinResonance/tabid/98/Default.aspx.

[1] M. D´az de Sihues, C. A. Durante-Rincon, and J. R. Fermin, J. Magn. Magn. Mater. 316,

[2] V. G. Gavriljuk, A. Dobrinsky, B. D. Shanina, and S. P. Kolesnik, J. Phys. Condens.

[5] V. P. Nascimento and E. Baggio Saitovitch, F. Pelegrinia, L. C. Figueiredo, A. Biondo,

[6] K. Lenz, E. Kosubek, T. Tolinski, J. Lindner, and K. Baberschke, J. Phys.: Condens.

[7] M. Oogane, T. Wakitani, S. Yakata, R. Yilgin, Y. Ando, A. Sakuma, T. Miyazaki, J. J.

[10] "Microwave Electronics: Measurement and Materials Characterization", L. F. Chen, C.

[12] "High-Frequency EPR Instrumentation", E.J. Reijerse, Appl Magn Reson (2010) 37:795–

[15] "*Quantitative EPR",* G. R. Eaton, S. S. Eaton, D. P. Barr, and R. T. Weber, Springer Wien,

[16] Marina Vroubel, Yan Zhuang, Behzad Rejaei, and Joachim N. Burghartz, J. Appl. Phys.

[17] C. Nistor, K. Sun, Z. Wang, M. Wu, C. Mathieu, and Matthew Hadley Appl. Phys. Lett.

[18] Y. Chen, D.S. Hung, Y.D. Yao, S.F. Lee, H.O. Ji, C. Yu, J. Appl. Phys. 101, 09C104 (2007)

[19] H. G LOWI'NSKI, J. DUBOWIK, Acta Physicae Superficierum , Vol. XII, 2012 [20] "The Art of Electronics", P. Horowitz, W. Hill, Cambridge Univ. Press, 2nd Ed. 1989 [21] Advanced Digital Signal Processing and Noise Reduction", S.V. Vaseghi, John Wiley &

[23] C.K. Lo, W.C. Lai, J.C. Cheng, Rev. Sci. Instruments, 82, 086114 (2011)

[24] R.W. Damon, Rev. Mod. Phys., 25, No.1, 239 ( 1953) [25] N. Bloembergen, S. Wang, Phys. Rev., 93 No.1 72 (1953)

[8] R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev. Lett. 87(21), 271204-1 (2009). [9] T. Kato, K. Nakazawa, R. Komiya, N. Nishizawa, S. Tsunashima, and S. Iwata, IEEE

[11] "Microwave Engineering", D.M. Pozar, John Wileys & Sons Ltd. (2012)

[14] See http://www.jeol.com/PRODUCTS/AnalyticalInstruments/Electron

[3] Z. Zhang, L. Zhou, P. E. Wigen, and K. Ounadjela, Phys. Rev. B 50(9), 6094 (1994). [4] B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas, R. Urban, and G. E. W. Bauer,

**5. References** 

818

New York, (2010)

99, 08P506 (2006)

95, 012504 (2009)

Sons Ltd., 2008 [22] Agilent VNA user manual

e462 (2007).

Matter 18, 7613 (2006).

Matter 15, 7175 (2003).

Appl. Phys. 50 (5A) 3889 (2006)

Due to nature of the magnetic materials, the experimental techniques to study their physical properties are generally sophisticated and expensive; and several techniques are used to obtain reliable information on the magnetic properties of these materials. One of the most employed techniques to characterize the magnetic materials is the electron magnetic resonance (EMR), also well-known as the ferromagnetic resonance (FMR) at temperatures below Curie temperature (Tc) and the electron paramagnetic resonance at temperatures above Tc. EMR is a powerful technique for studying the spin structure and magnetic properties in bulk samples, thin films and nanoparticles, being mainly characterized by means of two parameters: the resonant field (Hres) and the linewidth (HPP); these parameters reveal vital information on magnetic nature of the materials (Montiel et al., 2004, 2006; Alvarez et al., 2008, 2010). It is also necessary to mention that EMR is one of the most commonly used techniques to research the dependence of the magnetic anisotropy with respect to orientation of the sample (Montiel et al., 2007, 2008; G. Alvarez et al., 2008) and the temperature (Montiel et al., 2004; Alvarez et al., 2010); this technique is also applied to study magnetic relaxation in solid materials through their linewidth, and that it is due to conduction mechanisms and intrinsic relaxation.

In particular, EMR technique is employed with success to determine the onset of magnetic transitions (Okamura, 1951; Okamura et al., 1951; Healy, 1952; Montiel et al., 2004; Alvarez et al., 2006, 2010), such as the Néel transition (from a paramagnetic phase to antiferromagnetic ordering) and the Curie transition (from a ferromagnetic or ferrimagnetic order to paramagnetic phase); through changes in the spectral parameters. EMR around critical temperatures have been object of an active research, e.g. Okamura et al. (1951) have

© 2013 Montiel and Alvarez, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

studied the Néel transition in some materials by means of EMR technique. Fig. 1 shows the direct resonance absorption versus magnetic field for MnS in the 78-329 K temperature range. The height of the resonant absorption decreases and the linewidth is become broader with decreasing temperature, especially below the Néel temperature (TN= 160 K); where the position of the maximum is found at a constant magnetic field of 3510 G for all the temperatures, suggesting the existence of a strong local magnetic field in this material.

Detection of Magnetic Transitions by

Means of Ferromagnetic Resonance and Microwave Absorption Techniques 65

**Figure 2.** Direct resonance absorption versus dc magnetic field at 9 GHz in a sphere NiFe2O4 around

**Figure 3.** Direct resonance absorption versus dc magnetic field at 9.3 GHz for a disk NiFe2O4 at low

Curie transition (adapted from Healy, 1952).

temperature (adapted from Okamura, 1951).

The previous behavior is quite contrary to those of the ferromagnetic and ferrimagnetic materials around Curie transition. For example, Healy (1952) has employed EMR technique to study the nickel ferrite (NiFe2O4) in temperature range of 78 K to 861 K. In Fig. 2, the direct resonance spectra show that the linewidth decreases and a shift in resonant field were observed with increasing temperature, all these changes can be completely associated with the Curie transition (Tc= 858 K).

Okamura (1951) also carries out a characterization in the ferrite NiFe2O4 by means of EMR technique but at low temperature. Fig. 3 shows the direct resonance absorption with varying applied dc magnetic field at various temperatures in a disk NiFe2O4. In these spectra a double absorption were clearly observed, and both absorptions are dependent of the temperature; where one of the absorptions is near to zero magnetic field. In our EMR measurements, we also detected a second absorption mode around zero field in amorphous alloys (Montiel et al., 2005) and ferrites (Montiel et al., 2004; Alvarez et al., 2010); however, a detailed discussion on this absorption type is gathered to continuation.

**Figure 1.** Direct resonance absorption versus dc magnetic field at 9.3 GHz for MnS around Néel transition (adapted from Okamura et al., 1951).

the Curie transition (Tc= 858 K).

studied the Néel transition in some materials by means of EMR technique. Fig. 1 shows the direct resonance absorption versus magnetic field for MnS in the 78-329 K temperature range. The height of the resonant absorption decreases and the linewidth is become broader with decreasing temperature, especially below the Néel temperature (TN= 160 K); where the position of the maximum is found at a constant magnetic field of 3510 G for all the temperatures, suggesting the existence of a strong local magnetic field in this material.

The previous behavior is quite contrary to those of the ferromagnetic and ferrimagnetic materials around Curie transition. For example, Healy (1952) has employed EMR technique to study the nickel ferrite (NiFe2O4) in temperature range of 78 K to 861 K. In Fig. 2, the direct resonance spectra show that the linewidth decreases and a shift in resonant field were observed with increasing temperature, all these changes can be completely associated with

Okamura (1951) also carries out a characterization in the ferrite NiFe2O4 by means of EMR technique but at low temperature. Fig. 3 shows the direct resonance absorption with varying applied dc magnetic field at various temperatures in a disk NiFe2O4. In these spectra a double absorption were clearly observed, and both absorptions are dependent of the temperature; where one of the absorptions is near to zero magnetic field. In our EMR measurements, we also detected a second absorption mode around zero field in amorphous alloys (Montiel et al., 2005) and ferrites (Montiel et al., 2004; Alvarez et al., 2010); however, a

detailed discussion on this absorption type is gathered to continuation.

**Figure 1.** Direct resonance absorption versus dc magnetic field at 9.3 GHz for MnS around Néel

transition (adapted from Okamura et al., 1951).

**Figure 2.** Direct resonance absorption versus dc magnetic field at 9 GHz in a sphere NiFe2O4 around Curie transition (adapted from Healy, 1952).

**Figure 3.** Direct resonance absorption versus dc magnetic field at 9.3 GHz for a disk NiFe2O4 at low temperature (adapted from Okamura, 1951).

The non-resonant microwave absorption (NRMA) was used in 1987 to detect the transition between the normal state and superconducting state in high-Tc superconductor ceramics (Bhat et al., 1987; Blazey et al., 1987; Bohandy et al., 1987; Khachaturyan et al., 1987; Moorjani et al., 1987). This was followed by a large number of reports on not only high-Tc superconductor ceramics (Kim et al., 1993; Topacli, 1996, 1998; Velter-Stefanescu et al., 1998, 2005; Padam et al., 1999, 2010; Shaltiel et al., 2001; Alvarez & Zamorano, 2004), but also including organic superconductors (Zakhidov et al. 1991; Bele et al., 1994; Hirotake et al., 1997; Niebling et al., 1998; Stankowski et al., 2004), the conventional superconductors of type-I and type-II (Kheifets et al., 1990; Bhide et al., 2001; Owens et al., 2001; Andrzejewski et al., 2004) and the newly discovered iron pniticide (Panarina et al., 2010; Pascher et al., 2010). Researches on NRMA have shown that this phenomenon is highly sensitive to detection of a superconducting phase in a material under study. The NRMA is usually detected as a function of a dc applied magnetic field or temperature, where these two variants are historically known as the magnetically-modulated microwave absorption spectroscopy (MAMMAS) and the low-field microwave absorption (LFMA), respectively.

Detection of Magnetic Transitions by

Means of Ferromagnetic Resonance and Microwave Absorption Techniques 67

**Figure 4.** (a) MAMMAS response and (b) the LFMA spectra of a bulk sample of the ceramic

the magnetic transition from a ferromagnetic order to paramagnetic phase. They suggest that the LFMA signal is due to the presence of a domains structure in the magnetic material. LFMA signal is highly distorted at T=571 K and at T=587 K it is completely inverted. At T= 601 K, the line intensity reaches its maximum, see Fig. 5; and when increases the temperature their intensity and width (HLFMA) diminishes. For T 630.9 K, a LFMA line is observed and it agrees with the value of the Curie temperature given in the literature for nickel, 630 K; suggesting that HLFMA is determined by the magnetic anisotropy field and

**Figure 5.** LFMA spectra in nickel for selected temperatures around Curie transition (adapted from

superconductor Bi-Sr-Ca-Cu-O in the region of the superconducting transition.

the demagnetizing field.

Nabereznykh & Tsindlekht, 1982).

In Fig. 4(a) is shown the MAMMAS response for bulk sample of the ceramic superconductor Bi-Sr-Ca-Cu-O around the superconducting transition. This response shows a level of absorption constant from 300 K to Ton= 81.5 K, and suddenly this signal rises sharply until Tmax=63.6 K; at that temperature the superconducting transition has been completed. As temperature goes down from Tmax, the superconductor sample enters more and more into the mixed state with a rigid fluxon lattice, and a decrease in the microwave absorption is observed. Fig. 4(b) shows LFMA spectra for selected temperatures, in a bulk sample of the ceramic superconductor Bi-Sr-Ca-Cu-O, where a hysteresis loop is observed in the superconductive state; since the microwave induced dynamics of the fluxon is dissipative, the field sweep cycle in LFMA measurement shows a hysteresis. The physical meaning of this hysteresis has been amply discussed by Topacli (1998) and Padam et al. (1999). For T 63 K, a non-hysteretic LFMA signal is observed and which goes disappearing when increasing the temperature, i.e. when the sample enters to normal state. The abovementioned is a striking example of that the transition to the superconductive state in superconductor ceramics leads to NRMA. Additionally, it is recognized that NRMA is due mainly to the dissipative dynamics within Josephson junctions and/or to induced currents through weak links in superconductor materials.

Today, it is safe to assume that all superconductor materials exhibit a NRMA, and which has been experimentally confirmed; but the reverse statement, that any material that exhibits a NRMA is a superconductor, in general is not true. The NRMA may be caused not only by superconductivity, but also by any phenomena associated with magnetic-field-dependent microwave losses in the materials, and it can be employed to detect the magnetic transitions (Nabereznykh & Tsindlekht, 1982; Owens, 1997; Alvarez & Zamorano, 2004; Montiel et al., 2004; Alvarez et al., 2007, 2009, 2010). Some antecedent studies are given in the following. Nabereznykh & Tsindlekht (1982) have reported a study of NRMA in nickel near the Curie transition (Fig. 5), and in particular, they have employed LFMA measurements for detecting

#### Detection of Magnetic Transitions by Means of Ferromagnetic Resonance and Microwave Absorption Techniques 67

66 Ferromagnetic Resonance – Theory and Applications

through weak links in superconductor materials.

The non-resonant microwave absorption (NRMA) was used in 1987 to detect the transition between the normal state and superconducting state in high-Tc superconductor ceramics (Bhat et al., 1987; Blazey et al., 1987; Bohandy et al., 1987; Khachaturyan et al., 1987; Moorjani et al., 1987). This was followed by a large number of reports on not only high-Tc superconductor ceramics (Kim et al., 1993; Topacli, 1996, 1998; Velter-Stefanescu et al., 1998, 2005; Padam et al., 1999, 2010; Shaltiel et al., 2001; Alvarez & Zamorano, 2004), but also including organic superconductors (Zakhidov et al. 1991; Bele et al., 1994; Hirotake et al., 1997; Niebling et al., 1998; Stankowski et al., 2004), the conventional superconductors of type-I and type-II (Kheifets et al., 1990; Bhide et al., 2001; Owens et al., 2001; Andrzejewski et al., 2004) and the newly discovered iron pniticide (Panarina et al., 2010; Pascher et al., 2010). Researches on NRMA have shown that this phenomenon is highly sensitive to detection of a superconducting phase in a material under study. The NRMA is usually detected as a function of a dc applied magnetic field or temperature, where these two variants are historically known as the magnetically-modulated microwave absorption spectroscopy (MAMMAS) and the low-field microwave absorption (LFMA), respectively.

In Fig. 4(a) is shown the MAMMAS response for bulk sample of the ceramic superconductor Bi-Sr-Ca-Cu-O around the superconducting transition. This response shows a level of absorption constant from 300 K to Ton= 81.5 K, and suddenly this signal rises sharply until Tmax=63.6 K; at that temperature the superconducting transition has been completed. As temperature goes down from Tmax, the superconductor sample enters more and more into the mixed state with a rigid fluxon lattice, and a decrease in the microwave absorption is observed. Fig. 4(b) shows LFMA spectra for selected temperatures, in a bulk sample of the ceramic superconductor Bi-Sr-Ca-Cu-O, where a hysteresis loop is observed in the superconductive state; since the microwave induced dynamics of the fluxon is dissipative, the field sweep cycle in LFMA measurement shows a hysteresis. The physical meaning of this hysteresis has been amply discussed by Topacli (1998) and Padam et al. (1999). For T 63 K, a non-hysteretic LFMA signal is observed and which goes disappearing when increasing the temperature, i.e. when the sample enters to normal state. The abovementioned is a striking example of that the transition to the superconductive state in superconductor ceramics leads to NRMA. Additionally, it is recognized that NRMA is due mainly to the dissipative dynamics within Josephson junctions and/or to induced currents

Today, it is safe to assume that all superconductor materials exhibit a NRMA, and which has been experimentally confirmed; but the reverse statement, that any material that exhibits a NRMA is a superconductor, in general is not true. The NRMA may be caused not only by superconductivity, but also by any phenomena associated with magnetic-field-dependent microwave losses in the materials, and it can be employed to detect the magnetic transitions (Nabereznykh & Tsindlekht, 1982; Owens, 1997; Alvarez & Zamorano, 2004; Montiel et al., 2004; Alvarez et al., 2007, 2009, 2010). Some antecedent studies are given in the following. Nabereznykh & Tsindlekht (1982) have reported a study of NRMA in nickel near the Curie transition (Fig. 5), and in particular, they have employed LFMA measurements for detecting

**Figure 4.** (a) MAMMAS response and (b) the LFMA spectra of a bulk sample of the ceramic superconductor Bi-Sr-Ca-Cu-O in the region of the superconducting transition.

the magnetic transition from a ferromagnetic order to paramagnetic phase. They suggest that the LFMA signal is due to the presence of a domains structure in the magnetic material. LFMA signal is highly distorted at T=571 K and at T=587 K it is completely inverted. At T= 601 K, the line intensity reaches its maximum, see Fig. 5; and when increases the temperature their intensity and width (HLFMA) diminishes. For T 630.9 K, a LFMA line is observed and it agrees with the value of the Curie temperature given in the literature for nickel, 630 K; suggesting that HLFMA is determined by the magnetic anisotropy field and the demagnetizing field.

**Figure 5.** LFMA spectra in nickel for selected temperatures around Curie transition (adapted from Nabereznykh & Tsindlekht, 1982).

Other antecedent study is the detection of the Curie transition in a material with colossal magneto-resistance (CMR), as is La0.7Sr0.3MnO3 manganite (Owens, 1997). The presence of the NRMA is evidence of the existence of a ferromagnetic order, i.e. this signal is not present in the paramagnetic phase and emerges as the temperature is decreased below Curie temperature, see Fig. 6; providing a sensitive detector of ferromagnetism. In Fig. 6(a), MAMMAS response shows the appearance and the rapid increase of the microwave absorption at ferromagnetic transition. Additionally, the half LFMA spectrum at 144 K in the ferromagnetic phase of La0.7Sr0.3MnO3 is shown in Fig. 6(b). Owens (1997) suggests as a possible explanation of the origin of LFMA signal, the fact that the permeability in the ferromagnetic phase at constant temperature depends on the applied magnetic field, increasing at low fields to a maximum and then decreasing. Since the surface resistance of the material depends on the square root of the permeability, the microwave absorption depends non-linearly on the strength of the dc magnetic field, resulting in a NRMA centered at zero magnetic field.

Detection of Magnetic Transitions by

Means of Ferromagnetic Resonance and Microwave Absorption Techniques 69

In this work, the resonant and non-resonant microwave absorptions in several magnetic materials are studied around magnetic transitions. EMR technique measures the resonant microwave absorption as a function of dc magnetic field. Additionally, the NRMA measurements as a temperature function or an applied dc magnetic field are experimentally

The basic components of a standard EMR spectrometer are shown in Fig. 7. In a sample placed at center of a microwave cavity (see Fig. 8), an applied dc magnetic field is increased until that the energy difference between the spin-up and spin-down orientations, match the microwave frequency of the power supply; where a strong absorption is clearly detected. For this type of spectrometer, the derivative of microwave absorption (dP/dH) as a magnetic

We give a description detailed of the EMR spectrometer (Jeol JES-RES3X). In a T-magic bridge, the microwaves (Hmw), with power P0 and frequency mw= 8.8-9.8 GHz (X-band), are generated by a JEOL/ES-HX3 microwave unit and are fed to the cylindrical cavity (see Fig. 8) through a rectangular wave-guide; coupling adjuster of the cavity must be adjusted so that no wave is reflected from the cavity. The absorption of the microwave energy by a sample generates a change in the quality factor (Q) of the cavity; due to this change, the microwave bridge becomes non-balanced, causing that a wave is reflected from the cavity. The change in cavity Q-factor is due to changes in the energy absorption resonant by spins.

**2.1. Resonant microwave absorption measurement (EMR technique)** 

**2. Experimental methods** 

field function is plotted.

denominated as MAMMAS and LFMA, respectively.

**Figure 7.** The diagram to blocks of a standard EMR spectrometer.

**Figure 6.** (a) MAMMAS response and (b) the half LFMA spectrum at 144 K in a bulk sample of La0.7Sr0.3MnO3 manganite (adapted from Owens, 1997).

In this chapter, the changes in the EMR lineshape are studied for diverse magnetic materials in the 77-500 K temperature range; the different magnetic transitions are quantified by means of linewidth (Hpp) and the resonant field (Hres) as a function of temperature. Through these studies we can distinguish the kind of present magnetic transition in the materials. Also, we employed the LFMA and MAMMAS techniques to give a further knowledge on magnetic materials, studying the different types of magnetic transitions and showing their main characteristics highlighted; we distinguish distinctive features associated with the microwave absorption by the magnetic moments and discuss on usefulness of these techniques as powerful characterization methodologies.

### **2. Experimental methods**

68 Ferromagnetic Resonance – Theory and Applications

at zero magnetic field.

Other antecedent study is the detection of the Curie transition in a material with colossal magneto-resistance (CMR), as is La0.7Sr0.3MnO3 manganite (Owens, 1997). The presence of the NRMA is evidence of the existence of a ferromagnetic order, i.e. this signal is not present in the paramagnetic phase and emerges as the temperature is decreased below Curie temperature, see Fig. 6; providing a sensitive detector of ferromagnetism. In Fig. 6(a), MAMMAS response shows the appearance and the rapid increase of the microwave absorption at ferromagnetic transition. Additionally, the half LFMA spectrum at 144 K in the ferromagnetic phase of La0.7Sr0.3MnO3 is shown in Fig. 6(b). Owens (1997) suggests as a possible explanation of the origin of LFMA signal, the fact that the permeability in the ferromagnetic phase at constant temperature depends on the applied magnetic field, increasing at low fields to a maximum and then decreasing. Since the surface resistance of the material depends on the square root of the permeability, the microwave absorption depends non-linearly on the strength of the dc magnetic field, resulting in a NRMA centered

**Figure 6.** (a) MAMMAS response and (b) the half LFMA spectrum at 144 K in a bulk sample of

usefulness of these techniques as powerful characterization methodologies.

In this chapter, the changes in the EMR lineshape are studied for diverse magnetic materials in the 77-500 K temperature range; the different magnetic transitions are quantified by means of linewidth (Hpp) and the resonant field (Hres) as a function of temperature. Through these studies we can distinguish the kind of present magnetic transition in the materials. Also, we employed the LFMA and MAMMAS techniques to give a further knowledge on magnetic materials, studying the different types of magnetic transitions and showing their main characteristics highlighted; we distinguish distinctive features associated with the microwave absorption by the magnetic moments and discuss on

La0.7Sr0.3MnO3 manganite (adapted from Owens, 1997).

In this work, the resonant and non-resonant microwave absorptions in several magnetic materials are studied around magnetic transitions. EMR technique measures the resonant microwave absorption as a function of dc magnetic field. Additionally, the NRMA measurements as a temperature function or an applied dc magnetic field are experimentally denominated as MAMMAS and LFMA, respectively.

### **2.1. Resonant microwave absorption measurement (EMR technique)**

The basic components of a standard EMR spectrometer are shown in Fig. 7. In a sample placed at center of a microwave cavity (see Fig. 8), an applied dc magnetic field is increased until that the energy difference between the spin-up and spin-down orientations, match the microwave frequency of the power supply; where a strong absorption is clearly detected. For this type of spectrometer, the derivative of microwave absorption (dP/dH) as a magnetic field function is plotted.

**Figure 7.** The diagram to blocks of a standard EMR spectrometer.

We give a description detailed of the EMR spectrometer (Jeol JES-RES3X). In a T-magic bridge, the microwaves (Hmw), with power P0 and frequency mw= 8.8-9.8 GHz (X-band), are generated by a JEOL/ES-HX3 microwave unit and are fed to the cylindrical cavity (see Fig. 8) through a rectangular wave-guide; coupling adjuster of the cavity must be adjusted so that no wave is reflected from the cavity. The absorption of the microwave energy by a sample generates a change in the quality factor (Q) of the cavity; due to this change, the microwave bridge becomes non-balanced, causing that a wave is reflected from the cavity. The change in cavity Q-factor is due to changes in the energy absorption resonant by spins.

The reflected waves from the cavity (Pref) with the information of the microwave absorption by the sample are directed towards a detector crystal, which was previously biased to a 10% of the incident power in order to work in the linear regime; with this method of detectionhomodyne and lock-in amplification, a very high sensitivity in the measurements is guaranteed.

Detection of Magnetic Transitions by

Means of Ferromagnetic Resonance and Microwave Absorption Techniques 71

zero-value continuously from -1000 G to 8000 G. Hence, symmetric field-sweeps from ±0.1 G to ±1000 G are available and asymmetric field-sweeps up to –1000 G ≤ Hdc ≤ 8000 G are also available in order to detect possible hysteresis for NRMA signal, and which would point out to irreversible processes of microwave energy absorption. In this technique, the sample is zero field cooled or heated to the fixed temperature. For our studies, the temperature is maintained fixed with a maximum deviation of 1 K during the whole LFMA measurement (<8 min of sweeping). The magnetic field is swept following a cycle; the field sweep schemes have their analog in the magnetic hysteresis measurements. GPIB port of a PC receives the magnetic field coming from the Group3 DTM-141 teslameter, and it is displayed as the X-axis on the plot of the data being acquired; meanwhile, the voltmeter Vy receives the NRMA signal. Then, what is measured is not the microwave power absorption itself, but rather its derivative with respect to magnetic field (dP/dH). This allows us to distinguish the field-sensitive part of the microwave absorption from the part that does not depend on magnetic field, and record only the first; and also, to use narrow-band amplifier

to enhance the signal, which greatly increases the signal-to-noise ratio.

**Figure 9.** Block diagram of LFMA and MAMMAS techniques (adapted from Alvarez & Zamorano,

2004).

**Figure 8.** Distribution of (a) magnetic and (b) electric fields, and the sample location inside the TE011 cylindrical cavity in the JEOL JES-RES 3X spectrometer.

The sample is subjected to a dc magnetic field (Hdc), that it is produced by an electromagnet with truncated pole pieces, and a weak ac magnetic field (Hmod) is superimposed to Hdc. The Hmod is achieved by placing small Helmholtz-coils on each side of the cavity along the axis of the static field, which are fed and controlled by a sign generator. The amplitude of this field goes from 0.002 G to 20 G with a modulation frequency of 100 kHz, thus allowing, the microwave absorption registration at the modulation frequency. In EMR measurements, Hdc could be varied from 0 to 8000 G.

### **2.2. NRMA measurements (LFMA and MAMMAS techniques)**

The Jeol JES-RES3X spectrometer was modified (see Fig. 9), connecting the output of a digital voltmeter (signal Y) to a PC enabling digital data acquisition (Alvarez & Zamorano, 2004); where this electrical signal is proportional to NRMA from sample. The signal Y is fed to a 7½ digits - Keithley DMM-196 voltmeter. Hence, the reading of this voltmeter (Vy) carries the information of the microwaves absorption by sample.

LFMA technique measures the NRMA as a function of Hdc, this uniform field is produced by the same electromagnet, but which receives current regulated from two power supplies (JEOL JES-RE3X and ES-ZCS2); and they are synchronized to obtain a true zero-value of the magnetic field between the pole caps. The Jeol ESZCS2 zero-cross sweep unit compensates digitally for any remanence in the electromagnet, with a standard deviation of the measured field of less than 0.2 G, allowing measurements to be carried out by cycling the Hdc about its zero-value continuously from -1000 G to 8000 G. Hence, symmetric field-sweeps from ±0.1 G to ±1000 G are available and asymmetric field-sweeps up to –1000 G ≤ Hdc ≤ 8000 G are also available in order to detect possible hysteresis for NRMA signal, and which would point out to irreversible processes of microwave energy absorption. In this technique, the sample is zero field cooled or heated to the fixed temperature. For our studies, the temperature is maintained fixed with a maximum deviation of 1 K during the whole LFMA measurement (<8 min of sweeping). The magnetic field is swept following a cycle; the field sweep schemes have their analog in the magnetic hysteresis measurements. GPIB port of a PC receives the magnetic field coming from the Group3 DTM-141 teslameter, and it is displayed as the X-axis on the plot of the data being acquired; meanwhile, the voltmeter Vy receives the NRMA signal. Then, what is measured is not the microwave power absorption itself, but rather its derivative with respect to magnetic field (dP/dH). This allows us to distinguish the field-sensitive part of the microwave absorption from the part that does not depend on magnetic field, and record only the first; and also, to use narrow-band amplifier to enhance the signal, which greatly increases the signal-to-noise ratio.

70 Ferromagnetic Resonance – Theory and Applications

cylindrical cavity in the JEOL JES-RES 3X spectrometer.

could be varied from 0 to 8000 G.

guaranteed.

The reflected waves from the cavity (Pref) with the information of the microwave absorption by the sample are directed towards a detector crystal, which was previously biased to a 10% of the incident power in order to work in the linear regime; with this method of detectionhomodyne and lock-in amplification, a very high sensitivity in the measurements is

**Figure 8.** Distribution of (a) magnetic and (b) electric fields, and the sample location inside the TE011

**2.2. NRMA measurements (LFMA and MAMMAS techniques)** 

carries the information of the microwaves absorption by sample.

The sample is subjected to a dc magnetic field (Hdc), that it is produced by an electromagnet with truncated pole pieces, and a weak ac magnetic field (Hmod) is superimposed to Hdc. The Hmod is achieved by placing small Helmholtz-coils on each side of the cavity along the axis of the static field, which are fed and controlled by a sign generator. The amplitude of this field goes from 0.002 G to 20 G with a modulation frequency of 100 kHz, thus allowing, the microwave absorption registration at the modulation frequency. In EMR measurements, Hdc

The Jeol JES-RES3X spectrometer was modified (see Fig. 9), connecting the output of a digital voltmeter (signal Y) to a PC enabling digital data acquisition (Alvarez & Zamorano, 2004); where this electrical signal is proportional to NRMA from sample. The signal Y is fed to a 7½ digits - Keithley DMM-196 voltmeter. Hence, the reading of this voltmeter (Vy)

LFMA technique measures the NRMA as a function of Hdc, this uniform field is produced by the same electromagnet, but which receives current regulated from two power supplies (JEOL JES-RE3X and ES-ZCS2); and they are synchronized to obtain a true zero-value of the magnetic field between the pole caps. The Jeol ESZCS2 zero-cross sweep unit compensates digitally for any remanence in the electromagnet, with a standard deviation of the measured field of less than 0.2 G, allowing measurements to be carried out by cycling the Hdc about its

**Figure 9.** Block diagram of LFMA and MAMMAS techniques (adapted from Alvarez & Zamorano, 2004).

MAMMAS technique allows measurement as a temperature function, giving information on the temperature profile of NRMA response of each material and can also provide valuable information about the nature of magnetic ordering in the materials. The temperature of the sample is slowly varied (1 K/min) and controlled by flowing N2 gas through a double walled quartz tube, which is inserted through the center of the microwaves cavity, in the 77- 500 K temperature range. The temperature is measured by a copper constantan thermocouple placed inside the sample tube just outside the cavity. Its output signal is further digitized by means of the Stanford SR630 thermocouple monitor. In this technique, the temperature is plotted along the X-axis, and the microwaves absorbed by the sample are collected as Vy and are plotted along Y-axis.

Detection of Magnetic Transitions by

Means of Ferromagnetic Resonance and Microwave Absorption Techniques 73

**EMR spectrum** 

**0 2000 4000 6000**

**Figure 10.** The derivative of microwave absorption from -1000 G to 8000 G, and which consists of an

resonant condition implies that M=Ms with Hefec= Hdc+Hint, where Hint is the internal field. The saturation magnetization of the surface of the sample can be calculated from the resonance conditions as 4Ms = 4741 G; which is close to the bulk saturation magnetization 4Ms=5250 G, the difference can be attributed to the fact that FMR is probing only the surface of the sample. Additionally, this absorption shows no hysteresis between the

In Fig. 11, the temperature dependence of the EMR spectra can be observed. As temperature increases Hpp becomes wider, due to new magnetic processes; where the dominant process is the dipole-dipole interaction associated with the paramagnetic phase, while the exchange interaction of the ferromagnetic order disappears when increasing the temperature. The dipole-dipole interaction has the effect of increasing the linewidth, while exchange interaction tends to narrow the absorption line. Hpp as temperature function is shown in Fig. 12(a), the Curie temperature (Tc=482 K) is associated with the inflection point; where the derivative of the linewidth exhibited a maximum at Tc. A second EMR spectrum (SES) is detected after the magnetic transition, and it is associated with a second magnetic phase with a different Curie temperature; where this absorption mode is due to a nanocrystalline phase. The conductive behavior decreases due to the temperature increase, consequently, the absorption centers diminish and the EMR lineshape starts to become symmetrical. Also, the gain is one order of magnitude greater than at room temperature, and it is indicative of a reduced number of absorbing centers in the paramagnetic phase due to the high entropy of the system. Additionally, the temperature dependence of the resonant field is plotted in Fig. 12(b), where a shift in the resonant field is clearly observed. At 300 K, in ferromagnetic

EMR spectrum and a LFMA signal (adapted from Montiel et al., 2005).

**LFMA signal**

**Magnetic Field (Gauss)**

**DPPH pattern**

**-0,8**

forward and backward field sweeps.

**-0,4**

**0,0**

**dP/dH (a.u.)**

**0,4**

**0,8**

### **3. EMR and NRMA studies in magnetic transitions**

In this section, we show several EMR and NRMA studies in diverse magnetic materials through magnetic transitions. These examples include: the amorphous ribbon Co66Fe4B12Si13Nb4Cu, the ferrite Ni0.35Zn0.65Fe2O4 and the magnetoelectric Pb(Fe2/3W1/3)O3; highlighting their main characteristics and illustrating how magnetic transitions are manifested in this kind of measurements.

### **3.1. Curie transition (from a ferromagnetic order to paramagnetic phase)**

We show several studies in amorphous ribbons of nominal composition Co66Fe4B12Si13Nb4Cu and dimensions of 2 mm wide and 22 m thick, which were prepared by the melt-spinning method; where their initial amorphous state was checked by X-ray diffraction (XRD). All the measurements were performed in the 300-500 K temperature range.

### *3.1.1. EMR technique*

These measurements were carried out from -1000 G to 8000 G, with forward and backward Hdc sweeps in order to detect reversible and/or irreversible microwave absorption processes. At 300 K, two microwave absorptions were observed: the first absorption to high magnetic field (~1469 G) corresponding to an EMR spectrum, and other absorption at low magnetic fields around zero (LFMA signal). Fig. 10 shows the derivative of microwave absorption, and which consists of an EMR spectrum and a LFMA signal; additionally a DDPH pattern of paramagnetic nature is also included.

We will only center ourselves in EMR absorption, associated with the ferromagnetic resonance (FMR). This absorption satisfies the Larmor condition; when is applied to the case of a thin sheet with both negligible anisotropy field and the demagnetizing fields (Yildiz et al., 2002),

$$
\Delta w = \gamma [(4\pi M\_{\rm efc} + H\_{\rm efc}) \cdot H\_{\rm efc}]^{1/2} \tag{1}
$$

where is the microwave angular frequency (with = 2f and f = 9.4 GHz), is the gyromagnetic ratio, Hefec is the effective magnetic field and M is the magnetization. The

collected as Vy and are plotted along Y-axis.

manifested in this kind of measurements.

paramagnetic nature is also included.

*3.1.1. EMR technique* 

**3. EMR and NRMA studies in magnetic transitions** 

measurements were performed in the 300-500 K temperature range.

MAMMAS technique allows measurement as a temperature function, giving information on the temperature profile of NRMA response of each material and can also provide valuable information about the nature of magnetic ordering in the materials. The temperature of the sample is slowly varied (1 K/min) and controlled by flowing N2 gas through a double walled quartz tube, which is inserted through the center of the microwaves cavity, in the 77- 500 K temperature range. The temperature is measured by a copper constantan thermocouple placed inside the sample tube just outside the cavity. Its output signal is further digitized by means of the Stanford SR630 thermocouple monitor. In this technique, the temperature is plotted along the X-axis, and the microwaves absorbed by the sample are

In this section, we show several EMR and NRMA studies in diverse magnetic materials through magnetic transitions. These examples include: the amorphous ribbon Co66Fe4B12Si13Nb4Cu, the ferrite Ni0.35Zn0.65Fe2O4 and the magnetoelectric Pb(Fe2/3W1/3)O3; highlighting their main characteristics and illustrating how magnetic transitions are

We show several studies in amorphous ribbons of nominal composition Co66Fe4B12Si13Nb4Cu and dimensions of 2 mm wide and 22 m thick, which were prepared by the melt-spinning method; where their initial amorphous state was checked by X-ray diffraction (XRD). All the

These measurements were carried out from -1000 G to 8000 G, with forward and backward Hdc sweeps in order to detect reversible and/or irreversible microwave absorption processes. At 300 K, two microwave absorptions were observed: the first absorption to high magnetic field (~1469 G) corresponding to an EMR spectrum, and other absorption at low magnetic fields around zero (LFMA signal). Fig. 10 shows the derivative of microwave absorption, and which consists of an EMR spectrum and a LFMA signal; additionally a DDPH pattern of

We will only center ourselves in EMR absorption, associated with the ferromagnetic resonance (FMR). This absorption satisfies the Larmor condition; when is applied to the case of a thin sheet with both negligible anisotropy field and the demagnetizing fields (Yildiz et al., 2002),

where is the microwave angular frequency (with = 2f and f = 9.4 GHz), is the gyromagnetic ratio, Hefec is the effective magnetic field and M is the magnetization. The

1/2 [( ) 4 ] *w MH H efec efec* (1)

 

**3.1. Curie transition (from a ferromagnetic order to paramagnetic phase)** 

**Figure 10.** The derivative of microwave absorption from -1000 G to 8000 G, and which consists of an EMR spectrum and a LFMA signal (adapted from Montiel et al., 2005).

resonant condition implies that M=Ms with Hefec= Hdc+Hint, where Hint is the internal field. The saturation magnetization of the surface of the sample can be calculated from the resonance conditions as 4Ms = 4741 G; which is close to the bulk saturation magnetization 4Ms=5250 G, the difference can be attributed to the fact that FMR is probing only the surface of the sample. Additionally, this absorption shows no hysteresis between the forward and backward field sweeps.

In Fig. 11, the temperature dependence of the EMR spectra can be observed. As temperature increases Hpp becomes wider, due to new magnetic processes; where the dominant process is the dipole-dipole interaction associated with the paramagnetic phase, while the exchange interaction of the ferromagnetic order disappears when increasing the temperature. The dipole-dipole interaction has the effect of increasing the linewidth, while exchange interaction tends to narrow the absorption line. Hpp as temperature function is shown in Fig. 12(a), the Curie temperature (Tc=482 K) is associated with the inflection point; where the derivative of the linewidth exhibited a maximum at Tc. A second EMR spectrum (SES) is detected after the magnetic transition, and it is associated with a second magnetic phase with a different Curie temperature; where this absorption mode is due to a nanocrystalline phase. The conductive behavior decreases due to the temperature increase, consequently, the absorption centers diminish and the EMR lineshape starts to become symmetrical. Also, the gain is one order of magnitude greater than at room temperature, and it is indicative of a reduced number of absorbing centers in the paramagnetic phase due to the high entropy of the system. Additionally, the temperature dependence of the resonant field is plotted in Fig. 12(b), where a shift in the resonant field is clearly observed. At 300 K, in ferromagnetic

order, Hres corresponds to Hefec= Hdc+Hint, and as temperature increase Hint diminishes until Hint=0 at Tc=482 K in paramagnetic regime, with Hefec= Hdc.

Detection of Magnetic Transitions by

or

Means of Ferromagnetic Resonance and Microwave Absorption Techniques 75

LFMA signal in amorphous ribbon is shown with more detail in Fig. 13(a). This signal is centered at zero magnetic field and shows an opposite phase to the EMR spectrum. The opposite phase is undoubtedly indicating that the microwave absorption has a minimum value at zero magnetic field, in contrast to the maximum value for EMR spectrum. LFMA signal has been interpreted as due to low-field spin magnetization processes (Beach & Berkowitz, 1994; Domínguez et al., 2002). We have correlated the LFMA signal with magnetoimpedance (MI) phenomena (Montiel et al., 2005), where Fig. 13(b) shows the MI response at 50 MHz for amorphous ribbon. The double peak clearly indicates low-field surface magnetization processes (Beach & Berkowitz, 1994) originated by the change in transversal permeability. The peak-to-peak width in MI is associated with the anisotropy field (HK). In addition, Fig. 13(c) shows magnetometry measurements. The hysteresis loop is characterized by axial anisotropy, and a correlation between both experiments is observed on the basis of HK. We compare measurements of LFMA, MI and magnetometry, in Fig. 13. A significant decrease of the microwave absorption (from H=16 G down to zero) is observed in LFMA measurements, whereas at the same fields, the magnetoimpedance measurement show that MI response is approaching saturation at field lower than 20 G. As the field decreases, a maximum is reached by MI, which corresponds to the anisotropy field (H=15.6 G). A further decrease of impedance is observed at zero field. As it is well known, MI is due to changes in the skin depth as a consequence of changes in the transversal permeability under the influence of the external Hdc. The change in domain structure, and therefore in spin dynamics, is also produced by Hdc, in a direct interaction with the axial anisotropy of the material. Experimentally, the maxima in MI signal coincide with the minimum and maximum of the LFMA signal and it can be associated with a common origin for both

phenomena, where the magnetic processes in both phenomena are dependent of HK.

The hysteresis effect of the LFMA signal appears to be due to a non-uniform surface magnetization processes. A ferromagnetic conducting system can absorb electromagnetic radiation energy and the efficiency of this absorption depends on the particular conditions such as: the magnetic domain structure, magnetic anisotropy, the orientation of the incident propagation vector radiation, its conductivity, its frequency and amplitude. This absorption can easily be modified by Hdc, which changes the magnetic susceptibility, the penetration depth, the magnetization vector, the domain structure and spin dynamics. Such changes can show hysteresis, as normally occurs in a domain structure subjected to dc fields lower than the saturating field. By cycling the Hdc, different irreversible domain configurations occur, and therefore a hysteresis effect can be obtained. These results clearly suggest that MI effect and LFMA signal represent the same responses to an external Hdc. MI and LFMA are due to domain structure and spin dynamics, and they can be understood as the absorption of electromagnetic energy by spin systems that are modified by domain configuration and

Let us consider an electromagnetic wave with both electric (**E**) and magnetic (**H**) fields. The time-average density of the power absorption (P), for a ferromagnetic conductor at high frequencies, can be expressed by the complex Poynting vector as: P=½Re**EH**\*

*3.1.2. LFMA technique* 

strongly depends on HK.

**Figure 11.** EMR spectra at different temperatures in amorphous ribbon Co66Fe4B12Si13Nb4Cu; where the Curie temperature is detected at 482 K, and a decrease in microwave absorption is observed after magnetic transition.

**Figure 12.** Temperature dependence of (a) Hpp and (b) Hres in the amorphous ribbon Co66Fe4B12Si13Nb4Cu for EMR spectrum; solid lines are guides for the eye only. Also, Fig. 12(a) shows the derivative of Hpp with the temperature, showing a maximum to Curie temperature.

### *3.1.2. LFMA technique*

74 Ferromagnetic Resonance – Theory and Applications

**-0.9**

**1600**

**H**

**(Gauss)**

**res**

**2400**

**H**

**(Gauss)**

**pp**

**3200 Hres**

magnetic transition.

**-0.6**

**-0.3**

**SES**

**0.0**

**dP/dH (a.u.)**

**0.3**

**0.6**

Hint=0 at Tc=482 K in paramagnetic regime, with Hefec= Hdc.

order, Hres corresponds to Hefec= Hdc+Hint, and as temperature increase Hint diminishes until

**0 1000 2000 3000 4000 5000**

**Magnetic Field (Gauss)**

**gain 10 gain 50**

**Figure 11.** EMR spectra at different temperatures in amorphous ribbon Co66Fe4B12Si13Nb4Cu; where the Curie temperature is detected at 482 K, and a decrease in microwave absorption is observed after

**(a) <sup>0</sup>**

 **d H p p /dT**

**H p p**

**Figure 12.** Temperature dependence of (a) Hpp and (b) Hres in the amorphous ribbon

the derivative of Hpp with the temperature, showing a maximum to Curie temperature.

**300 350 400 450 500**

**Tem perature (K)**

Co66Fe4B12Si13Nb4Cu for EMR spectrum; solid lines are guides for the eye only. Also, Fig. 12(a) shows

**gain 100**

**0**

**T C**

**(b)**

**4**

**8**

**1 2**

**dHPP)/dT**

**1 6**

 **300 K 329 K 373 K 414 K 442 K 482 K 500 K 512 K** LFMA signal in amorphous ribbon is shown with more detail in Fig. 13(a). This signal is centered at zero magnetic field and shows an opposite phase to the EMR spectrum. The opposite phase is undoubtedly indicating that the microwave absorption has a minimum value at zero magnetic field, in contrast to the maximum value for EMR spectrum. LFMA signal has been interpreted as due to low-field spin magnetization processes (Beach & Berkowitz, 1994; Domínguez et al., 2002). We have correlated the LFMA signal with magnetoimpedance (MI) phenomena (Montiel et al., 2005), where Fig. 13(b) shows the MI response at 50 MHz for amorphous ribbon. The double peak clearly indicates low-field surface magnetization processes (Beach & Berkowitz, 1994) originated by the change in transversal permeability. The peak-to-peak width in MI is associated with the anisotropy field (HK). In addition, Fig. 13(c) shows magnetometry measurements. The hysteresis loop is characterized by axial anisotropy, and a correlation between both experiments is observed on the basis of HK. We compare measurements of LFMA, MI and magnetometry, in Fig. 13. A significant decrease of the microwave absorption (from H=16 G down to zero) is observed in LFMA measurements, whereas at the same fields, the magnetoimpedance measurement show that MI response is approaching saturation at field lower than 20 G. As the field decreases, a maximum is reached by MI, which corresponds to the anisotropy field (H=15.6 G). A further decrease of impedance is observed at zero field. As it is well known, MI is due to changes in the skin depth as a consequence of changes in the transversal permeability under the influence of the external Hdc. The change in domain structure, and therefore in spin dynamics, is also produced by Hdc, in a direct interaction with the axial anisotropy of the material. Experimentally, the maxima in MI signal coincide with the minimum and maximum of the LFMA signal and it can be associated with a common origin for both phenomena, where the magnetic processes in both phenomena are dependent of HK.

The hysteresis effect of the LFMA signal appears to be due to a non-uniform surface magnetization processes. A ferromagnetic conducting system can absorb electromagnetic radiation energy and the efficiency of this absorption depends on the particular conditions such as: the magnetic domain structure, magnetic anisotropy, the orientation of the incident propagation vector radiation, its conductivity, its frequency and amplitude. This absorption can easily be modified by Hdc, which changes the magnetic susceptibility, the penetration depth, the magnetization vector, the domain structure and spin dynamics. Such changes can show hysteresis, as normally occurs in a domain structure subjected to dc fields lower than the saturating field. By cycling the Hdc, different irreversible domain configurations occur, and therefore a hysteresis effect can be obtained. These results clearly suggest that MI effect and LFMA signal represent the same responses to an external Hdc. MI and LFMA are due to domain structure and spin dynamics, and they can be understood as the absorption of electromagnetic energy by spin systems that are modified by domain configuration and strongly depends on HK.

Let us consider an electromagnetic wave with both electric (**E**) and magnetic (**H**) fields. The time-average density of the power absorption (P), for a ferromagnetic conductor at high frequencies, can be expressed by the complex Poynting vector as: P=½Re**EH**\* or

**Figure 13.** (a) LFMA signal, (b) MI signal at frequency of 50 MHz, and (c) VSM hysteresis loop for the amorphous ribbon Co66Fe4B12Si13Nb4Cu (adapted from Montiel et al., 2005).

P=½Re[E·H\* ], where **H**\* is the complex conjugate of **H**, and Re[*x*] the real part of the operator. Additionally, the ac surface impedance for a ferromagnetic conductor material is defined as the ratio of the fields at the surface: Z=Es/Hs. Then the time-average density of the microwave power absorption can be written as P=½Hs 2Re(Z). The ac magnetic field Hs, in a ferromagnetic conductor at high frequency, is generated by a uniform current j=Es (with the electrical conductivity) induced by the ac electric field Es; and therefore Hs is constant to changes of an applied static magnetic field.

Therefore, we can establish a relation between the field derivative (dP/dH) of the microwave power absorption and the rate of change of Re (Z) with an applied static magnetic field, H:

$$\text{dIP/dH} = \left(\text{H}\_s^{2}/2\right) \left[d\text{Re}\left(Z\right)/dH\right] \tag{2}$$

Detection of Magnetic Transitions by

Means of Ferromagnetic Resonance and Microwave Absorption Techniques 77

**-100 -80 -60 -40 -20 0 20 40 60 80 100**

**394 K**

**416 K**

**444 K**

**485 K**

**500 K**

**Magnetic Field (Gauss)**

of the ferromagnetic transition (Montiel et al., 2004; Alvarez et al., 2010; Gavi et al., 2012). Therefore, it is possible to establish that for temperatures above the Curie temperature, the long-range magnetic order is completely lost and LFMA signal disappears. LFMA signal shows a decrease in the intensity as temperature is approached to Tc, and finally disappeared at T>Tc. Fig. 14 shows the temperature dependence of the LFMA signal, it is necessary to mention that LFMA signal is located around zero field for all temperatures. At room temperature, the LFMA signal has a phase opposite to EMR spectrum. As temperature increases, for T373 K, LFMA signal invests its phase until disappearing. This behavior is correlated with the long-range order in the ferromagnetic state and with the temperature dependence of the anisotropy field. The phase change has been observed previously in nickel around Curie transition, Nabereznykh & Tsindlekht (1982), and it can be explained by means of magnetic fluctuations; and they are associated with the electric properties of the material.

LFMA signal showed a decrease in HLFMA and hysteresis remains until Tc is reached.

**300 K**

485 K the LFMA signal disappears, indicating the transition from the ferromagnetic order to

**Figure 14.** LFMA signals at different temperatures in the amorphous ribbon Co66Fe4B12Si13Nb4Cu. At

**3.2. Curie and Yafet-Kittel transitions (from ferrimagnetic order to a Yafet-Kittel-**

The polycrystalline Ni-Zn ferrites (Ni1-xZnxFe2O4, 0 x 1) are an important family of solid solutions with a remarkable variety of magnetic properties and applications

**-0.4 0.0 0.4 0.8 -0.4 0.0 0.4 0.8 -0.4 0.0 0.4 0.8 -0.10 -0.05 0.00 0.05 0.10 -0.10 -0.05 0.00 0.05 0.10**

**dP/dH (a.u.)**

**313 K**

**330 K**

**350 K**

**373 K**

**-100 -80 -60 -40 -20 0 20 40 60 80 100**

**Magnetic Field (Gauss)**

**-0.4 0.0 0.4 0.8 -0.4 0.0 0.4 0.8 -0.4 0.0 0.4 0.8 -0.4 0.0 0.4 0.8 -0.4 0.0 0.4 0.8**

paramagnetic phase.

**type ordering)** 

**dP/dH (a.u.)**

For a good magnetic conductor Z= (1+j)/, with the classical skin depth 1/= (0/2)½ and the permeability. The magnetoimpedance is defined as the change of the impedance of a magnetic conductor subjected to an ac excitation current, under the application of a static magnetic field Hdc; it is a very similar phenomenon to the one involved in the microwave power absorption. At high-frequencies (microwaves) and due to the skin depth effect, only the surface impedance is involved.

LFMA signal can be used to detect magnetic order and to determine Curie temperature, because the appearance of LFMA signal has been widely accepted as a signature of the onset of the ferromagnetic transition (Montiel et al., 2004; Alvarez et al., 2010; Gavi et al., 2012). Therefore, it is possible to establish that for temperatures above the Curie temperature, the long-range magnetic order is completely lost and LFMA signal disappears. LFMA signal shows a decrease in the intensity as temperature is approached to Tc, and finally disappeared at T>Tc. Fig. 14 shows the temperature dependence of the LFMA signal, it is necessary to mention that LFMA signal is located around zero field for all temperatures. At room temperature, the LFMA signal has a phase opposite to EMR spectrum. As temperature increases, for T373 K, LFMA signal invests its phase until disappearing. This behavior is correlated with the long-range order in the ferromagnetic state and with the temperature dependence of the anisotropy field. The phase change has been observed previously in nickel around Curie transition, Nabereznykh & Tsindlekht (1982), and it can be explained by means of magnetic fluctuations; and they are associated with the electric properties of the material. LFMA signal showed a decrease in HLFMA and hysteresis remains until Tc is reached.

76 Ferromagnetic Resonance – Theory and Applications

**-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5**

microwave power absorption can be written as P=½Hs

**dP/dH (a. u.)**

], where **H**\*

changes of an applied static magnetic field.

the surface impedance is involved.

P=½Re[E·H\*

**(%)**

**emu/g**

**(a)**

amorphous ribbon Co66Fe4B12Si13Nb4Cu (adapted from Montiel et al., 2005).

**(b)**

**(c)**

**-100 -80 -60 -40 -20 0 20 40 60 80 100**

**Magnetic Field (Gauss)**

is the complex conjugate of **H**, and Re[*x*] the real part of the

<sup>2</sup> / /2 / *<sup>s</sup> dP dH H dRe Z dH* (2)

2Re(Z). The ac magnetic field Hs, in a

**Figure 13.** (a) LFMA signal, (b) MI signal at frequency of 50 MHz, and (c) VSM hysteresis loop for the

operator. Additionally, the ac surface impedance for a ferromagnetic conductor material is defined as the ratio of the fields at the surface: Z=Es/Hs. Then the time-average density of the

ferromagnetic conductor at high frequency, is generated by a uniform current j=Es (with the electrical conductivity) induced by the ac electric field Es; and therefore Hs is constant to

Therefore, we can establish a relation between the field derivative (dP/dH) of the microwave power absorption and the rate of change of Re (Z) with an applied static magnetic field, H:

For a good magnetic conductor Z= (1+j)/, with the classical skin depth 1/= (0/2)½ and the permeability. The magnetoimpedance is defined as the change of the impedance of a magnetic conductor subjected to an ac excitation current, under the application of a static magnetic field Hdc; it is a very similar phenomenon to the one involved in the microwave power absorption. At high-frequencies (microwaves) and due to the skin depth effect, only

LFMA signal can be used to detect magnetic order and to determine Curie temperature, because the appearance of LFMA signal has been widely accepted as a signature of the onset

**Figure 14.** LFMA signals at different temperatures in the amorphous ribbon Co66Fe4B12Si13Nb4Cu. At 485 K the LFMA signal disappears, indicating the transition from the ferromagnetic order to paramagnetic phase.

### **3.2. Curie and Yafet-Kittel transitions (from ferrimagnetic order to a Yafet-Kitteltype ordering)**

The polycrystalline Ni-Zn ferrites (Ni1-xZnxFe2O4, 0 x 1) are an important family of solid solutions with a remarkable variety of magnetic properties and applications

(Ravindaranathan et al., 1987). This solid solution crystallizes in a cubic spinel-type structure, see Fig. 15(a), where Zn ions normally are located in tetrahedral sites (A-sites) and Ni ions have a marked preference to occupy the octahedral sites (B-sites), while Fe ions are distributed among both sites types. The antiferromagnetic superexchange interaction (A-O-B) is the main cause of the cooperative behavior of the magnetic moments in Ni-Zn ferrites below their Curie temperature. In a great variety of experimental observations (Satya Murthy et al., 1969; Pong et al., 1997; Akther Hossain et al., 2004) have found that for x0.5 the resultant of the magnetic moments in the A and B sites has a classic collinear arrangement, see Fig. 15(b); while that for x>0.5 a non-collinear arrangement of the magnetic moments is employed to explain these behaviors. The previous behaviors are because the superexchange interaction B-O-B begins to be comparable with A-O-B interaction, and the arrangement of the magnetic moments shows a Yafet-Kittel-type canting (Yafet & Kittel et al., 1952). Also, the transition temperature from a ferrimagnetic ordering (collinear arrangement) to a Yafet-Kittel-type magnetic ordering (non-collinear arrangement) is called Yafet-Kittel temperature (TYK); in particular, Ni0.35Zn0.65Fe2O4 ferrite has a TYK smaller than the Curie temperature (Satya Murthy et al., 1969; Akther Hossain et al., 2004). The polycrystalline Ni0.35Zn0.65Fe2O4 ferrite was prepared by two different methods: the conventional classical ceramic method known as the solid-state reaction, and the coprecipitation method.

Detection of Magnetic Transitions by

**239 K 262 K 300 K 321 K 351 K 382 K 425 K**

(3)

Means of Ferromagnetic Resonance and Microwave Absorption Techniques 79

with the transition from a ferrimagnetic order to a paramagnetic phase; i.e. the evolution from a FMR spectrum to an EPR spectrum is used to determine the Curie temperature (Tc) in Ni-Zn ferrites (Montiel et al., 2004; Wu et al., 2006; Priyadharsini et al., 2009; Alvarez et al., 2010). Additionally, EMR spectra exhibit an additional absorption at low magnetic field, this new absorption mode is a LFMA signal which will be discussed with more detail in the

**0 1000 2000 3000 4000**

**Magnetic Field (Gauss)**

**Figure 16.** EMR spectra of the Ni-Zn ferrite prepared by solid-state reaction, for selected temperatures

In a polycrystalline magnetic material, the resonance condition for FMR signal is expressed

*w Hres* 

with Hres=Hdc+Hint, and Hint is the internal field which is the combination of several factors associated with the long-range order in the ferrite (Schlomann, 1958): the anisotropy field (HK), the porosity field (Hp), the field due to eddy currents (He) and the demagnetization field (Hd). Additionally, the inhomogeneities in Ni-Zn ferrites also can contribute to internal field, and they are associated with differences in sites occupancy by cations. Other source of inhomogeneity in the internal field is the disorder in the site occupancy. Even if the occupancy of sites is well determined (i.e., in Ni-Zn ferrite, all Zn cations on A sites, all Ni cations on B sites), there can be an inhomogeneous distribution of each of them on the sites. EMR spectra can be slightly different when this occupancy of sites is not strictly homogeneous, since some terms of Hint are not exactly the same for all the absorbing centers. It is possible to change the cations distribution in the ferrites by means of thermal treatments or when preparing samples of the same composition but with different synthesis methods.

following section.

**-0,9**

as:

**-0,6**

**-0,3**

**0,0**

**dP/dH (a.u.)**

**0,3**

**LFMA signal**

in the 239-425 K temperature range; circle shows LFMA signal.

**EMR spectrum** 

**0,6**

**0,9**

**Figure 15. (**a) Schematic representation of unit cell structure for Ni-Zn ferrites; where A and B are tetrahedral and octahedral sites, respectively. (b) Spin orientation on the A and B sites, for the collinear (to left) and the non-collinear (to right) model, in Ni-Zn ferrites; where YK is the Yafet-Kittel angle (adapted from Alvarez et al., 2010).

#### *3.2.1. EMR technique*

Fig. 16 shows the EMR spectra of the Ni-Zn ferrite prepared by solid-state reaction at different temperatures. For all temperature range, EMR spectra exhibit a broad signal, but their lineshape change with a shift in Hres when varying the temperature. Beginning to low temperature, an asymmetric mode (FMR signal) is observed and it gradually changes to a symmetric mode (EPR signal) when increasing the temperature. This change is associated with the transition from a ferrimagnetic order to a paramagnetic phase; i.e. the evolution from a FMR spectrum to an EPR spectrum is used to determine the Curie temperature (Tc) in Ni-Zn ferrites (Montiel et al., 2004; Wu et al., 2006; Priyadharsini et al., 2009; Alvarez et al., 2010). Additionally, EMR spectra exhibit an additional absorption at low magnetic field, this new absorption mode is a LFMA signal which will be discussed with more detail in the following section.

78 Ferromagnetic Resonance – Theory and Applications

precipitation method.

(adapted from Alvarez et al., 2010).

*3.2.1. EMR technique* 

(Ravindaranathan et al., 1987). This solid solution crystallizes in a cubic spinel-type structure, see Fig. 15(a), where Zn ions normally are located in tetrahedral sites (A-sites) and Ni ions have a marked preference to occupy the octahedral sites (B-sites), while Fe ions are distributed among both sites types. The antiferromagnetic superexchange interaction (A-O-B) is the main cause of the cooperative behavior of the magnetic moments in Ni-Zn ferrites below their Curie temperature. In a great variety of experimental observations (Satya Murthy et al., 1969; Pong et al., 1997; Akther Hossain et al., 2004) have found that for x0.5 the resultant of the magnetic moments in the A and B sites has a classic collinear arrangement, see Fig. 15(b); while that for x>0.5 a non-collinear arrangement of the magnetic moments is employed to explain these behaviors. The previous behaviors are because the superexchange interaction B-O-B begins to be comparable with A-O-B interaction, and the arrangement of the magnetic moments shows a Yafet-Kittel-type canting (Yafet & Kittel et al., 1952). Also, the transition temperature from a ferrimagnetic ordering (collinear arrangement) to a Yafet-Kittel-type magnetic ordering (non-collinear arrangement) is called Yafet-Kittel temperature (TYK); in particular, Ni0.35Zn0.65Fe2O4 ferrite has a TYK smaller than the Curie temperature (Satya Murthy et al., 1969; Akther Hossain et al., 2004). The polycrystalline Ni0.35Zn0.65Fe2O4 ferrite was prepared by two different methods: the conventional classical ceramic method known as the solid-state reaction, and the co-

**Figure 15. (**a) Schematic representation of unit cell structure for Ni-Zn ferrites; where A and B are tetrahedral and octahedral sites, respectively. (b) Spin orientation on the A and B sites, for the collinear (to left) and the non-collinear (to right) model, in Ni-Zn ferrites; where YK is the Yafet-Kittel angle

Fig. 16 shows the EMR spectra of the Ni-Zn ferrite prepared by solid-state reaction at different temperatures. For all temperature range, EMR spectra exhibit a broad signal, but their lineshape change with a shift in Hres when varying the temperature. Beginning to low temperature, an asymmetric mode (FMR signal) is observed and it gradually changes to a symmetric mode (EPR signal) when increasing the temperature. This change is associated

**Figure 16.** EMR spectra of the Ni-Zn ferrite prepared by solid-state reaction, for selected temperatures in the 239-425 K temperature range; circle shows LFMA signal.

In a polycrystalline magnetic material, the resonance condition for FMR signal is expressed as:

$$
\omega \mathbf{w} = \mathbf{\mathcal{Y}} \mathbf{H}\_{\text{res}} \tag{3}
$$

with Hres=Hdc+Hint, and Hint is the internal field which is the combination of several factors associated with the long-range order in the ferrite (Schlomann, 1958): the anisotropy field (HK), the porosity field (Hp), the field due to eddy currents (He) and the demagnetization field (Hd). Additionally, the inhomogeneities in Ni-Zn ferrites also can contribute to internal field, and they are associated with differences in sites occupancy by cations. Other source of inhomogeneity in the internal field is the disorder in the site occupancy. Even if the occupancy of sites is well determined (i.e., in Ni-Zn ferrite, all Zn cations on A sites, all Ni cations on B sites), there can be an inhomogeneous distribution of each of them on the sites. EMR spectra can be slightly different when this occupancy of sites is not strictly homogeneous, since some terms of Hint are not exactly the same for all the absorbing centers. It is possible to change the cations distribution in the ferrites by means of thermal treatments or when preparing samples of the same composition but with different synthesis methods.

Hres as a function of temperature for the conventional and co-precipitate Ni-Zn ferrites are shown in Fig. 17(a). The values for conventional ferrite are lightly higher than those of the co-precipitate ferrite. This difference can be due to inhomogeneities associated with a different distribution of cations, and that it is originated by synthesis method. For both samples, the increment of Hres as temperature increases is due to decrement of the internal field, i.e. in the ferrimagnetic order Hint is added to the applied field and the resonance condition is reached at low values of Hdc. In contrast, in the paramagnetic phase, the necessary magnetic field to satisfy the resonance condition has to be supplied entirely by the applied field, Hint= 0 and Hres= Hdc; i.e. when increasing the temperature the progressive disappearance of Hint is associate with the lost long-range order.

Detection of Magnetic Transitions by

Means of Ferromagnetic Resonance and Microwave Absorption Techniques 81

dominant. For both ferrites, the decrease in Hpp as temperature increases is associated to a weakening of the magneto-crystalline anisotropy as T approaches Tc (Byun et al., 2000). The magnetic transition (ferri-paramagnetic) appears as an inflection point in plot Hpp vs. temperature, as is shown in Fig. 17(b). The inflection points at Tc1~ 408 K (conventional) and Tc2~ 430 K (co-precipitate) are associated with Curie temperature of each sample. We observed that the Curie transition is higher in the co-precipitate Ni-Zn ferrite. As Tc is an intrinsic property, that it depends entirely on the ferrite composition, the difference in Curie temperature between both ferrites also suggests a different occupancy between the A and B sites. To high temperature, the long-range magnetic order is completely lost except for some short range order islands in the material that contribute strongly in the broadening of the EMR spectrum. On the other hand, in Fig. 17(b), a second inflection point is clearly observed at TY1= 262 K and TY2= 240 K, in conventional and co-precipitate ferrites respectively. This behavior is attributed to a non-collinear arrangement of the magnetic moments in the A and B sites, i.e. it is due to a Yafet-Kittel-type ordering of the magnetic moments in both samples; the difference of temperatures (*T*Y1>*T*Y2) between both ferrites is indicative of an

In Fig. 18(a), we show LFMA spectra for conventional ferrite in the 208-408 K temperature range. This microwave absorption, around zero field, is far from the resonance condition given by eq.(3), i.e. the sample is in an unsaturated state; therefore, this absorption is associated with interaction between the microwave field and the dynamics of the magnetic domains structure in the sample. For T>TC1 (= 408 K), LFMA spectra exhibit a linear behavior with a positive slope and non-hysteretic traces, and which is associate with paramagnetic phase, i.e. the long-range order has completely disappeared in the sample. For TYK1(= 261 K) ≤T≤ TC1, LFMA spectra show an antisymmetrical shape around zero field, displaying a clear hysteresis upon cycling the field, see Fig. 18(a); and they have the same phase of the EMR spectra, indicating that this absorption has a maximum value at zero field. For this temperature range, it can be observed that HLFMA increases when the temperature decreases, as can be seen in Fig. 18(b), for conventional and co-precipitate ferrites; this behavior indicates an increase in the ferromagnetically coupled superexchange interactions in the samples, suggesting that HLFMA is determined by the magnetic anisotropy and by the demagnetizing field (Montiel et al., 2004, 2005; Alvarez et al., 2008, 2010). But between both samples, the widths difference can be due to different distributions of cations in B sites, originating changes in the antiferromagnetic superexchange interactions; being more intense

For T<TYK1, an additional absorption mode also centered at zero field is observed in Fig. 18(a), and that it is suggested by high distortion of the LFMA signal at low temperature; this new absorption mode is more evident in co-precipitate ferrite (Alvarez et al., 2010). This signal exhibits an opposite phase (out-of-phase) with regard to EMR spectra; indicating that this microwave absorption has a minimum value at zero field. The presence of an out-ofphase signal has been correlated with the occurrence of a ferromagnetic order. It can be

inhomogeneous distribution on occupancy of the B sites.

*3.2.2. LFMA technique* 

in the conventional Ni-Zn ferrite.

Hpp can be due to several factors (Srivastava & Patni, 1974), in particular for polycrystalline samples the linewidth is due to: the sample porosity (Hpor), the magnetic anisotropy (HK), the eddy currents (Heddy), and the magnetic demagnetization (Hdes). It is necessary to mention that there is a broadening due to variations in cations distribution on the A and B sites, and it is highly dependent of the preparation method. In Fig. 17(b), we show the behavior of Hpp with temperature for the conventional and co-precipitate Ni-Zn ferrites. For all the temperatures, Hpp in co-precipitate ferrite is higher than for the conventional ferrite. This behavior can be due to differences in the microscopic magnetic interactions inside the samples, mainly the interparticle magnetic dipole interaction and the superexchange interaction; and they are originated by different cations distributions in the samples, due to synthesis method. The magneto-crystalline anisotropy has a strong contribution to Hpp and we can have the following approximation Hpp=HK=K1/2MS, i.e. for a system of randomly oriented crystallites, the contribution of the anisotropy field is

**Figure 17.** (a) Temperature dependence of Hres in the 200-440 K temperature range for the (○) conventional and (□) co-precipitate Ni-Zn ferrites. (b) Temperature dependence of Hpp in the same temperature range for the (●) conventional and (■) co-precipitate Ni-Zn ferrites. The curves connecting points are only guides for the eye.

dominant. For both ferrites, the decrease in Hpp as temperature increases is associated to a weakening of the magneto-crystalline anisotropy as T approaches Tc (Byun et al., 2000). The magnetic transition (ferri-paramagnetic) appears as an inflection point in plot Hpp vs. temperature, as is shown in Fig. 17(b). The inflection points at Tc1~ 408 K (conventional) and Tc2~ 430 K (co-precipitate) are associated with Curie temperature of each sample. We observed that the Curie transition is higher in the co-precipitate Ni-Zn ferrite. As Tc is an intrinsic property, that it depends entirely on the ferrite composition, the difference in Curie temperature between both ferrites also suggests a different occupancy between the A and B sites. To high temperature, the long-range magnetic order is completely lost except for some short range order islands in the material that contribute strongly in the broadening of the EMR spectrum. On the other hand, in Fig. 17(b), a second inflection point is clearly observed at TY1= 262 K and TY2= 240 K, in conventional and co-precipitate ferrites respectively. This behavior is attributed to a non-collinear arrangement of the magnetic moments in the A and B sites, i.e. it is due to a Yafet-Kittel-type ordering of the magnetic moments in both samples; the difference of temperatures (*T*Y1>*T*Y2) between both ferrites is indicative of an inhomogeneous distribution on occupancy of the B sites.

### *3.2.2. LFMA technique*

80 Ferromagnetic Resonance – Theory and Applications

disappearance of Hint is associate with the lost long-range order.

**200 240 280 320 360 400 440**

**Temperature (K)**

**1600**

points are only guides for the eye.

**2000**

**2400**

**Hres (Gauss)**

**2800**

**3200**

**(a)**

Hres as a function of temperature for the conventional and co-precipitate Ni-Zn ferrites are shown in Fig. 17(a). The values for conventional ferrite are lightly higher than those of the co-precipitate ferrite. This difference can be due to inhomogeneities associated with a different distribution of cations, and that it is originated by synthesis method. For both samples, the increment of Hres as temperature increases is due to decrement of the internal field, i.e. in the ferrimagnetic order Hint is added to the applied field and the resonance condition is reached at low values of Hdc. In contrast, in the paramagnetic phase, the necessary magnetic field to satisfy the resonance condition has to be supplied entirely by the applied field, Hint= 0 and Hres= Hdc; i.e. when increasing the temperature the progressive

Hpp can be due to several factors (Srivastava & Patni, 1974), in particular for polycrystalline samples the linewidth is due to: the sample porosity (Hpor), the magnetic anisotropy (HK), the eddy currents (Heddy), and the magnetic demagnetization (Hdes). It is necessary to mention that there is a broadening due to variations in cations distribution on the A and B sites, and it is highly dependent of the preparation method. In Fig. 17(b), we show the behavior of Hpp with temperature for the conventional and co-precipitate Ni-Zn ferrites. For all the temperatures, Hpp in co-precipitate ferrite is higher than for the conventional ferrite. This behavior can be due to differences in the microscopic magnetic interactions inside the samples, mainly the interparticle magnetic dipole interaction and the superexchange interaction; and they are originated by different cations distributions in the samples, due to synthesis method. The magneto-crystalline anisotropy has a strong contribution to Hpp and we can have the following approximation Hpp=HK=K1/2MS, i.e. for a system of randomly oriented crystallites, the contribution of the anisotropy field is

**200 240 280 320 360 400 440**

**Temperature (K)**

**TY1= 262 K**

**TC2= 430 K**

**(b)**

**TC1= 408 K**

**150**

**300**

**450**

**Hpp (Gauss)**

**Figure 17.** (a) Temperature dependence of Hres in the 200-440 K temperature range for the (○) conventional and (□) co-precipitate Ni-Zn ferrites. (b) Temperature dependence of Hpp in the same temperature range for the (●) conventional and (■) co-precipitate Ni-Zn ferrites. The curves connecting

**600**

**750**

**<sup>900</sup> TY2= 240 K**

In Fig. 18(a), we show LFMA spectra for conventional ferrite in the 208-408 K temperature range. This microwave absorption, around zero field, is far from the resonance condition given by eq.(3), i.e. the sample is in an unsaturated state; therefore, this absorption is associated with interaction between the microwave field and the dynamics of the magnetic domains structure in the sample. For T>TC1 (= 408 K), LFMA spectra exhibit a linear behavior with a positive slope and non-hysteretic traces, and which is associate with paramagnetic phase, i.e. the long-range order has completely disappeared in the sample. For TYK1(= 261 K) ≤T≤ TC1, LFMA spectra show an antisymmetrical shape around zero field, displaying a clear hysteresis upon cycling the field, see Fig. 18(a); and they have the same phase of the EMR spectra, indicating that this absorption has a maximum value at zero field. For this temperature range, it can be observed that HLFMA increases when the temperature decreases, as can be seen in Fig. 18(b), for conventional and co-precipitate ferrites; this behavior indicates an increase in the ferromagnetically coupled superexchange interactions in the samples, suggesting that HLFMA is determined by the magnetic anisotropy and by the demagnetizing field (Montiel et al., 2004, 2005; Alvarez et al., 2008, 2010). But between both samples, the widths difference can be due to different distributions of cations in B sites, originating changes in the antiferromagnetic superexchange interactions; being more intense in the conventional Ni-Zn ferrite.

For T<TYK1, an additional absorption mode also centered at zero field is observed in Fig. 18(a), and that it is suggested by high distortion of the LFMA signal at low temperature; this new absorption mode is more evident in co-precipitate ferrite (Alvarez et al., 2010). This signal exhibits an opposite phase (out-of-phase) with regard to EMR spectra; indicating that this microwave absorption has a minimum value at zero field. The presence of an out-ofphase signal has been correlated with the occurrence of a ferromagnetic order. It can be

assumed that a ferromagnetic arrangement is related with this signal, while a ferrimagnetic structure leads to the opposite result. Therefore, this out-of-phase signal can be associated with the appearance of a ferromagnetic arrangement of the magnetic moments in the sample; where a Yafet-Kittel-type canting of the magnetic moments in the B sites can provide this ferromagnetic component. Additionally, in this temperature region, HLFMA increases continuously with temperature decrease, but now with a higher change rate; this quick broaden is due to a build-up of the short-range magnetic correlations preceding to magnetic transition. For the above-mentioned, we propose that the phase transition at low temperature is due to a Yafet-Kittel-type magnetic ordering of the moments in the B sites, with onset at TYK1; i.e. for T<TYK1, the parallel arrangement of the B sites is modified, and a non-collinear arrangement of the magnetic moments in the A and B sites appears, leading to a change in the microwave absorption regime. It is necessary to mention that a similar behavior is observed in co-precipitate ferrite but with T<TYK2 (= 239 K), see Fig. 18(b). The relevant temperatures are different but the dynamics of the magnetic moments is similar, detecting the Yafet-Kittel-type magnetic ordering around TYK2, in a good correspondence with EMR technique. The difference of temperatures (TYK1>TYK2) between both ferrite is also indicative of a different occupancy of cations in the A and B sites.

Detection of Magnetic Transitions by

Means of Ferromagnetic Resonance and Microwave Absorption Techniques 83

**360 380 400 420 440 0.091**

**Temperature (K)**

**Tp2= 390 K**

**TH2= 424 K**

**TH1= 396 K** 

p1), the MAMMAS response decreases and another magnetic process

p1=355 K). As temperature increases (decreases)

heating and cooling the samples; with the purpose of looking a change associated with ferriparamagnetic transition. For the sample of solid-state reaction, during heating (cooling), this signal increases monotonically as temperature increases (decreases) from 320 K (440 K),

sets-in, modifying its microwave absorption and this suggests a magnetic transition. An observed interesting feature in MAMMAS response is the absence of thermal hysteresis in the heating and cooling cycles, and it is only observed when TTH1(= 396 K); this merging point (TH1) indicates the onset of the magnetic ordering. The TH1 value is in a very good agreement with the value of Curie temperature detected by the EMR and LFMA measurements. A similar MAMMAS response has been observed in co-precipitate ferrite in the 350-440 K temperature range, see the inset of the Fig. 19. The relevant temperatures are different but the dynamics of the magnetic moments is similar, detecting the ferriparamagnetic transition around TH2=424 K, in a good correspondence with EMR and LFMA

**320 340 360 380 400 420 440**

**Temperature (K)**

**Tp1= 379 K**

**Figure 19.** MAMMAS response of the conventional Ni-Zn ferrite in the 320-440 K temperature range; the inset shows the MAMMAS response of the co-precipitate Ni-Zn ferrite in the 350-440 K temperature

MAMMAS responses for these two ferrites, in 150-300 K temperature range, give us additional information on the magnetic transition at low temperature (see Fig. 20). Beginning to 300 K, these responses exhibit a continuous decrease to a minimum value at Tm1= 260 K and Tm2= 240 K, in conventional and co-precipitate ferrites respectively; and these absorptions increases when continuing diminishing the temperature. This feature points to a change in the microwave absorption regime due to a change in the magnetic structure

**0.092**

**0.093**

**MAMMAS response (a.u.)**

**0.094**

**0.095**

**T\* p2= 375 K**

reaching a maximum value at Tp1=379 K (T\*

**0.240**

**0.245**

**0.250**

**MAMMAS response (a.u.)**

**0.255**

**T\***

**p1= 355 K**

**0.260**

further, T>Tp1 (T<T\*

techniques.

range.

**Figure 18.** (a) LFMA spectra of the conventional Ni-Zn ferrite in the 362-408 K (down) and 208-261 K (up) temperature ranges. (b) Behavior of HLFMA for the (●) conventional and (■) co-precipitate Ni-Zn ferrites, as a function of temperature in the 150-450 K temperature range.

#### *3.2.3. MAMMAS technique*

We used MAMMAS technique to detect the ferri-paramagnetic transition in both ferrites sample. MAMMAS responses are shown in Fig. 19, where the measurements are carried out heating and cooling the samples; with the purpose of looking a change associated with ferriparamagnetic transition. For the sample of solid-state reaction, during heating (cooling), this signal increases monotonically as temperature increases (decreases) from 320 K (440 K), reaching a maximum value at Tp1=379 K (T\* p1=355 K). As temperature increases (decreases) further, T>Tp1 (T<T\* p1), the MAMMAS response decreases and another magnetic process sets-in, modifying its microwave absorption and this suggests a magnetic transition. An observed interesting feature in MAMMAS response is the absence of thermal hysteresis in the heating and cooling cycles, and it is only observed when TTH1(= 396 K); this merging point (TH1) indicates the onset of the magnetic ordering. The TH1 value is in a very good agreement with the value of Curie temperature detected by the EMR and LFMA measurements. A similar MAMMAS response has been observed in co-precipitate ferrite in the 350-440 K temperature range, see the inset of the Fig. 19. The relevant temperatures are different but the dynamics of the magnetic moments is similar, detecting the ferriparamagnetic transition around TH2=424 K, in a good correspondence with EMR and LFMA techniques.

82 Ferromagnetic Resonance – Theory and Applications

**-1000 -500 0 500 1000**

*3.2.3. MAMMAS technique* 

**Magnetic Field (Gauss)**

**dP/dH (a.u.)**

**(a)**

assumed that a ferromagnetic arrangement is related with this signal, while a ferrimagnetic structure leads to the opposite result. Therefore, this out-of-phase signal can be associated with the appearance of a ferromagnetic arrangement of the magnetic moments in the sample; where a Yafet-Kittel-type canting of the magnetic moments in the B sites can provide this ferromagnetic component. Additionally, in this temperature region, HLFMA increases continuously with temperature decrease, but now with a higher change rate; this quick broaden is due to a build-up of the short-range magnetic correlations preceding to magnetic transition. For the above-mentioned, we propose that the phase transition at low temperature is due to a Yafet-Kittel-type magnetic ordering of the moments in the B sites, with onset at TYK1; i.e. for T<TYK1, the parallel arrangement of the B sites is modified, and a non-collinear arrangement of the magnetic moments in the A and B sites appears, leading to a change in the microwave absorption regime. It is necessary to mention that a similar behavior is observed in co-precipitate ferrite but with T<TYK2 (= 239 K), see Fig. 18(b). The relevant temperatures are different but the dynamics of the magnetic moments is similar, detecting the Yafet-Kittel-type magnetic ordering around TYK2, in a good correspondence with EMR technique. The difference of temperatures (TYK1>TYK2) between both ferrite is also

indicative of a different occupancy of cations in the A and B sites.

**362 K**

**261 K 238 K 217 K 208 K**

**372 K**

**395 K**

**408 K**

ferrites, as a function of temperature in the 150-450 K temperature range.

**0**

**Figure 18.** (a) LFMA spectra of the conventional Ni-Zn ferrite in the 362-408 K (down) and 208-261 K (up) temperature ranges. (b) Behavior of HLFMA for the (●) conventional and (■) co-precipitate Ni-Zn

We used MAMMAS technique to detect the ferri-paramagnetic transition in both ferrites sample. MAMMAS responses are shown in Fig. 19, where the measurements are carried out

**200**

**HLFMA (Gauss)**

**TYK2= 239 K**

**400**

**600**

**800**

**1000**

**150 200 250 300 350 400 450**

**Temperature (K)**

**TYK1= 261 K**

**TC1= 408 K**

**TC2= 430 K**

**(b)**

**Figure 19.** MAMMAS response of the conventional Ni-Zn ferrite in the 320-440 K temperature range; the inset shows the MAMMAS response of the co-precipitate Ni-Zn ferrite in the 350-440 K temperature range.

MAMMAS responses for these two ferrites, in 150-300 K temperature range, give us additional information on the magnetic transition at low temperature (see Fig. 20). Beginning to 300 K, these responses exhibit a continuous decrease to a minimum value at Tm1= 260 K and Tm2= 240 K, in conventional and co-precipitate ferrites respectively; and these absorptions increases when continuing diminishing the temperature. This feature points to a change in the microwave absorption regime due to a change in the magnetic structure

(Alvarez et al., 2010); revealing the appearance of a new population of absorbing centers, and which it also is suggested from EMR and LFMA measurements. All this profile shows that this change appears progressively as temperature is diminished, i.e. it is not a sharp change as could be expected from a structural phase transition. We associate this behavior with the transition from the collinear magnetic structure, T>Tm1 and T>Tm2, to the noncollinear (Yafet-Kittel-type) structure. Therefore, the MAMMAS responses depend on the thermal dependence of the magnetic moments dynamics, and the intensity of these signals follow the variations on the number of absorption centers (as is suggested by the EMR parameters), in turn is controlled by the establishment of the Yafet-Kittel-type canting of magnetic moments in the B sites at low temperature.

Detection of Magnetic Transitions by

**423 K**

**355 K**

**323 K**

**294 K**

**0 2000 4000 6000**

**M agnetic Field (G auss)**

**Figure 21.** EMR spectra of PFW powders for selected temperatures; the solid lines correspond to the fits

obtained from a functional form similar to the one used by Ivanshin et al. (2000).

**Signal 1**

**Signal 2**

Means of Ferromagnetic Resonance and Microwave Absorption Techniques 85

Fig. 21 shows the EMR spectra recorded in the 294-423 K temperature range. For 355 K <T 423 K, we observed a single broad Lorentzian line due to spin of the Fe+3 ions. In this lineshape kind, the derivative of the microwave power absorption with respect to the static field (dP/dH) can be fitted into two-component Lorentzian, accounting for the contributions from the clockwise and anticlockwise rotating components of the microwave magnetic field (Joshi et al., 2002). For T 355 K, when the temperature goes diminishing, the absorption mode changes toward a broad asymmetric line of Dyson-type (Dyson, 1955; Feher & Kip, 1955) and their intensity diminishes; this Dyson lineshape is associated with a conductive contribution. This lineshape is a combination of an absorption component and other dispersion component of a symmetric Lorentzian mode, originating an additional parameter: the A/B ratio, i.e. the ratio of the amplitude of the left peak to that of the right peak of the EMR spectrum. Thus, EMR spectra are fitted to a functional form similar to the one used by Ivanshin et al. (2000). Additionally, for this temperature range, a second EMR mode (signal 2) is also observed; being more evident at room temperature, see Fig. 21.

*3.3.1. EMR technique* 

**-3.0 -1.5 0.0 1.5 3.0**

**-0.50 -0.25 0.00 0.25**

**-0.10 -0.05 0.00 0.05**

**-0.05 0.00 0.05 0.10**

**dP/dH (a.u.)**

**Figure 20.** MAMMAS response of the conventional Ni-Zn ferrite in the 150-300 K temperature range; the inset shows the MAMMAS response of the co-precipitate Ni-Zn ferrite for same temperature range.

#### **3.3. Néel transition (from a paramagnetic phase to antiferromagnetic ordering)**

Lead iron tungstate, Pb(Fe2/3W1/3)O3 (PFW), shows an Néel transition around 350-380 K (Smolenskii et al., 1964; Feng et al., 2002; Ivanova et al., 2004); this magnetic ordering is due to superexchange interaction between the Fe ions through the O ions. For this study, PFW powders has been prepared through columbite precursor method (Zhou et al., 2000), where the purity of the powders was checked by means of XRD; all observed reflection lines are indexed as a cubic perovskite-type structure, in a good agreement with the standard data for PFW powders.

### *3.3.1. EMR technique*

84 Ferromagnetic Resonance – Theory and Applications

magnetic moments in the B sites at low temperature.

**-2.4**

PFW powders.

**-1.6**

**-0.2 0.0 0.2 0.4**

**MAMMAS response (a.u.)**

**-0.8**

**MAMMAS response (a.u.)**

**0.0**

**0.8**

(Alvarez et al., 2010); revealing the appearance of a new population of absorbing centers, and which it also is suggested from EMR and LFMA measurements. All this profile shows that this change appears progressively as temperature is diminished, i.e. it is not a sharp change as could be expected from a structural phase transition. We associate this behavior with the transition from the collinear magnetic structure, T>Tm1 and T>Tm2, to the noncollinear (Yafet-Kittel-type) structure. Therefore, the MAMMAS responses depend on the thermal dependence of the magnetic moments dynamics, and the intensity of these signals follow the variations on the number of absorption centers (as is suggested by the EMR parameters), in turn is controlled by the establishment of the Yafet-Kittel-type canting of

**150 175 200 225 250 275 300**

**Temperature (K)** 

**Figure 20.** MAMMAS response of the conventional Ni-Zn ferrite in the 150-300 K temperature range; the inset shows the MAMMAS response of the co-precipitate Ni-Zn ferrite for same temperature range.

**3.3. Néel transition (from a paramagnetic phase to antiferromagnetic ordering)** 

Lead iron tungstate, Pb(Fe2/3W1/3)O3 (PFW), shows an Néel transition around 350-380 K (Smolenskii et al., 1964; Feng et al., 2002; Ivanova et al., 2004); this magnetic ordering is due to superexchange interaction between the Fe ions through the O ions. For this study, PFW powders has been prepared through columbite precursor method (Zhou et al., 2000), where the purity of the powders was checked by means of XRD; all observed reflection lines are indexed as a cubic perovskite-type structure, in a good agreement with the standard data for

**Tm1= 260 K**

**150 175 200 225 250 275 300 -0.4**

**Temperature (K)**

**Tm2= 240 K**

Fig. 21 shows the EMR spectra recorded in the 294-423 K temperature range. For 355 K <T 423 K, we observed a single broad Lorentzian line due to spin of the Fe+3 ions. In this lineshape kind, the derivative of the microwave power absorption with respect to the static field (dP/dH) can be fitted into two-component Lorentzian, accounting for the contributions from the clockwise and anticlockwise rotating components of the microwave magnetic field (Joshi et al., 2002). For T 355 K, when the temperature goes diminishing, the absorption mode changes toward a broad asymmetric line of Dyson-type (Dyson, 1955; Feher & Kip, 1955) and their intensity diminishes; this Dyson lineshape is associated with a conductive contribution. This lineshape is a combination of an absorption component and other dispersion component of a symmetric Lorentzian mode, originating an additional parameter: the A/B ratio, i.e. the ratio of the amplitude of the left peak to that of the right peak of the EMR spectrum. Thus, EMR spectra are fitted to a functional form similar to the one used by Ivanshin et al. (2000). Additionally, for this temperature range, a second EMR mode (signal 2) is also observed; being more evident at room temperature, see Fig. 21.

**Figure 21.** EMR spectra of PFW powders for selected temperatures; the solid lines correspond to the fits obtained from a functional form similar to the one used by Ivanshin et al. (2000).

In Fig. 22, the temperature dependences of the EPR parameters for signal 1 are plotted. Hpp as a temperature function is shown in Fig. 22(a) for PFW powders. Starting from 423 K, as temperature decreases, Hpp decreases continuously until 323 K; exhibiting a minimum at this temperature. This narrowing in Hpp indicates an increase in the superexchange interactions in the sample, because the superexchange mechanism tend to narrow the absorption line; and it can be due to magnetic fluctuations, i.e. fluctuations in the establishment of the long-range order that precedes the transition to antiferromagnetic order. When continuing diminishing the temperature, T<323 K, Hpp shows a weak increase until 294 K, this increase is indicative of a weak ferromagnetic behavior in the PFW; similar behaviors are observed in others magnetoelectric materials (Alvarez et al., 2006, 2010, 2012).

Detection of Magnetic Transitions by

Means of Ferromagnetic Resonance and Microwave Absorption Techniques 87

interactions with Fe2+ ions. Additionally, a change in slope to 334 K is also observed; see the inset of the Fig. 22(a), where this feature can be associated with the para-antiferromagnetic

**300 320 340 360 380 400 420**

**Temperature (K)**

**Figure 22.** Temperature dependence of (a) Hpp, (b) the g-factor and (c) the A/B ration of PFW powders for the signal 1; the inset of the Fig. 22(a) shows the temperature dependence of Hpp for the signal 2.

In Fig. 23(a), we show the LFMA signal in the 294-423 K temperature range. For all temperatures, LFMA signal exhibits two antisymmetric peaks about zero magnetic field, with opposed phase to EMR spectrum, and a clear hysteresis of this signal appears on cycling the field. This strongly contrasts with the LFMA signal observed in others

**(c)**

**(b)**

**(a)**

**0.00 0.25 0.50 0.75 1.00 1.850**

**A/B**

Solid lines are guides for the eye only.

*3.3.2. LFMA technique* 

**g-factor**

**H**

**PP (Gauss)**

**1.875**

**300 315 330 345 360 <sup>900</sup>**

**Temperature (K)**

**HPP (Gauss)**

transition.

Fig. 22(b) shows the behavior of the g-factor vs. temperature for the signal 1, which is estimated from Hres, with g= h/BHres; where h is the Planck constant, is the frequency and B is the Bohr magneton. EMR spectra give g-factor smaller than for a free electron (= 2.0023), along the entire temperature range. This behavior can be explained through the spin value of the Fe3+ ions (S=5/2) and to changes in the spin-orbit coupling; where the effective gfactor (geff) of a paramagnetic center is given by geff=g(1±/), where is the crystal-field splitting and is the spin-orbit coupling constant. The g-factor shows a weak decrease in the 423-365 K temperature range and then the g-factor continues diminishing, but this time with a higher change rate, reaching a minimum value at 344 K (gmin= 1.8491). This fast decrease can be due to build-up of magnetic correlations preceding the transition to the long-range antiferromagnetic ordering at Néel temperature (TN 344 K). For T<TN, the g-factor increases until 294 K and this behavior is an indication of a weak ferromagnetism in the PFW; this behavior can be due to a canting of the sublattices of Fe3+ ions in the antiferromagnetic matrix, generating an effective magnetic moment.

The temperature dependence of the A/B ratio is shown in Fig. 22(c). From this plot can be seen that, starting from 423 K to close to 378 K, the A/B ratio remains essentially constant at a value of 1. This value indicates that the paramagnetic centers are static and also suggests a strong dipolar interaction between Fe3+ ions. Further, the A/B ratio continually diminishes until a minimum value to 344 K; indicating a dispersion contribution and it suggests a conduction effect in the sample. As the temperature is decreased further, 294 K≤T<344 K, the A/B ratio increases toward a near value of 1, due to a decrease of the conductivity in sample. Recently, electric mensurations were carried out on PFW samples (Eiras et al. 2010; Fraygola et al. 2011), and which indicate a conductive contribution associated with an electronic-hoping mechanism, in a good correspondence with Dyson lineshape of the EMR spectra.

A second absorption mode (signal 2) is clearly observed at room temperature and it is associated with the presence of a fraction of Fe2+ ions, where oxygen atoms deficiency generate a state of mixed valency; originating a strong magnetic dipolar interaction between Fe2+ and Fe3+ ions, and that produces a broaden in absorption mode. In inset of the Fig. 22(a), we show the Hpp behavior as a temperature function for the signal 2. For T<378 K, when diminishing the temperature, the broadening of signal 2 is indicative of an increase of the interactions with Fe2+ ions. Additionally, a change in slope to 334 K is also observed; see the inset of the Fig. 22(a), where this feature can be associated with the para-antiferromagnetic transition.

**Figure 22.** Temperature dependence of (a) Hpp, (b) the g-factor and (c) the A/B ration of PFW powders for the signal 1; the inset of the Fig. 22(a) shows the temperature dependence of Hpp for the signal 2. Solid lines are guides for the eye only.

#### *3.3.2. LFMA technique*

86 Ferromagnetic Resonance – Theory and Applications

matrix, generating an effective magnetic moment.

lineshape of the EMR spectra.

In Fig. 22, the temperature dependences of the EPR parameters for signal 1 are plotted. Hpp as a temperature function is shown in Fig. 22(a) for PFW powders. Starting from 423 K, as temperature decreases, Hpp decreases continuously until 323 K; exhibiting a minimum at this temperature. This narrowing in Hpp indicates an increase in the superexchange interactions in the sample, because the superexchange mechanism tend to narrow the absorption line; and it can be due to magnetic fluctuations, i.e. fluctuations in the establishment of the long-range order that precedes the transition to antiferromagnetic order. When continuing diminishing the temperature, T<323 K, Hpp shows a weak increase until 294 K, this increase is indicative of a weak ferromagnetic behavior in the PFW; similar behaviors are observed in others magnetoelectric materials (Alvarez et al., 2006, 2010, 2012). Fig. 22(b) shows the behavior of the g-factor vs. temperature for the signal 1, which is estimated from Hres, with g= h/BHres; where h is the Planck constant, is the frequency and B is the Bohr magneton. EMR spectra give g-factor smaller than for a free electron (= 2.0023), along the entire temperature range. This behavior can be explained through the spin value of the Fe3+ ions (S=5/2) and to changes in the spin-orbit coupling; where the effective gfactor (geff) of a paramagnetic center is given by geff=g(1±/), where is the crystal-field splitting and is the spin-orbit coupling constant. The g-factor shows a weak decrease in the 423-365 K temperature range and then the g-factor continues diminishing, but this time with a higher change rate, reaching a minimum value at 344 K (gmin= 1.8491). This fast decrease can be due to build-up of magnetic correlations preceding the transition to the long-range antiferromagnetic ordering at Néel temperature (TN 344 K). For T<TN, the g-factor increases until 294 K and this behavior is an indication of a weak ferromagnetism in the PFW; this behavior can be due to a canting of the sublattices of Fe3+ ions in the antiferromagnetic

The temperature dependence of the A/B ratio is shown in Fig. 22(c). From this plot can be seen that, starting from 423 K to close to 378 K, the A/B ratio remains essentially constant at a value of 1. This value indicates that the paramagnetic centers are static and also suggests a strong dipolar interaction between Fe3+ ions. Further, the A/B ratio continually diminishes until a minimum value to 344 K; indicating a dispersion contribution and it suggests a conduction effect in the sample. As the temperature is decreased further, 294 K≤T<344 K, the A/B ratio increases toward a near value of 1, due to a decrease of the conductivity in sample. Recently, electric mensurations were carried out on PFW samples (Eiras et al. 2010; Fraygola et al. 2011), and which indicate a conductive contribution associated with an electronic-hoping mechanism, in a good correspondence with Dyson

A second absorption mode (signal 2) is clearly observed at room temperature and it is associated with the presence of a fraction of Fe2+ ions, where oxygen atoms deficiency generate a state of mixed valency; originating a strong magnetic dipolar interaction between Fe2+ and Fe3+ ions, and that produces a broaden in absorption mode. In inset of the Fig. 22(a), we show the Hpp behavior as a temperature function for the signal 2. For T<378 K, when diminishing the temperature, the broadening of signal 2 is indicative of an increase of the

In Fig. 23(a), we show the LFMA signal in the 294-423 K temperature range. For all temperatures, LFMA signal exhibits two antisymmetric peaks about zero magnetic field, with opposed phase to EMR spectrum, and a clear hysteresis of this signal appears on cycling the field. This strongly contrasts with the LFMA signal observed in others

magnetoelectric materials (Alvarez et al., 2007, 2010, 2012), where these absorptions are lineal and non-hysteretic. The hysteresis feature has been associated with low field magnetization processes in ferromagnetic materials (Montiel et al., 2004; Alvarez et al., 2010; Gavi et al., 2012), suggesting the presence of a magnetic component in the material (Fraygola et al., 2011). With the help of a reference line, clearly one can observe that the LFMA signal has a lineal absorption component, see Fig. 23(a); where their slope is a temperature function. Fig. 23(b) shows the slope behavior of the lineal component for the 294-423 K temperature range. The slope increases monotonically when the temperature decreases from 423 K, reaching a maximum value at 365 K; this increasing behavior is characteristic of a paramagnetic phase. As the temperature is decreased further, T<365 K, the slope decreases very fast with decreasing temperature until Tmin= 334 K; in this region the quantity of absorbing centers diminishes considerably due to the process of antiparallel spin alignment. Below Tmin, the slope has an approximately lineal increase, where this behavior is a signature of the weak ferromagnetism in this temperature region (Alvarez et al., 2007, 2010, 2012).

Detection of Magnetic Transitions by

Means of Ferromagnetic Resonance and Microwave Absorption Techniques 89

In this work is shown that EMR is the most powerful spectroscopic method available to determine the magnetic transitions in the materials. LFMA and MAMMAS techniques provide information on the dependences in temperature and magnetic field of the nonresonant microwave absorption. More important, these techniques can distinguish between different dissipative dynamics of microwave absorbing centers, providing valuable information about the nature of magnetic ordering within materials. We have shown that all these techniques are powerful tools for the research of magnetic materials at microwave

*Centro de Ciencias Aplicadas y Desarrollo Tecnológico, Universidad Nacional Autónoma de México,* 

*Escuela Superior de Física y Matemáticas del IPN, U.P.A.L.M, Edificio 9, San Pedro Zacatenco,* 

G. Alvarez acknowledges research support in the laboratory of magnetic mensurations and biophysics of ESFM-IPN-Mexico. The authors would like to thank R. Zamorano by the use of the EMR spectrometer. Support from project PAPIIT-UNAM No. IN111111 is gratefully

Akther Hossain A.K.M., Seki M., Kawai T., Tabata H., (2004) J. Appl. Phys. 96, 1273.

Alvarez G., Font R., Portelles J., Valenzuela R., Zamorano R., (2006) Physica B 384, 322. Alvarez G., Font R., Portelles J., Zamorano R., Valenzuela R., (2007) J. Phys. Chem. Solids 68,

Alvarez G., Montiel H., Cos D., García-Arribas A., Zamorano R., Barandiarán J.M.,

Alvarez, G., Cruz M.P., Durán A.C., Montiel H., Zamorano R., (2010) Solid State Commun.

Alvarez G., Font R., Portelles J., Raymond O., Zamorano R., (2009) Solid State Sci. 11, 881. Alvarez G., Montiel H., Barron J.F., Gutierrez M.P., Zamorano R., (2010) J. Magn. Magn.

Alvarez G., Zamorano R., (2004) J. Alloys Compd. 369, 231.

Valenzuela R., (2008) J. Non-Cryst. Solids 354, 5195.

**4. Conclusions** 

frequencies.

H. Montiel

G. Alvarez

**Author details** 

*México DF 07738, México* 

**Acknowledgement** 

acknowledged.

**5. References** 

1436.

Mater. 322, 348.

150, 1597.

*Del. Coyoacán, México DF 04510, México* 

**Figure 23.** (a) LFMA signal for selected temperatures of PFW powders, where the straight lines are only a help to visualize the lineal component of the LFMA signal. (b) The slope temperature dependence of the lineal component for LFMA signal in the 294-423 K temperature range; the solid lines are guides for the eye only.

### **4. Conclusions**

88 Ferromagnetic Resonance – Theory and Applications

**0.10 423 K**

**0.10 355 K**

**-1000 -500 0 500 1000**

**Magnetic Field (Gauss)**

**344 K**

**294 K**

**-1.5**

**0.0**

**1.5**

**3.0**

**Slope (x10-5)**

**Figure 23.** (a) LFMA signal for selected temperatures of PFW powders, where the straight lines are only a help to visualize the lineal component of the LFMA signal. (b) The slope temperature dependence of the lineal component for LFMA signal in the 294-423 K temperature range; the solid lines are guides for

**4.5**

**6.0**

**(b)**

**300 320 340 360 380 400 420**

**Temperature (K)**

2010, 2012).

**-0.05 0.00 0.05** **(a)**

**-0.05 0.00 0.05**

**dP/dH (a.u.)**

**-0.05 0.00 0.05 0.10**

**-0.05 0.00 0.05 0.10**

the eye only.

magnetoelectric materials (Alvarez et al., 2007, 2010, 2012), where these absorptions are lineal and non-hysteretic. The hysteresis feature has been associated with low field magnetization processes in ferromagnetic materials (Montiel et al., 2004; Alvarez et al., 2010; Gavi et al., 2012), suggesting the presence of a magnetic component in the material (Fraygola et al., 2011). With the help of a reference line, clearly one can observe that the LFMA signal has a lineal absorption component, see Fig. 23(a); where their slope is a temperature function. Fig. 23(b) shows the slope behavior of the lineal component for the 294-423 K temperature range. The slope increases monotonically when the temperature decreases from 423 K, reaching a maximum value at 365 K; this increasing behavior is characteristic of a paramagnetic phase. As the temperature is decreased further, T<365 K, the slope decreases very fast with decreasing temperature until Tmin= 334 K; in this region the quantity of absorbing centers diminishes considerably due to the process of antiparallel spin alignment. Below Tmin, the slope has an approximately lineal increase, where this behavior is a signature of the weak ferromagnetism in this temperature region (Alvarez et al., 2007,

In this work is shown that EMR is the most powerful spectroscopic method available to determine the magnetic transitions in the materials. LFMA and MAMMAS techniques provide information on the dependences in temperature and magnetic field of the nonresonant microwave absorption. More important, these techniques can distinguish between different dissipative dynamics of microwave absorbing centers, providing valuable information about the nature of magnetic ordering within materials. We have shown that all these techniques are powerful tools for the research of magnetic materials at microwave frequencies.

### **Author details**

H. Montiel

*Centro de Ciencias Aplicadas y Desarrollo Tecnológico, Universidad Nacional Autónoma de México, Del. Coyoacán, México DF 04510, México* 

### G. Alvarez

*Escuela Superior de Física y Matemáticas del IPN, U.P.A.L.M, Edificio 9, San Pedro Zacatenco, México DF 07738, México* 

### **Acknowledgement**

G. Alvarez acknowledges research support in the laboratory of magnetic mensurations and biophysics of ESFM-IPN-Mexico. The authors would like to thank R. Zamorano by the use of the EMR spectrometer. Support from project PAPIIT-UNAM No. IN111111 is gratefully acknowledged.

### **5. References**

Akther Hossain A.K.M., Seki M., Kawai T., Tabata H., (2004) J. Appl. Phys. 96, 1273.

Alvarez G., Zamorano R., (2004) J. Alloys Compd. 369, 231.


Alvarez G., Peña J.A., Castellanos M.A., Montiel H., Zamorano R., (2012) Rev. Mex. Fis. S 58(2), 24.

Detection of Magnetic Transitions by

Means of Ferromagnetic Resonance and Microwave Absorption Techniques 91

Montiel H., Alvarez G., Betancourt I., Zamorano R., Valenzuela R., (2006) Physica B 384,

Montiel H., Alvarez G., Zamorano R., Valenzuela R., (2007) J. Non-Cryst. Solids 353, 908. Montiel H., Alvarez G., Zamorano R., Valenzuela R., (2008) J. Non-Cryst. Solids

Niebling U., Steinl J., Schweitzer D., Strunz W., (1998) Solid State Commun. 106, 505.

Pacher N., Deisenhofer J., Krung von Nidda H.-A., Hemmida M., Jeevan H.S., Gegenwart P.,

Padam G.K., Ekbote S.N., Tripathy M.R., Srivastava G.P., Das B.K., (1999) Physica C

Panarina N.Y., Talanov Y.I., Shaposhnikova T.S., Beysengulov N.R., Vavilona E., Behr G., Kondrat A., Hess C., Leps N., Wurmehl S., Klinger R., Kataev V., Büchner B., (2010)

Pong W.F., Chang Y.K., Su M.H., Tseng P.K., Lin H.J., Ho G.H., Tsang K.L., Chen C.T.,

Priyadharsini P., Pradeep A., Sambasiva Rao P., Chandrasekaran G., (2009) Mater. Chem.

Satya Murthy N.S., Natera M.G., Youssef S.I., Begum R.J., Srivastava C.M., (1969) Phys. Rev.

Shaltiel D., Bezalel M., Revaz B., Walker E., Tamegai T., Ooi S., (2001) Physica C 349, 139.

Stankowski J., Piekara-Sady L., Kempinski W., (2004) J. Phys. Chem. Solids 65, 321.

Velter-Stefanescu M., Duliu O.G., Mihalache V., (2005) J. Optoelectron. Adv. M. 7, 1557. Wu K.H., Shin Y.M., Yang C.C., Wang G.P., Horng D.N., (2006) Mater. Lett. 60, 2707.

Yildiz F., Rameev B.Z., Tarapov S.I., Tagirov L.R., Aktas B., (2002) J. Magn. Magn. Mater.

Velter-Stefanescu M., Totovana A., Sandu V., (1998) J. Supercond. 11, 327.

Moorjani K., Bohandy J., Adrian F.J., Kim B.F., (1987) Phys. Rev. B 36, 4036.

Padam G.K., Arora N.K., Ekbote S.N., (2010) Mater. Chem. Phys. 123, 752.

Nabereznykh V.P., Tsindlekht M.I., (1982) JETP Lett. 36, 157.

Okamura T., Torizuka Y., Kojima Y., (1951) Phys. Rev. 82, 285.

Ravindaranathan P., Patil K.C., (1987) J. Mater. Sci. 22, 3261.

Smolenskii G.A., Bokov V.A., (1964) J. Appl. Phys. 35, 915. Srivastava C.M., Patni M.J., (1974) J. Magn. Res. 15, 359.

Schlomann E., (1958) J. Phys. Chem. Solids 6, 257.

297.

354, 5192.

315, 45.

Okamura T., (1951) Nature 168, 162.

Owens F.J., (2001) Physica C 363, 202.

Phys. Rev. B 81, 224509.

Phys. 116, 207.

181, 969.

247, 222.

(1997) Phys. Rev. B 55, 11409.

Topacli C., (1996) J. Supercond. 9, 263. Topacli C., (1998) Physica C 301, 92.

Yafet Y., Kittel C., (1952) Phys. Rev. 87, 290.

Owens F.J., (1997) J. Phys. Chem. Solids 58, 1311.

Loidl A., (2010) Phys. Rev. B 82, 054525.


Beach R.S., Berkowitz A. E., (1994) J. Appl. Phys. 76, 6209.

Sahni V.C., (2001) Supercond. Sci. Technol. 14, 572.

Matacotta F.C., (1987) Phys. Rev. B 36, 7241.

Magn. Magn. Mater. 249, 117. Dyson F.J., (1955) Phys. Rev. 98, 349.

Feher G., Kip A.F., (1955) Phys. Rev. 98, 337.

Feng L., Ye Z.G., (2002) J. Solid State Chem. 163, 484.

(2012) J. Magn. Magn. Mater. 324, 1172.

Eremin M.V., (2000) Phys. Rev. B 61, 6213.

Kheifets A.S., Veinger A.I., (1990) Physica C 165, 491.

Healy D.W., (1952) Phys. Rev. 86, 1009.

024410.

8309.

133.

Compd. 369, 141.

86, 072503.

Yoshino, (1997) Physica C 277, 277.

58(2), 24.

623.

L559.

Alvarez G., Peña J.A., Castellanos M.A., Montiel H., Zamorano R., (2012) Rev. Mex. Fis. S

Andrzejewski B., Kowalczyk A., Stankowski J., Szlaferek A., (2004) J. Phys. Chem. Solids 65,

Bhat S.V., Ganguly P., Ramakrishnan T.V., Rao C.N.R., (1987) J. Phys. C: Solid State Phys. 20,

Bhide M.K., Kadam R.M., Sastry M.D., Ajay Singh, Shashwati Sen, Aswal D.K., Gupta S.K.,

Blazey K.W., Müller K.A., Bednorz J.G., Berlinger W., Amoretti G., Buluggiu E., Vera A.,

Domínguez M., García-Beneytez J.M., Vázquez M., Lofland S.E., Baghat S.M., (2002) J.

Gavi H., Ngomb B.D., Beye A.C., Strydom A.M., Srinivasu V.V., Chaker M., Manyala N.,

Hirotake Kajii, Hisashi Araki, Zakhidov A.A., Kazuya Tada, Kyuya Yakushi, Katsumi

Joshi J.P., Gupta R., Sood A.K., Bhat S.V., Raju A.R., Rao C.N.R., (2002) Phys. Rev. B 65,

Khachaturyan K., Weber E.R., Tejedor P., Stacy A.M., Portis A.M., (1987) Phys. Rev. B 36,

Kim B.F., Moorjani K., Adrian F.J., Bohandy J., (1993) Materials Science Forum 137-139,

Montiel H., Alvarez G., Gutierrez M.P., Zamorano R., Valenzuela R., (2004) J. Alloys

Montiel H., Alvarez G., Betancourt I., Zamorano R., Valenzuela R., (2005) Appl. Phys. Lett.

Ivanova S.A., Eriksson S.G., Tellgrend R., Rundlöf H., (2004) Mater. Res. Bull. 39, 2317. Ivanshin V.A., Deisenhofer J., Krug von Nidda H.-A., Loidl A., Mukhin A., Balbashov J.,

Bohandy J., Suter J., Kim B.F., Moorjani K., Adrian F.J., (1987) Appl. Phys. Lett. 51, 2161.

Eiras J.A., Fraygola B.M., Garcia D., (2010) Key Engineering Materials 434-435, 307.

Fraygola B.M., Coelho A.A., Garcia D., Eiras J.A., (2011) Materials Research 14, 434.

Byun T.Y., Byeon S.C., Hong K.S., Kyung C., (2000) J. Appl. Phys. 87, 6220.

Bele P., Brunner H., Schweitzer D., J. Keller H., (1994) Solid State Commun. 92, 189.


Zakhidov A.A., Ugawa A., Imaeda K., Yakushi K., Inokuchi H., (1991) Solid State Commun. 79, 939.

**Chapter 4** 

© 2013 Sharma et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**FMR Measurements of Magnetic Nanostructures** 

Ferromagnetic nanowires showed solitary and tunable magnetization properties due to their inherent shape anisotropy. The fabrication of such nanowires in polycarbonate track-etched and anodic alumina membranes have been widely studied during the last 15 years [1-2]. Their potential applications might be explored in spintronic devices and more specifically in magnetic random access memory (MRAM) and magnetic logic devices [3-5]. Furthermore, microwave devices, such as circulators or filters for wireless communication and automotive systems can be fabricated on ferromagnetic nanowires embedded in AAO substrates [6-9].

This chapter begins with a brief overview of the historical development of the theory of ferromagnetic resonance in magnetic nanostructures. State-of-the-art calculations for resonance frequency in ferromagnetic nanowires (solid and hollow) and multilayer nanowires are presented. In addition, experimental approach to synthesis such structures and detecting material properties using various techniques will be discussed in brief. Recently, due to the development of spintronics, there have been increasing interests in the microwave dynamics of one-dimensional structures such as nanowires and two dimensional structures like multilayer magnetic films. The most important parameters that control dynamic behaviors are the internal fields and damping constant. The ferromagnetic nanowires in anodic alumina (AAO) templates seem to be attractive substrates for microwave applications. Since they have high aspect ratio, electromagnetic waves can easily penetrate through them. They exhibit ferromagnetic resonance (FMR) even at zero bias fields and, due to their high saturation magnetization, operating frequency can be tuned

FMR is a useful technique in the measurement of magnetic properties of ferromagnetic materials. It has been applied to a range of materials from bulk ferromagnetic materials to nano-scale magnetic thin films and now a day's people have started research to characterise nanoparticles and nanowires systems. The dynamic properties of magnetic materials can be easily perturbed by ferromagnetic resonance (FMR), as they can excite standing spin waves

and reproduction in any medium, provided the original work is properly cited.

Manish Sharma, Sachin Pathak and Monika Sharma

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/56615

**1. Introduction** 

with DC fields.

Zhou L., Vilarinho P.M., Bptista J.L., Fortunato E., (2000) J. Eur. Ceram. Soc. 20, 1035.

## **FMR Measurements of Magnetic Nanostructures**

Manish Sharma, Sachin Pathak and Monika Sharma

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/56615

### **1. Introduction**

92 Ferromagnetic Resonance – Theory and Applications

79, 939.

Zakhidov A.A., Ugawa A., Imaeda K., Yakushi K., Inokuchi H., (1991) Solid State Commun.

Zhou L., Vilarinho P.M., Bptista J.L., Fortunato E., (2000) J. Eur. Ceram. Soc. 20, 1035.

Ferromagnetic nanowires showed solitary and tunable magnetization properties due to their inherent shape anisotropy. The fabrication of such nanowires in polycarbonate track-etched and anodic alumina membranes have been widely studied during the last 15 years [1-2]. Their potential applications might be explored in spintronic devices and more specifically in magnetic random access memory (MRAM) and magnetic logic devices [3-5]. Furthermore, microwave devices, such as circulators or filters for wireless communication and automotive systems can be fabricated on ferromagnetic nanowires embedded in AAO substrates [6-9].

This chapter begins with a brief overview of the historical development of the theory of ferromagnetic resonance in magnetic nanostructures. State-of-the-art calculations for resonance frequency in ferromagnetic nanowires (solid and hollow) and multilayer nanowires are presented. In addition, experimental approach to synthesis such structures and detecting material properties using various techniques will be discussed in brief. Recently, due to the development of spintronics, there have been increasing interests in the microwave dynamics of one-dimensional structures such as nanowires and two dimensional structures like multilayer magnetic films. The most important parameters that control dynamic behaviors are the internal fields and damping constant. The ferromagnetic nanowires in anodic alumina (AAO) templates seem to be attractive substrates for microwave applications. Since they have high aspect ratio, electromagnetic waves can easily penetrate through them. They exhibit ferromagnetic resonance (FMR) even at zero bias fields and, due to their high saturation magnetization, operating frequency can be tuned with DC fields.

FMR is a useful technique in the measurement of magnetic properties of ferromagnetic materials. It has been applied to a range of materials from bulk ferromagnetic materials to nano-scale magnetic thin films and now a day's people have started research to characterise nanoparticles and nanowires systems. The dynamic properties of magnetic materials can be easily perturbed by ferromagnetic resonance (FMR), as they can excite standing spin waves

© 2013 Sharma et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

due to magnetic pinning [10-11,31,32]. It also yields direct information about the uniform precession mode of the nanowires which can be related to the average anisotropy magnitude [12-15]. Several measurement techniques which have been used to characterise magnetization dynamics such as femtosecond spectroscopy [16-18], pulse inductive microwave magnetometer [19], FMR force microscopy [20], network analyzer FMR [21,31,32] and high-frequency electrical measurements of magnetodynamics [22]. All these techniques can be used for modern application for example in telecommunications and data storage systems.

FMR Measurements of Magnetic Nanostructures 95

*<sup>M</sup> M H,* (2)

 

Nx, Ny, Nz are the demagnetization factors in the x, y, z directions, and so on, and where *g* is the spectroscopic splitting factor (Lande factor) and *µB* = *eħ/*2me is the Bohr magneton. The demagnetizing factors affect the shape anisotropy of the magnetic material depending upon

Part of the classical approach to ferromagnetism is to replace the spins by a classical microspin vector *M* magnetization. The time-dependence of the magnetization can be obtained

> *dt*

where = *gµb*/*ħ* is a gyromagnetic ratio. This equation represents an undamped precession of the magnetization. From experiments actual changes of the magnetization are known to decay in a finite time. The occurrence of a damping mechanism leads to reversal of the magnetization towards the direction of *H* within several nanoseconds. The damping is just

*eff eff*

films. Eq. 3 is known as Landau-Lifshitz-Gilbert equation after Gilbert introduces the damping term. *Heff* is the total effective magnetic field which is a sum of static applied magnetic field (*H*), dynamic magnetic field (*hrf*) and internal magnetic field (*Hin*). Internal field constitutes various magnetic anisotropies such as magnetocrystalline anisotropy, shape anisotropy, and magnetoelastic anisotropy etc. This equation therefore describes torque acting on *M*. This torque leads to a rotation of the magnetization towards the direction of the external magnetic field. The damping causes decay in precessional motion which by

**Figure 1.** (a) Torque components exerted on the magnetization *M* by rotational field *H* (b) Motion of *M*

*dd d dt M dt M dt*

applying a dynamic magnetic field becomes continuous as shown in Fig. 1.

*s S*

*MM M M H MH M* (3)

 

is the dimensionless Gilbert damping constant, of order 10-2 in ferromagnetic thin

directly by calculating the torque acting on *M* by an effective field *Heff*,

*d*

its geometry to be ellipsoid, sphere, thin film etc.

added as a phenomenological term to Eq. 2.

where 

for constant *H*.

In the present chapter, we deal with magnetic nanostructures such as nanowires and multilayered nanodiscs and rings which exhibit unique FMR responses since their various anisotropy energies are strongly influenced by size and shape [33]. In addition, interactions between multilayered segments can be tailored such that the FMR response is not only angle-dependent but also influenced strongly at certain frequencies [32]. In this chapter, we shall be describing three different aspects of this topic:


### **2. Fundamental theory of ferromagnetic resonance**

Ferromagnetic resonance (FMR) is a very powerful experimental technique in the study of ferromagnetic nanomaterials. The precessional motion of a magnetization *M* of ferromagnetic material about the applied external magnetic field *H* is known as the Ferromagnetic resonance (FMR). In the physical process of resonance, the energy is absorbed from rf transverse magnetic field *hrf*, which occurred when frequency matched with precessional frequency (). The precession frequency depends on the orientation of the material and the strength of the magnetic field. It allows us to measure all the most important parameters of the material: Curie temperature, total magnetic moment, relaxation mechanism, elementary excitations and others.

A single domain magnetic particle with ellipsoid shape was considered as in [Kittel] to drive the resonance condition for the phenomenon of ferromagnetic resonance. A uniform, static magnetic field H is applied along the z-axis and set the sample in a microwave cavity. A resonance is observed at a frequency given by

$$
\hbar \rho = \g \,\mu\_B \sqrt{[H + (N\_x - N\_z)M][H + (N\_y - N\_z)M]}.\tag{1}
$$

Nx, Ny, Nz are the demagnetization factors in the x, y, z directions, and so on, and where *g* is the spectroscopic splitting factor (Lande factor) and *µB* = *eħ/*2me is the Bohr magneton. The demagnetizing factors affect the shape anisotropy of the magnetic material depending upon its geometry to be ellipsoid, sphere, thin film etc.

94 Ferromagnetic Resonance – Theory and Applications

shall be describing three different aspects of this topic:

storage systems.

discussed.

due to magnetic pinning [10-11,31,32]. It also yields direct information about the uniform precession mode of the nanowires which can be related to the average anisotropy magnitude [12-15]. Several measurement techniques which have been used to characterise magnetization dynamics such as femtosecond spectroscopy [16-18], pulse inductive microwave magnetometer [19], FMR force microscopy [20], network analyzer FMR [21,31,32] and high-frequency electrical measurements of magnetodynamics [22]. All these techniques can be used for modern application for example in telecommunications and data

In the present chapter, we deal with magnetic nanostructures such as nanowires and multilayered nanodiscs and rings which exhibit unique FMR responses since their various anisotropy energies are strongly influenced by size and shape [33]. In addition, interactions between multilayered segments can be tailored such that the FMR response is not only angle-dependent but also influenced strongly at certain frequencies [32]. In this chapter, we

1. We first develop the theory of FMR response of densely packed nanowire arrays that can be treated as two-dimensional periodic nanostructures. We then extend the theory to three-dimensional structures, which can be made using multilayered nanowires. 2. We then describe how to synthesize such nanowire arrays and also direct measurements of such structures. Several different experimental techniques are

3. We then continue the treatment to describe the use of such periodic nanowire arrays in microwave devices to exhibit nonlinear responses and also for circulators and isolators.

Ferromagnetic resonance (FMR) is a very powerful experimental technique in the study of ferromagnetic nanomaterials. The precessional motion of a magnetization *M* of ferromagnetic material about the applied external magnetic field *H* is known as the Ferromagnetic resonance (FMR). In the physical process of resonance, the energy is absorbed from rf transverse magnetic field *hrf*, which occurred when frequency matched with precessional frequency (). The precession frequency depends on the orientation of the material and the strength of the magnetic field. It allows us to measure all the most important parameters of the material: Curie temperature, total magnetic moment, relaxation

A single domain magnetic particle with ellipsoid shape was considered as in [Kittel] to drive the resonance condition for the phenomenon of ferromagnetic resonance. A uniform, static magnetic field H is applied along the z-axis and set the sample in a microwave cavity. A

[ ( ) ][ ( ) ]. *B xz yz*

*g H N NMH N NM* (1)

Although the effects seen till now are weak, these are still quite promising.

**2. Fundamental theory of ferromagnetic resonance** 

mechanism, elementary excitations and others.

resonance is observed at a frequency given by

  Part of the classical approach to ferromagnetism is to replace the spins by a classical microspin vector *M* magnetization. The time-dependence of the magnetization can be obtained directly by calculating the torque acting on *M* by an effective field *Heff*,

$$\frac{d\mathbf{M}}{dt} = -\gamma \mathbf{M} \times \mathbf{H}\_r \tag{2}$$

where = *gµb*/*ħ* is a gyromagnetic ratio. This equation represents an undamped precession of the magnetization. From experiments actual changes of the magnetization are known to decay in a finite time. The occurrence of a damping mechanism leads to reversal of the magnetization towards the direction of *H* within several nanoseconds. The damping is just added as a phenomenological term to Eq. 2.

$$\frac{d\mathbf{M}}{dt} = -\gamma \mathbf{M} \times \left(\mathbf{H}\_{e\sharp} - \frac{\alpha}{\gamma M\_s} \frac{d\mathbf{M}}{dt}\right) = -\gamma \mathbf{M} \times \mathbf{H}\_{e\sharp} + \frac{\alpha}{M\_S} \mathbf{M} \times \frac{d\mathbf{M}}{dt} \tag{3}$$

where is the dimensionless Gilbert damping constant, of order 10-2 in ferromagnetic thin films. Eq. 3 is known as Landau-Lifshitz-Gilbert equation after Gilbert introduces the damping term. *Heff* is the total effective magnetic field which is a sum of static applied magnetic field (*H*), dynamic magnetic field (*hrf*) and internal magnetic field (*Hin*). Internal field constitutes various magnetic anisotropies such as magnetocrystalline anisotropy, shape anisotropy, and magnetoelastic anisotropy etc. This equation therefore describes torque acting on *M*. This torque leads to a rotation of the magnetization towards the direction of the external magnetic field. The damping causes decay in precessional motion which by applying a dynamic magnetic field becomes continuous as shown in Fig. 1.

**Figure 1.** (a) Torque components exerted on the magnetization *M* by rotational field *H* (b) Motion of *M* for constant *H*.

### **3. FMR dispersion relation for nanostructures**

Due to miniaturization from bulk to the nanoscale, material properties shows a drastic change like surface-to-volume ratio, electron transport, thermodynamic fluctuations, defects etc. The nanomagnetic material demonstrates distinct magnetic response due to various anisotropies. The one-dimensional nanostructures such as nanowires, nanotubes, nanorods and nanorings are the area of recent research for data storage applications, sensors, biomedical drugs, microwave devices. To perturb the dynamic magnetization in these structures, ferromagnetic resonance is an effective tool.

FMR Measurements of Magnetic Nanostructures 97

(6)

(7)

*<sup>H</sup> <sup>H</sup> H eff K* (8)

 

> 

(10)

(10a)

(9)

2

<sup>2</sup> 2 3 *d r* 

where d is the diameter of the pore and r is the centre-to-centre inter-wire distance between the pores. The equilibrium values for polar angles are obtained by minimizing the energy term

> <sup>0</sup> *<sup>E</sup> <sup>E</sup>*

 

2

γ

eq (5)

where, *<sup>m</sup>*

cases:

*nm m*

*Case 1*: H|| to the wire and Heff>0

*Case 2*: H|| to the wire and Heff<0

*Case 3*: H easy axis and H<Heff

 

> 

From eq (8) we retrieve the dispersion relation which can be written as

cos sin cos sin cos sin 2 0

Here, 2 *eff eff S* is the effective anisotropy field that comes from a combination of effects including shape, magnetocrystalline and magnetoelastic anisotropy. Therefore from

2 13 2 *<sup>U</sup>*

2 1 3{1 (1 )} 2 *<sup>U</sup>*

multilayer section. If *hnm* = 0 (i.e. no non-magnetic spacer layer), the above equation represents the case of a single-element nanowire. Depending upon the direction of the external magnetic field along the easy axis of the nanowires, we can determine the various

*H Heff*

*H H eff*

*<sup>K</sup> M fP <sup>M</sup>*

*<sup>h</sup> <sup>f</sup> h h* , *hm* and *hnm* represents magnetic and non-magnetic thicknesses in

*<sup>K</sup> M P*

 

2

<sup>ω</sup> Hcos H cos Hcos H cos2

 

 

H eff H eff

 

*S*

*S*

(11)

(12)

*M*

 

*eff S*

For the case of multilayer nanowires, the effective anisotropy field *Heff* is given by

*eff S*

 

In case of a system which incorporates an array of nanowires dipole-dipole interaction, shape anisotropy, crystalline anisotropy and Zeeman energy interaction plays a complex role. Therefore the total energy will be the sum of all internal energies.

**Figure 2.** Coordinate system for an array of nanowires

Fig. 2 shows the schematic of an array of nanowires with relative orientation of the magnetization *M* and the applied magnetic field *H w.r.t* nanowire axis in spherical coordinate systems. The free energy density equation for an array of magnetic nanowires in the presence of external magnetic field at angles ( *<sup>H</sup>*) from nanowires axis can be written as

$$\mathbf{E} \approx -\mathbf{M} \mathbf{H} \left( \sin \theta \sin \theta\_H \cos \left( \phi - \phi\_H \right) + \cos \theta \cos \theta\_H \right) + K\_{e\sharp} \sin^2 \theta \tag{4}$$

where *Keff* is the effective uniaxial anisotropy which can be written as

$$\mathbf{K}\_{eff} = \pi \mathbf{M}^2 \left( \mathbf{I} - \Im \mathbf{P} \right) + \mathbf{K}\_{\mathrm{ul}} \tag{5}$$

The first term includes the dipole-dipole interaction between the nanowires and second term represents second-order uniaxial anisotropy along the wire axis. P is the porosity which can be obtained from

FMR Measurements of Magnetic Nanostructures 97

$$\mathbf{P} = \frac{\pi}{2\sqrt{3}} \frac{d^2}{r^2} \tag{6}$$

where d is the diameter of the pore and r is the centre-to-centre inter-wire distance between the pores. The equilibrium values for polar angles are obtained by minimizing the energy term

$$E\_{\theta} = \frac{\partial E}{\partial \theta} = 0 \tag{7}$$

$$\mathbf{E}\_{\theta} = -\mathbf{M} \mathbf{H} \left( \cos \theta \sin \theta\_{H} \cos \left( \phi - \phi\_{H} \right) - \sin \theta \cos \theta\_{H} \right) + K\_{\text{eff}} \sin 2\theta = 0 \tag{8}$$

From eq (8) we retrieve the dispersion relation which can be written as

$$\left(\frac{\omega}{\gamma}\right)^2 = \left[\text{H}\cos\left(\theta - \theta\_{\text{H}}\right) + \text{H}\_{\text{eff}}\cos^2\theta\right] \left[\text{H}\cos\left(\theta - \theta\_{\text{H}}\right) + \text{H}\_{\text{eff}}\cos2\theta\right] \tag{9}$$

Here, 2 *eff eff S* is the effective anisotropy field that comes from a combination of effects including shape, magnetocrystalline and magnetoelastic anisotropy. Therefore from eq (5)

$$\mathcal{H}\_{\rm eff} = 2\pi M\_S \left( 1 - 3P \right) + 2 \frac{K\_U}{M\_S} \tag{10}$$

For the case of multilayer nanowires, the effective anisotropy field *Heff* is given by

$$\mathcal{H}\_{\rm eff} = 2\pi M\_S \left( 1 - 3\{1 - f(1 - P)\} \right) + 2\frac{K\_{\underline{U}}}{M\_S} \tag{10a}$$

where, *<sup>m</sup> nm m <sup>h</sup> <sup>f</sup> h h* , *hm* and *hnm* represents magnetic and non-magnetic thicknesses in

multilayer section. If *hnm* = 0 (i.e. no non-magnetic spacer layer), the above equation represents the case of a single-element nanowire. Depending upon the direction of the external magnetic field along the easy axis of the nanowires, we can determine the various cases:

*Case 1*: H|| to the wire and Heff>0

96 Ferromagnetic Resonance – Theory and Applications

**3. FMR dispersion relation for nanostructures** 

structures, ferromagnetic resonance is an effective tool.

**Figure 2.** Coordinate system for an array of nanowires

the presence of external magnetic field at angles ( *<sup>H</sup>*

which can be obtained from

 

where *Keff* is the effective uniaxial anisotropy which can be written as

role. Therefore the total energy will be the sum of all internal energies.

Due to miniaturization from bulk to the nanoscale, material properties shows a drastic change like surface-to-volume ratio, electron transport, thermodynamic fluctuations, defects etc. The nanomagnetic material demonstrates distinct magnetic response due to various anisotropies. The one-dimensional nanostructures such as nanowires, nanotubes, nanorods and nanorings are the area of recent research for data storage applications, sensors, biomedical drugs, microwave devices. To perturb the dynamic magnetization in these

In case of a system which incorporates an array of nanowires dipole-dipole interaction, shape anisotropy, crystalline anisotropy and Zeeman energy interaction plays a complex

Fig. 2 shows the schematic of an array of nanowires with relative orientation of the magnetization *M* and the applied magnetic field *H w.r.t* nanowire axis in spherical coordinate systems. The free energy density equation for an array of magnetic nanowires in

<sup>2</sup> sin sin cos cos cos sin

 <sup>2</sup> 1 3 *eff <sup>U</sup>* 

The first term includes the dipole-dipole interaction between the nanowires and second term represents second-order uniaxial anisotropy along the wire axis. P is the porosity

   

) from nanowires axis can be written as

*<sup>H</sup> <sup>H</sup> H eff K* (4)

 

(5)

$$\frac{d\phi}{d\gamma} = H + H\_{\text{eff}} \tag{11}$$

*Case 2*: H|| to the wire and Heff<0

$$\frac{d\theta}{d\gamma} = H\_{\text{eff}} - H$$

*Case 3*: H easy axis and H<Heff

$$
\left(\frac{\alpha}{\mathcal{I}}\right)^2 = \left(H^2{}\_{eff} - H^2\right) \tag{13}
$$

FMR Measurements of Magnetic Nanostructures 99

**5. Growth of nanostructures** 

**Polycarbonate track etched (PCTE)** 

to 1012 per cm2) in case of PCTE. **Anodic aluminum oxide (AAO)** 

SEM micrograph of AAO templates.

anodic alumina (cross-section view).

Nanowires were grown by using two different types of templates (commercially available) polycarbonate track etched (PCTE) and aluminum oxide (AAO) from Whatman. These templates are completely different from each other due to their qualities as well as the preparation methods. The methods used for fabrication and properties of templates are

In PCTE template, the pore size varies from 20nm to 200nm and the thickness of ~6µm. To fabricate the PCTE template, initially high energy particle are used to bombard it to produce the path. Later these paths are etched in different chemical bath. The size of the pores is determined by the etching process. Fig.4(a) shows a SEM picture of a commercial PCTE membrane, with a reported pore size of 100 nm. Although the pores seem to have similar diameters, the pore placement is random. The pores density is quite low (in the range of 1010

In 1995, Masuda and Fukuda reported the method to fabricate highly ordered nanohole arrays on aluminum foil. Double anodization of aluminium in acidic solution is adopted to fabricate porous alumina template. The beauty of these templates is their cylindrical pores of uniform diameter, arranged in hexagonal arrays with a thin oxide layer exist at the bottom. Anodic Aluminum Oxide (AAO) templates have been successfully used as templates for the growth of nanowires by electrodeposition. In order to fabricate the AAO template the aluminum is cleaned in the acidic medium to remove the surface impurities. Further the cleaned aluminium processed by two step anodization described by Masuda. The pore size and spacing between pore in the alumina templates are controlled by the anodization voltage.

Temperature of electrolyte also plays a important role in pore parameters. The pores in AAO template cylindrical and highly dense which makes it a good candidate to study the interaction effect between the nanowires on magnetic properties. Fig. 4(b)-(c) shows the

**Figure 4.** SEM micrograph of membranes (a) Polycarbonate (b) Anodic alumina (Top view) and (c)

**Templates** 

given below;

*Case 4*: H easy axis, H>Heff>0

$$\left(\frac{\alpha}{\mathcal{Y}}\right)^2 = H(H - H\_{\text{eff}}) \tag{14}$$

The frequency-field characteristics can be studied from these relations for various cases of the direction of the applied field and corresponding angular variation with resonance field of the nanowires as shown in Fig. 3. The horizontal line shows the intersection of the dispersion relation and indicates where in an FMR spectrum the resonance lines would be found for a fixed frequency in Fig. 3(a).

**Figure 3.** Simulated plots (a) dispersion relation for Ni nanowires for various angles in which the external magnetic field is applied (b) resonance field as a function of the externally applied field angle *θH*.

### **4. Sample preparation and characterization**

During last decade magnetic nanowires have attracted enormous research attention in many areas of advanced nanotechnology, including patterned magnetic recording media, materials for optical and microwave applications. Template assisted growth of nanostructures under constant potential in several electrolytes has been carried out by researchers for over 30 years. In comparison to other deposition techniques (sputtering and MBE) electrodeposition is a low cost and simple technique to fabricate magnetic nanowires and multilayers. Arrays of Co nanowires were fabricated by template assisted electrochemical deposition into the nanometer-sized pores.

### **5. Growth of nanostructures**

### **Templates**

98 Ferromagnetic Resonance – Theory and Applications

found for a fixed frequency in Fig. 3(a).

*θH*.

*Case 4*: H easy axis, H>Heff>0

( ) *HH Heff*

The frequency-field characteristics can be studied from these relations for various cases of the direction of the applied field and corresponding angular variation with resonance field of the nanowires as shown in Fig. 3. The horizontal line shows the intersection of the dispersion relation and indicates where in an FMR spectrum the resonance lines would be

a b

**4. Sample preparation and characterization** 

electrochemical deposition into the nanometer-sized pores.

**Figure 3.** Simulated plots (a) dispersion relation for Ni nanowires for various angles in which the external magnetic field is applied (b) resonance field as a function of the externally applied field angle

During last decade magnetic nanowires have attracted enormous research attention in many areas of advanced nanotechnology, including patterned magnetic recording media, materials for optical and microwave applications. Template assisted growth of nanostructures under constant potential in several electrolytes has been carried out by researchers for over 30 years. In comparison to other deposition techniques (sputtering and MBE) electrodeposition is a low cost and simple technique to fabricate magnetic nanowires and multilayers. Arrays of Co nanowires were fabricated by template assisted

(13)

(14)

2 2 *H H eff*

2

2

Nanowires were grown by using two different types of templates (commercially available) polycarbonate track etched (PCTE) and aluminum oxide (AAO) from Whatman. These templates are completely different from each other due to their qualities as well as the preparation methods. The methods used for fabrication and properties of templates are given below;

### **Polycarbonate track etched (PCTE)**

In PCTE template, the pore size varies from 20nm to 200nm and the thickness of ~6µm. To fabricate the PCTE template, initially high energy particle are used to bombard it to produce the path. Later these paths are etched in different chemical bath. The size of the pores is determined by the etching process. Fig.4(a) shows a SEM picture of a commercial PCTE membrane, with a reported pore size of 100 nm. Although the pores seem to have similar diameters, the pore placement is random. The pores density is quite low (in the range of 1010 to 1012 per cm2) in case of PCTE.

### **Anodic aluminum oxide (AAO)**

In 1995, Masuda and Fukuda reported the method to fabricate highly ordered nanohole arrays on aluminum foil. Double anodization of aluminium in acidic solution is adopted to fabricate porous alumina template. The beauty of these templates is their cylindrical pores of uniform diameter, arranged in hexagonal arrays with a thin oxide layer exist at the bottom. Anodic Aluminum Oxide (AAO) templates have been successfully used as templates for the growth of nanowires by electrodeposition. In order to fabricate the AAO template the aluminum is cleaned in the acidic medium to remove the surface impurities. Further the cleaned aluminium processed by two step anodization described by Masuda. The pore size and spacing between pore in the alumina templates are controlled by the anodization voltage.

Temperature of electrolyte also plays a important role in pore parameters. The pores in AAO template cylindrical and highly dense which makes it a good candidate to study the interaction effect between the nanowires on magnetic properties. Fig. 4(b)-(c) shows the SEM micrograph of AAO templates.

**Figure 4.** SEM micrograph of membranes (a) Polycarbonate (b) Anodic alumina (Top view) and (c) anodic alumina (cross-section view).

### **Template assisted electrodeposition technique**

Electrodeposition process involves the electric current to reduce cations from electrolyte and deposited that material as a thin film onto a conducting substrate. At the cathode the metal reduction takes place and metal deposits according to:

FMR Measurements of Magnetic Nanostructures 101

**Steps involved in the template assisted synthesis of nanowires** 

**Step 4:** The growth of nanowires is carried at the optimized potential.

electrode in a three-electrode electrochemical cell.

pores.

condition for deposition.

electrodeposition process.

oxide and dichloromethane respectively.

**Step 1:** The porous anodic alumina (AAO) /polycarbonate templates were taken and one side of the AAO were sputtered with Au by RF sputtering, which acted as the working

**Step 2:** The electrodeposition solution was restrained to the other side of the membrane so that deposition was initiated onto the Au layer within the pores. The array of Co nanowires was deposited from a solution of 25 gm/L CoSO4.7H2O, 5 gm/L H3BO3 and sodium Lauryl Sulphate (SLS), which is used to reduce the surface tension of water for proper wetting of

**Step 3:** The cyclic voltammetry is used to figure out the constant deposition potential of the working electrode with respect to a standard reference electrode **RE** to get the favourable

**Step 5:** For further investigation of freely standing nanowires, the templates is used to dissolve in an appropriate solution. Etching solution for AAO and PCTE are sodium hydro-

**Figure 6.** Schematic illustration of the growth of magnetic nanowire in alumina template by

$$\mathbf{M} \mathbf{n}^+ + \mathbf{n} \text{ e }^- = \mathbf{M}(\mathbf{s})$$

In order to form the nanowires, the cations from electrolyte move through the nonconducting template (AAO/ PCTE) having nanosized pores and deposited on the conducting substrate.

The desired material properties depends upon the various process parameters like electrolyte composition, bath pH, mode of deposition (DC, pulse and AC) and deposition temperature.

**Figure 5.** Schematic representation of three-electrode electrochemical cell setup employed. AAO template mounted electrodes act as a working electrodes (WE), platinum foil counter electrode (CE) and Saturated calomel electrode (SCE), reference electrode (RE).

Fig.5 illustrates the three-electrode cell set-up used in this study. A platinum foil and a saturated calomel electrode (SCE) were used as the anode (or counter electrode) and as a reference electrode respectively. The steps of preparation of nanostructure are as followed;

### **Steps involved in the template assisted synthesis of nanowires**

100 Ferromagnetic Resonance – Theory and Applications

conducting substrate.

temperature.

**Template assisted electrodeposition technique** 

reduction takes place and metal deposits according to:

Electrodeposition process involves the electric current to reduce cations from electrolyte and deposited that material as a thin film onto a conducting substrate. At the cathode the metal

**Mn n e M s** 

In order to form the nanowires, the cations from electrolyte move through the nonconducting template (AAO/ PCTE) having nanosized pores and deposited on the

The desired material properties depends upon the various process parameters like electrolyte composition, bath pH, mode of deposition (DC, pulse and AC) and deposition

**Figure 5.** Schematic representation of three-electrode electrochemical cell setup employed. AAO template mounted electrodes act as a working electrodes (WE), platinum foil counter electrode (CE) and

Fig.5 illustrates the three-electrode cell set-up used in this study. A platinum foil and a saturated calomel electrode (SCE) were used as the anode (or counter electrode) and as a reference electrode respectively. The steps of preparation of nanostructure are as followed;

Saturated calomel electrode (SCE), reference electrode (RE).

**Step 1:** The porous anodic alumina (AAO) /polycarbonate templates were taken and one side of the AAO were sputtered with Au by RF sputtering, which acted as the working electrode in a three-electrode electrochemical cell.

**Step 2:** The electrodeposition solution was restrained to the other side of the membrane so that deposition was initiated onto the Au layer within the pores. The array of Co nanowires was deposited from a solution of 25 gm/L CoSO4.7H2O, 5 gm/L H3BO3 and sodium Lauryl Sulphate (SLS), which is used to reduce the surface tension of water for proper wetting of pores.

**Step 3:** The cyclic voltammetry is used to figure out the constant deposition potential of the working electrode with respect to a standard reference electrode **RE** to get the favourable condition for deposition.

**Step 4:** The growth of nanowires is carried at the optimized potential.

**Step 5:** For further investigation of freely standing nanowires, the templates is used to dissolve in an appropriate solution. Etching solution for AAO and PCTE are sodium hydrooxide and dichloromethane respectively.

**Figure 6.** Schematic illustration of the growth of magnetic nanowire in alumina template by electrodeposition process.

FMR Measurements of Magnetic Nanostructures 103

**Figure 9.** Scanning electron micrograph (SEM) of electrodeposited nanostructure in (a) Polycarbonate

Transmission Electron Microscope (TEM) is a widely used instrument for characterizing the interior structure of materials. For TEM, the template was completely etched and rinsed several times with deionised water to clean the residual template part. Template etching is an important step of sample preparation of nanowires for TEM characterization. Cobalt nanowires were scratched from the substrate and ultrasonicated in acetone for 15 min so that the nanowires could disperse properly. Few drops of the suspension were then transferred on to a carbon coated copper grid and the microstructures were analyzed by high resolution TEM (HRTEM: Technai G20 S-Twin model) operating at 200 kV. To get the actual diameter of the nanowire, the complete dissolving of residual template from surrounding the nanowires is very important. Typical TEM results obtained are shown in Figure 10. While nanowires grown in PCTE templates typically have a tapered cross-section due to the non-uniform diameters of the pores, the AAO template-based nanowires grow in

**Figure 10.** Transmission electron micrograph (TEM) of electrodeposited nanostructure in (a)

and (b) & (c) Anodic alumina.

a more uniform manner.

Polycarbonate and (b) Anodic alumina.

**Transmission electron microscopy** 

**Figure 7.** Typical choronoamperometery plot during potentiostatic electrodeposition taken during the fabrication of Co nanowires. The various stage of pore filling during deposition is shown as insets at the respective current-time positions.

**Figure 8.** Scanning electron micrograph (SEM) of empty and filled surface morphology of the template.

### **6. Characterization of nanostructures**

#### **Scanning electron microscopy**

For scanning electron microscopy (SEM), the template containing nanowires is partially released from their template by appropriate solution. To remove the residual part of template the etched sample is cleaned by deionised water. To carry out the surface analysis secondary electron imaging in scanning electron microscope has been utilized. Surface morphology of Co nanowire was investigated by scanning electron microscope (SEM: ZEISS EVO 50) operating at 20 kV accelerating voltage by secondary electron imaging. Fig. 8 shows the SEM micrograph of empty AAO and filled with nanowires. The morphology of grown nanowires in PCTE template can be clearly seen in Fig. 9(a). Side and top view of AAO assisted nanowires are presented in Fig. 9(b & c). The growth density of nanowires in AAO template is more as compared to PCTE template.

**Figure 9.** Scanning electron micrograph (SEM) of electrodeposited nanostructure in (a) Polycarbonate and (b) & (c) Anodic alumina.

#### **Transmission electron microscopy**

102 Ferromagnetic Resonance – Theory and Applications

respective current-time positions.

**6. Characterization of nanostructures** 

template is more as compared to PCTE template.

**Scanning electron microscopy** 

**Figure 7.** Typical choronoamperometery plot during potentiostatic electrodeposition taken during the fabrication of Co nanowires. The various stage of pore filling during deposition is shown as insets at the

**Figure 8.** Scanning electron micrograph (SEM) of empty and filled surface morphology of the template.

For scanning electron microscopy (SEM), the template containing nanowires is partially released from their template by appropriate solution. To remove the residual part of template the etched sample is cleaned by deionised water. To carry out the surface analysis secondary electron imaging in scanning electron microscope has been utilized. Surface morphology of Co nanowire was investigated by scanning electron microscope (SEM: ZEISS EVO 50) operating at 20 kV accelerating voltage by secondary electron imaging. Fig. 8 shows the SEM micrograph of empty AAO and filled with nanowires. The morphology of grown nanowires in PCTE template can be clearly seen in Fig. 9(a). Side and top view of AAO assisted nanowires are presented in Fig. 9(b & c). The growth density of nanowires in AAO Transmission Electron Microscope (TEM) is a widely used instrument for characterizing the interior structure of materials. For TEM, the template was completely etched and rinsed several times with deionised water to clean the residual template part. Template etching is an important step of sample preparation of nanowires for TEM characterization. Cobalt nanowires were scratched from the substrate and ultrasonicated in acetone for 15 min so that the nanowires could disperse properly. Few drops of the suspension were then transferred on to a carbon coated copper grid and the microstructures were analyzed by high resolution TEM (HRTEM: Technai G20 S-Twin model) operating at 200 kV. To get the actual diameter of the nanowire, the complete dissolving of residual template from surrounding the nanowires is very important. Typical TEM results obtained are shown in Figure 10. While nanowires grown in PCTE templates typically have a tapered cross-section due to the non-uniform diameters of the pores, the AAO template-based nanowires grow in a more uniform manner.

**Figure 10.** Transmission electron micrograph (TEM) of electrodeposited nanostructure in (a) Polycarbonate and (b) Anodic alumina.

### **7. Electron paramagnetic resonance**

EPR/FMR measurements obtained from ferromagnetic nanowire arrays give detailed information on the size of the nanowires. The spectra can be used to calculate the interwire magnetic interactions quite accurately [31]. In Fig. 11 are shown typical spectra obtained from Co nanowire arrays. Here, both single-Co nanowire arrays and multilayered Co/Pd nanowire arrays are compared. There is clearly an angular dependence of the applied microwave field with respect to the easy axis of the nanowires, as has been studied previously [32].

FMR Measurements of Magnetic Nanostructures 105

**Figure 12.** Electron Spin resonance (ESR/FMR) spectra of multilayered Co/Pd nanowire arrays in parallel orientation with temperature range (90K-290K). (Inset) Peak height change with temperature.

The increasing demand for higher frequency magnetic microwave structures triggered a tremendous development in the field of magnetization dynamics over the past decade. In order to develop smaller and faster devices, a better understanding of the complex magnetization precessional dynamics, the magnetization anisotropy, and the sources of spin scattering at the nanoscale is necessary [23-25]. Magnetic data storage with its promise of non-volatility, robustness, high speed and low energy dissipation attracted long back. It has been encountered in a number of applications such as smart cards, hard drives, thin film read heads or video tapes [26-28]. Industry activities are also directed to replace semiconductor random access memories by magnet based memory devices. Logical 0 and 1 are encoded by the direction of the magnetization of a small magnetic element, such as a giant magnetoresistance (GMR) or tunneling magnetoresistance (TMR) element. For data manipulation *i.e.* reading and writing, the magnetization has to be switched between the two equilibrium positions. In order to push forward data transfer rates which are already in the GHz range, spin waves transportation mechanism has to be employed in place of current semiconductor based devices [29-32]. To this end an understanding of the dynamic

On the road towards the size reduction of microwave devices, ferromagnetic nanowires embedded into porous templates have proven to be an interesting route to ferrite based materials. The main advantages of what we call ferromagnetic nanowired (FMNW) substrates are that they present a zero-field microwave absorption frequency that can be easily tuned over a large range of frequencies, as well as being low cost and fast to produce

**8. Microwave devices** 

motion and the mode spectrum is necessary.

**Figure 11.** Electron paramagnetic resonance (EPR/FMR) spectra of single element Co nanowire arrays in parallel and perpendicular orientation at room temperature.

Fig.12 shows the FMR spectra of Co/Pd MLNW arrays for the temperature range from 90K to 290K. It is clear from the data that two different peaks in the FMR spectrum have been observed. The peak present at the ~ 3310 G is corresponding to the main peak of FMR and it is due to the quantum confinement in the nanostructures which is more in case of Co/Pd multilayered nanowires. The spin waves are confined in the magnetic nanostructure and it is found to be dominant when we reduce the third dimension (in z-direction) of the nanostructure. This main FMR peak present at all the temperature and the variation of resonance field at different temperature is very slow. As we go down to the 90K with decrease of 20K from room temperature the resonance field increases slowly. This slow variation can be explain as temperature decreases the Ms (T) increase and resultant head-totail alignment between FM segments increases, which further reduces the effective field as a result slow increase in the resonance field. Inset in Fig.12 shows the variation of resonance field with temperature.

**Figure 12.** Electron Spin resonance (ESR/FMR) spectra of multilayered Co/Pd nanowire arrays in parallel orientation with temperature range (90K-290K). (Inset) Peak height change with temperature.

### **8. Microwave devices**

104 Ferromagnetic Resonance – Theory and Applications

previously [32].

**7. Electron paramagnetic resonance** 

EPR/FMR measurements obtained from ferromagnetic nanowire arrays give detailed information on the size of the nanowires. The spectra can be used to calculate the interwire magnetic interactions quite accurately [31]. In Fig. 11 are shown typical spectra obtained from Co nanowire arrays. Here, both single-Co nanowire arrays and multilayered Co/Pd nanowire arrays are compared. There is clearly an angular dependence of the applied microwave field with respect to the easy axis of the nanowires, as has been studied

**Figure 11.** Electron paramagnetic resonance (EPR/FMR) spectra of single element Co nanowire arrays

Fig.12 shows the FMR spectra of Co/Pd MLNW arrays for the temperature range from 90K to 290K. It is clear from the data that two different peaks in the FMR spectrum have been observed. The peak present at the ~ 3310 G is corresponding to the main peak of FMR and it is due to the quantum confinement in the nanostructures which is more in case of Co/Pd multilayered nanowires. The spin waves are confined in the magnetic nanostructure and it is found to be dominant when we reduce the third dimension (in z-direction) of the nanostructure. This main FMR peak present at all the temperature and the variation of resonance field at different temperature is very slow. As we go down to the 90K with decrease of 20K from room temperature the resonance field increases slowly. This slow variation can be explain as temperature decreases the Ms (T) increase and resultant head-totail alignment between FM segments increases, which further reduces the effective field as a result slow increase in the resonance field. Inset in Fig.12 shows the variation of resonance

in parallel and perpendicular orientation at room temperature.

field with temperature.

The increasing demand for higher frequency magnetic microwave structures triggered a tremendous development in the field of magnetization dynamics over the past decade. In order to develop smaller and faster devices, a better understanding of the complex magnetization precessional dynamics, the magnetization anisotropy, and the sources of spin scattering at the nanoscale is necessary [23-25]. Magnetic data storage with its promise of non-volatility, robustness, high speed and low energy dissipation attracted long back. It has been encountered in a number of applications such as smart cards, hard drives, thin film read heads or video tapes [26-28]. Industry activities are also directed to replace semiconductor random access memories by magnet based memory devices. Logical 0 and 1 are encoded by the direction of the magnetization of a small magnetic element, such as a giant magnetoresistance (GMR) or tunneling magnetoresistance (TMR) element. For data manipulation *i.e.* reading and writing, the magnetization has to be switched between the two equilibrium positions. In order to push forward data transfer rates which are already in the GHz range, spin waves transportation mechanism has to be employed in place of current semiconductor based devices [29-32]. To this end an understanding of the dynamic motion and the mode spectrum is necessary.

On the road towards the size reduction of microwave devices, ferromagnetic nanowires embedded into porous templates have proven to be an interesting route to ferrite based materials. The main advantages of what we call ferromagnetic nanowired (FMNW) substrates are that they present a zero-field microwave absorption frequency that can be easily tuned over a large range of frequencies, as well as being low cost and fast to produce over a large area as compared to standard ferrite devices. Conventional ferrite circulators need to be biased by a magnetic field to operate. This biasing field is generally provided by permanent magnet and in view of the volume reduction, we need to think of FMNW substrates which work at zero fields. A rigorous theory of microwave devices is very cumbersome due to ferromagnetic resonance (FMR) and spin-wave phenomena.

FMR Measurements of Magnetic Nanostructures 107

The transmission response of Ni NWs placed on a co-planar waveguide transmission line (Fig. 14(a)) for different static magnetic field is shown in Fig. 14(b). A small but perceptible and repeatable change was observed when the applied magnetic field was turned on/off. The change is however too small to show up in the plot of Fig. 14(b). This is partly due to masking of the shifts due to reflections from soldering and other undesirable metal deposits near the patterned transmission line and partly due to the relatively small interaction between the RF fields (largely confined to the substrate holding the CPW line) and the nanowires. It is thus expected that clearer shifts can be seen with more careful preparation and patterning of the substrate and ultimately by printing the CPW line on the nanowire

**Figure 14.** (a) CPW geometery for magnetic field applied along perpendicular direction (b)

(a) (b)

**10. Substrate integrated magnetic microwave devices** 

transmission response as a function of frequency of Ni nanowires for various applied magnetic field

Newly developed techniques enable to characterize and re-arrange matter at nanometer scale. Now a day, automotive and wireless communication requires reduction in dimensions of nanodevices for yielding higher cut-off frequencies. Here, we propose integrated

**Figure 15.** schematic of band stop filter (b) S-parameter of the DUT for three different samples

template itself.

VNA-FMR is an important technique for the investigation of magnetization dynamics of lowdimensional magnetic structures and patterned microwave devices. Also, broadband flip-chip technique can be used to measure the material intrinsic and extrinsic properties by applying external magnetic field. In this chapter, we will demonstrate both techniques to have better understanding of ferromagnetic resonance. We propose fabrication and measurements of various non reciprocal microwave devices like band-stop filter, isolator and electromagnetic band-gap (EBG) structures using FMNW substrates using Ni and Co nanowires.

## **9. Conventional Ferromagnetic Resonance – Flip-chip based technique (FMR)**

In the frequency domain measurements, the magnetic excitation is sinusoidal magnetic field *hrf* and the response of the sample is detected by vector network analyzer. The magnetic field can be applied along parallel and perpendicular direction of the nanowires satisfying the FMR condition. The main components of the experimental setup are shown in Fig. 13. The VNA is connected to a coplanar waveguide (CPW) having a characteristic impedance of 50 Ω using coaxial cables and microwave connectors. For such radio frequency connections often coaxial cables with Teflon insulation and SMA connectors are employed. They are comparably low priced and offer a bandwidth of typically 18 GHz. The used cables should not have a metallic reinforcement.

Flip-chip based measurements are done to extract the material parameters, which are technologically important for data storage applications [31]. Fig. 13 shows the schematic of the set-up used for flip-chip measurements. Magnetic field is applied parallel to the nanowires and perpendicular to RF magnetic field, so that resonance condition must be satisfied.

**Figure 13.** Schematic representation of VNA-FMR system

The transmission response of Ni NWs placed on a co-planar waveguide transmission line (Fig. 14(a)) for different static magnetic field is shown in Fig. 14(b). A small but perceptible and repeatable change was observed when the applied magnetic field was turned on/off. The change is however too small to show up in the plot of Fig. 14(b). This is partly due to masking of the shifts due to reflections from soldering and other undesirable metal deposits near the patterned transmission line and partly due to the relatively small interaction between the RF fields (largely confined to the substrate holding the CPW line) and the nanowires. It is thus expected that clearer shifts can be seen with more careful preparation and patterning of the substrate and ultimately by printing the CPW line on the nanowire template itself.

**Figure 14.** (a) CPW geometery for magnetic field applied along perpendicular direction (b) transmission response as a function of frequency of Ni nanowires for various applied magnetic field

### **10. Substrate integrated magnetic microwave devices**

106 Ferromagnetic Resonance – Theory and Applications

**(FMR)** 

not have a metallic reinforcement.

**Figure 13.** Schematic representation of VNA-FMR system

over a large area as compared to standard ferrite devices. Conventional ferrite circulators need to be biased by a magnetic field to operate. This biasing field is generally provided by permanent magnet and in view of the volume reduction, we need to think of FMNW substrates which work at zero fields. A rigorous theory of microwave devices is very

VNA-FMR is an important technique for the investigation of magnetization dynamics of lowdimensional magnetic structures and patterned microwave devices. Also, broadband flip-chip technique can be used to measure the material intrinsic and extrinsic properties by applying external magnetic field. In this chapter, we will demonstrate both techniques to have better understanding of ferromagnetic resonance. We propose fabrication and measurements of various non reciprocal microwave devices like band-stop filter, isolator and electromagnetic

cumbersome due to ferromagnetic resonance (FMR) and spin-wave phenomena.

band-gap (EBG) structures using FMNW substrates using Ni and Co nanowires.

**9. Conventional Ferromagnetic Resonance – Flip-chip based technique** 

In the frequency domain measurements, the magnetic excitation is sinusoidal magnetic field *hrf* and the response of the sample is detected by vector network analyzer. The magnetic field can be applied along parallel and perpendicular direction of the nanowires satisfying the FMR condition. The main components of the experimental setup are shown in Fig. 13. The VNA is connected to a coplanar waveguide (CPW) having a characteristic impedance of 50 Ω using coaxial cables and microwave connectors. For such radio frequency connections often coaxial cables with Teflon insulation and SMA connectors are employed. They are comparably low priced and offer a bandwidth of typically 18 GHz. The used cables should

Flip-chip based measurements are done to extract the material parameters, which are technologically important for data storage applications [31]. Fig. 13 shows the schematic of the set-up used for flip-chip measurements. Magnetic field is applied parallel to the nanowires

and perpendicular to RF magnetic field, so that resonance condition must be satisfied.

Newly developed techniques enable to characterize and re-arrange matter at nanometer scale. Now a day, automotive and wireless communication requires reduction in dimensions of nanodevices for yielding higher cut-off frequencies. Here, we propose integrated

**Figure 15.** schematic of band stop filter (b) S-parameter of the DUT for three different samples

magnetic band-stop filter fabricated on FMNWS and studying their microwave properties like permittivity and permeability using CPW. Fig. 15(a) shows the schematic of the device under test which is a coplanar waveguide on NW substrate. The transmission coefficient of the device for three samples having bare AAO, Ni and Co NWs are shown in Fig. 15(b) at zero biasing. It is observed that by applying the magnetic field the resonance frequency shift towards higher range, so that we can tune our operating frequency of the device. We also observe that material properties also influence the Device-Under-Test (DUT).

FMR Measurements of Magnetic Nanostructures 109

[2] S. Shamaila, R. Sharif, S. Riaz, M. Khaleeq-ur-Rahman and X. F. Han, "Fabrication and magnetic characterization of CoxPt1-x nanowire arrays", Appl. Phys. A, 92 (2008) 687-

[3] Z. Z. Sun and J. Schliemann, "Fast domain wall propagation under an optimal field

[4] M. Yan, A. Kakay, S. Gliga and R. Hertel, "Beating the walker limit with massless

[5] C. T. Boone, J. A. Katine, M. Carey, J. R. Childress, X. Cheng and I. N. Krivorotov, "Rapid domain wall motion in permalloy nanowires excited by a spin-polarized current

[6] B. K. Kuanr, V. Veerakumar, R. Marson, S. Mishra, R. E. Camley and Z. Celinski, "Nonreciprocal microwave devices based on magnetic nanowires", Appl. Phys. Lett. 94

[7] M. Darques, J. De La T. Medina, L. Piraux, L. Cagnon and I. Huynen, "Microwave circulator based on ferromagnetic nanowires in alumina template", Nanotechnology 21

[8] J. De La T. Medina, J. Spiegel, M. Darques, L. Piraux and I. Huynen, "Differential phase shift in nonreciprocal microstrip lines on magnetic nanowired substrates", Appl. Phys.

[9] C. Kittel, "On the theory of ferromagnetic resonance absorption", Phys. Rev. 73 (1948)

[10] C. Kittel, "Excitation of spin waves in a ferromagnet by a uniform rf field", Phys. Rev.

[11] L. Kraus, G. Infante, Z. Frait and M. Vazquez, "Ferromagnetic resonance in microwires

[12] J. De La T. Medina, L. Piraux, J. M. Olais Govea and A. Encinas, "Double ferromagnetic resonanace and configuration-dependent dipolar coupling in unsaturated arrays of

[13] A. Encinas-Oropesa, M. Demand, L. Piraux, I. Huynen and U. Ebels, "Dipolar interactions in arrays of nickel nanowires studied by ferromagnetic resonance", Phys.

[14] C. A. Ramos, M. Vazquez, K. Nielsch, K. Pirota, R. B. Rivas, R. B. Wherspohn, M. Tovar, R. D. Sanchez and U. Gosele, "FMR characterization of hexagonal arrays of Ni

[15] C. A. Ramos, E. Vassallo Brigneti and M. Vazquez, "Self-organized nanowires: evidence of dipolar interactions from ferromagnetic resonance measurements", Physica B 354

[16] E. Beaurepaire, J. C. Merle, A. Daunois and J. Y. Bigot, "Ultrafast spin dynamics in

[17] M. R. Freeman, and W. K. Hiebert, "Stroboscopic microscopy of magnetic dynamics", In: B. Hillebrands, K. Ounadjela (Eds.), Spin Dynamics in Confined Magnetic Structures

domain walls in cylindrical nanowires", Phys. Rev. Lett. 104 (2010) 057201.

applied perpendicular to the nanowire", Phys. Rev. Lett. 104 (2010) 097203.

pulse in magnetic nanowires", Phys. Rev. Lett. 104 (2010) 037206.

691.

(2009) 202505.

(2010) 145208.

155.

Lett. 96 (2010) 072508.

110 (1958) 1295-1297.

Rev. B 63 (2001) 104415.

I. Springer, Berlin, pp. 93-126.

(2004) 195-197.

and nanowires", Phys. Rev. B 83 (2011) 174438.

bistable magnetic nanowires", Phys. Rev. B 81 (2010) 144411.

nanowires", J. Magn. Magn. Mater. 272-276 (2004) 1652-1653.

ferromagnetic nickel", Phys. Rev. Lett. 76 (1996) 4250-4253.

### **11. Non-reciprocal devices**

It is important to note that it is possible to obtain non-reciprocal structures also using nanowire-based devices. Essentially, the microwave properties like permittivity and permeability can be made to be asymmetric. This is useful to obtain non reciprocal devices such as isolators and circulators. Prototypes of such devices are presently under fabrication. Some of the proposed devices are shown in Fig. 16 below. The detailed studies of these devices are out of scope of this chapter.

**Figure 16.** Possible non-reciprocal structures using nanowires.

### **Author details**

Manish Sharma, Sachin Pathak and Monika Sharma *Indian Institute of Technology Delhi, India* 

### **12. References**

[1] A. Fert and L. Piraux, "Magnetic nanowires", J. Magn. Magn. Mater. 200 (1999) 338-358.

[2] S. Shamaila, R. Sharif, S. Riaz, M. Khaleeq-ur-Rahman and X. F. Han, "Fabrication and magnetic characterization of CoxPt1-x nanowire arrays", Appl. Phys. A, 92 (2008) 687- 691.

108 Ferromagnetic Resonance – Theory and Applications

**11. Non-reciprocal devices** 

devices are out of scope of this chapter.

**Figure 16.** Possible non-reciprocal structures using nanowires.

Manish Sharma, Sachin Pathak and Monika Sharma

*Indian Institute of Technology Delhi, India* 

**Author details** 

**12. References** 

magnetic band-stop filter fabricated on FMNWS and studying their microwave properties like permittivity and permeability using CPW. Fig. 15(a) shows the schematic of the device under test which is a coplanar waveguide on NW substrate. The transmission coefficient of the device for three samples having bare AAO, Ni and Co NWs are shown in Fig. 15(b) at zero biasing. It is observed that by applying the magnetic field the resonance frequency shift towards higher range, so that we can tune our operating frequency of the device. We also

It is important to note that it is possible to obtain non-reciprocal structures also using nanowire-based devices. Essentially, the microwave properties like permittivity and permeability can be made to be asymmetric. This is useful to obtain non reciprocal devices such as isolators and circulators. Prototypes of such devices are presently under fabrication. Some of the proposed devices are shown in Fig. 16 below. The detailed studies of these

[1] A. Fert and L. Piraux, "Magnetic nanowires", J. Magn. Magn. Mater. 200 (1999) 338-358.

observe that material properties also influence the Device-Under-Test (DUT).

	- [18] W. K. Hiebert, A. Stankiewicz, and M. R. Freeman, "Direct observation of magnetic relaxation in a small permalloy disk by time-resolved scanning Kerr microscopy", Phys. Rev. Lett. 79 (1997) 1134-1137.

**Chapter 5** 

© 2013 Singh and Saipriya, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**FMR Studies of [SnO2/Cu-Zn Ferrite] Multilayers** 

The artificially structured multilayers (ML) have opened a new field of interface magnetism. It is possible to fabricate a multilayer sample with a specific design according to the requirement consisting of magnetic, non-magnetic, metallic or non-metallic components. Various types of interfaces may be synthesized in ML by combining a magnetic component with a non-magnetic one or from two different magnetic components. The interface between two non-magnetic components is particularly interesting if a magnetic anomaly happens to

The oscillations of magnetic parameters as a function of the number of interfaces, spacer layer or magnetic layer thickness in the ML have been gaining attention in the recent years. Bruno et al [1] employed RKKY model and Edwards et al [2] used spin dependent confinement of electrons in the quantum well provided by the spacer layer (quantum interference) to explain the oscillations initially observed in metallic ML. Exchange coupling in ML is also affected by direct dipolar coupling like the correlation of spins at rough interfaces (orange peel coupling), which is magnetostatic in origin and occurs due to the interaction of the dipoles which appear due to the roughness of the material [3]. This coupling favors parallel or anti-parallel alignment of spins depending upon the interplay

The oscillations in exchange coupling are also observed in oxide ML apart from the metallic ML [4]. Oxide materials are more stable and their ML can form magnetic structures which do not exist in bulk form. There are reports on ML of ferrites used along with NiO [5,6],

The mixed spinel structure of Cu-Zn ferrites with composition Cu0.6Zn0.4Fe3O4 (CZF) gives high magnetization when deposited in Ar environment [13]. The properties of non-magnetic (NM) SnO2 are sensitive to oxygen vacancies [14]. The interfacial region between these two materials may give rise to interesting magnetic phenomena. The objective of the present

and reproduction in any medium, provided the original work is properly cited.

R. Singh and S. Saipriya

http://dx.doi.org/10.5772/56330

**1. Introduction** 

Additional information is available at the end of the chapter

be induced at the interface atom layer due to some interface effect.

between the magnetostatic exchange and anisotropy energy.

MgO [7] , CoO [8] and SnO2 [ 9-12].


## **FMR Studies of [SnO2/Cu-Zn Ferrite] Multilayers**

R. Singh and S. Saipriya

110 Ferromagnetic Resonance – Theory and Applications

Rev. Lett. 79 (1997) 1134-1137.

7862.

6495.

(2003) 1157.

218-221.

Canada 65 (2009).

magnetic composite materials".

AIP Conf. Proc. 1347 (2011).

Applied Physics 103, 093915 (2008).

[18] W. K. Hiebert, A. Stankiewicz, and M. R. Freeman, "Direct observation of magnetic relaxation in a small permalloy disk by time-resolved scanning Kerr microscopy", Phys.

[19] T. J. Silva, C. S. Lee, T. M. Crawford, and C. T. Rogers, "Inductive measurement of ultrafast magnetization dynamics in thin-film permalloy", J. Appl. Phys. 85 (1999) 7849-

[20] M. M. Midzor, P. E. Wigen, D. Pelekhov, W. Chen, P. C. Hammel, and M. L. Roukes, "Imaging mechanisms of force detected FMR microscopy", J. App. Phys. 87 (2000) 6493-

[21] O. Mosendz, B. Kardasz, D. S. Schmool, and B. Heinrich, "Spin dynamics at low microwave frequencies in crystalline Fe ultrathin film double layers using co-planar

[22] N. Mecking, Y. S. Gui, and C. M. Hu, "Microwave photovoltage and photoresistance

[23] B. K. Kuanr, M. Buchmeier, D. E. Buergler, P. Gruenberg, R. E. Camley, and Z, Celinski, "Dynamic and static measurements on epitaxial Fe-Si-Fe", J. Vac. Sci. Technol. A 24

[24] J. Curiale, R. D. Sanchez, C. A. Ramos, A. G. Leyva, A. Butera, "Dynamic response of magnetic nanoparticles arranged in a tubular shape", J. Magn. Magn. Mater. 320 (2008)

[25] Can-Ming Hu, "Recent progress in spin dynamics research in Canada", La Physique Au

[26] X. Kou, X. Fan, R. K. Dumas, Q. Lu, Y. Zhang, H. Zhu, X. Zhang, K. Liu and J. Q. Xiao, "Magnetic effects in magnetic nanowire arrays", Adv. Mater. 23 (2011) 1393-1397. [27] T. Schrefl, J. Fidler, D. Suss, and W. Scholz, "Hysteresis and switching dynamics of

[28] X. Kou, X. Fan, H. Zhu and J. Q. Xiao, "Ferromagnetic resonance and memory effects in

[29] K. Nagai, Y. Cao, T. Tanaka and K. Matsuyama, "Binary data coding with domain wall for spin wave based logic devices", J. Appl. Phys. 111 (2012) 07D1301-07D1304. [30] M. Sharma, S. Pathak, S. Singh, M. Sharma and A. Basu, "Highly ordered magnetic nickel nanowires: structural properties and ferromagnetic resonance measurements",

[31] O. Yalçn, et al, Ferromagnetic resonance studies of Co nanowire arrays, Journal of

[32] G. Kartopu, O. Yalçn, et al. Size effects and origin of easy-axis in nickel nanowire

[33] G. Kartopu, O. Yalçn, M. Es-Souni, and A. C. Başaran, Magnetization behavior of ordered and high density Co nanowire arrays with varying aspect ratio, Journal of

transmission lines", J. Magn. Magn. Mater. 300 (2006) 174-178.

patterned magnetic elements", Phys. B 275 (2000) 55-58.

Magnetism and Magnetic Materials 272–276 (2004) 1684–1685.

arrays, Journal of Applied Physics 109, 033909 (2011).

effects in ferromagnetic microstrips", Phys. Rev. B 76 (2007) 224430.

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/56330

### **1. Introduction**

The artificially structured multilayers (ML) have opened a new field of interface magnetism. It is possible to fabricate a multilayer sample with a specific design according to the requirement consisting of magnetic, non-magnetic, metallic or non-metallic components. Various types of interfaces may be synthesized in ML by combining a magnetic component with a non-magnetic one or from two different magnetic components. The interface between two non-magnetic components is particularly interesting if a magnetic anomaly happens to be induced at the interface atom layer due to some interface effect.

The oscillations of magnetic parameters as a function of the number of interfaces, spacer layer or magnetic layer thickness in the ML have been gaining attention in the recent years. Bruno et al [1] employed RKKY model and Edwards et al [2] used spin dependent confinement of electrons in the quantum well provided by the spacer layer (quantum interference) to explain the oscillations initially observed in metallic ML. Exchange coupling in ML is also affected by direct dipolar coupling like the correlation of spins at rough interfaces (orange peel coupling), which is magnetostatic in origin and occurs due to the interaction of the dipoles which appear due to the roughness of the material [3]. This coupling favors parallel or anti-parallel alignment of spins depending upon the interplay between the magnetostatic exchange and anisotropy energy.

The oscillations in exchange coupling are also observed in oxide ML apart from the metallic ML [4]. Oxide materials are more stable and their ML can form magnetic structures which do not exist in bulk form. There are reports on ML of ferrites used along with NiO [5,6], MgO [7] , CoO [8] and SnO2 [ 9-12].

The mixed spinel structure of Cu-Zn ferrites with composition Cu0.6Zn0.4Fe3O4 (CZF) gives high magnetization when deposited in Ar environment [13]. The properties of non-magnetic (NM) SnO2 are sensitive to oxygen vacancies [14]. The interfacial region between these two materials may give rise to interesting magnetic phenomena. The objective of the present

© 2013 Singh and Saipriya, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

work is to study the magnetic properties of SnO2 and Cu-Zn Ferrite multilayers and interpret them in terms of suitable magnetization mechanism.

FMR Studies of [SnO2/Cu-Zn Ferrite] Multilayers 113

**Figure 1.** FESEM images of [SnO2 (46nm) CZF (x nm)]5 for x = 249 (a), 83(b) and 42 nm (c).

This chapter is about the Ferromagnetic Resonance (FMR) studies of (SnO2/Cu-Zn Ferrrite) ML as a function of CZF layer thickness and SnO2 layer thickness.

## **2. Experimental**

Alternate layers of SnO2 and Cu0.6Zn0.4Fe3O4 (CZF) were deposited on quartz substrates at room temperature (RT) using rf-magnetron sputtering in Ar environment from SnO2 and CZF targets at a power of 50W and 70W respectively. The rate of deposition was estimated by depositing thin films of SnO2 and CZF separately. The synthesis method of targets and the rf sputtering system are described in our earlier work [15]. The field emission scanning electron microscopy (FESEM) studies were carried out to observe the multilayer structure. The FMR studies were carried out in the temperature range 100 - 450 K using JEOL x-band spectrometer. FMR studies were carried out at different temperatures and by varying the angle θH between the thin film normal and the direction of the applied field. The peak-to-peak FMR signal intensity (Ipp) which is proportional to the power absorbed was measured between the positive and negative peak points along the y-axis. The peak-to-peak linewidth (∆H) was measured as a function of temperature.

### **3. Ferromagnetic Resonance (FMR) studies**

### **3.1. Effect of CZF layer thickness**

The ML samples [SnO2 (46 nm)/CFZ(x nm)]5 where x = 42 nm, 83 nm and 249 nm were synthesized. A final capping layer of 46 nm of SnO2 was deposited on all the ML samples. The error in the thickness is ± 5 nm. The FESEM images (figure 1) portray a stack of alternate dark and light layers corresponding to CZF and SnO2 respectively as confirmed by the EDAX data. The CZF layer exhibits a columnar growth as observed in many of the sputtered films [16]. The protrusions of CZF column into SnO2 layers visible from the FESEM images indicate that the interfaces are diffused.

FMR studies were carried out at different temperatures and by varying the angle θ<sup>H</sup> between the thin film normal and the direction of the applied field. Figure 2 shows the room temperature FMR spectra of the ML in θH = 90° (parallel) and 0° (perpendicular) configuration. In the parallel geometry, the FMR signal for the ML with x = 249 nm is highly asymmetric. The steep increase in the low field side and a slow increase in the high field side indicate negative anisotropy in the ML [17].

The line shape of FMR signal is sensitive to magnetic interactions in the material and hence a change in line shape is an indication of change in the magnetic interactions in the ML. Many a times the observed single FMR signal may arise from overlapping of 2 or more FMR signals.

**2. Experimental** 

temperature.

signals.

interpret them in terms of suitable magnetization mechanism.

**3. Ferromagnetic Resonance (FMR) studies** 

**3.1. Effect of CZF layer thickness** 

indicate that the interfaces are diffused.

side indicate negative anisotropy in the ML [17].

ML as a function of CZF layer thickness and SnO2 layer thickness.

work is to study the magnetic properties of SnO2 and Cu-Zn Ferrite multilayers and

This chapter is about the Ferromagnetic Resonance (FMR) studies of (SnO2/Cu-Zn Ferrrite)

Alternate layers of SnO2 and Cu0.6Zn0.4Fe3O4 (CZF) were deposited on quartz substrates at room temperature (RT) using rf-magnetron sputtering in Ar environment from SnO2 and CZF targets at a power of 50W and 70W respectively. The rate of deposition was estimated by depositing thin films of SnO2 and CZF separately. The synthesis method of targets and the rf sputtering system are described in our earlier work [15]. The field emission scanning electron microscopy (FESEM) studies were carried out to observe the multilayer structure. The FMR studies were carried out in the temperature range 100 - 450 K using JEOL x-band spectrometer. FMR studies were carried out at different temperatures and by varying the angle θH between the thin film normal and the direction of the applied field. The peak-to-peak FMR signal intensity (Ipp) which is proportional to the power absorbed was measured between the positive and negative peak points along the y-axis. The peak-to-peak linewidth (∆H) was measured as a function of

The ML samples [SnO2 (46 nm)/CFZ(x nm)]5 where x = 42 nm, 83 nm and 249 nm were synthesized. A final capping layer of 46 nm of SnO2 was deposited on all the ML samples. The error in the thickness is ± 5 nm. The FESEM images (figure 1) portray a stack of alternate dark and light layers corresponding to CZF and SnO2 respectively as confirmed by the EDAX data. The CZF layer exhibits a columnar growth as observed in many of the sputtered films [16]. The protrusions of CZF column into SnO2 layers visible from the FESEM images

FMR studies were carried out at different temperatures and by varying the angle θ<sup>H</sup> between the thin film normal and the direction of the applied field. Figure 2 shows the room temperature FMR spectra of the ML in θH = 90° (parallel) and 0° (perpendicular) configuration. In the parallel geometry, the FMR signal for the ML with x = 249 nm is highly asymmetric. The steep increase in the low field side and a slow increase in the high field

The line shape of FMR signal is sensitive to magnetic interactions in the material and hence a change in line shape is an indication of change in the magnetic interactions in the ML. Many a times the observed single FMR signal may arise from overlapping of 2 or more FMR

**Figure 1.** FESEM images of [SnO2 (46nm) CZF (x nm)]5 for x = 249 (a), 83(b) and 42 nm (c).

FMR Studies of [SnO2/Cu-Zn Ferrite] Multilayers 115

The asymmetry parameter, A/B ratio, was estimated by dividing the intensity of the upper part of the FMR signal by that of the lower part (figure 3). A/B > 1 indicates that the intensity of the upper half of the FMR signal is higher than the lower one and vice versa. A/B = 1

The A/B ratio at room temperature in the parallel configuration is 1.27, 1.12 and 0.69 for x = 249, 83 and 42 nm respectively. A/B > 1 gives Dysonian line shape to the spectrum which results from lower skin depth and higher conductivity of the sample [18]. Hence the possibility of occurrence of dispersive component due to diffusion of electrons into and out

The asymmetric nature of the FMR spectra of x = 249 nm may also be attributed to the interlayer coupling in the ML. The asymmetry decreases with decrease in CFZ layer thickness presumably due to the decrease in the interlayer coupling strength between the layers. As a consequence of interlayer coupling, the coupled spins precess out of phase with the uncoupled ones. Hence the FMR spectra of spins which precess at different frequencies overlap to give an asymmetric signal [19]. The peak to peak FMR signal intensity (Ipp) is measured between the positive and negative peak points along the y-axis and is proportional to the power absorbed. In the parallel configuration Ipp values, normalized to the volume of the CZF layers, increases as the magnetic layer (CZF) thickness increases from x = 42 to 83 nm followed by a decrease for x = 249 nm. The resonance field (Hres) decreases with increase in the CFZ layer thickness in this geometry. This is attributed to the increase in internal field with increasing CZF layer (magnetic layer) thickness because of which the magnetization also increases. The FMR parameters for both parallel and perpendicular

The intrinsic line shape and linewidth of an FMR signal is sensitive to variation of magnetization at the interfaces, surface pits, grain boundaries etc. The peak to peak linewidth (∆H) is sensitive to inhomogeneity, surface roughness, internal field etc. ΔH values for the ML reported in this study initially decreases from 631 to 541 Oe as x changes from 42 to 83 nm followed by an increase to 704 Oe for x = 249 nm. The ratio of the active layer thickness to dead layer thickness is low for ML with x = 42 nm. As x changes from 42 to 83 nm, the contribution from the dead layer becomes relatively negligible due to large active layer thickness. The increase in ΔH for x = 249 nm may be due to the presence of

The ML with x = 249 nm exhibits multiple resonances in θH = 0° configuration at RT (figure 2). The resonances occur at 363 and 418 mT for this ML. The occurrence of multiple resonances may be attributed to the excitation of spin waves due to the inter layer coupling of CZF layers mediated by the non-magnetic SnO2 layers [21]. This implies that the entire ML is coupled by the interlayer exchange coupling as a single magnetic entity. The spin waves propagate through the SnO2 layers and are sustained by the entire multilayer film. This confirms that for x = 249 nm the CZF layer thickness is ample enough to polarize SnO2 layer. With decreasing x it loses its capacity to polarize SnO2 layer. The occurrence of

indicates a signal with both its upper and lower half with equal intensity.

of the skin region cannot be ruled out for the ML with x = 249 and 83 nm.

configuration are listed in table 1.

interlayer coupling [20].

**Figure 2.** RT FMR spectra of [SnO2 (46 nm) CZF(x nm)]5 in the parallel and perpendicular configuration for x = 249 (a), 83(b) and 42 nm (c).

**Figure 3.** A representative FMR plot depicting the A/B ratio.

The asymmetry parameter, A/B ratio, was estimated by dividing the intensity of the upper part of the FMR signal by that of the lower part (figure 3). A/B > 1 indicates that the intensity of the upper half of the FMR signal is higher than the lower one and vice versa. A/B = 1 indicates a signal with both its upper and lower half with equal intensity.

114 Ferromagnetic Resonance – Theory and Applications

for x = 249 (a), 83(b) and 42 nm (c).

**Figure 3.** A representative FMR plot depicting the A/B ratio.

**Figure 2.** RT FMR spectra of [SnO2 (46 nm) CZF(x nm)]5 in the parallel and perpendicular configuration

The A/B ratio at room temperature in the parallel configuration is 1.27, 1.12 and 0.69 for x = 249, 83 and 42 nm respectively. A/B > 1 gives Dysonian line shape to the spectrum which results from lower skin depth and higher conductivity of the sample [18]. Hence the possibility of occurrence of dispersive component due to diffusion of electrons into and out of the skin region cannot be ruled out for the ML with x = 249 and 83 nm.

The asymmetric nature of the FMR spectra of x = 249 nm may also be attributed to the interlayer coupling in the ML. The asymmetry decreases with decrease in CFZ layer thickness presumably due to the decrease in the interlayer coupling strength between the layers. As a consequence of interlayer coupling, the coupled spins precess out of phase with the uncoupled ones. Hence the FMR spectra of spins which precess at different frequencies overlap to give an asymmetric signal [19]. The peak to peak FMR signal intensity (Ipp) is measured between the positive and negative peak points along the y-axis and is proportional to the power absorbed. In the parallel configuration Ipp values, normalized to the volume of the CZF layers, increases as the magnetic layer (CZF) thickness increases from x = 42 to 83 nm followed by a decrease for x = 249 nm. The resonance field (Hres) decreases with increase in the CFZ layer thickness in this geometry. This is attributed to the increase in internal field with increasing CZF layer (magnetic layer) thickness because of which the magnetization also increases. The FMR parameters for both parallel and perpendicular configuration are listed in table 1.

The intrinsic line shape and linewidth of an FMR signal is sensitive to variation of magnetization at the interfaces, surface pits, grain boundaries etc. The peak to peak linewidth (∆H) is sensitive to inhomogeneity, surface roughness, internal field etc. ΔH values for the ML reported in this study initially decreases from 631 to 541 Oe as x changes from 42 to 83 nm followed by an increase to 704 Oe for x = 249 nm. The ratio of the active layer thickness to dead layer thickness is low for ML with x = 42 nm. As x changes from 42 to 83 nm, the contribution from the dead layer becomes relatively negligible due to large active layer thickness. The increase in ΔH for x = 249 nm may be due to the presence of interlayer coupling [20].

The ML with x = 249 nm exhibits multiple resonances in θH = 0° configuration at RT (figure 2). The resonances occur at 363 and 418 mT for this ML. The occurrence of multiple resonances may be attributed to the excitation of spin waves due to the inter layer coupling of CZF layers mediated by the non-magnetic SnO2 layers [21]. This implies that the entire ML is coupled by the interlayer exchange coupling as a single magnetic entity. The spin waves propagate through the SnO2 layers and are sustained by the entire multilayer film. This confirms that for x = 249 nm the CZF layer thickness is ample enough to polarize SnO2 layer. With decreasing x it loses its capacity to polarize SnO2 layer. The occurrence of

multiple resonances may also be attributed to the existence of two different magnetic phases pertaining to the interfacial region and the bulk of the magnetic layer.

FMR Studies of [SnO2/Cu-Zn Ferrite] Multilayers 117

**Figure 5.** A representative plot of **a**ngular dependence of A/B ratio of [SnO2(46 nm)CFZ(249 nm)]5 ML.

Figure 6 depicts the angular variation of Hres, ΔH and Ipp. For all the ML there is no significant change in Hres and ΔH the FMR spectra for θH ≥ 60°. Below 60°, Hres and ΔH increase continuously. This shows that the ML is anisotropic in nature. The Ipp value initially increases from 0° till 40° followed by a decrease with further increase in θH. Thus the ML is

When an applied field is oriented along the film plane, Hres increases with decreasing thickness of the CZF layers. But Hres decreases with decreasing thickness of the CZF layers when the external magnetic field is oriented perpendicular to the film plane. This behavior is attributed to the decrease of the perpendicular anisotropy energy with decreasing CZF thickness [22]. The difference between the resonance fields in the parallel and perpendicular configuration decreases with decreasing CZF thickness evidencing the fact that the ML with

In the perpendicular configuration, the ML with x = 249 nm exhibits multiple resonances presumably due to the presence of spin waves. The spin wave resonance spectrum at RT and HT for ML with x = 249 nm were initially analyzed using the dispersion relation

> గ ቁ ଶ

݊ଶ (1)

x = 42 nm is relatively less anisotropic compared to the ML with x = 249 and 83 nm.

ቀ ܯ߉ െ ܪ ൌ ܪ

homogeneous when placed at an orientation of 40°.

developed by Kittel as follows [23]

where ߉ ൌ ଶ

ெబ మ

The number of resonances decreases with decreasing CZF thickness in the perpendicular geometry. As the thickness of CZF layer increases, the ML provides sufficient length for the spin wave modes to be formed.

**Figure 4.** Room temperature FMR spectra of [SnO2(46 nm)CFZ(249 nm)]5 at different orientations.

The FMR spectra of [SnO2(46 nm)/CZF(249 nm)]5 recorded at different orientations is shown in figure 4. For θH > 20° there is only single resonance in the FMR spectra. Small shoulder appears at θH = 20°. The multiple resonances appear at θH < 20°. The relative intensity of the FMR signals is also angle dependent. As θH approaches 0°, the amplitude of the shoulder peaks increases at the cost of that of the main peak. The separation between the resonances also increases and the intensity of the FMR signal decreases as θH approaches 0°. This may be due to the prominent demagnetization effects in the perpendicular configuration. A similar behavior is exhibited by the ML with x = 86 nm. There are no multiple resonances for the ML with x = 42 nm.

Figure 5 is a representative plot of the angular dependence of A/B ratio. A/B is close to 2.1 for the perpendicular configuration and decreases as the ML approaches parallel configuration. A/B = 1 for θH ~ 30° showing that the spectra is symmetric at an orientation of 30° of the sample normal with the applied field. Below 30° A/B is < 1 and above 30° A/B is > 1 evidencing the variation in the intensity of the lower and upper part of the FMR spectra with θH.

**Figure 5.** A representative plot of **a**ngular dependence of A/B ratio of [SnO2(46 nm)CFZ(249 nm)]5 ML.

Figure 6 depicts the angular variation of Hres, ΔH and Ipp. For all the ML there is no significant change in Hres and ΔH the FMR spectra for θH ≥ 60°. Below 60°, Hres and ΔH increase continuously. This shows that the ML is anisotropic in nature. The Ipp value initially increases from 0° till 40° followed by a decrease with further increase in θH. Thus the ML is homogeneous when placed at an orientation of 40°.

When an applied field is oriented along the film plane, Hres increases with decreasing thickness of the CZF layers. But Hres decreases with decreasing thickness of the CZF layers when the external magnetic field is oriented perpendicular to the film plane. This behavior is attributed to the decrease of the perpendicular anisotropy energy with decreasing CZF thickness [22]. The difference between the resonance fields in the parallel and perpendicular configuration decreases with decreasing CZF thickness evidencing the fact that the ML with x = 42 nm is relatively less anisotropic compared to the ML with x = 249 and 83 nm.

In the perpendicular configuration, the ML with x = 249 nm exhibits multiple resonances presumably due to the presence of spin waves. The spin wave resonance spectrum at RT and HT for ML with x = 249 nm were initially analyzed using the dispersion relation developed by Kittel as follows [23]

$$H\_n = H\_0 - \Lambda M\_0 \left(\frac{\pi}{L}\right)^2 n^2 \tag{1}$$

where ߉ ൌ ଶ ெబ మ

116 Ferromagnetic Resonance – Theory and Applications

spin wave modes to be formed.

the ML with x = 42 nm.

with θH.

multiple resonances may also be attributed to the existence of two different magnetic phases

The number of resonances decreases with decreasing CZF thickness in the perpendicular geometry. As the thickness of CZF layer increases, the ML provides sufficient length for the

**Figure 4.** Room temperature FMR spectra of [SnO2(46 nm)CFZ(249 nm)]5 at different orientations.

The FMR spectra of [SnO2(46 nm)/CZF(249 nm)]5 recorded at different orientations is shown in figure 4. For θH > 20° there is only single resonance in the FMR spectra. Small shoulder appears at θH = 20°. The multiple resonances appear at θH < 20°. The relative intensity of the FMR signals is also angle dependent. As θH approaches 0°, the amplitude of the shoulder peaks increases at the cost of that of the main peak. The separation between the resonances also increases and the intensity of the FMR signal decreases as θH approaches 0°. This may be due to the prominent demagnetization effects in the perpendicular configuration. A similar behavior is exhibited by the ML with x = 86 nm. There are no multiple resonances for

Figure 5 is a representative plot of the angular dependence of A/B ratio. A/B is close to 2.1 for the perpendicular configuration and decreases as the ML approaches parallel configuration. A/B = 1 for θH ~ 30° showing that the spectra is symmetric at an orientation of 30° of the sample normal with the applied field. Below 30° A/B is < 1 and above 30° A/B is > 1 evidencing the variation in the intensity of the lower and upper part of the FMR spectra

pertaining to the interfacial region and the bulk of the magnetic layer.

FMR Studies of [SnO2/Cu-Zn Ferrite] Multilayers 119

� (2)

Here Hn is the position of the nth mode of the spin wave resonance, H0 is the position of the FMR resonance, L is the thickness of the sample, M0 is the magnetization of the film, n is an odd integer and A is the exchange constant. This relation is valid for uniformly magnetized film. The results obtained do not fit well into the Kittel model where Hn is plotted against n2.

According to Portis model [24] for films with non uniform magnetic profile, the magnetization shows a parabolic drop away from the center of the film. The spin wave

� �������� �� � �

�

�� � �� <sup>−</sup> ���

**Figure 7.** Variation of resonance field with the mode number according to Portis model.

Portis considered that the pinning in the film arises due to non uniform magnetization in the film. This model proposes a linear relation between Hn and n. Also from the above equation, the difference between the resonance modes decreases with decrease in magnetization.

Figure 7 shows the linear fits corresponding to Portis model. The exchange constant is proportional to the slope of the line. The slope of the line decreases with increasing temperature. Hence the exchange constant decreases with increasing temperature leading to

They rather exhibit a linear behavior with n.

modes are given by the following relation

Here � is the distortion parameter in the film.

**Figure 6.** Angle dependence of resonance field (1), resonance linewidth (2) and amplitude (3) of [SnO2(46 nm)CFZ(x nm)]5 for x = 249 (a), 83(b) and 42 nm (c).

Here Hn is the position of the nth mode of the spin wave resonance, H0 is the position of the FMR resonance, L is the thickness of the sample, M0 is the magnetization of the film, n is an odd integer and A is the exchange constant. This relation is valid for uniformly magnetized film. The results obtained do not fit well into the Kittel model where Hn is plotted against n2. They rather exhibit a linear behavior with n.

According to Portis model [24] for films with non uniform magnetic profile, the magnetization shows a parabolic drop away from the center of the film. The spin wave modes are given by the following relation

$$H\_n = H\_0 - \frac{4M\_0}{L} (4\pi\epsilon)^{1/2} \left(n + \frac{1}{2}\right) \tag{2}$$

Here � is the distortion parameter in the film.

118 Ferromagnetic Resonance – Theory and Applications

1

2

3

**Figure 6.** Angle dependence of resonance field (1), resonance linewidth (2) and amplitude (3) of

[SnO2(46 nm)CFZ(x nm)]5 for x = 249 (a), 83(b) and 42 nm (c).

**Figure 7.** Variation of resonance field with the mode number according to Portis model.

Portis considered that the pinning in the film arises due to non uniform magnetization in the film. This model proposes a linear relation between Hn and n. Also from the above equation, the difference between the resonance modes decreases with decrease in magnetization.

Figure 7 shows the linear fits corresponding to Portis model. The exchange constant is proportional to the slope of the line. The slope of the line decreases with increasing temperature. Hence the exchange constant decreases with increasing temperature leading to

weak coupling and hence lower magnetization. Also, the separation between the resonance modes decreases at higher temperatures, indicating a decreasing magnetization.

The effective magnetization (4πMeff) and effective g value (geff) calculated using the Kittel's relations are listed in table1. At room temperature 4πMeff increases and geff decreases with increasing CZF thickness. This may be due to increasing FM layer thickness [25]. The various factors which can account for deviation of geff from that of the free electron are spin orbit coupling, coupling at the interfaces etc [26].


**Table 1.** FMR parameters for [SnO2(46 nm)/ CZF (x)]5 ML for various values of x.

The perpendicular anisotropy field HK is obtained from the following equation [27]

$$H\_K = 4\pi M\_S - 4\pi M\_{eff} \tag{3}$$

FMR Studies of [SnO2/Cu-Zn Ferrite] Multilayers 121

**Figure 8.** (K..x) vs x plots for [SnO2(46 nm)CFZ(x nm)]5 ML for x = 43, 83 and 249 nm.

thickness indicates the increase in the damping of spins with increasing thickness.

42 0.049 0.057 2.027 86 0.041 0.045 -1.989 249 0.053 0.146 -2.845,

**Table 2.** Damping and anisotropy parameters for or [SnO2(46 nm)/ CZF (x)]5 ML for various values of x.

The temperature dependent FMR studies in the temperature range 133- 473K were carried out on the ML. Figure 9 shows the FMR spectra of the ML at various temperatures. The change in

Figure 10 shows the variation of Ipp with temperature in the parallel configuration. Ipp increases with increasing temperature for all the ML. Ipp of the ML with x= 42 nm exhibits a linear trend. Whereas for ML with x=83 and 249 nm, the increase is not linear. It increases slowly from 133K to 298K (RT), followed by a rapid increase with temperature in the

line shape indicates change in the magnetic interaction with increasing temperature.

**x (nm)** 

Gilbert damping is a spin relaxation phenomenon in magnetic systems which controls the rate at which the spins reach equilibrium. Spin-orbit coupling [32], non-local spin relaxations like spin wave dissipation [33,34] and disorder present in the materials are the major factors causing Gilbert damping. The damping parameter α values are 0.0096, 0.0083 and 0.0108 for x = 42, 83 and 249 nm respectively. The increase in α value with CZF

> **α K1 (103 erg/cm3** θH = 90° θH = 0° **)**

> > -1.254

The values of K are listed in table 2. In multilayers, there are two factors which contribute to the effective anisotropy – anisotropy due to the interfacial region (KS) and anisotropy due to the bulk volume of the magnetic layer (KV). The relation is given as follows [28]

$$K = K\_V + 2\frac{K\_S}{\varkappa} \tag{4}$$

Where x is the thickness of the magnetic layer. The factor of 2 in the second term on the (r. h.s) of the equation is due to an assumption that the magnetic layer is bound by identical interfaces on either side. When (K.x) is plotted against t (figure 8), the slope gives the value of KV and the intercept gives the value of KS. The negative slope indicates a negative volume anisotropy which tends to confine the magnetic moment in the plane of the film [29]. The intercept at x = 0 gives positive value of KS which tends to align the magnetic moment perpendicular to the surface of the film. The intercept on the x axis gives the thickness at which the contribution from KV outweighs that from KS. From the figure, the intercept on x axis is found to be 56 nm. Thus below 56nm, major contribution comes from interfaces. The KS and KV values obtained from the plot are 194980 and -3671 erg/cm3

The value of K changes from positive to negative as the thickness of CZF increases (figure 8). Negative values of K indicates in-plane anisotropy [30]. Thus with increasing CZF thickness, the direction of magnetization is confined to the plane of the sample surface. The thickness value at which the anisotropy changes sign is 56 nm. The perpendicular anisotropy for small values of x may be due to the lattice mismatch at the interfaces [31].

**Figure 8.** (K..x) vs x plots for [SnO2(46 nm)CFZ(x nm)]5 ML for x = 43, 83 and 249 nm.

orbit coupling, coupling at the interfaces etc [26].

4.173

ΔHres (kG)

1.107,

**Table 1.** FMR parameters for [SnO2(46 nm)/ CZF (x)]5 ML for various values of x.

The perpendicular anisotropy field HK is obtained from the following equation [27]

the bulk volume of the magnetic layer (KV). The relation is given as follows [28]

KS and KV values obtained from the plot are 194980 and -3671 erg/cm3

values of x may be due to the lattice mismatch at the interfaces [31].

Hres (kG)

249 2.526 3.633,

x (nm)

weak coupling and hence lower magnetization. Also, the separation between the resonance

The effective magnetization (4πMeff) and effective g value (geff) calculated using the Kittel's relations are listed in table1. At room temperature 4πMeff increases and geff decreases with increasing CZF thickness. This may be due to increasing FM layer thickness [25]. The various factors which can account for deviation of geff from that of the free electron are spin

g (± 0.01)

42 2.977 3.165 0.188 2.21 2.08 125.76 2.16 631 750 86 2.649 3.720 1.071 2.48 1.77 728.39 2.19 541 590

1.647 2.60 1.81, 1.58 754.41,

The values of K are listed in table 2. In multilayers, there are two factors which contribute to the effective anisotropy – anisotropy due to the interfacial region (KS) and anisotropy due to

���� + 2 ��

Where x is the thickness of the magnetic layer. The factor of 2 in the second term on the (r. h.s) of the equation is due to an assumption that the magnetic layer is bound by identical interfaces on either side. When (K.x) is plotted against t (figure 8), the slope gives the value of KV and the intercept gives the value of KS. The negative slope indicates a negative volume anisotropy which tends to confine the magnetic moment in the plane of the film [29]. The intercept at x = 0 gives positive value of KS which tends to align the magnetic moment perpendicular to the surface of the film. The intercept on the x axis gives the thickness at which the contribution from KV outweighs that from KS. From the figure, the intercept on x axis is found to be 56 nm. Thus below 56nm, major contribution comes from interfaces. The

The value of K changes from positive to negative as the thickness of CZF increases (figure 8). Negative values of K indicates in-plane anisotropy [30]. Thus with increasing CZF thickness, the direction of magnetization is confined to the plane of the sample surface. The thickness value at which the anisotropy changes sign is 56 nm. The perpendicular anisotropy for small

θH = 90° θH = 0° θH = 90° θH = 0° θH = 90° θH = 0°

4πMeff (Oe) geff

1132.88

�� � ���� � ������ (3)

2.28,

� (4)

ΔH (G)

2.14 703 1930

modes decreases at higher temperatures, indicating a decreasing magnetization.

Gilbert damping is a spin relaxation phenomenon in magnetic systems which controls the rate at which the spins reach equilibrium. Spin-orbit coupling [32], non-local spin relaxations like spin wave dissipation [33,34] and disorder present in the materials are the major factors causing Gilbert damping. The damping parameter α values are 0.0096, 0.0083 and 0.0108 for x = 42, 83 and 249 nm respectively. The increase in α value with CZF thickness indicates the increase in the damping of spins with increasing thickness.


**Table 2.** Damping and anisotropy parameters for or [SnO2(46 nm)/ CZF (x)]5 ML for various values of x.

The temperature dependent FMR studies in the temperature range 133- 473K were carried out on the ML. Figure 9 shows the FMR spectra of the ML at various temperatures. The change in line shape indicates change in the magnetic interaction with increasing temperature.

Figure 10 shows the variation of Ipp with temperature in the parallel configuration. Ipp increases with increasing temperature for all the ML. Ipp of the ML with x= 42 nm exhibits a linear trend. Whereas for ML with x=83 and 249 nm, the increase is not linear. It increases slowly from 133K to 298K (RT), followed by a rapid increase with temperature in the

temperature range RT- 473K. From 133K to 250K there is no significant difference in amplitude for ML with x= 83 and 249 nm. At temperatures above RT the amplitude of ML with x= 83 nm is higher than that of ML with x = 249 nm.

FMR Studies of [SnO2/Cu-Zn Ferrite] Multilayers 123

**Figure 10.** Temperature dependent amplitude of [SnO2(46 nm)CFZ(x nm)]5 for x = 249(a), 83(b) and 42

Variation of ΔH with temperature in the parallel configuration is displayed in figure 11. The width of the FMR spectra increases continuously with decreasing temperature for x= 42 nm. For x = 83 nm it decreases with temperature and saturates at ~ 215 K . Whereas for x = 249 nm it begins saturating at ~ RT presumably due to freezing of spins. The Hres increases with increasing temperature (figure 12) in all the ML presumably due to increasing internal field at low temperatures. The ML with x = 249 nm exhibits a linear dependence in the entire temperature range whereas the ML with x = 83 and 42 nm exhibit saturating tendency at higher temperatures in the parallel configuration. This may be an interfacial effect. For x = 42 nm the ratio of spins at the interfaces to the spins in the bulk is high. Whereas for x = 83 and 249 nm the ratio is lower. Hence the contribution from the bulk of the film becomes significant than that from the interfaces. The saturating tendency at high temperatures for the ML with x = 83 and 42 nm shows that even at higher temperatures, the thermal energy is

Similar behavior is observed for CZF thin films [12]. The increase in saturation temperature

The change of slope of ΔH indicates a change in the kind of magnetic interactions in the ML. The line shape of the FMR spectra also changes around the same temperature. The curvature of the temperature dependent of ΔH plot changes as x increases from 42 to 249 presumably due to increase in active to dead layer ratio. Decreasing trend in Ipp and Hres and increasing trend in ΔH with decreasing temperature is a signature of superparamagnetism [35]. The resonance field of a given particle includes contribution from magneto crystalline anisotropy

with increase in CZF thickness could be due to the interlayer coupling.

not sufficient to unlock the spins at the interfaces.

field and demagnetizing field.

nm (c).

**Figure 9.** FMR spectra of [SnO2(46 nm)CFZ(x nm)]5 ML at various temperatures for x = 249 (a), 83 (b) and 42 nm (c)in the parallel configuration.

with x= 83 nm is higher than that of ML with x = 249 nm.

temperature range RT- 473K. From 133K to 250K there is no significant difference in amplitude for ML with x= 83 and 249 nm. At temperatures above RT the amplitude of ML

**Figure 9.** FMR spectra of [SnO2(46 nm)CFZ(x nm)]5 ML at various temperatures for x = 249 (a), 83 (b)

and 42 nm (c)in the parallel configuration.

**Figure 10.** Temperature dependent amplitude of [SnO2(46 nm)CFZ(x nm)]5 for x = 249(a), 83(b) and 42 nm (c).

Variation of ΔH with temperature in the parallel configuration is displayed in figure 11. The width of the FMR spectra increases continuously with decreasing temperature for x= 42 nm. For x = 83 nm it decreases with temperature and saturates at ~ 215 K . Whereas for x = 249 nm it begins saturating at ~ RT presumably due to freezing of spins. The Hres increases with increasing temperature (figure 12) in all the ML presumably due to increasing internal field at low temperatures. The ML with x = 249 nm exhibits a linear dependence in the entire temperature range whereas the ML with x = 83 and 42 nm exhibit saturating tendency at higher temperatures in the parallel configuration. This may be an interfacial effect. For x = 42 nm the ratio of spins at the interfaces to the spins in the bulk is high. Whereas for x = 83 and 249 nm the ratio is lower. Hence the contribution from the bulk of the film becomes significant than that from the interfaces. The saturating tendency at high temperatures for the ML with x = 83 and 42 nm shows that even at higher temperatures, the thermal energy is not sufficient to unlock the spins at the interfaces.

Similar behavior is observed for CZF thin films [12]. The increase in saturation temperature with increase in CZF thickness could be due to the interlayer coupling.

The change of slope of ΔH indicates a change in the kind of magnetic interactions in the ML. The line shape of the FMR spectra also changes around the same temperature. The curvature of the temperature dependent of ΔH plot changes as x increases from 42 to 249 presumably due to increase in active to dead layer ratio. Decreasing trend in Ipp and Hres and increasing trend in ΔH with decreasing temperature is a signature of superparamagnetism [35]. The resonance field of a given particle includes contribution from magneto crystalline anisotropy field and demagnetizing field.

FMR Studies of [SnO2/Cu-Zn Ferrite] Multilayers 125

For a system containing randomly oriented particles, the magnetic resonance signal is broad. At high temperatures where *k*b*T* is much greater than the anisotropy barriers the thermal fluctuations of the magnetic moments reduces the angular anisotropy of the resonance fields and the linewidth of individual nanoparticle [36] resulting in a superparamagnetic resonance (SPR) spectra in the case of nanoparticles. At low temperatures the particle moments are unable to overcome the local anisotropy barriers and, thus, become trapped in metastable states (blocking phenomenon). This results in broad

A characteristic feature of SPR is that at low temperatures, ΔH is almost equal to the value of Hres. For the ML with x = 42 nm around 40% of CZF layer is dead. The diffused region consists of fine particles of CZF dispersed in SnO2 matrix. Hence the SPR phenomenon is pronounced for this ML. Whereas for ML with x = 83 and 249 nm, the thickness of the diffused region is small compared to that of the bulk CZF layer. Hence the value of ΔH at

FMR susceptibility is estimated by calculating the double integrated intensity i.e. area under the absorption curve in FMR is proportional to the concentration of spins. Figure 13 shows temperature dependence of ESR susceptibility in the parallel configuration for *[SnO2(46 nm)CFZ(x nm)]5* ML. It initially increases with decreasing temperature, reaches a maximum and then decreases with further decrease in temperature evidencing the superparamagnetic behavior. The width of the transition is large due to the wide spread of blocking

**Figure 13.** Variation of FMR susceptibility, χFMR with temperature of [SnO2(46 nm)CFZ(x nm)]5 ML for x

For ML with x = 42 nm, χFMR decreases continuously with decrease in temperature down till 183 K. The negative absorption is due to the fact that the area under the lower part of the

resonance spectra with linewidth comparable to that of the resonance field.

low temperatures is not same as that of Hres.

temperatures.

= 249 (a), 83 (b) and 42 nm (c).

**Figure 11.** Temperature dependent resonance linewidth of [SnO2(46 nm)CFZ(x nm)]5 for x = 249(a), 83(b) and 42 nm (c).

**Figure 12.** Temperature dependent resonance field, Hres, in the parallel and perpendicular configuration of [SnO2(46 nm)CFZ(x nm)]5 for x = 249(a), 83(b) and 42 nm (c).

For a system containing randomly oriented particles, the magnetic resonance signal is broad. At high temperatures where *k*b*T* is much greater than the anisotropy barriers the thermal fluctuations of the magnetic moments reduces the angular anisotropy of the resonance fields and the linewidth of individual nanoparticle [36] resulting in a superparamagnetic resonance (SPR) spectra in the case of nanoparticles. At low temperatures the particle moments are unable to overcome the local anisotropy barriers and, thus, become trapped in metastable states (blocking phenomenon). This results in broad resonance spectra with linewidth comparable to that of the resonance field.

124 Ferromagnetic Resonance – Theory and Applications

83(b) and 42 nm (c).

**Figure 11.** Temperature dependent resonance linewidth of [SnO2(46 nm)CFZ(x nm)]5 for x = 249(a),

**Figure 12.** Temperature dependent resonance field, Hres, in the parallel and perpendicular

configuration of [SnO2(46 nm)CFZ(x nm)]5 for x = 249(a), 83(b) and 42 nm (c).

A characteristic feature of SPR is that at low temperatures, ΔH is almost equal to the value of Hres. For the ML with x = 42 nm around 40% of CZF layer is dead. The diffused region consists of fine particles of CZF dispersed in SnO2 matrix. Hence the SPR phenomenon is pronounced for this ML. Whereas for ML with x = 83 and 249 nm, the thickness of the diffused region is small compared to that of the bulk CZF layer. Hence the value of ΔH at low temperatures is not same as that of Hres.

FMR susceptibility is estimated by calculating the double integrated intensity i.e. area under the absorption curve in FMR is proportional to the concentration of spins. Figure 13 shows temperature dependence of ESR susceptibility in the parallel configuration for *[SnO2(46 nm)CFZ(x nm)]5* ML. It initially increases with decreasing temperature, reaches a maximum and then decreases with further decrease in temperature evidencing the superparamagnetic behavior. The width of the transition is large due to the wide spread of blocking temperatures.

**Figure 13.** Variation of FMR susceptibility, χFMR with temperature of [SnO2(46 nm)CFZ(x nm)]5 ML for x = 249 (a), 83 (b) and 42 nm (c).

For ML with x = 42 nm, χFMR decreases continuously with decrease in temperature down till 183 K. The negative absorption is due to the fact that the area under the lower part of the

FMR curve is greater than that of the upper part. The Curie- Weiss law does not fit 1/ χ FMR vs T plot due to the continuous curvature in the entire temperatures range.

FMR Studies of [SnO2/Cu-Zn Ferrite] Multilayers 127

**Figure 15.** FMR spectra of [SnO2(46 nm)CFZ(x nm)]5 at various temperatures in the perpendicular

configuration for x = 249 (a), 83 (b) and 42 nm (c).

**Figure 14.** Variation A/B ratio with temperature of [SnO2(46 nm)CFZ(x nm)]5 ML for x = 249(a), 83(b) and 42 nm (c).

Figure 14 shows the A/B ratio with temperature for these ML. It decreases almost linearly with decreasing temperature from 400 to 200 K for the ML with x = 42 nm. Below 200 K there is a change in slope. The temperature dependence of A/B ratio exhibits different slopes in different temperature region for the ML with x = 83 nm. Whereas for the ML with x = 249 nm, there is no significant change in the A/B ratio in the temperature range between 500 and 350 K. Below 350 K the ratio decreases continuously. The temperature at which the slope for A/B ratio changes matches with that of temperature dependence of linewidth.

Temperature dependence of FMR spectra of these ML in the perpendicular configurations is displayed in figure 15. The resonance field shifts towards higher values with decreasing temperature. This is accompanied with decrease in intensity and increase in line width. This indicates that AFM interactions dominate as temperature decreases in the perpendicular configuration.

#### FMR Studies of [SnO2/Cu-Zn Ferrite] Multilayers 127

126 Ferromagnetic Resonance – Theory and Applications

and 42 nm (c).

configuration.

FMR curve is greater than that of the upper part. The Curie- Weiss law does not fit 1/ χ FMR

**Figure 14.** Variation A/B ratio with temperature of [SnO2(46 nm)CFZ(x nm)]5 ML for x = 249(a), 83(b)

Figure 14 shows the A/B ratio with temperature for these ML. It decreases almost linearly with decreasing temperature from 400 to 200 K for the ML with x = 42 nm. Below 200 K there is a change in slope. The temperature dependence of A/B ratio exhibits different slopes in different temperature region for the ML with x = 83 nm. Whereas for the ML with x = 249 nm, there is no significant change in the A/B ratio in the temperature range between 500 and 350 K. Below 350 K the ratio decreases continuously. The temperature at which the slope for

Temperature dependence of FMR spectra of these ML in the perpendicular configurations is displayed in figure 15. The resonance field shifts towards higher values with decreasing temperature. This is accompanied with decrease in intensity and increase in line width. This indicates that AFM interactions dominate as temperature decreases in the perpendicular

A/B ratio changes matches with that of temperature dependence of linewidth.

vs T plot due to the continuous curvature in the entire temperatures range.

**Figure 15.** FMR spectra of [SnO2(46 nm)CFZ(x nm)]5 at various temperatures in the perpendicular configuration for x = 249 (a), 83 (b) and 42 nm (c).

Figure 15(a) also shows the variation in the positions of the multiple resonances for x = 249. They are narrow and closely spaced at 473 K. As temperature decreases, the FMR peaks broaden and move away from each other with decreasing temperature. The peak at the lower field moves further towards lower field and the one at the higher field moves further towards higher field. This is a clear indication of existence of both FM and AFM phase. The relative intensity of the peak at higher field decreases and that of the one at lower field increases with decreasing temperature. The distance between the peaks increases with decreasing temperature. This indicates increase in phase separation with decrease in temperature.

FMR Studies of [SnO2/Cu-Zn Ferrite] Multilayers 129

(a)

(b)

(c)

**Figure 17.** Temperature dependent resonance field, Hres, in the parallel and perpendicular

configuration of [SnO2(46 nm)CFZ(x nm)]5 for x = 249 (a), 83 (b) and 42 nm (c).

Ipp value in the perpendicular configuration is lower than that in the parallel configuration at all temperatures. However, the trend remains the same in both parallel and perpendicular configurations. The ΔH value in the perpendicular configuration is not very much different from that in the parallel configuration for the ML with x = 42 and 83 nm (figure 16). On the other hand, ΔH for x = 249 ML is much higher in the perpendicular configuration.

**Figure 16.** Temperature dependent resonance linewidth in the parallel (full circle) and perpendicular (open circle) configuration of [SnO2 (46 nm)CFZ(x nm)]5 for x = 249(a), 83(b) and 42 nm (c).

Figure 17 shows the variation of Hres in the parallel and perpendicular configuration. Unlike the parallel configuration, the resonance field increases with decreasing temperature in the perpendicular configuration for ML with x = 249 and 83 nm. Hres remains insensitive to temperature in the range 300- 473K. Below 300 K it increases with further decrease in temperature.

temperature.

temperature.

Figure 15(a) also shows the variation in the positions of the multiple resonances for x = 249. They are narrow and closely spaced at 473 K. As temperature decreases, the FMR peaks broaden and move away from each other with decreasing temperature. The peak at the lower field moves further towards lower field and the one at the higher field moves further towards higher field. This is a clear indication of existence of both FM and AFM phase. The relative intensity of the peak at higher field decreases and that of the one at lower field increases with decreasing temperature. The distance between the peaks increases with decreasing temperature. This indicates increase in phase separation with decrease in

Ipp value in the perpendicular configuration is lower than that in the parallel configuration at all temperatures. However, the trend remains the same in both parallel and perpendicular configurations. The ΔH value in the perpendicular configuration is not very much different from that in the parallel configuration for the ML with x = 42 and 83 nm (figure 16). On the

**Figure 16.** Temperature dependent resonance linewidth in the parallel (full circle) and perpendicular

Figure 17 shows the variation of Hres in the parallel and perpendicular configuration. Unlike the parallel configuration, the resonance field increases with decreasing temperature in the perpendicular configuration for ML with x = 249 and 83 nm. Hres remains insensitive to temperature in the range 300- 473K. Below 300 K it increases with further decrease in

(open circle) configuration of [SnO2 (46 nm)CFZ(x nm)]5 for x = 249(a), 83(b) and 42 nm (c).

other hand, ΔH for x = 249 ML is much higher in the perpendicular configuration.

**Figure 17.** Temperature dependent resonance field, Hres, in the parallel and perpendicular configuration of [SnO2(46 nm)CFZ(x nm)]5 for x = 249 (a), 83 (b) and 42 nm (c).

Whereas for the ML with x = 42 nm, initially there is a small increase in Hres as temperature decreases from 473 to 433 K. Below 433 K it decreases continuously. At all temperatures, Hres is less sensitive to temperature in the perpendicular configuration than in the parallel configuration.

FMR Studies of [SnO2/Cu-Zn Ferrite] Multilayers 131

decreasing temperature. This shows increasing anisotropy with decreasing temperature. The multiple resonance for the ML with x = 249 is prominent at 298 and 473 K (figure 20(a)). For ML with x < 249 nm, the absence of multiple resonances even at high temperatures indicates absence of interlayer coupling. At all temperatures the ΔHres is highest for the ML with x =

**Figure 19.** Representative plot of the FMR susceptibility, FMR of [SnO2 (46 nm)CFZ(249 nm)]5 ML in

At room temperature, ΔH shows a minimum ~ 30-40° (figure 21). Ipp shows maximum around the same value. Hence the sample has an easy axis of magnetization at that inclination. At 473 and 298 K its intensity initially increases, reaches a maximum and then decreases with decreasing θH. Whereas at 133 K, Ipp values remains constant initially then

Another evidence for decreasing anisotropy with decreasing x is shown in figure 21. At 473 K, ΔH for the ML with x = 249 nm initially increases, exhibits a minimum ~ 40° and then increases continuously with further increase in θH . Whereas for the ML with x = 83 nm, with increasing θH, ΔH value remains constant till certain angle then decreases continuously. On the other hand the ΔH value for the ML with x = 42 nm, the ΔH value continuously decrease reaches a minimum and then increases with increasing θH. The occurrence of minimum in ΔH may be due to existence of regions with different effective magnetization values due to inhomogeneous nature of the sample [37]. The minimum in ΔH shifts towards higher angle with decreasing x. Similar behavior is observed at 298 K for all the ML. With decreasing

parallel and perpendicular configuration.

decreases continuously with decreasing value of θH.

249 nm. These multiple resonances disappear with decreasing temperature.

For all the ML, the FMR susceptibility increases with decreasing temperature in the perpendicular configuration. The effective magnetization is a measure of net magnetization within the ML. Figure 18 shows the variation of effective magnetization with temperature. It increases with decreasing temperature for all ML.

**Figure 18.** Variation of effective magnetization with temperature for [SnO2 (46 nm)CFZ(x nm)]5 ML for x = 249(a), 83(b) and 42 nm (c).

Figure 19 is a representative graph of [SnO2 (46 nm)/ CZF(249 nm)]5 ML showing the variation of FMR susceptibility with temperature in both parallel and perpendicular configurations. Unlike the parallel configuration, χFMR in the perpendicular configuration increases continuously with decreasing temperature.

The FMR spectra were recorded at different orientations of the normal to the sample with the applied field (θH). The rotational dependence of FMR spectra was measured at 133, 298 (room temperature) and 473 K. At all temperatures the resonance field increases continuously as θH decreases exhibiting uniaxial anisotropy. For the ML, the difference between ΔH values in the parallel and perpendicular configuration increases with decreasing temperature. This shows increasing anisotropy with decreasing temperature. The multiple resonance for the ML with x = 249 is prominent at 298 and 473 K (figure 20(a)). For ML with x < 249 nm, the absence of multiple resonances even at high temperatures indicates absence of interlayer coupling. At all temperatures the ΔHres is highest for the ML with x = 249 nm. These multiple resonances disappear with decreasing temperature.

130 Ferromagnetic Resonance – Theory and Applications

increases with decreasing temperature for all ML.

configuration.

x = 249(a), 83(b) and 42 nm (c).

increases continuously with decreasing temperature.

Whereas for the ML with x = 42 nm, initially there is a small increase in Hres as temperature decreases from 473 to 433 K. Below 433 K it decreases continuously. At all temperatures, Hres is less sensitive to temperature in the perpendicular configuration than in the parallel

For all the ML, the FMR susceptibility increases with decreasing temperature in the perpendicular configuration. The effective magnetization is a measure of net magnetization within the ML. Figure 18 shows the variation of effective magnetization with temperature. It

**Figure 18.** Variation of effective magnetization with temperature for [SnO2 (46 nm)CFZ(x nm)]5 ML for

Figure 19 is a representative graph of [SnO2 (46 nm)/ CZF(249 nm)]5 ML showing the variation of FMR susceptibility with temperature in both parallel and perpendicular configurations. Unlike the parallel configuration, χFMR in the perpendicular configuration

The FMR spectra were recorded at different orientations of the normal to the sample with the applied field (θH). The rotational dependence of FMR spectra was measured at 133, 298 (room temperature) and 473 K. At all temperatures the resonance field increases continuously as θH decreases exhibiting uniaxial anisotropy. For the ML, the difference between ΔH values in the parallel and perpendicular configuration increases with

**Figure 19.** Representative plot of the FMR susceptibility, FMR of [SnO2 (46 nm)CFZ(249 nm)]5 ML in parallel and perpendicular configuration.

At room temperature, ΔH shows a minimum ~ 30-40° (figure 21). Ipp shows maximum around the same value. Hence the sample has an easy axis of magnetization at that inclination. At 473 and 298 K its intensity initially increases, reaches a maximum and then decreases with decreasing θH. Whereas at 133 K, Ipp values remains constant initially then decreases continuously with decreasing value of θH.

Another evidence for decreasing anisotropy with decreasing x is shown in figure 21. At 473 K, ΔH for the ML with x = 249 nm initially increases, exhibits a minimum ~ 40° and then increases continuously with further increase in θH . Whereas for the ML with x = 83 nm, with increasing θH, ΔH value remains constant till certain angle then decreases continuously. On the other hand the ΔH value for the ML with x = 42 nm, the ΔH value continuously decrease reaches a minimum and then increases with increasing θH. The occurrence of minimum in ΔH may be due to existence of regions with different effective magnetization values due to inhomogeneous nature of the sample [37]. The minimum in ΔH shifts towards higher angle with decreasing x. Similar behavior is observed at 298 K for all the ML. With decreasing

FMR Studies of [SnO2/Cu-Zn Ferrite] Multilayers 133

(a)

(c)

**Figure 21.** Rotational dependence of resonance linewidth, ΔH, at various temperatures for [SnO2 (46

(b)

nm)CFZ(x nm)]5 ML for x = 42(a), 83(b) and 249 nm (c).

**Figure 20.** Rotational dependence of resonance field, Hres, at various temperatures for [SnO2 (46 nm)CFZ(x nm)]5 ML where x = 249(a), 83(b) and 42 nm (c).

**Figure 20.** Rotational dependence of resonance field, Hres, at various temperatures for [SnO2 (46

nm)CFZ(x nm)]5 ML where x = 249(a), 83(b) and 42 nm (c).

**Figure 21.** Rotational dependence of resonance linewidth, ΔH, at various temperatures for [SnO2 (46 nm)CFZ(x nm)]5 ML for x = 42(a), 83(b) and 249 nm (c).

temperature, the curvature of the angle dependent linewidth changes. At 133 K, for x = 249 nm, the minima vanishes and ΔH increases continuously with increasing θH. For the ML with x = 83 nm, the curvature is reversed at 133 K. It increases with increasing θH, attains a maximum value at ~ 40° and then decreases. On the contrary there is no significant change in ΔH for the ML with x = 42 nm. Thus at 133 K, x = 249 nm ML is more anisotropic than the ML with x = 42 and 83 nm.

FMR Studies of [SnO2/Cu-Zn Ferrite] Multilayers 135

FMR studies were carried out at different temperatures and by varying the angle θ<sup>H</sup> between the film normal and the direction of the applied field. Figure 23 shows the first derivative of the FMR absorption spectra of the ML at room temperature in parallel (θH = 90°) and perpendicular (θH = 0°) geometry, normalized to the volume of the CZF layers. Ipp increases as the spacer (SnO2) thickness increases from x = 46 to 115 nm followed by a decrease for x = 184 nm. The peak to peak linewidth (∆H) is sensitive to inhomogenieties, surface roughness, internal field etc. The χ FMR is estimated by calculating double integrated intensity (DI). Both ∆H and DI follow a trend similar to that of Ipp with increase in spacer layer thickness. This behavior may be attributed to the oscillation in the exchange coupling across the NM layer due to quantum interference of the electron waves reflected from the interfaces. The A/B ratio is 0.82, 0.74 and 0.79 for x = 46, 115 and 184 nm respectively. The asymmetry ratio is not very different from each other. For the ML with x = 115 nm, the upper part of the FMR signal rises steeply whereas the rise is relatively slow for the lower

**Figure 23.** Room temperature FMR spectra of [SnO2(x nm) CZF(83 nm)]5 ML in the parallel and

table 3. The effective magnetization decreases with increasing SnO2 thickness.

The resonance field (Hres) increases from 2.65 to 2.78 kG and the g value decreases from 2.47 to 2.36 with increase in SnO2 layer thickness in parallel geometry. The shift in Hres to higher values might be due to the decrease in interlayer coupling strength. The Hres decreases and g values increases from 1.77 to 1.96 with increasing spacer layer thickness in the perpendicular geometry. This might be attributed to lower perpendicular anisotropy for well separated films [38]. The FMR parameters in the parallel and perpendicular configurations are listed in

perpendicular configuration for x = 46 (a), 115 (b) and 184 nm (c)..

part indicating a negative anisotropy.

### **3.2. Effect of spacer layer thickness**

The effect of thickness of spacer (SnO2) layer was studied on 3 sets of samples with varying thickness of the SnO2 layer. The total number of bilayers is 5 and individual CZF layer thickness is 83 nm for all the samples. The ML samples were [SnO2(x nm) CZF(83 nm)]5 ML for x = 46 , 115 and 184 nm .

Stack of alternate dark and light layers are evident from the FESEM images (figure 22). The dark columnar layers correspond to CZF and the light layers correspond to SnO2. The variation in the thickness ratio between SnO2 and CZF is also clearly visible from these images. The absence of fringes in XRR spectra may be due to large thickness of the ML stack.

**Figure 22.** FESEM images of [SnO2(x nm) CZF(83 nm)]5 ML for x = 46 (a), 115 (b) and 184 nm (c).

FMR studies were carried out at different temperatures and by varying the angle θ<sup>H</sup> between the film normal and the direction of the applied field. Figure 23 shows the first derivative of the FMR absorption spectra of the ML at room temperature in parallel (θH = 90°) and perpendicular (θH = 0°) geometry, normalized to the volume of the CZF layers. Ipp increases as the spacer (SnO2) thickness increases from x = 46 to 115 nm followed by a decrease for x = 184 nm. The peak to peak linewidth (∆H) is sensitive to inhomogenieties, surface roughness, internal field etc. The χ FMR is estimated by calculating double integrated intensity (DI). Both ∆H and DI follow a trend similar to that of Ipp with increase in spacer layer thickness. This behavior may be attributed to the oscillation in the exchange coupling across the NM layer due to quantum interference of the electron waves reflected from the interfaces. The A/B ratio is 0.82, 0.74 and 0.79 for x = 46, 115 and 184 nm respectively. The asymmetry ratio is not very different from each other. For the ML with x = 115 nm, the upper part of the FMR signal rises steeply whereas the rise is relatively slow for the lower part indicating a negative anisotropy.

134 Ferromagnetic Resonance – Theory and Applications

**3.2. Effect of spacer layer thickness** 

ML with x = 42 and 83 nm.

for x = 46 , 115 and 184 nm .

temperature, the curvature of the angle dependent linewidth changes. At 133 K, for x = 249 nm, the minima vanishes and ΔH increases continuously with increasing θH. For the ML with x = 83 nm, the curvature is reversed at 133 K. It increases with increasing θH, attains a maximum value at ~ 40° and then decreases. On the contrary there is no significant change in ΔH for the ML with x = 42 nm. Thus at 133 K, x = 249 nm ML is more anisotropic than the

The effect of thickness of spacer (SnO2) layer was studied on 3 sets of samples with varying thickness of the SnO2 layer. The total number of bilayers is 5 and individual CZF layer thickness is 83 nm for all the samples. The ML samples were [SnO2(x nm) CZF(83 nm)]5 ML

Stack of alternate dark and light layers are evident from the FESEM images (figure 22). The dark columnar layers correspond to CZF and the light layers correspond to SnO2. The variation in the thickness ratio between SnO2 and CZF is also clearly visible from these images. The absence of fringes in XRR spectra may be due to large thickness of the ML stack.

**Figure 22.** FESEM images of [SnO2(x nm) CZF(83 nm)]5 ML for x = 46 (a), 115 (b) and 184 nm (c).

**Figure 23.** Room temperature FMR spectra of [SnO2(x nm) CZF(83 nm)]5 ML in the parallel and perpendicular configuration for x = 46 (a), 115 (b) and 184 nm (c)..

The resonance field (Hres) increases from 2.65 to 2.78 kG and the g value decreases from 2.47 to 2.36 with increase in SnO2 layer thickness in parallel geometry. The shift in Hres to higher values might be due to the decrease in interlayer coupling strength. The Hres decreases and g values increases from 1.77 to 1.96 with increasing spacer layer thickness in the perpendicular geometry. This might be attributed to lower perpendicular anisotropy for well separated films [38]. The FMR parameters in the parallel and perpendicular configurations are listed in table 3. The effective magnetization decreases with increasing SnO2 thickness.


FMR Studies of [SnO2/Cu-Zn Ferrite] Multilayers 137

The angle dependent FMR spectra are shown in figure 24. The symmetry in the FMR spectra increases as θH decreases from 90° to 40°. For θH > 40° the spectra becomes asymmetric and the single FMR peak splits into two resonance peaks. With decreasing θH the separation of

Figure 25 is a representative graph of the angular dependence of ∆H and Ipp. For all ML, ∆H value initially decreases till θH = 40° and then increases till θH = 0°. Whereas, Ipp increases till θH = 40° and then decreases till θH = 0°. This indicates that at θH = 40°, the susceptibility of the film increases resulting in higher magnetization. This may be due to existence of regions with two different effective magnetizations. For all the samples Hres shifts to higher fields as the angle between the sample normal and the direction of the field (θH) changes from 90° to

**Figure 25.** Variation of room temperature FMR linewidth (∆H) and intensity (Ipp) of [SnO2 (46 nm)

FMR spectra recorded at various temperatures between 133 and 473 K are shown in figure 26. The change in the lineshape with decreasing temperature evidences an evolution of a

The FMR signal is asymmetric at high temperatures and the symmetry increases with decreasing temperature. In the high temperature region A/B is greater than 1 indicating that the upper part of the FMR spectrum is of higher intensity than the lower half and indicates the presence of dispersive component in the ML. Figure 27 shows the variation of A/B with

0°. This indicates that the ML exhibits uniaxial anisotropy.

CZF(83 nm)]5 ML at different orientations of the film.

different type of magnetic interaction.

temperature.

the peaks increases.

**Table 3.** FMR parameters of [SnO2(x nm) CZF(83 nm)]5 multilayers for various spacer layer thickness

The ML with x = 46 nm exhibits multiple peaks in the FMR spectrum recorded in the perpendicular configuration. This may be due to the interlayer coupling of the CZF layers mediated by the non magnetic spacer layer. When two spins are coupled across the spacer layer, they tend to precess at a frequency different from that of the rest. Hence the resonance for the coupled spins occurs at a different field. The absence of multiple splitting in the other two samples indicates lack of interlayer coupling in the magnetic layers. Apart from the multiple resonances, the asymmetry in the FMR signal in the perpendicular configuration also decreases with the increase in SnO2 layer thickness. As the thickness of the spacer increases, the distance between the CZF layers increases and hence the coupling between them decreases leading to symmetrical FMR spectra. The effective magnetization value at room temperature was calculated using the Kittel's relations and are 710, 485 and 406 Oe for x = 46, 115 and 184 nm respectively.

**Figure 24.** Room temperature FMR spectra of [SnO2(46 nm)CFZ(83 nm)]5 at different orientations.

The angle dependent FMR spectra are shown in figure 24. The symmetry in the FMR spectra increases as θH decreases from 90° to 40°. For θH > 40° the spectra becomes asymmetric and the single FMR peak splits into two resonance peaks. With decreasing θH the separation of the peaks increases.

136 Ferromagnetic Resonance – Theory and Applications

**ΔHres (kG)**  **g (±0.01)**

2.652 3.721 1.069 2.48 1.77 710 541 595 50 2.733 3.452 0.719 2.41 1.90 485 693 733 100 2.784 3.356 0.576 2.36 1.96 406 675 736 30 **Table 3.** FMR parameters of [SnO2(x nm) CZF(83 nm)]5 multilayers for various spacer layer thickness

The ML with x = 46 nm exhibits multiple peaks in the FMR spectrum recorded in the perpendicular configuration. This may be due to the interlayer coupling of the CZF layers mediated by the non magnetic spacer layer. When two spins are coupled across the spacer layer, they tend to precess at a frequency different from that of the rest. Hence the resonance for the coupled spins occurs at a different field. The absence of multiple splitting in the other two samples indicates lack of interlayer coupling in the magnetic layers. Apart from the multiple resonances, the asymmetry in the FMR signal in the perpendicular configuration also decreases with the increase in SnO2 layer thickness. As the thickness of the spacer increases, the distance between the CZF layers increases and hence the coupling between them decreases leading to symmetrical FMR spectra. The effective magnetization value at room temperature was calculated using the Kittel's relations and are 710, 485 and 406 Oe for

**Figure 24.** Room temperature FMR spectra of [SnO2(46 nm)CFZ(83 nm)]5 at different orientations.

**CZF)** θH = 90° <sup>θ</sup>H = 0° θH = 90° <sup>θ</sup>H = 0°

**4πMeff (Oe)** 

**<sup>Δ</sup>H (G) MS** 

θH = 90° θH = 0°

**(emu/cc** 

**Hres (kG)**

x = 46, 115 and 184 nm respectively.

**x (nm)** 

> Figure 25 is a representative graph of the angular dependence of ∆H and Ipp. For all ML, ∆H value initially decreases till θH = 40° and then increases till θH = 0°. Whereas, Ipp increases till θH = 40° and then decreases till θH = 0°. This indicates that at θH = 40°, the susceptibility of the film increases resulting in higher magnetization. This may be due to existence of regions with two different effective magnetizations. For all the samples Hres shifts to higher fields as the angle between the sample normal and the direction of the field (θH) changes from 90° to 0°. This indicates that the ML exhibits uniaxial anisotropy.

**Figure 25.** Variation of room temperature FMR linewidth (∆H) and intensity (Ipp) of [SnO2 (46 nm) CZF(83 nm)]5 ML at different orientations of the film.

FMR spectra recorded at various temperatures between 133 and 473 K are shown in figure 26. The change in the lineshape with decreasing temperature evidences an evolution of a different type of magnetic interaction.

The FMR signal is asymmetric at high temperatures and the symmetry increases with decreasing temperature. In the high temperature region A/B is greater than 1 indicating that the upper part of the FMR spectrum is of higher intensity than the lower half and indicates the presence of dispersive component in the ML. Figure 27 shows the variation of A/B with temperature.

138 Ferromagnetic Resonance – Theory and Applications

FMR Studies of [SnO2/Cu-Zn Ferrite] Multilayers 139

The Ipp value decreases with decreasing temperature. The increase in ∆H at lower temperatures indicates the large spin relaxation times and hence large coercivity at low

The decreasing trend in Ipp and Hres and increasing trend in ΔH with decreasing temperature is a signature of superparamagnetism. Figure 29 shows variation of FMR susceptibility, χ FMR as a function of temperature. It initially increases, reaches a maximum and then decreases with decreasing temperature confirming the superparamagnetic nature of the ML. Figure 30 shows the temperature dependent FMR spectra in the perpendicular configuration. All the ML samples exhibit multiple resonances. The resonances broaden and the separation between them increases with decreasing temperature. The intensity of the peak at lower field side increases and that at higher field side decreases with decreasing

The behavior of ∆H and Ipp is same as that of θH = 90°. The difference between ∆H in both the configurations is negligible in the HT region for all the samples. The substantial difference occurs at low temperatures (figure 31(a)). This may be due to the presence of magnetic inhomogeneities in the ML [39]. In the parallel configuration Hres decreases with decrease in temperature indicating strengthening of FM interactions. Whereas in the perpendicular configuration it is less sensitive to temperature in the HT- RT range, then

**Figure 27.** Variation of A/B ratio of [SnO2 (x nm) CZF (83 nm)]5 with temperature for x = 46 (a), 115 (b)

increases slowly with decrease in temperature (figure 31 (b)).

temperatures (figure 28(a)).

temperature.

and 184 (c).

**Figure 26.** FMR spectra of [SnO2 (x nm) CZF(83 nm)]5 ML recorded at various temperatures in parallel configuration for x = 46 (a), 115 (b) and 184 (c).

The Ipp value decreases with decreasing temperature. The increase in ∆H at lower temperatures indicates the large spin relaxation times and hence large coercivity at low temperatures (figure 28(a)).

138 Ferromagnetic Resonance – Theory and Applications

**Figure 26.** FMR spectra of [SnO2 (x nm) CZF(83 nm)]5 ML recorded at various temperatures in parallel

(c)

(b)

(a)

configuration for x = 46 (a), 115 (b) and 184 (c).

The decreasing trend in Ipp and Hres and increasing trend in ΔH with decreasing temperature is a signature of superparamagnetism. Figure 29 shows variation of FMR susceptibility, χ FMR as a function of temperature. It initially increases, reaches a maximum and then decreases with decreasing temperature confirming the superparamagnetic nature of the ML. Figure 30 shows the temperature dependent FMR spectra in the perpendicular configuration. All the ML samples exhibit multiple resonances. The resonances broaden and the separation between them increases with decreasing temperature. The intensity of the peak at lower field side increases and that at higher field side decreases with decreasing temperature.

The behavior of ∆H and Ipp is same as that of θH = 90°. The difference between ∆H in both the configurations is negligible in the HT region for all the samples. The substantial difference occurs at low temperatures (figure 31(a)). This may be due to the presence of magnetic inhomogeneities in the ML [39]. In the parallel configuration Hres decreases with decrease in temperature indicating strengthening of FM interactions. Whereas in the perpendicular configuration it is less sensitive to temperature in the HT- RT range, then increases slowly with decrease in temperature (figure 31 (b)).

**Figure 27.** Variation of A/B ratio of [SnO2 (x nm) CZF (83 nm)]5 with temperature for x = 46 (a), 115 (b) and 184 (c).

FMR Studies of [SnO2/Cu-Zn Ferrite] Multilayers 141

**Figure 29.** Variation of FMR susceptibility, χFMR with temperature of [SnO2(x nm)CFZ(83 nm)]5 ML for

x = 46 (a), 115 (b) and 184 nm (c).

**Figure 28.** Variation of FMR resonance linewidth (top) and resonance field (bottom) with temperature of [SnO2 (x nm) CZF(83 nm)]5 for x = 46 (a), 115 (b) and 184 nm (c).

**Figure 28.** Variation of FMR resonance linewidth (top) and resonance field (bottom) with temperature

of [SnO2 (x nm) CZF(83 nm)]5 for x = 46 (a), 115 (b) and 184 nm (c).

**Figure 29.** Variation of FMR susceptibility, χFMR with temperature of [SnO2(x nm)CFZ(83 nm)]5 ML for x = 46 (a), 115 (b) and 184 nm (c).

142 Ferromagnetic Resonance – Theory and Applications

FMR Studies of [SnO2/Cu-Zn Ferrite] Multilayers 143

(a)

(b)

**Figure 31.** Variation of resonance field, Hres (a) and resonance linewidth, ΔH (b) with temperature in

The effective magnetization is found to increase with decreasing temperature. Major contribution to the effective magnetization comes from the perpendicular anisotropy. This implies that as the temperature decreases the perpendicular anisotropy relaxes. χFMR shows a different behavior as a function of temperature in the perpendicular configuration. Figure 32 is a representative plot of variation of χFMR with temperature in parallel and perpendicular configuration. It increases continuously with decreasing temperature for all

parallel and perpendicular configurations for [SnO2 (46 nm) CZF(83 nm)]5 ML .

the ML in the perpendicular configuration.

**Figure 30.** FMR spectra of [SnO2 (x nm) CZF(83 nm)]5 ML recorded at various temperatures in perpendicular configuration for x = 46 (a), 115 (b) and 184 (c).

**Figure 30.** FMR spectra of [SnO2 (x nm) CZF(83 nm)]5 ML recorded at various temperatures in

perpendicular configuration for x = 46 (a), 115 (b) and 184 (c).

**Figure 31.** Variation of resonance field, Hres (a) and resonance linewidth, ΔH (b) with temperature in parallel and perpendicular configurations for [SnO2 (46 nm) CZF(83 nm)]5 ML .

The effective magnetization is found to increase with decreasing temperature. Major contribution to the effective magnetization comes from the perpendicular anisotropy. This implies that as the temperature decreases the perpendicular anisotropy relaxes. χFMR shows a different behavior as a function of temperature in the perpendicular configuration. Figure 32 is a representative plot of variation of χFMR with temperature in parallel and perpendicular configuration. It increases continuously with decreasing temperature for all the ML in the perpendicular configuration.

FMR Studies of [SnO2/Cu-Zn Ferrite] Multilayers 145

[5] D. M. Lind, S. D. Berry, G. Chern, H. Mathias, and L. R. Testardi, *Phys. Rev. B* 45 (1992)

[6] M. A. James, F.C. Voogt, L. Niesen, O.C. Rogojanu and T. Hibma, *Surf. Sci.* 402 (1998)

[7] P. J. Van der Zaag, R.M. Wolf, A.R. Ball, C. Bordel, L.F. Feiner and R. Jungblut, *J. Magn.* 

[17] Janhavi P. Joshi, Rajeev Gupta, A. K. Sood and S. V. Bhat, *Phys. Rev. B* 65 (2001) 024410 [18] Y. Gong, Z. Cevher, M. Ebrahim, J. Lou, C. Pettiford, N. X. Sun and Y. H. Ren, *J. Appl.* 

[21] S.S. Kang, J.W. Feng, G.J. Jin, M. Lu, X.N. Xu, A. Hu, S.S. Jiang and H. Xia, *J. Magn.* 

[25] R. D. McMichael, M. D. Stiles, P. J. Chen, and W. F. Egelhoff, Jr., *Phys. Rev. B* 58 (1998)

[28] M. T. Johnson, P. J. H. Bloemen, F. J. A. den Broeder and J. J. de Vriesy, *Rep. Prog. Phys*.

[32] Y. Tserkovnyak, Arne Brataas, Gerrit E. W. Bauer and Bertrand I. Halperin, *Rev. Mod.* 

[34] R. Rai, K. Verma, S. Sharma, Swapna S. Nair, M. A. Valente, A. L. Kholkin and N. A.

[36] M. R. Diehl, J-Y Yu, J. R. Heath, G. A. Held, H. Doyle, S. Sun and C. B. Murray, *J. Phys.* 

[20] A. Ohtomo, D. A. Muller, J. L. Grazul and H. Y. Hwang, *Nature* 419 (2002) 378

[24] S. J. Yuan, L. Wang, R. Shan and S.M. Zhou, *Appl. Phys. A* 79 (2004) 701

[26] M. Sultan and R. Singh, J. Phys: Conf. Ser. 200, 072090 (2010). [27] U. Gradmann and A Mueller, *Phys. Status. Solidi.* 27 (1968) 313

[29] J. H. Jung, S. H. Lim and S. R. Lee, *J. Appl. Phys.* 108 (2010) 113902

[31] M. C. Hickey and S. J. Moodera, *Phys. Rev. Lett.* 102 (2009) 137601

[35] M. Marysko and J. Simsova, *Phys. Stat. Sol. A,* 33 (1976) K133-K136

[33] G. Eliers, M. Lüttich and M. Münzenberg, *Phys. Rev. B.* 74 (2006) 054411

[30] J. Pelzl et al, *J. Phys.: Condens. Matter* 15 (2003) S451

Sobolev, *J. Phys. Chem. Solids* 72 (2011) 862

[8] S. Saipriya, Joji Kurian and R. Singh, *IEEE Trans Mag.*147 (2011) 10 [9] S. Saipriya, Joji Kurian and R. Singh, AIP Conf. Proc. 1451 (2012) 67

[11] S. Saipriya, Joji Kurian and R. Singh, J. Appl. Phys 111 (2012) 07C110

[10] S. Saipriya and R. Singh, *Mater. Lett.* 71 (2012) 157

[19] S. S. Yan, *Acta Metullurgica Sinica* 9 (1996) 283

[12] M. Sultan and R. Singh, J. Appl. Phys. 107 (2010) 09A510 [13] S. Saipriya, M. Sultan and R. Singh, Physica B 406 (2011) 812 [14] M. Sultan and R. Singh, *J. Phys. D: Appl. Phys*. 42 (2009) 115306 [15] G. S. Bales and A. Zangwill, J. Vac. Sci. Technol A 9 (1991) 145 [16] O. S. Josyulu and J. Shobanadri, *J. Mat. Sci.* 20 (1985) 2750

1838.

332

*Magn. Mater*.148 (1995) 346

*Phys.* 106 (2009) 063916

*Magn. Mater.* 166 (1997) 277 [22] C. Kittel, *Phys. Rev.* 110 (1958) 1295 [23] A. M. Portis, *Appl. Phys. Lett.* 2 (1963) 69

8605

59 (1996) 1409

*Phys.* 77 (2005) 1375

*Chem. B* 105 (2001) 7913

**Figure 32.** Representative plot showing variation of FMR susceptibility with temperature in parallel and perpendicular configurations for [SnO2 (46 nm) CZF(83 nm)]5 ML .

### **4. Conclusions**

The temperature dependent ferromagnetic resonance (FMR) studies on [SnO2/ Cu-Zn ferrite]n as function of magnetic layer and spacer layer thickness were carried out in the temperature range 100- 475 K. The study of temperature dependence of FMR lineshape, peak-to-peak linewidth, peak-to –peak line intensity and resonance field provide an insight into the interfacial effects. The inplane and out of plane FMR studies provide information about the magnetic anisotropies of the ML.

### **Author details**

R.Singh and S.Saipriya

*School of Physics, University of Hyderabad, Central University P.O., Hyderabad, India* 

### **5. References**


**4. Conclusions** 

**Author details** 

**5. References** 

R.Singh and S.Saipriya

*Matter* 11 (1999) 81

Phys.Rev. Lett. 85 (2000) 3728

**Figure 32.** Representative plot showing variation of FMR susceptibility with temperature in parallel

The temperature dependent ferromagnetic resonance (FMR) studies on [SnO2/ Cu-Zn ferrite]n as function of magnetic layer and spacer layer thickness were carried out in the temperature range 100- 475 K. The study of temperature dependence of FMR lineshape, peak-to-peak linewidth, peak-to –peak line intensity and resonance field provide an insight into the interfacial effects. The inplane and out of plane FMR studies provide information

*School of Physics, University of Hyderabad, Central University P.O., Hyderabad, India* 

[2] D.M.Edwards, J.Mathon, R.B.Muniz, and M.S. Phan, *Phys. Rev. Lett.* 67 (1991) 493

[3] C.H.Marrows, Nathan Wiser, B. J. Hickey, T.P.A Hase and B.K. Tanner, *J. Phys.: Cond.* 

[4] K. R. Nikolaev, A. Yu. Dobin, I. N. Krivorotov, W. K. Cooley, A. Bhattacharya, A. L. Kobrinskii, L. I. Glazman, R. M. Wentzovitch, E. Dan Dahlberg, and A. M. Goldman

[1] P. Bruno and C. Chappert, *Phys. Rev. Lett. B* 67 (1991) 1602

and perpendicular configurations for [SnO2 (46 nm) CZF(83 nm)]5 ML .

about the magnetic anisotropies of the ML.

	- [37] R. Topkaya, M. Erkovan, A. Öztürk, O. Öztürk, B. Aktaş and M. Özdemir, *J. Appl. Phys.*108 023910

**Chapter 6** 

© 2013 Widuch et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**Dynamic and Rotatable Exchange Anisotropy** 

Stefania Widuch, Robert L. Stamps, Danuta Skrzypek and Zbigniew Celinski

Exchange bias with natural antiferromagnets is typically investigated for systems with only a single magnetic interface [1-7]. Of these, one of the best understood in terms of atomic level spin configurations is the epitaxial Fe/FeF2 film system. The present work explores exchange bias in a structure in which an epitaxially grown KNiF3 film is used to separate the Fe and FeF2. The uniaxial magnetic anisotropy of the KNiF3 is much weaker than that of the FeF2, and the corresponding exchange bias and exchange anisotropies are also different.

By changing the thickness of the KNiF3, we show that it is possible to change the exchange anisotropy and bias from that of the FeF2 to that of the KNiF3. In the limit of very thick KNiF3, we observe unidirectional, uniaxial and rotatable exchange anisotropies. Competing effects from the FeF2 are observed as the KNiF3 thickness is reduced, allowing us to probe

We used ferromagnetic resonance (FMR) to obtain values for exchange anisotropies and bias. The FMR response is provided by the Fe in our Fe/KNiF3/FeF2 systems, and magnetic anisotropies and other parameters are obtained by fitting the raw data to well known resonance conditions for thin films. In this chapter we show that the fitted values reveal an exchange anisotropy that appears to be dependent on applied field orientation relative to the magnetic anisotropy symmetry axes. Throughout the remainder of the chapter we label this anisotropy *AFM Hdyn* . The onset of *AFM Hdyn* with temperature is similar to that of a rotatable

anisotropy observed previously by us for KNiF3 systems [8, 9], and we suggest that *AFM Hdyn* in the present chapter consists of an isotropic 'rotatable' component *Hrot* in addition to an

It is useful at this point to summarize the relevant magnetic properties of the antiferromagnets and their associated exchange bias phenomena. KNiF3 is a Heisenberg

and reproduction in any medium, provided the original work is properly cited.

magnetic lengthscales associated with spin ordering near the KNiF3 interfaces.

**in Fe/KNiF3/FeF2 Trilayers** 

Additional information is available at the end of the chapter

orientation dependent, unidirectional component *HE*.

http://dx.doi.org/10.5772/56019

**1. Introduction** 

[38] B. X. Gu and X. Wang, *Acta Metalurgica Sinica(English letters)*12 (1999) 181

## **Dynamic and Rotatable Exchange Anisotropy in Fe/KNiF3/FeF2 Trilayers**

Stefania Widuch, Robert L. Stamps, Danuta Skrzypek and Zbigniew Celinski

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/56019

### **1. Introduction**

146 Ferromagnetic Resonance – Theory and Applications

*Phys.*108 023910

[37] R. Topkaya, M. Erkovan, A. Öztürk, O. Öztürk, B. Aktaş and M. Özdemir, *J. Appl.* 

[38] B. X. Gu and X. Wang, *Acta Metalurgica Sinica(English letters)*12 (1999) 181

Exchange bias with natural antiferromagnets is typically investigated for systems with only a single magnetic interface [1-7]. Of these, one of the best understood in terms of atomic level spin configurations is the epitaxial Fe/FeF2 film system. The present work explores exchange bias in a structure in which an epitaxially grown KNiF3 film is used to separate the Fe and FeF2. The uniaxial magnetic anisotropy of the KNiF3 is much weaker than that of the FeF2, and the corresponding exchange bias and exchange anisotropies are also different.

By changing the thickness of the KNiF3, we show that it is possible to change the exchange anisotropy and bias from that of the FeF2 to that of the KNiF3. In the limit of very thick KNiF3, we observe unidirectional, uniaxial and rotatable exchange anisotropies. Competing effects from the FeF2 are observed as the KNiF3 thickness is reduced, allowing us to probe magnetic lengthscales associated with spin ordering near the KNiF3 interfaces.

We used ferromagnetic resonance (FMR) to obtain values for exchange anisotropies and bias. The FMR response is provided by the Fe in our Fe/KNiF3/FeF2 systems, and magnetic anisotropies and other parameters are obtained by fitting the raw data to well known resonance conditions for thin films. In this chapter we show that the fitted values reveal an exchange anisotropy that appears to be dependent on applied field orientation relative to the magnetic anisotropy symmetry axes. Throughout the remainder of the chapter we label this anisotropy *AFM Hdyn* . The onset of *AFM Hdyn* with temperature is similar to that of a rotatable anisotropy observed previously by us for KNiF3 systems [8, 9], and we suggest that *AFM Hdyn* in the present chapter consists of an isotropic 'rotatable' component *Hrot* in addition to an orientation dependent, unidirectional component *HE*.

It is useful at this point to summarize the relevant magnetic properties of the antiferromagnets and their associated exchange bias phenomena. KNiF3 is a Heisenberg

antiferromagnet with cubic perovskite structure and Nèel temperature TN=250K [10-13]. The FeF2 has a rutile-type crystal structure and Nèel temperature TN=79K [14-17]. KNiF3 has a very small uniaxial anisotropy in comparison to FeF2 [12-17].

Dynamic and Rotatable Exchange Anisotropy in Fe/KNiF3/FeF2 Trilayers 149

cells and the fluorides were grown using e-beam evaporation at room temperature. A 0.6 nm thick Fe seed layer was followed by a 75 nm thick Ag buffer layer. At this point the structure was annealed at 3500C for 12h. From RHEED we confirmed that annealing produced an average terrace size of ~13 nm. Next an Fe(001) layer was grown at room temperature followed by a KNiF3 layer and topped with a 50 nm FeF2 layer. Series with different KNiF3 thicknesses between 0 and 90nm were prepared. The structures were capped with a 2.5 nm thick Au layer

**Figure 1.** (a) Fragment of the RHEED intensity oscillations measured at the Fe specular spot during the growth; (b) a schematic diagram of the two grown Fe/KNiF3/FeF2 structures with 2.8nm- and 3.6nmthick KNiF3; (c) geometry for FMR measurement; (d) resonance field for different orientations of the static applied field relative to a magnetocrystalline anisotropy axis for Fe thin film (2.59nm). Unfilled circles denote the data taken at RT. Solid line exhibits the fit. Fe easy axis [100] (denoted as EA) for H =

450 and Fe hard axis [110] (denoted as HA) are shown.

to protect samples during measurements in ambient conditions.

Results of magnetometry and FMR studies of bilayered thin films of single crystal Fe(001) and either single crystal or polycrystalline KNiF3 are given in references [8, 9]. The smallest lattice mismatch to the Fe is 1.2%, assuming that the Fe/KNiF3 interface coincides at the (100) face of both materials. This good lattice match between the ferromagnetic and antiferromagnetic layers preserves the cubic structure of both.

The (100) plane of KNiF3 should be compensated with both sublattices present in equal numbers. This should also be true for polycrystalline KNiF3 on Fe since all possible growth planes are reasonably well lattice matched with Fe. The most striking feature observed was a blocking temperature in the range 50 K to 80 K. This temperature is much lower than the 250K Néel temperature expected for bulk KNiF3. Particularly relevant for the present work was the observation (by FMR) of a rotatable anisotropy that was an order of magnitude larger than the exchange bias.

As noted earlier, the Fe/FeF2 system has been particularly well studied. Nogués et al. [1] have shown that the exchange bias depends strongly on the spin structure at the interface and in particular, on the angle between the FM and AFM spins. Work by Fitzsimmons et al. [18-22] using polarized neutron reflectometry and magnetometry show that spin configurations at the FeF2 interface differ from the bulk, and can be linked to exchange anisotropies and bias phenomena.

We have used MOKE microscopy to observe domains in a Fe/FeF2 system to identify interface regions in which spin arrangements are distinct from either AFM or FM magnetic spin arrangements [23]. Results indicate that the crystallographic arrangement affects the value of the exchange bias but not the temperature dependence. Measurements of the temperature dependence of domain density in conjunction with coercivity field enhancement and exchange bias suggest that exchange bias and coercivity are different in origin in the sense that the unpinned magnetic moments at the AFM/FM interface are responsible for the enhancement of the coercivity field while pinned moments shifts the hysteresis loop.

In what follows we first discuss the sample structure and characterization in Section 2. This is followed in Section 3 by a discussion of reference FM/AFM1 and FM/AFM2 bilayers, and FM/AFM1/AFM2 trilayer results. In Section 4 we present our micromagnetic model and discuss the possibility of different thickness regimes. Results and discussion are summarized in Section 5.

### **2. Sample preparation and structural characterization**

Samples were grown on GaAs(001) substrates using Molecular Beam Epitaxy. Wafers were annealed at elevated temperatures (~4500C) and sputtered with Ar ions until 4x6 reconstruction on the surface of GaAs was clearly visible. The Fe was deposited using K-

cells and the fluorides were grown using e-beam evaporation at room temperature. A 0.6 nm thick Fe seed layer was followed by a 75 nm thick Ag buffer layer. At this point the structure was annealed at 3500C for 12h. From RHEED we confirmed that annealing produced an average terrace size of ~13 nm. Next an Fe(001) layer was grown at room temperature followed by a KNiF3 layer and topped with a 50 nm FeF2 layer. Series with different KNiF3 thicknesses between 0 and 90nm were prepared. The structures were capped with a 2.5 nm thick Au layer to protect samples during measurements in ambient conditions.

148 Ferromagnetic Resonance – Theory and Applications

larger than the exchange bias.

anisotropies and bias phenomena.

hysteresis loop.

summarized in Section 5.

very small uniaxial anisotropy in comparison to FeF2 [12-17].

antiferromagnetic layers preserves the cubic structure of both.

antiferromagnet with cubic perovskite structure and Nèel temperature TN=250K [10-13]. The FeF2 has a rutile-type crystal structure and Nèel temperature TN=79K [14-17]. KNiF3 has a

Results of magnetometry and FMR studies of bilayered thin films of single crystal Fe(001) and either single crystal or polycrystalline KNiF3 are given in references [8, 9]. The smallest lattice mismatch to the Fe is 1.2%, assuming that the Fe/KNiF3 interface coincides at the (100) face of both materials. This good lattice match between the ferromagnetic and

The (100) plane of KNiF3 should be compensated with both sublattices present in equal numbers. This should also be true for polycrystalline KNiF3 on Fe since all possible growth planes are reasonably well lattice matched with Fe. The most striking feature observed was a blocking temperature in the range 50 K to 80 K. This temperature is much lower than the 250K Néel temperature expected for bulk KNiF3. Particularly relevant for the present work was the observation (by FMR) of a rotatable anisotropy that was an order of magnitude

As noted earlier, the Fe/FeF2 system has been particularly well studied. Nogués et al. [1] have shown that the exchange bias depends strongly on the spin structure at the interface and in particular, on the angle between the FM and AFM spins. Work by Fitzsimmons et al. [18-22] using polarized neutron reflectometry and magnetometry show that spin configurations at the FeF2 interface differ from the bulk, and can be linked to exchange

We have used MOKE microscopy to observe domains in a Fe/FeF2 system to identify interface regions in which spin arrangements are distinct from either AFM or FM magnetic spin arrangements [23]. Results indicate that the crystallographic arrangement affects the value of the exchange bias but not the temperature dependence. Measurements of the temperature dependence of domain density in conjunction with coercivity field enhancement and exchange bias suggest that exchange bias and coercivity are different in origin in the sense that the unpinned magnetic moments at the AFM/FM interface are responsible for the enhancement of the coercivity field while pinned moments shifts the

In what follows we first discuss the sample structure and characterization in Section 2. This is followed in Section 3 by a discussion of reference FM/AFM1 and FM/AFM2 bilayers, and FM/AFM1/AFM2 trilayer results. In Section 4 we present our micromagnetic model and discuss the possibility of different thickness regimes. Results and discussion are

Samples were grown on GaAs(001) substrates using Molecular Beam Epitaxy. Wafers were annealed at elevated temperatures (~4500C) and sputtered with Ar ions until 4x6 reconstruction on the surface of GaAs was clearly visible. The Fe was deposited using K-

**2. Sample preparation and structural characterization** 

**Figure 1.** (a) Fragment of the RHEED intensity oscillations measured at the Fe specular spot during the growth; (b) a schematic diagram of the two grown Fe/KNiF3/FeF2 structures with 2.8nm- and 3.6nmthick KNiF3; (c) geometry for FMR measurement; (d) resonance field for different orientations of the static applied field relative to a magnetocrystalline anisotropy axis for Fe thin film (2.59nm). Unfilled circles denote the data taken at RT. Solid line exhibits the fit. Fe easy axis [100] (denoted as EA) for H = 450 and Fe hard axis [110] (denoted as HA) are shown.

In figure 1(a), example RHEED intensity oscillations measured at the specular spot during the growth are shown, confirming layer by layer growth. Also, from RHEED we determined that the Fe layer is monocrystalline, and the fluorides layers are polycrystalline. Note that in order to reduce variations in growth conditions within the series, two thicknesses of KNiF3 were grown on each Fe film, thereby forming two samples. The geometry is sketched in figure 1(b) for the 2.8 and 3.6 nm samples from the series. For each pair of samples the same roughness at the Fe/KNiF3 interface is expected.

Dynamic and Rotatable Exchange Anisotropy in Fe/KNiF3/FeF2 Trilayers 151

RT 24 K

*4πMeff*  (kOe)

*H||*  (kOe)

*H||*  (kOe)

are typically an order of magnitude larger than Fe fourfold

*<sup>E</sup>*. Both terms *Hrot* and *HE* are temperature dependent. As noted

*4πMeff*  (kOe)

Fe(2.59nm) 15.85 0.41 16.17 0.61 Fe(5.19nm) 18.91 0.53 18.65 0.68

Fe(2.60nm)/FeF2(50nm) 14.15 0.44 12.88 0.68 Fe(2.62nm)/KNiF3(16nm) 14.98 0.45 14.04 0.67 Fe(2.62nm)/KNiF3(50nm) 15.17 0.43 14.50 0.69

Fe(2.60nm)/KNiF3(0.8nm)/FeF2(50nm) 14.18 0.43 12.94 0.65 Fe(2.61nm)/KNiF3(1.2nm)/FeF2 15.92 0.48 14.79 0.71 Fe(2.60nm)/KNiF3(1.7nm)/FeF2 14.44 0.47 13.15 0.76 Fe(2.58nm)/KNiF3(2.0nm)/FeF2 15.62 0.43 15.15 0.645 Fe(2.59nm)/KNiF3(2.8nm)/FeF2 14.76 0.45 13.84 0.66 Fe(2.59nm)/KNiF3(3.6nm)/FeF2 14.90 0.46 13.76 0.68 Fe(2.68nm)/KNiF3(4.0nm)/FeF2 15.62 0.42 15.52 0.63 Fe(2.61nm)/KNiF3(5.2nm)/FeF2 15.51 0.45 14.08 0.77 Fe(2.60nm)/KNiF3(6.0nm)/FeF2 15.00 0.49 13.37 0.77 Fe(2.58nm)/KNiF3(91.2nm)/FeF2 17.27 0.43 14.89 0.64 Fe(5.29nm)/KNiF3(0.8nm)/FeF2(50nm) 19.22 0.54 19.095 0.73 Fe(5.29nm)/KNiF3(2.8nm)/FeF2 19.11 0.55 18.99 0.75 Fe(5.15nm)/KNiF3(4.0nm)/FeF2 18.96 0.53 19.07 0.68

**Table 1.** Values of the effective demagnetizing field 4πMeff (±0.06 kOe) and the fourfold iron anisotropy field H|| (±0.01 kOe) magnitudes at 24 K from linear fitting extrapolations for trilayer systems of

Fe/KNiF3/FeF2(50nm) in comparison to single Fe layer and bilayers Fe/FeF2, Fe/KNiF3.

where the first term represents a constant shift in the resonance field baseline. This shift is

anisotropy. The second term consists of an induced shift in the position of the resonance line due to exchange coupling which has been attributed to rotatable anisotropy. The third and fourth terms contain the angular variation of *Hres* caused by the Fe fourfold and coupling induced unidirectional anisotropies, respectively. The easy direction of the unidirectional

A summary of results for effective magnetization and fourfold anisotropy fields determined

The fourfold anisotropy field of the 18 ML Fe films increased from ~440 Oe (average value for all measured samples) at room temperature to ~680 Oe at 24 K. For 36 ML thick Fe layers, a similar increase of the fourfold anisotropy field was observed (~540 Oe to ~710 Oe from

above, Eq. (1) is valid only for *Hres* large enough to saturate the magnetization.

due to the fact that *4*

anisotropy is given by

room temperature to 24 K ).

Fe

Bilayers

Trilayers

Sample

*Meff* and

from fits of *Hres* to Eq. (1) are summarized in table 1.

### **3. Experimental details and results**

Ferromagnetic resonance was made at 24 GHz at temperatures between 24K and 300K, using the TE011 mode of a cylindrical cavity. The temperature dependence measurements were carried out in a dewar equipped with a closed-cycle helium refrigerator. The temperature of the sample cavity was monitored with two E-type thermocouples. The dc signal was measured on a diode, and the first derivative of the power absorption signal with respect to the applied field was detected. A lock-in amplifier technique was used employing a weak, 0.5 Oe, 155 Hz ac modulation field superimposed on the applied dc magnetic field. The FMR absorption spectra were fit with standard Lorentzian function, providing directly the resonance field, *Hres*, and FMR linewidth, *ΔH*.

A sketch of the experimental geometry is shown in figure 1(c). In-plane angles for magnetization and applied field (*F* and *<sup>H</sup>*, respectively) are defined relative to the Fe[110] direction. Measurements were made by varying the angle *<sup>H</sup>*. An applied magnetic field was swept between 1 – 6 kOe. These fields were large enough to saturate the samples to within 20 for all angles of applied field orientation. The samples were first measured at a room temperature, then were field cooled to 24K in the cooling field *Hcf* = 0.87kOe and *Hcf* = 2.07kOe for Fe easy axis (denoted as EA; *H* = 450) and Fe hard axis (denoted as HA; *<sup>H</sup>* = 00), respectively. These values of the cooling fields are significantly larger than field needed to saturate FM ( 600 Oe along hard axis).

The effective magnetization *4πMeff*, and a fourfold anisotropy field, || || 2 *S K H M* can be

determined with FMR by measuring *Hres* for different orientations of the static applied field *Happ* relative to a magnetocrystalline anisotropy axis. An example for Fe thin film (2.59nm) is shown in figure 1(d). All FMR measurements were fit to the following resonance condition equation [8,9]:

$$H\_{res} = \frac{1}{2} \left[ \left[ \left( 4 \pi M\_{eff} + \frac{3H\_{\parallel \perp}}{4} (1 + \cos 4\phi\_H) \right)^2 + \left( \frac{2\alpha}{\mathcal{I}} \right)^2 \right]^{1/2} - 4 \pi M\_{eff} \right] \tag{1}$$

$$-H\_{rot} + \frac{H\_{\parallel \perp}}{8} (5 \cos 4\phi\_H - 3) - H\_E \cos \left( \phi\_H - \phi\_E \right)$$

where the first term represents a constant shift in the resonance field baseline. This shift is due to the fact that *4Meff* and are typically an order of magnitude larger than Fe fourfold

150 Ferromagnetic Resonance – Theory and Applications

roughness at the Fe/KNiF3 interface is expected.

the resonance field, *Hres*, and FMR linewidth, *ΔH*.

*F* and 

direction. Measurements were made by varying the angle

magnetization and applied field (

2.07kOe for Fe easy axis (denoted as EA;

saturate FM ( 600 Oe along hard axis).

equation [8,9]:

**3. Experimental details and results** 

In figure 1(a), example RHEED intensity oscillations measured at the specular spot during the growth are shown, confirming layer by layer growth. Also, from RHEED we determined that the Fe layer is monocrystalline, and the fluorides layers are polycrystalline. Note that in order to reduce variations in growth conditions within the series, two thicknesses of KNiF3 were grown on each Fe film, thereby forming two samples. The geometry is sketched in figure 1(b) for the 2.8 and 3.6 nm samples from the series. For each pair of samples the same

Ferromagnetic resonance was made at 24 GHz at temperatures between 24K and 300K, using the TE011 mode of a cylindrical cavity. The temperature dependence measurements were carried out in a dewar equipped with a closed-cycle helium refrigerator. The temperature of the sample cavity was monitored with two E-type thermocouples. The dc signal was measured on a diode, and the first derivative of the power absorption signal with respect to the applied field was detected. A lock-in amplifier technique was used employing a weak, 0.5 Oe, 155 Hz ac modulation field superimposed on the applied dc magnetic field. The FMR absorption spectra were fit with standard Lorentzian function, providing directly

A sketch of the experimental geometry is shown in figure 1(c). In-plane angles for

swept between 1 – 6 kOe. These fields were large enough to saturate the samples to within 20 for all angles of applied field orientation. The samples were first measured at a room temperature, then were field cooled to 24K in the cooling field *Hcf* = 0.87kOe and *Hcf* =

respectively. These values of the cooling fields are significantly larger than field needed to

determined with FMR by measuring *Hres* for different orientations of the static applied field *Happ* relative to a magnetocrystalline anisotropy axis. An example for Fe thin film (2.59nm) is shown in figure 1(d). All FMR measurements were fit to the following resonance condition

*res eff H eff*

5cos4 3 cos

*rot H E HE*

1 2 <sup>3</sup> <sup>4</sup> 1 cos4 <sup>4</sup>

*<sup>H</sup> H M <sup>M</sup>*

The effective magnetization *4πMeff*, and a fourfold anisotropy field, ||


*<sup>H</sup> H H*


8

2 4

*<sup>H</sup>*, respectively) are defined relative to the Fe[110]

*<sup>H</sup>*. An applied magnetic field was


*H*

*<sup>H</sup>* = 00),

*M* can be

(1)

*S K*

*H* = 450) and Fe hard axis (denoted as HA;

1/2 <sup>2</sup> <sup>2</sup>

  anisotropy. The second term consists of an induced shift in the position of the resonance line due to exchange coupling which has been attributed to rotatable anisotropy. The third and fourth terms contain the angular variation of *Hres* caused by the Fe fourfold and coupling induced unidirectional anisotropies, respectively. The easy direction of the unidirectional anisotropy is given by *<sup>E</sup>*. Both terms *Hrot* and *HE* are temperature dependent. As noted above, Eq. (1) is valid only for *Hres* large enough to saturate the magnetization.

A summary of results for effective magnetization and fourfold anisotropy fields determined from fits of *Hres* to Eq. (1) are summarized in table 1.

The fourfold anisotropy field of the 18 ML Fe films increased from ~440 Oe (average value for all measured samples) at room temperature to ~680 Oe at 24 K. For 36 ML thick Fe layers, a similar increase of the fourfold anisotropy field was observed (~540 Oe to ~710 Oe from room temperature to 24 K ).


**Table 1.** Values of the effective demagnetizing field 4πMeff (±0.06 kOe) and the fourfold iron anisotropy field H|| (±0.01 kOe) magnitudes at 24 K from linear fitting extrapolations for trilayer systems of Fe/KNiF3/FeF2(50nm) in comparison to single Fe layer and bilayers Fe/FeF2, Fe/KNiF3.

The effective demagnetizing field (*4πMeff* ) value is less than 21.4 kOe at room temperature. A possible explanation for this is the existence of surface anisotropy and uniaxial anisotropy fields, *Hs* and *Hu* respectively, such that *4πMeff = 4πMS – Hu – Hs.* It is interesting to note that the *4πMeff* for the thinner Fe layers decreases slightly with decreasing temperature from~15.1 kG to ~14.1 kG between room temperature and 24 K. The thicker Fe layers show *4πMeff* nearly constant (~19.0 kG) between room temperature and 24 K. This is suggestive of an out of plane anisotropy induced by the KNiF3 interface.

Dynamic and Rotatable Exchange Anisotropy in Fe/KNiF3/FeF2 Trilayers 153

**Figure 2.** (a) Temperature dependence of the resonance field for bilayer Fe(2.60nm)/FeF2(50.1nm). The inset in (a) shows the linear temperature dependence of the reference Fe layer, (b) Temperature

dependence of *AFM Hdyn* for bilayer Fe/FeF2 and (c) Fe/KNiF3. Filled squares denote measurements along Fe easy axis (EA; H = 450) and unfilled circles along Fe hard axis (HA; H = 00). The cooling field was Hcf = 0.87kOe and Hcf = 2.07kOe for Fe easy and hard axis, respectively. The lines are guides to the eye.

### **3.1. Fe/FeF2 and Fe/KNiF3 reference structures**

The resonance field for the reference bilayers is less than that of the single Fe film data. This reduction is attributed to the AFM-induced dynamic anisotropy field *AFM Hdyn* . The temperature dependence of the resonance field for easy and hard axis field orientation for bilayer Fe/FeF2 is given in figure 2(a) and the inset shows the experimental results for the single Fe film. In figure 2(b) and (c) the temperature dependence of *AFM Hdyn* for easy and hard axis field orientation for reference bilayers Fe/FeF2 and Fe/KNiF3 is presented. The values of *AFM Hdyn* were obtained as the difference between the resonance field data of bilayers and the linear fit extrapolated to low temperature. For each sample the linear fit becomes from the high-temperature (i.e. greater than blocking temperature) resonance field dependence.

The values for the exchange bias *HE* were obtained from scans with positive and negative applied magnetic fields. According to Eq. (1) the asymmetry between resonance fields at each 1800 interval is 2*HE*. For Fe(2.60 nm)/FeF2(50.1nm) structure value of *HE* at 24K was found to be 185 Oe and is significally smaller (by a factor of four) than *AFM Hdyn* determined for this sample (see figure2(b)). At the same time, the FMR results show a low value of *AFM Hdyn* for Fe(2.62nm)/KNiF3(16.0nm) bilayer (see figure2(c)). An exchange bias field of *HE* = 19 Oe was found, in agreement with the value reported by Wee et al.[8, 9].

### **3.2. Fe/KNiF3/FeF2**

An example of how *AFM Hdyn* varies with temperature for trilayers is shown in figure 3. Values extracted for *AFM Hdyn* display several interesting features. The first is that *AFM Hdyn* is a nonmonotonic function of field orientation. Figure 3 shows the temperature dependence of the *AFM Hdyn* for Fe(2.6 nm)/KNiF3(0.8 nm)/FeF2(50 nm) structure for three different orientations of the applied field: along an easy axis (EA), a hard axis (HA) and midway between (IA; *<sup>φ</sup>H*=22.50 ). Values of *AFM Hdyn* are similar for hard and intermediate axis orientations and are significantly larger than that determined for easy axis orientation (at 24K: 1040 Oe, 1048 Oe and 757 Oe, respectively).

of plane anisotropy induced by the KNiF3 interface.

**3.1. Fe/FeF2 and Fe/KNiF3 reference structures** 

The effective demagnetizing field (*4πMeff* ) value is less than 21.4 kOe at room temperature. A possible explanation for this is the existence of surface anisotropy and uniaxial anisotropy fields, *Hs* and *Hu* respectively, such that *4πMeff = 4πMS – Hu – Hs.* It is interesting to note that the *4πMeff* for the thinner Fe layers decreases slightly with decreasing temperature from~15.1 kG to ~14.1 kG between room temperature and 24 K. The thicker Fe layers show *4πMeff* nearly constant (~19.0 kG) between room temperature and 24 K. This is suggestive of an out

The resonance field for the reference bilayers is less than that of the single Fe film data. This reduction is attributed to the AFM-induced dynamic anisotropy field *AFM Hdyn* . The temperature dependence of the resonance field for easy and hard axis field orientation for bilayer Fe/FeF2 is given in figure 2(a) and the inset shows the experimental results for the single Fe film. In figure 2(b) and (c) the temperature dependence of *AFM Hdyn* for easy and hard axis field orientation for reference bilayers Fe/FeF2 and Fe/KNiF3 is presented. The values of *AFM Hdyn* were obtained as the difference between the resonance field data of bilayers and the linear fit extrapolated to low temperature. For each sample the linear fit becomes from the high-temperature (i.e. greater than blocking temperature) resonance field dependence.

The values for the exchange bias *HE* were obtained from scans with positive and negative applied magnetic fields. According to Eq. (1) the asymmetry between resonance fields at each 1800 interval is 2*HE*. For Fe(2.60 nm)/FeF2(50.1nm) structure value of *HE* at 24K was found to be 185 Oe and is significally smaller (by a factor of four) than *AFM Hdyn* determined for

this sample (see figure2(b)). At the same time, the FMR results show a low value of *AFM Hdyn* for Fe(2.62nm)/KNiF3(16.0nm) bilayer (see figure2(c)). An exchange bias field of *HE* = 19 Oe

An example of how *AFM Hdyn* varies with temperature for trilayers is shown in figure 3. Values

extracted for *AFM Hdyn* display several interesting features. The first is that *AFM Hdyn* is a nonmonotonic function of field orientation. Figure 3 shows the temperature dependence of the *AFM Hdyn* for Fe(2.6 nm)/KNiF3(0.8 nm)/FeF2(50 nm) structure for three different orientations of the applied field: along an easy axis (EA), a hard axis (HA) and midway between (IA; *<sup>φ</sup>H*=22.50 ). Values of *AFM Hdyn* are similar for hard and intermediate axis orientations and are significantly larger than that determined for easy axis orientation (at 24K: 1040 Oe, 1048 Oe

was found, in agreement with the value reported by Wee et al.[8, 9].

**3.2. Fe/KNiF3/FeF2** 

and 757 Oe, respectively).

**Figure 2.** (a) Temperature dependence of the resonance field for bilayer Fe(2.60nm)/FeF2(50.1nm). The inset in (a) shows the linear temperature dependence of the reference Fe layer, (b) Temperature dependence of *AFM Hdyn* for bilayer Fe/FeF2 and (c) Fe/KNiF3. Filled squares denote measurements along Fe easy axis (EA; H = 450) and unfilled circles along Fe hard axis (HA; H = 00). The cooling field was Hcf = 0.87kOe and Hcf = 2.07kOe for Fe easy and hard axis, respectively. The lines are guides to the eye.

Dynamic and Rotatable Exchange Anisotropy in Fe/KNiF3/FeF2 Trilayers 155

Another interesting feature is associated with the dependence of *AFM Hdyn* on KNiF3 thickness.

In figure 4 we show *AFM Hdyn* as a function of the KNiF3 film thickness at 24 K for structures with a 2.6 nm-thick Fe layer. Note that the value of this anisotropy initially decreases very rapidly to almost zero for the 2nm-thick KNiF3 layer. For samples with a thicker KNiF3 layers (> 2 nm), the value of *AFM Hdyn* increases and exhibits a non-monotonic dependence on thickness. For a fairly thick KNiF3 layer (91.2nm) the value of the AFM-induced dynamic anisotropy reaches saturation at approximately 100 Oe. Note that even in this regime a difference of 40 Oe persists for values with the field measured along easy and hard axes.

We also mention results from field cooling. It has repeatedly been shown that the value of the unidirectional anisotropy HE depends upon the cooling field strength. A stronger cooling field leads to more pinned spins and larger values for the exchange bias. We have tested the dependence of *AFM Hdyn* on cooling field strength and determined that there is no

such dependence in fields up to 1T. Instead, the magnitude of *AFM Hdyn* for different

Finally, we demonstrate that *AFM Hdyn* depends on the KNiF3 thickness. Values of *AFM Hdyn* are

Fe/KNiF3(0.8nm)/FeF2 706 966 301 429 2.34 2.25 Fe/KNiF3(2.8nm)/FeF2 502 625 244 333 2.06 1.88

**Table 2.** Comparison of the *AFM Hdyn* values obtained for trilayer structures with two different

the results displayed in figure 4 may be related to spin structure *within* the KNiF3.

Doubling the thickness of the Fe reduces *AFM Hdyn* by approximately a factor of two for each KNiF3 thicknesses, as one would expect for an interface effect. We therefore conclude that

Ferromagnetic resonance linewidth is the primary parameter, outside resonance field, considered in the experiment. In Figure 5(a,b,c) is shown FMR linewidth ΔH as function of temperature for selected reference Fe/FeF2 and Fe/KNiF3 samples and Fe/KNiF3/FeF2 trilayers. The FMR linewidth for the single Fe layer displays a linear dependence on temperature. On the contrary, the observed temperature changes in linewidth of the exchange coupled systems are strong and non-linear. As an instance, *ΔH* of Fe(2.60nm)/KNiF3(0.8nm)

*AFM Hdyn* (tFe=2.6nm) *AFM Hdyn* (tFe=5.29nm) ( )

EA (Oe) HA (Oe) EA (Oe) HA (Oe) along EA along HA

( )

*AFM dyn AFM dyn*

*H thinFe H thick Fe*

orientations probably depends on magnitude of the AFM anisotropy field.

tabulated in table 2 for two different thicknesses of Fe in Fe/KNiF3/FeF2.

thicknesses of FM layer.

**3.3. FMR linewidth** 

**Figure 3.** Temperature dependence of the *AFM Hdyn* for trilayer structure Fe(2.6 nm)/KNiF3(0.8 nm)/FeF2(50 nm) for three different orientations of the applied field: along an easy axis (EA), hard axis (HA) and midway between (IA; φH=22.50 ).

**Figure 4.** *AFM Hdyn* as a function of the KNiF3 film thickness at 24 K for structures with a 2.6 nm-thick Fe layer.

Another interesting feature is associated with the dependence of *AFM Hdyn* on KNiF3 thickness. In figure 4 we show *AFM Hdyn* as a function of the KNiF3 film thickness at 24 K for structures with a 2.6 nm-thick Fe layer. Note that the value of this anisotropy initially decreases very rapidly to almost zero for the 2nm-thick KNiF3 layer. For samples with a thicker KNiF3 layers (> 2 nm), the value of *AFM Hdyn* increases and exhibits a non-monotonic dependence on thickness. For a fairly thick KNiF3 layer (91.2nm) the value of the AFM-induced dynamic anisotropy reaches saturation at approximately 100 Oe. Note that even in this regime a difference of 40 Oe persists for values with the field measured along easy and hard axes.

We also mention results from field cooling. It has repeatedly been shown that the value of the unidirectional anisotropy HE depends upon the cooling field strength. A stronger cooling field leads to more pinned spins and larger values for the exchange bias. We have tested the dependence of *AFM Hdyn* on cooling field strength and determined that there is no such dependence in fields up to 1T. Instead, the magnitude of *AFM Hdyn* for different orientations probably depends on magnitude of the AFM anisotropy field.

Finally, we demonstrate that *AFM Hdyn* depends on the KNiF3 thickness. Values of *AFM Hdyn* are tabulated in table 2 for two different thicknesses of Fe in Fe/KNiF3/FeF2.


**Table 2.** Comparison of the *AFM Hdyn* values obtained for trilayer structures with two different thicknesses of FM layer.

Doubling the thickness of the Fe reduces *AFM Hdyn* by approximately a factor of two for each KNiF3 thicknesses, as one would expect for an interface effect. We therefore conclude that the results displayed in figure 4 may be related to spin structure *within* the KNiF3.

### **3.3. FMR linewidth**

154 Ferromagnetic Resonance – Theory and Applications

(HA) and midway between (IA; φH=22.50 ).

layer.

**Figure 3.** Temperature dependence of the *AFM Hdyn* for trilayer structure Fe(2.6 nm)/KNiF3(0.8

nm)/FeF2(50 nm) for three different orientations of the applied field: along an easy axis (EA), hard axis

**Figure 4.** *AFM Hdyn* as a function of the KNiF3 film thickness at 24 K for structures with a 2.6 nm-thick Fe

Ferromagnetic resonance linewidth is the primary parameter, outside resonance field, considered in the experiment. In Figure 5(a,b,c) is shown FMR linewidth ΔH as function of temperature for selected reference Fe/FeF2 and Fe/KNiF3 samples and Fe/KNiF3/FeF2 trilayers.

The FMR linewidth for the single Fe layer displays a linear dependence on temperature. On the contrary, the observed temperature changes in linewidth of the exchange coupled systems are strong and non-linear. As an instance, *ΔH* of Fe(2.60nm)/KNiF3(0.8nm) /FeF2(50.1nm) increases 12 times with decreasing temperature to 24K when the magnetic field was applied along an easy axis. When the FMR measurements were carried out with the field applied along hard axis, the observed increment is eightfold.

Dynamic and Rotatable Exchange Anisotropy in Fe/KNiF3/FeF2 Trilayers 157

We proposed on explanation for FMR linewidth behavior that depends upon of inhomogeneity existing in the local fields acting on the ferromagnet. Some evidence for this may come from our MOKE magnetometer studies on Fe/FeF2 structure [23] and the observations made by some of us for Fe/KNiF3 system [8, 9]. For these reference bilayers a close correspondence between linewidth and coercive field *HC* (see figure 5(a)) was noted. Similarly to FMR linewidth theory, the spatial inhomogeneities in the local internal fields

The temperature dependence of *AFM Hdyn* described above behaves generally like that observed previously for Fe/KNiF3 bilayers. A rotatable anisotropy was able to describe well FMR results in the previous work. Possible mechanisms underlying the formation of the rotatable anisotropy were discussed in references [8] and [9], and argued to be consistent with measured linewidths. The essential idea is that the Fe exchange couples to different regions of the KNiF3 interface, and coupling between these regions can produce a rotatable

Similar arguments should apply in the present case. In the case of trilayers additional measurements were made in relation to reference bilayers, namely for Fe/KNiF3(0.8nm)/FeF2 sample temperature dependence of the *AFM Hdyn* for three field orientations (EA, HA, IA) was

studied. It turned out that the *AFM Hdyn* is a non-monotonic function of the field orientation. We have at present no evidence to suggest a mechanism for this anisotropy, but can speculate that the presence of the second interface with high anisotropy FeF2 introduces an

It seems that a lot of information to verify the introduced model of *AFM Hdyn* could be obtained from studies of Fe/AFM1/AFM2 with single crystalline AFMs. However, the studies of Fe/AFM bilayers, where AFM was deposited onto single crystal of Fe, either in the polycrystalline or single crystal form do not give a clear picture. The systematic study of the influence of in-plane crystalline quality of the antiferromagnet on anisotropies in Fe/FeF2 structure were examined by Fitzsimmons et al [22]. In this study three types of samples were investigated with polycrystalline ferromagnetic Fe thin films and antiferromagnetic FeF2 as: (i) untwinned single crystal; (ii) twinned single crystal, and (iii) textured polycrystal. The results obtained suggest that the value of exchange bias depends on many conditions, such as: the orientation between the spins in AFM and FM during field cooling; the choice of cooling field orientation relative to the AFM; engineering the AFM microstructure and so on. The bilayer Fe/AFM structures consisting of single-crystal Fe and KNiF3, or KCoF3, or KFeF3 films were investigated by some of us [8,9,24,25]. All antiferromagnets in the samples had single crystalline structure or polycrystalline one with a high degree of texture. The crystalline quality of the antiferromagnets significantly affects the size of training effects, the magnitude

additional competition that may affect alignment of regions in the KNiF3.

can have a large effect on the coercive fields and magnetization loop widths.

anisotropy and coercivity as observed experimentally.

**4. Discussion** 

**Figure 5.** (a) Ferromagnetic resonance linewidths H for Fe/FeF2. For comparison, the temperature dependence of H for single Fe is shown. Temperature dependence of coercive field HC for Fe/FeF2 is presented on the right-hand side scale. (b) Ferromagnetic resonance linewidths H for Fe/KNiF3 and (c) trilayer Fe/KNiF3/FeF2 as functions of temperature.

We proposed on explanation for FMR linewidth behavior that depends upon of inhomogeneity existing in the local fields acting on the ferromagnet. Some evidence for this may come from our MOKE magnetometer studies on Fe/FeF2 structure [23] and the observations made by some of us for Fe/KNiF3 system [8, 9]. For these reference bilayers a close correspondence between linewidth and coercive field *HC* (see figure 5(a)) was noted. Similarly to FMR linewidth theory, the spatial inhomogeneities in the local internal fields can have a large effect on the coercive fields and magnetization loop widths.

### **4. Discussion**

156 Ferromagnetic Resonance – Theory and Applications

/FeF2(50.1nm) increases 12 times with decreasing temperature to 24K when the magnetic field was applied along an easy axis. When the FMR measurements were carried out with

**Figure 5.** (a) Ferromagnetic resonance linewidths H for Fe/FeF2. For comparison, the temperature dependence of H for single Fe is shown. Temperature dependence of coercive field HC for Fe/FeF2 is presented on the right-hand side scale. (b) Ferromagnetic resonance linewidths H for Fe/KNiF3 and (c)

(c)

trilayer Fe/KNiF3/FeF2 as functions of temperature.

the field applied along hard axis, the observed increment is eightfold.

(a)

(b)

The temperature dependence of *AFM Hdyn* described above behaves generally like that observed previously for Fe/KNiF3 bilayers. A rotatable anisotropy was able to describe well FMR results in the previous work. Possible mechanisms underlying the formation of the rotatable anisotropy were discussed in references [8] and [9], and argued to be consistent with measured linewidths. The essential idea is that the Fe exchange couples to different regions of the KNiF3 interface, and coupling between these regions can produce a rotatable anisotropy and coercivity as observed experimentally.

Similar arguments should apply in the present case. In the case of trilayers additional measurements were made in relation to reference bilayers, namely for Fe/KNiF3(0.8nm)/FeF2 sample temperature dependence of the *AFM Hdyn* for three field orientations (EA, HA, IA) was studied. It turned out that the *AFM Hdyn* is a non-monotonic function of the field orientation. We have at present no evidence to suggest a mechanism for this anisotropy, but can speculate that the presence of the second interface with high anisotropy FeF2 introduces an additional competition that may affect alignment of regions in the KNiF3.

It seems that a lot of information to verify the introduced model of *AFM Hdyn* could be obtained from studies of Fe/AFM1/AFM2 with single crystalline AFMs. However, the studies of Fe/AFM bilayers, where AFM was deposited onto single crystal of Fe, either in the polycrystalline or single crystal form do not give a clear picture. The systematic study of the influence of in-plane crystalline quality of the antiferromagnet on anisotropies in Fe/FeF2 structure were examined by Fitzsimmons et al [22]. In this study three types of samples were investigated with polycrystalline ferromagnetic Fe thin films and antiferromagnetic FeF2 as: (i) untwinned single crystal; (ii) twinned single crystal, and (iii) textured polycrystal. The results obtained suggest that the value of exchange bias depends on many conditions, such as: the orientation between the spins in AFM and FM during field cooling; the choice of cooling field orientation relative to the AFM; engineering the AFM microstructure and so on. The bilayer Fe/AFM structures consisting of single-crystal Fe and KNiF3, or KCoF3, or KFeF3 films were investigated by some of us [8,9,24,25]. All antiferromagnets in the samples had single crystalline structure or polycrystalline one with a high degree of texture. The crystalline quality of the antiferromagnets significantly affects the size of training effects, the magnitude of the parameters such as: FMR linewidth, the blocking temperature, the exchange bias. *ΔHs* of the samples with the single crystal fluorides were reduced compared to the polycrystalline structures. At the same time, the observed changes of the blocking temperature, the exchange bias, the coercivity vs the crystallity of the antiferromagnet film were different for different fluorides. It therefore seems indisputable that only the FMR linewidth is a parameter that uniquely altered depending on poly- or single crystalline nature of AFMs.

Dynamic and Rotatable Exchange Anisotropy in Fe/KNiF3/FeF2 Trilayers 159

**Figure 6.** The difference in energies for the θH=0o and θH=180o configurations as a function of KNiF3

This asymmetry in energies disappears when the KNiF3 is thicker than 4 ML. This thickness is characteristic of the depth over which a twist can develop in the KNiF3. Above 4 ML, the twist can extend to 180o, and behave like the 'partial' wall in the Stiles and McMichael

On the basis of this, we suggest that *AFM Hdyn* is associated with the gradient of the energy as a function of θH, and that this energy is different for small and large *θH* for KNiF3 thicknesses less than a domain wall width. KNiF3 thicknesses sufficiently larger than this support formation of a partial wall that effectively isolates the Fe from the KNiF3 interface. This results in an energy whose gradient is the same for 0 and 180o orientation of the field. As such, the *AFM Hdyn* arises in this model as a kind of susceptibility of the spin configuration to the orientation of the Fe magnetization (which is controlled by the external applied field).

In this model for *AFM Hdyn* , the thinnest KNiF3 thickness region creates asymmetry in energies at 0 and 180o because the KNiF3 is not wide enough to support a complete partial wall. In this region the effect of FeF2 is apparent. The thickest KNiF3 films support a complete partial wall, and the resulting *AFM Hdyn* are associated with energy gradients that are symmetric with respect to reversal of the applied field. The intermediate thickness region allows spin configurations that are strongly thickness dependent with strongly distorted partial walls. The *AFM Hdyn* in this region vary strongly with KNiF3 thickness, in a manner determined by details of configurational energies of the distorted partial wall. Lastly, we note that this picture can also explain the dependence of *AFM Hdyn* on field orientation with respect to easy

and hard directions. This follows because the magnitude of *AFM Hdyn* will depend on

orientation of the Fe with respect to anisotropy symmetry axes.

thickness, where 1ML=a(KNiF3)=0.4013nm (Muller et al.[10]).

picture of exchange bias[28, 29].

Perhaps the most curious feature observed was the complex dependence of *AFM Hdyn* on KNiF3

thickness as illustrated in figure 4. Three different behaviors were observed, corresponding to three thickness regions: 0 to 2 nm, 2 to 6nm and everything greater than 6 nm. The first two regions involve only a few monolayers of KNiF3. Some insight into reasons underlying the existence of these three regions can be obtained by considering possible spin configurations in a model trilayer. The reason is that KNiF3 is a weak antiferromagnet, and so we might expect that strong interlayer coupling to the Fe and FeF2 might result in large modifications of the spin ordering near the interfaces. Characteristic length scales would correspond to a few nanometers. We explore this idea in some detail with a model described in the appendix. The model makes use of an iterative energy minimization scheme and was used to examine static equilibrium configurations of spins in a thin film trilayer using material parameters appropriate to the Fe/KNiF3/FeF2 trilayers studied in this work. Details of the model are left for the appendix, and we discuss below only the essential results and implications for understanding the dependence of *AFM Hdyn* on KNiF3 thickness.

According to our calculations, for example for trilayer structure Fe/KNiF3(3ML)/FeF2, we find that for angles *H* (the direction of the applied field is characterized by the angle *<sup>H</sup>*) from 00 to –950 the net moments in the first layers of both AFM's oppose the direction of the applied field. Between –950 and –960 a sudden change in spin orientation appears in the first layer of the FeF2. In order to understand how these results may illuminate our measured results for *AFM Hdyn* , we note first that the canted spin configuration calculated for all KNiF3 thicknesses is suggestive of Koon's model [26] for exchange bias. In this model, a net moment at the interface is created by spin canting at a compensated interface of the antiferromagnet in contact with the ferromagnet. It was later pointed out by Schulthess and Butler [27] and discussed by Stiles and McMichael [28] that this canting was not itself sufficient to produce bias due to instabilities in the canted configuration. The instabilities allow the canted moment to reverse into a configuration whose energy is the same as the original field cooled orientation energy.

The same principle applies in our case for the largest thickness KNiF3 films. However for small KNiF3 thicknesses, the energy of the reversed state is not the same as for the field cooled orientation. The difference in energies for the *θH*=0 and *θH*=180o configurations can be sizable, and is shown in figure 6 as a function of KNiF3 thickness where 1ML=a(KNiF3)=0.4013nm (Muller et al. [10]).

thickness.

find that for angles

original field cooled orientation energy.

1ML=a(KNiF3)=0.4013nm (Muller et al. [10]).

of the parameters such as: FMR linewidth, the blocking temperature, the exchange bias. *ΔHs* of the samples with the single crystal fluorides were reduced compared to the polycrystalline structures. At the same time, the observed changes of the blocking temperature, the exchange bias, the coercivity vs the crystallity of the antiferromagnet film were different for different fluorides. It therefore seems indisputable that only the FMR linewidth is a parameter that

Perhaps the most curious feature observed was the complex dependence of *AFM Hdyn* on KNiF3 thickness as illustrated in figure 4. Three different behaviors were observed, corresponding to three thickness regions: 0 to 2 nm, 2 to 6nm and everything greater than 6 nm. The first two regions involve only a few monolayers of KNiF3. Some insight into reasons underlying the existence of these three regions can be obtained by considering possible spin configurations in a model trilayer. The reason is that KNiF3 is a weak antiferromagnet, and so we might expect that strong interlayer coupling to the Fe and FeF2 might result in large modifications of the spin ordering near the interfaces. Characteristic length scales would correspond to a few nanometers. We explore this idea in some detail with a model described in the appendix. The model makes use of an iterative energy minimization scheme and was used to examine static equilibrium configurations of spins in a thin film trilayer using material parameters appropriate to the Fe/KNiF3/FeF2 trilayers studied in this work. Details of the model are left for the appendix, and we discuss below only the essential results and implications for understanding the dependence of *AFM Hdyn* on KNiF3

According to our calculations, for example for trilayer structure Fe/KNiF3(3ML)/FeF2, we

from 00 to –950 the net moments in the first layers of both AFM's oppose the direction of the applied field. Between –950 and –960 a sudden change in spin orientation appears in the first layer of the FeF2. In order to understand how these results may illuminate our measured results for *AFM Hdyn* , we note first that the canted spin configuration calculated for all KNiF3 thicknesses is suggestive of Koon's model [26] for exchange bias. In this model, a net moment at the interface is created by spin canting at a compensated interface of the antiferromagnet in contact with the ferromagnet. It was later pointed out by Schulthess and Butler [27] and discussed by Stiles and McMichael [28] that this canting was not itself sufficient to produce bias due to instabilities in the canted configuration. The instabilities allow the canted moment to reverse into a configuration whose energy is the same as the

The same principle applies in our case for the largest thickness KNiF3 films. However for small KNiF3 thicknesses, the energy of the reversed state is not the same as for the field cooled orientation. The difference in energies for the *θH*=0 and *θH*=180o configurations can be sizable, and is shown in figure 6 as a function of KNiF3 thickness where

*H* (the direction of the applied field is characterized by the angle

*<sup>H</sup>*)

uniquely altered depending on poly- or single crystalline nature of AFMs.

**Figure 6.** The difference in energies for the θH=0o and θH=180o configurations as a function of KNiF3 thickness, where 1ML=a(KNiF3)=0.4013nm (Muller et al.[10]).

This asymmetry in energies disappears when the KNiF3 is thicker than 4 ML. This thickness is characteristic of the depth over which a twist can develop in the KNiF3. Above 4 ML, the twist can extend to 180o, and behave like the 'partial' wall in the Stiles and McMichael picture of exchange bias[28, 29].

On the basis of this, we suggest that *AFM Hdyn* is associated with the gradient of the energy as a function of θH, and that this energy is different for small and large *θH* for KNiF3 thicknesses less than a domain wall width. KNiF3 thicknesses sufficiently larger than this support formation of a partial wall that effectively isolates the Fe from the KNiF3 interface. This results in an energy whose gradient is the same for 0 and 180o orientation of the field. As such, the *AFM Hdyn* arises in this model as a kind of susceptibility of the spin configuration to the orientation of the Fe magnetization (which is controlled by the external applied field).

In this model for *AFM Hdyn* , the thinnest KNiF3 thickness region creates asymmetry in energies at 0 and 180o because the KNiF3 is not wide enough to support a complete partial wall. In this region the effect of FeF2 is apparent. The thickest KNiF3 films support a complete partial wall, and the resulting *AFM Hdyn* are associated with energy gradients that are symmetric with respect to reversal of the applied field. The intermediate thickness region allows spin configurations that are strongly thickness dependent with strongly distorted partial walls. The *AFM Hdyn* in this region vary strongly with KNiF3 thickness, in a manner determined by details of configurational energies of the distorted partial wall. Lastly, we note that this picture can also explain the dependence of *AFM Hdyn* on field orientation with respect to easy and hard directions. This follows because the magnitude of *AFM Hdyn* will depend on orientation of the Fe with respect to anisotropy symmetry axes.

### **5. Conclusions**

In summary, we show that a ferromagnet exchange coupled to an antiferromagnetic bilayer can allow the character and strength of exchange anisotropy to be modified. We have studied using FMR exchange bias and exchange anisotropy for Fe exchange coupled to KNiF3/FeF2 . The trilayer was grown by MBE. The temperature dependence of the ferromagnetic resonance peak position shows a characteristic negative shift of the resonance from that of a single Fe layer. This negative shift is a direct result of the exchange coupling between ferromagnetic and antiferromagnetic layers and results in a dynamically induced field *AFM Hdyn* .

Dynamic and Rotatable Exchange Anisotropy in Fe/KNiF3/FeF2 Trilayers 161

*HH H AFM* int (A.1)

1,2 1, 2 *H HH AFM a b* (A.2)

anisotropy energies, so that the energy of the spins in any given layer depends on the orientation and magnitude of the spins in the nearby layers. The orientation of a spin in layer "*i*" is characterized by the angle *θn* made with respect to the applied field. A low energy magnetic configuration is obtained when the energy is minimized [30] i.e. the angle *θn* is determined by rotating the spin into the direction of a local effective field calculated from the gradient of the energy at the spin site. The procedure determines the configuration in the following manner: (i) a given layer is randomly chosen and the spins in that layer are rotated to be parallel to the local effective field; (ii) the process is continued until one has a self-consistent, stable state where all the spins are aligned with the local effective fields

In this model, the multilayer is treated as an effective one-dimensional system with one spin representing an entire sublattice in a given layer. Only nearest-neighbor exchange coupling

where *HAFM* is the Hamiltonian representing a given antiferromagnet with its both

The geometry is defined in figure 7. The direction normal to the layer planes is *y*. Each *xz*

Both AFMs, KNiF3 and FeF2, exhibit uniaxial anisotropy with easy axes in the *+z* and *–z*  directions. The first AFM is KNiF3 and is known *G-type* meaning that the nearest neighbor coupling is antiferromagnetic. The nearest neighbor exchange constants for each AFM1 layer (*JNNP*) and between AFM1 layers (*JNN*) are defined as negative values. Each layer of the KNiF3 is assumed to be compensated such that both AFM1 sublattices are present within a given layer. The second AFM is FeF2 and is regarded as completely uncompensated within a given layer. In this case, the *JNNP2* exchange coupling constant along *x* is defined as positive.

( 1) ( )( 1) ( 1) ( 1)

( 1) ( )( 1) ( 1) ( 1)

4

*b b b a NN n AFM n m AFM NNP n AFM n AFM*

**SS SS**

(A.3)

(A.4)

4

*a b a b NN n AFM n m AFM NNP n AFM n AFM*

**SS SS**

sublattices, and *Hint* is the Hamiltonian describing the interfaces contribution.

plane, for a given value of *y*, depicts one layer of an AFM.

The complete Hamiltonian can be written as follows:

*a a app n AFM*

1

*a u n AFM*

**S**

1

*b u n AFM*

**S**

*b b app n AFM*

*m*

*K*

*m*

*K*

**H S**

**H S**

1

1

*H*

*H*

( 1)

( 1)

( 1)

2

2

*J J*

*J J*

( 1)

produced by the neighboring spins.

is considered. The energy is defined by:

The value of *AFM Hdyn* is different for measurements performed along ferromagnet easy and hard axes for all the bilayer and trilayer samples. The largest values were found for the field along Fe hard axis. The dependence of the *AFM Hdyn* on the thickness of the KNiF3 is nonmonotonic, reaching a minimum for 2.0nm thick KNiF3. For structures with thinner KNiF3, the magnitude of the *AFM Hdyn* increases with decreasing thickness, reaching the maximum expected for the bilayer Fe/FeF2.

Results from a calculational model for the equilibrium configuration of spins in a ferromagnet/antiferromagnet/antiferromagn trilayer suggests that a form of spin canting may occur at the antiferromagnetic interfaces. For sufficiently thin KNiF3, significant spin canting at the Fe and FeF2 interfaces occurs due to exchange coupling. Effects of the FeF2 are effectively isolated by KNiF3 thick enough to support a partial magnetic domain wall. As a general result, we suggest that *AFM Hdyn* is a measure of the susceptibility of the interface spin ordering to interface coupling in the KNiF3.

## **Appendix**

A numerical mean field model for equilibrium spin orientations in a ferromagnet/ antiferromagnet/antiferromagnet is described in this appendix. The spins are treated as classical vectors on a lattice.

The energy of a spin configuration is determined in the following way. An atomistic approach is employed in a mean field approximation. A cubic lattice of vector spins is considered consisting of N layers representing the first antiferromagnet (called AFM1 and representing the KNiF3) and M layers representing a second antiferromagnet (called AFM2 and representing the FeF2). The outermost layer of AFM1 is in contact with a block spin representing the FM. Each layer is described by a unit cell of spins representing the two antiferromagnetic sublattices. Periodic boundary conditions are assumed in the plane of each layer.

The ground state spin configuration is found as follows. The applied magnetic field is set, and a spin site in the FM/AFM1/AFM2 trilayer is chosen. The algorithm is begun by picking a set of values for the initial configuration of spins in each layer corresponding to a field cooled orientation. Exchange coupling between spins is taken into account in addition to anisotropy energies, so that the energy of the spins in any given layer depends on the orientation and magnitude of the spins in the nearby layers. The orientation of a spin in layer "*i*" is characterized by the angle *θn* made with respect to the applied field. A low energy magnetic configuration is obtained when the energy is minimized [30] i.e. the angle *θn* is determined by rotating the spin into the direction of a local effective field calculated from the gradient of the energy at the spin site. The procedure determines the configuration in the following manner: (i) a given layer is randomly chosen and the spins in that layer are rotated to be parallel to the local effective field; (ii) the process is continued until one has a self-consistent, stable state where all the spins are aligned with the local effective fields produced by the neighboring spins.

In this model, the multilayer is treated as an effective one-dimensional system with one spin representing an entire sublattice in a given layer. Only nearest-neighbor exchange coupling is considered. The energy is defined by:

$$H = H\_{AFM} + H\_{\text{int}} \tag{A.1}$$

$$H\_{AFM} = H\_a^{1,2} + H\_b^{1,2} \tag{A.2}$$

where *HAFM* is the Hamiltonian representing a given antiferromagnet with its both sublattices, and *Hint* is the Hamiltonian describing the interfaces contribution.

The geometry is defined in figure 7. The direction normal to the layer planes is *y*. Each *xz* plane, for a given value of *y*, depicts one layer of an AFM.

Both AFMs, KNiF3 and FeF2, exhibit uniaxial anisotropy with easy axes in the *+z* and *–z*  directions. The first AFM is KNiF3 and is known *G-type* meaning that the nearest neighbor coupling is antiferromagnetic. The nearest neighbor exchange constants for each AFM1 layer (*JNNP*) and between AFM1 layers (*JNN*) are defined as negative values. Each layer of the KNiF3 is assumed to be compensated such that both AFM1 sublattices are present within a given layer. The second AFM is FeF2 and is regarded as completely uncompensated within a given layer. In this case, the *JNNP2* exchange coupling constant along *x* is defined as positive.

The complete Hamiltonian can be written as follows:

160 Ferromagnetic Resonance – Theory and Applications

expected for the bilayer Fe/FeF2.

**Appendix** 

classical vectors on a lattice.

ordering to interface coupling in the KNiF3.

In summary, we show that a ferromagnet exchange coupled to an antiferromagnetic bilayer can allow the character and strength of exchange anisotropy to be modified. We have studied using FMR exchange bias and exchange anisotropy for Fe exchange coupled to KNiF3/FeF2 . The trilayer was grown by MBE. The temperature dependence of the ferromagnetic resonance peak position shows a characteristic negative shift of the resonance from that of a single Fe layer. This negative shift is a direct result of the exchange coupling between ferromagnetic and

The value of *AFM Hdyn* is different for measurements performed along ferromagnet easy and hard axes for all the bilayer and trilayer samples. The largest values were found for the field along Fe hard axis. The dependence of the *AFM Hdyn* on the thickness of the KNiF3 is nonmonotonic, reaching a minimum for 2.0nm thick KNiF3. For structures with thinner KNiF3, the magnitude of the *AFM Hdyn* increases with decreasing thickness, reaching the maximum

Results from a calculational model for the equilibrium configuration of spins in a ferromagnet/antiferromagnet/antiferromagn trilayer suggests that a form of spin canting may occur at the antiferromagnetic interfaces. For sufficiently thin KNiF3, significant spin canting at the Fe and FeF2 interfaces occurs due to exchange coupling. Effects of the FeF2 are effectively isolated by KNiF3 thick enough to support a partial magnetic domain wall. As a general result, we suggest that *AFM Hdyn* is a measure of the susceptibility of the interface spin

A numerical mean field model for equilibrium spin orientations in a ferromagnet/ antiferromagnet/antiferromagnet is described in this appendix. The spins are treated as

The energy of a spin configuration is determined in the following way. An atomistic approach is employed in a mean field approximation. A cubic lattice of vector spins is considered consisting of N layers representing the first antiferromagnet (called AFM1 and representing the KNiF3) and M layers representing a second antiferromagnet (called AFM2 and representing the FeF2). The outermost layer of AFM1 is in contact with a block spin representing the FM. Each layer is described by a unit cell of spins representing the two antiferromagnetic sublattices.

The ground state spin configuration is found as follows. The applied magnetic field is set, and a spin site in the FM/AFM1/AFM2 trilayer is chosen. The algorithm is begun by picking a set of values for the initial configuration of spins in each layer corresponding to a field cooled orientation. Exchange coupling between spins is taken into account in addition to

Periodic boundary conditions are assumed in the plane of each layer.

antiferromagnetic layers and results in a dynamically induced field *AFM Hdyn* .

**5. Conclusions** 

$$\begin{split} H\_{a}^{1} &= -\mathbf{H}\_{app} \cdot \mathbf{S}\_{n(AFM1)}^{a} \\ &+ \sum\_{m=\pm 1} \left[ J\_{NN} \mathbf{S}\_{n(AFM1)}^{a} \cdot \mathbf{S}\_{\{n+m\}(AFM1)}^{b} + 4J\_{NNP} \mathbf{S}\_{n(AFM1)}^{a} \cdot \mathbf{S}\_{n(AFM1)}^{b} \right] \\ &- \mathbf{K}\_{u} \left( \mathbf{S}\_{n(AFM1)}^{a} \right)^{2} \end{split} \tag{A.3}$$

$$\begin{aligned} H\_b^1 &= -\mathbf{H}\_{\textit{app}} \cdot \mathbf{S}\_{n(\textit{AFM1})}^b \\ &+ \sum\_{m=\pm 1} \left[ J\_{\textit{NN}} \mathbf{S}\_{n(\textit{AFM1})}^b \cdot \mathbf{S}\_{(n+m)(\textit{AFM1})}^b + \mathbf{4} J\_{\textit{NNP}} \mathbf{S}\_{n(\textit{AFM1})}^b \cdot \mathbf{S}\_{n(\textit{AFM1})}^a \right] \\ &- \mathbf{K}\_u \left( \mathbf{S}\_{n(\textit{AFM1})}^b \right)^2 \end{aligned} \tag{A.4}$$

Dynamic and Rotatable Exchange Anisotropy in Fe/KNiF3/FeF2 Trilayers 163

(A.6)

(A.7)

2 ( 2) ( )( 2) 2 ( 2) '( 2)

*b a b b NN n AFM n m AFM NN P n AFM n AFM*

**S S S S**

12 max( 1) 1( 2) 12 max( 1) 1( 2)

*aa ba n AFM n AFM n AFM n AFM*

**SS SS**

*n n* **S S** are the spin vectors at layer "*n*" for sublattice "*a*" and "*b*" of the AFM1 and AFM2, respectively. No out-of-plane spin component is considered within this model.

*n n* **S S** are the spin vectors at layer "*n*" for sublattice "*a*" and "*b*" , respectively of the AFM2. Here, the different spin orientation is allowed within one *n*-th layer but again,

*JNN, JNN2* are the exchange interaction constants between spins at "*n*" and "*n+m*" layers of

*JNNP, JNNP2* are the exchange interaction constants within *n*-th layer of AFM1 and AFM2,

*JFAF* is the interface exchange coupling constant between the FM layer and the first

*J12* is the interface exchange coupling constant between the first and the second

Note that the sign of these exchange interactions has been introduced through the numerical values of the parameters. All the values of anisotropy and exchange coupling are given in field units and are summarized in table 3. All calculations were made for 50 atomic layers in

An example arrangement of spins in the AFM1 and AFM2 is depicted in figure A1. Bulk values of the anisotropy and the exchange coupling constants for KNiF3 and FeF2 have been used in all calculations. The interface exchange coupling constant between AFMs (*J12*) have been assumed to be a geometrical mean of the exchange coupling constants of the adjacent layers. The interface exchange coupling between FM and AFM1 is defined as *ZJNN*, i.e. the exchange coupling within the first AFM1 multiplied by the number of nearest neighbors of

2 ( 2)

*b u n AFM*

*H J J*

**S**

1 ' 2

*J J*

*a b FAF FM n AFM FAF FM n AFM*

*m n*

int 1( 1) 1( 1)

**SS SS**

The only spin arrangement is to lie within *xz* plane i.e. (*x, 0, z*),

no out-of-plane spin component is allowed to appear within this model, *Ku, Ku2* are site anisotropies for spins at layer "*n*" of AFM1 and AFM2, respectively.

*J J*

( 2)

*b b app n AFM*

*K*

Finally, the interface term with the FM is given by:

The exchange coupling constants are defined as:

AFM1 and AFM2, respectively,

respectively,

AFM2.

each Fe spin site.

antiferromagnet,

AFM1/AFM2 layer.

2

**H S**

*H*

The notation is defined by:

, *a b*

 ' ' , *a b*

*Happ* is the applied magnetic field,

**Figure 7.** Geometry of the system (a) defining an angle of the applied field within the film plane θH, (b) the direction normal to the layer planes is y. Each xz plane, for a given value of y, depicts one layer of an AFM.

Similar terms can be written for the second antiferromagnet AFM2. Here, however, a given AFM2 plane is regarded, as completely compensated and only one sublattice is present in the interface:

$$\begin{aligned} H\_a^2 &= -\mathbf{H}\_{app} \cdot \mathbf{S}\_{n(AFM2)}^a \\ &+ \sum\_{m=\pm 1} f\_{NN2} \mathbf{S}\_{n(AFM2)}^a \cdot \mathbf{S}\_{(n+m)(AFM2)}^b + \sum\_{n^\*} f\_{NN2P} \mathbf{S}\_{n(AFM2)}^a \cdot \mathbf{S}\_{n^\*(AFM2)}^a \\ &- \mathbf{K}\_{n2} \left( \mathbf{S}\_{n(AFM2)}^a \right)^2 \end{aligned} \tag{A.5}$$

$$\begin{aligned} H\_b^2 &= -\mathbf{H}\_{app} \cdot \mathbf{S}\_{n(AFM2)}^b \\ &+ \sum\_{m=\pm 1} f\_{NN2} \mathbf{S}\_{n(AFM2)}^b \cdot \mathbf{S}\_{\{n+m\}(AFM2)}^a + \sum\_{n'} f\_{NN2} \mathbf{S}\_{n(AFM2)}^b \cdot \mathbf{S}\_{n'(AFM2)}^b \\ &- \mathbf{K}\_{u2} \left( \mathbf{S}\_{n(AFM2)}^b \right)^2 \end{aligned} \tag{A.6}$$

Finally, the interface term with the FM is given by:

$$\begin{aligned} \mathbf{^jH}\_{\text{int}} &= f\_{\text{FAF}} \mathbf{S}\_{\text{FM}} \cdot \mathbf{S}^a\_{\text{n=1(AFM1)}} - f\_{\text{FAF}} \mathbf{S}\_{\text{FM}} \cdot \mathbf{S}^b\_{\text{n=1(AFM1)}} \\ &+ f\_{12} \mathbf{S}^a\_{\text{n=max(AFM1)}} \cdot \mathbf{S}^a\_{\text{n=1(AFM2)}} - f\_{12} \mathbf{S}^b\_{\text{n=max(AFM1)}} \cdot \mathbf{S}^a\_{\text{n=1(AFM2)}} \end{aligned} \tag{A.7}$$

The notation is defined by:

162 Ferromagnetic Resonance – Theory and Applications

an AFM.

the interface:

2

**H S**

*H*

**Figure 7.** Geometry of the system (a) defining an angle of the applied field within the film plane θH, (b) the direction normal to the layer planes is y. Each xz plane, for a given value of y, depicts one layer of

Similar terms can be written for the second antiferromagnet AFM2. Here, however, a given AFM2 plane is regarded, as completely compensated and only one sublattice is present in

> 1 ' 2

*J J*

*m n*

2 ( 2) ( )( 2) 2 ( 2) '( 2)

(A.5)

*a b a a NN n AFM n m AFM NN P n AFM n AFM*

**S S S S**

2 ( 2)

*a u n AFM*

**S**

( 2)

*a a app n AFM*

*K*


The exchange coupling constants are defined as:


Note that the sign of these exchange interactions has been introduced through the numerical values of the parameters. All the values of anisotropy and exchange coupling are given in field units and are summarized in table 3. All calculations were made for 50 atomic layers in AFM2.

An example arrangement of spins in the AFM1 and AFM2 is depicted in figure A1. Bulk values of the anisotropy and the exchange coupling constants for KNiF3 and FeF2 have been used in all calculations. The interface exchange coupling constant between AFMs (*J12*) have been assumed to be a geometrical mean of the exchange coupling constants of the adjacent layers. The interface exchange coupling between FM and AFM1 is defined as *ZJNN*, i.e. the exchange coupling within the first AFM1 multiplied by the number of nearest neighbors of each Fe spin site.


Dynamic and Rotatable Exchange Anisotropy in Fe/KNiF3/FeF2 Trilayers 165

**Figure 8.** (a) The total energy per spin site as a function of the angle of the applied magnetic field θH for different thicknesses of the first AFM1. The horizontal axis (θH (deg)) is the same for all calculated structures, but the total calculated energy is not to scale in order to visibly depict the calculated results for each thickness of AFM1, (b) The values of angle θH for which the discontinuity appears (θHdis) for

thicknesses of AFM1 ranging from 1ML to 9ML. The lines are guides to the eye.

a M.E. Lines, Phys. Rev. **164**, 736 (1967).

b H. Yamaguchi, K. Katsumata, M, Hagiwara, M. Tokunaga, H. L. Liu, A Zibold, D. B. Tanner, and Y.J.Wang, Phys. Rev. B **59,** 6021 (1999).

c M.L. Silva, A.L. Dantas, and A.S. Carrico, Solid State Commun. **135**, 769 (2005).

d D.P. Belanger, P. Nordblad, A.R. King, V. Jaccarino, L. Lundgren, and O.Beckman, J. Magn. Magn. Mater. **31-34,** 1095 (1983).

**Table 3.** Used bulk values of the exchange and anisotropy fields for the KNiF3 and FeF2.

After converging to a stable configuration, the total energy per spin site was calculated as a function of the angle of the applied magnetic field, *θH*. Example results are shown in figure 8(a) for different thicknesses of the first AFM1.

Note that the horizontal axis (*θH* (deg)) is the same for all calculated structures, but the total calculated energy is not to scale in order to visibly depict the calculated results for each thickness of AFM1. A discontinuity in the total energy per spin site appears for thicknesses of AFM1 ranging from 1ML to 9ML. The values of angle *θH* for which the discontinuity appears (*θHdis*) is presented in Fig 8(b). Three regions can be distinguished in these results: 1ML to 4ML, 5ML to 9ML, and > 9ML.

The different regions correspond to different spin arrangements within each AFM layer. An example for the 3ML thick AFM1 is shown in figure 9. The spin arrangements are presented in terms of the angle that each spin vector makes with the *z*-direction. Here, blue arrow represents the applied field, and red arrows are used for the spins in the three layers of AFM1 and green arrows represent the first few spin sites of AFM2.

The comparison between a spin arrangement for *θH*=-950 (top diagrams) and for *θH*=-960 (bottom diagrams) is shown. The central diagram in figure 9 shows the spin arrangement when the field is applied along *+z* direction. The first layer of AFM1 experiences spincanting. An energy minimum for the fully compensated interface has been obtained for perpendicular interfacial coupling between the FM and AFM spins. The spin-canting and the resulting net moment induced within the first layer of the AFM1 always opposes the direction of the applied field. Moreover, spin-canting appears also in the second antiferromagnet AFM2 for the first two layers. It is important to note that the direction of the net moment induced in the first layer of AFM2 does not correlate directly to the direction of the applied magnetic field.

a M.E. Lines, Phys. Rev. **164**, 736 (1967).

Rev. B **59,** 6021 (1999).

c

(1983).

AFM Exchange field

8(a) for different thicknesses of the first AFM1.

1ML to 4ML, 5ML to 9ML, and > 9ML.

direction of the applied magnetic field.

(kOe)

AFM2: FeF2 *Hex2* = 434c *HKu2* = 149c

M.L. Silva, A.L. Dantas, and A.S. Carrico, Solid State Commun. **135**, 769 (2005).

AFM1 and green arrows represent the first few spin sites of AFM2.

AFM1: KNiF3 *Hex* = 3500a *HKu* = 0.080b 2.4·10-5

b H. Yamaguchi, K. Katsumata, M, Hagiwara, M. Tokunaga, H. L. Liu, A Zibold, D. B. Tanner, and Y.J.Wang, Phys.

d D.P. Belanger, P. Nordblad, A.R. King, V. Jaccarino, L. Lundgren, and O.Beckman, J. Magn. Magn. Mater. **31-34,** 1095

After converging to a stable configuration, the total energy per spin site was calculated as a function of the angle of the applied magnetic field, *θH*. Example results are shown in figure

Note that the horizontal axis (*θH* (deg)) is the same for all calculated structures, but the total calculated energy is not to scale in order to visibly depict the calculated results for each thickness of AFM1. A discontinuity in the total energy per spin site appears for thicknesses of AFM1 ranging from 1ML to 9ML. The values of angle *θH* for which the discontinuity appears (*θHdis*) is presented in Fig 8(b). Three regions can be distinguished in these results:

The different regions correspond to different spin arrangements within each AFM layer. An example for the 3ML thick AFM1 is shown in figure 9. The spin arrangements are presented in terms of the angle that each spin vector makes with the *z*-direction. Here, blue arrow represents the applied field, and red arrows are used for the spins in the three layers of

The comparison between a spin arrangement for *θH*=-950 (top diagrams) and for *θH*=-960 (bottom diagrams) is shown. The central diagram in figure 9 shows the spin arrangement when the field is applied along *+z* direction. The first layer of AFM1 experiences spincanting. An energy minimum for the fully compensated interface has been obtained for perpendicular interfacial coupling between the FM and AFM spins. The spin-canting and the resulting net moment induced within the first layer of the AFM1 always opposes the direction of the applied field. Moreover, spin-canting appears also in the second antiferromagnet AFM2 for the first two layers. It is important to note that the direction of the net moment induced in the first layer of AFM2 does not correlate directly to the

**Table 3.** Used bulk values of the exchange and anisotropy fields for the KNiF3 and FeF2.

Anisotropy field (kOe)

*anisotropy field exchange field*

0.33d

**Figure 8.** (a) The total energy per spin site as a function of the angle of the applied magnetic field θH for different thicknesses of the first AFM1. The horizontal axis (θH (deg)) is the same for all calculated structures, but the total calculated energy is not to scale in order to visibly depict the calculated results for each thickness of AFM1, (b) The values of angle θH for which the discontinuity appears (θHdis) for thicknesses of AFM1 ranging from 1ML to 9ML. The lines are guides to the eye.

Dynamic and Rotatable Exchange Anisotropy in Fe/KNiF3/FeF2 Trilayers 167

**6. References** 

(2004).

3186 (1996).

5510 (1999).

Phys. Rep. 422, 65 (2005).

(Springer, New York, 2006).

Heidelberg-Berlin, 1974).

[14] R.A. Erickson, Phys. Rev. 90, 779 (1953).

Schuller, Phys. Rev. B 75, 214412 (2007).

J. Dura, Phys. Rev. Lett. 84, 3986 (2000).

[26] N. C. Koon, Phys. Rev. Lett. 78, 4865 (1997).

Phys. Rev. B 65, 134436 (2002).

Phys. Rev. B.77, 184433 (2008).

Appl. Phys. 93, 6835 (2003)

thesis, Univ. of Chicago, 1964.

[1] J. Nogués, and I. K. Schuller, J. Magn. Magn. Mater. 192, 203 (1999). [2] A.E. Berkowitz, and K. Takano, J. Magn. Magn. Mater. 200, 552 (1999).

[5] J. Nogués, J. Sort, V. Langlais, V. Skumryev, S. Suriñach, J.S. Muñoz, and M.D. Baró,

[6] A.E. Berkowitz, and R.H. Kodama, in *Contemporary Concepts of Condensed Matter Science*, editted by D.L. Mills and J.A.C. Bland (Elsevier B.V. 2006), Chapter 5, p. 115. [7] J. Stöhr, and H.C. Siegmann, *Magnetism – From Fundamentals to Nanoscale Dynamics* 

[10] O. Muller, and R. Roy, *The Major Ternary Structural Families* (Springer, NewYork-

[16] W. Jauch, A. Palmer and A.J. Schultz, Acta Cryst. B49, 984 (1993); K. Haefner, Ph.D.

[18] M.R. Fitzsimmons, B.J. Kirby. S. Roy, Zhi-Pan Li, Igor V. Roshchin, S.K. Sinha, and I.K.

[19] J. Nogués, D. Lederman, T.J. Moran, I.K. Schuller, and K.V. Rao, Appl. Phys. Lett. 68,

[22] M. R. Fitzsimmons, C. Leighton, J. Nogués, A. Hoffmann, Kai Liu, C. F. Majkrzak, J.A. Dura, J. R. Groves, R. W. Springer, P. N. Arendt, V. Leiner, H. Lauter, and I.K. Schuller,

[23] S. Widuch, Z. Celinski, K. Balin, R. Schäfer, L. Schultz, D. Skrzypek, and J. McCord,

[24] L.Malkinski, T.O'Keevan, R.E.Camley, Z.Celinski, L.Wee, R.L.Stamps, D.Skrzypek, J.

[25] W.Pang, R.L.Stamps, L.Malkinski, Z.Celinski, D.Skrzypek, J. Appl. Phys. 95, 7309 (2004)

[27] T. C. Schulthess, and W. H. Butler, Phys. Rev. Lett. 81, 4516 (1998); J. Appl. Phys. 85,

[20] J. Nogués, T.J. Moran, D. Lederman, and I.K. Schuller, Phys. Rev. B 59, 6984 (1999). [21] M.R. Fitzsimmons, P. Yashar, C. Leighton, I.K Schuller, J. Nogués, C.F. Majchrzak, and

[8] L. Wee, R.L. Stamps, L. Malkinski, and Z. Celinski, Phys. Rev. B 69, 134426 (2004). [9] L. Wee, R.L. Stamps, L. Malkinski, Z. Celinski, and D. Skrzypek, Phys. Rev. B 69, 134425

[11] V. Scatturin, L. Corliss, N Elliott, and J. Hastings, Acta Cryst. 14, 19 (1961). [12] K. Hirakawa, K. Hirakawa, and T. Hashimoto, J. Phys. Soc. Jpn. 15, 2063 (1960).

[17] M.T. Hutchings, B.D.Rainford, and H.J. Guggenheim, J. Phys. C 3, 307 (1970).

[13] Z. Celinski, and D. Skrzypek, Acta Phys. Pol. A 65, 149 (1984).

[15] W. Stout and S.A. Reed, J. Am. Chem. Soc. 76, 5279 (1954).

[3] R.L. Stamps, J. Phys. D: Appl. Phys. 33, R247 (2000). [4] M. Kiwi, J. Magn. Magn. Mater. 234, 584 (2001).

**Figure 9.** The spin arrangements are presented in terms of the angle that each spin vector makes with the z-direction for the 3ML-thick AFM1. Blue represents the applied field Ha, red arrows are used for the spins in the three layers of AFM1 and green represent the first few spin sites of AFM2.

### **Author details**

Stefania Widuch *Center for Magnetism and Magnetic Nanostructures, University of Colorado, Colorado Springs, Colorado, USA School of Physics (M013), University of Western Australia (UWA), Crawley, WA, Australia Institute of Physics, University of Silesia, Katowice, Poland* 

Robert L. Stamps *School of Physics and Astronomy, Kelvin Building, University of Glasgow, Scotland, UK* 

Danuta Skrzypek *Institute of Physics, University of Silesia, Katowice, Poland* 

Zbigniew Celinski *Center for Magnetism and Magnetic Nanostructures, University of Colorado, Colorado Springs, Colorado, USA* 

## **Acknowledgement**

RLS acknowledges the Australian Research Council. The work at UCCS was supported by the National Science Foundation Grants (DMR 0605629 and DMR 0907053). SW acknowledges funding through the IEEE for the visit at the UWA Perth.

#### **6. References**

166 Ferromagnetic Resonance – Theory and Applications

**Author details** 

Stefania Widuch

Robert L. Stamps

Danuta Skrzypek

Zbigniew Celinski

**Acknowledgement** 

*Colorado, USA* 

*Colorado, USA* 

**Figure 9.** The spin arrangements are presented in terms of the angle that each spin vector makes with the z-direction for the 3ML-thick AFM1. Blue represents the applied field Ha, red arrows are used for

*Center for Magnetism and Magnetic Nanostructures, University of Colorado, Colorado Springs,* 

*School of Physics (M013), University of Western Australia (UWA), Crawley, WA, Australia* 

*School of Physics and Astronomy, Kelvin Building, University of Glasgow, Scotland, UK* 

*Center for Magnetism and Magnetic Nanostructures, University of Colorado, Colorado Springs,* 

RLS acknowledges the Australian Research Council. The work at UCCS was supported by the National Science Foundation Grants (DMR 0605629 and DMR 0907053). SW

acknowledges funding through the IEEE for the visit at the UWA Perth.

*Institute of Physics, University of Silesia, Katowice, Poland* 

*Institute of Physics, University of Silesia, Katowice, Poland* 

the spins in the three layers of AFM1 and green represent the first few spin sites of AFM2.

	- [28] M. D. Stiles, and R. D. McMichael, Phys Rev. B 59, 3722 (1999).
	- [29] M. D. Stiles, and R. D. McMichael, Phys. Rev. B 60, 12950 (1999).
	- [30] R. Camley, and R.L. Stamps, J. Phys.: Condens. Matter 5, 3727 (1993).

**Chapter 7** 

© 2013 Valenzuela et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**Microwave Absorption** 

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/56058

**1. Introduction** 

*Magnetic Nanoparticles* 

adsorption from aqueous systems.

**in Nanostructured Spinel Ferrites** 

Gabriela Vázquez-Victorio, Ulises Acevedo-Salas and Raúl Valenzuela

Magnetic nanoparticles (MNPs) are playing a crucial role in an extensive number of potential applications and science fields. Nanotechnology industry is rapidly growing with the promise that it will lead to significant economic and scientific impacts on a wide range of areas, such as health care, nanoelectronics, environmental remediation. MNPs are mostly ferrites, i.e, transition metal oxides with ferric ions as main constituent. Although the magnetic properties of ferrites [1] are less intense than metal's, especially saturation magnetization, ferrites possess a large chemical stability (corrosion resistance), high

The use of MNPs for biological and clinical applications [2] is undoubtedly one of the most challenging research areas in the field of nanomaterials, involving the organized collaboration of research teams formed by physicists, chemists, biologists, physiciens. The advantages of MNPs are based on their nanoscale size, large surface area, tailoring of magnetic properties and negligible side effects in living tissues. These applications include drug delivery [3], magnetic hyperthermia [4], magnetic resonance imaging [5], biosensors [6]. A field related with microwave absorption is electromagnetic interference EMI [7], as the number of electromagnetic radiation sources has growth at an exponential rate. MNPs have found applications also in environmental fields, such as soil remediation [8] and heavy metal removal [9], as MNPs provide high surface area and specific affinity for heavy metal

The reduction in scale leads to strong changes in macroscopic properties. The main reason can be attributed to the enhanced importance of the surface atom fraction as compared with core atom fraction, as the material becomes a nanoparticle. A simple estimate reveals that

and reproduction in any medium, provided the original work is properly cited.

electrical resistivity, and extended applicability at high magnetic field frequencies.

## **Microwave Absorption in Nanostructured Spinel Ferrites**

Gabriela Vázquez-Victorio, Ulises Acevedo-Salas and Raúl Valenzuela

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/56058

### **1. Introduction**

168 Ferromagnetic Resonance – Theory and Applications

[28] M. D. Stiles, and R. D. McMichael, Phys Rev. B 59, 3722 (1999). [29] M. D. Stiles, and R. D. McMichael, Phys. Rev. B 60, 12950 (1999). [30] R. Camley, and R.L. Stamps, J. Phys.: Condens. Matter 5, 3727 (1993).

### *Magnetic Nanoparticles*

Magnetic nanoparticles (MNPs) are playing a crucial role in an extensive number of potential applications and science fields. Nanotechnology industry is rapidly growing with the promise that it will lead to significant economic and scientific impacts on a wide range of areas, such as health care, nanoelectronics, environmental remediation. MNPs are mostly ferrites, i.e, transition metal oxides with ferric ions as main constituent. Although the magnetic properties of ferrites [1] are less intense than metal's, especially saturation magnetization, ferrites possess a large chemical stability (corrosion resistance), high electrical resistivity, and extended applicability at high magnetic field frequencies.

The use of MNPs for biological and clinical applications [2] is undoubtedly one of the most challenging research areas in the field of nanomaterials, involving the organized collaboration of research teams formed by physicists, chemists, biologists, physiciens. The advantages of MNPs are based on their nanoscale size, large surface area, tailoring of magnetic properties and negligible side effects in living tissues. These applications include drug delivery [3], magnetic hyperthermia [4], magnetic resonance imaging [5], biosensors [6]. A field related with microwave absorption is electromagnetic interference EMI [7], as the number of electromagnetic radiation sources has growth at an exponential rate. MNPs have found applications also in environmental fields, such as soil remediation [8] and heavy metal removal [9], as MNPs provide high surface area and specific affinity for heavy metal adsorption from aqueous systems.

The reduction in scale leads to strong changes in macroscopic properties. The main reason can be attributed to the enhanced importance of the surface atom fraction as compared with core atom fraction, as the material becomes a nanoparticle. A simple estimate reveals that

© 2013 Valenzuela et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

for a ~100 nm nanoparticle, surface atom fraction is about 6% of the total NP atoms, while for a ~5 nm NP this fraction can attain 78% [10]. Surface layer of materials exhibits different properties simply because these atoms have a very different structure than the core's. Surface atoms, for instance, have a reduced coordination number (unsatisfied bonding), crystal defects and modified crystal planes ("broken symmetry"). In the case of magnetically ordered materials (ferro, ferri, antiferromagnetic phases), additionally, several magnetic properties critically change at the nanometric scale. These properties are, for instance, the change from multidomain to single domain magnetic structure, domain wall thickness, the decrease in anisotropy energy giving rise to superparamagnetic phenomena. MNPs can thus exhibit many property changes with the reduction in size. Last (but no least), MNPs can show important macroscopic effects of interparticle magnetic interactions, which can involve additive forces (exchange), or attraction/repulsion (dipole).

Microwave Absorption in Nanostructured Spinel Ferrites 171

100 nm in size) with high densities. The behavior of electron paramagnetic resonance (EPR) of some relevant magnetically disordered MNPs systems is also presented. We devote a part of this review to the emerging low field microwave absorption technique (LFMA), which is a non-resonant method providing valuable information on magnetically-ordered materials. Results on MNPs in different aggregation states, as well as SPS-sintered materials are briefly

As FMR [16] and EPR [17] techniques are well known, so no additional treatment of them is

One of the most studied phenomena on MNP's is the change of their FMR spectra as a function of temperature. The main parameters describing FMR signal are plotted versus temperature and eventually compared with the bulk counterpart, as an attempt to characterize the changes associated with the nanometric scale. Such parameters are generally peak-to-peak resonance linewidth, ∆*H*pp, resonance field, *H*res, and the intensity or height of the resonant absorption signal. Often, a simple linear dependence with *T* is observed [18-23]. Magnetic and structural phase transitions appear as a discontinuous event

The resonance field behavior with temperature for bulk and MNP's decreases with decreasing temperature, as a consequence of the enhancement of the contributions to the internal field associated with magnetic ordering (mainly exchange and anisotropy). This effect is stronger for small particles (figure 1) [25], revealing additional contributions to the internal field at low temperatures as a consequence of the size decreasing. These contributions can be assumed as an extra unidirectional internal field arising from surface disorder, where the magnetization processes are presumably to be isotropic and causes an extra shift of the resonance field. Therefore, as the surface area increases with decreasing the particle size, the isotropic effects on the magnetic resonance behavior are more pronounced and an additional distribution of energy barriers, promoted by surface isotropic disorder,

As observed in many works [23,25-27], the FMR spectra for MNPs at intermediate temperatures results to be a mixing of two lines: a broad component corresponding to typical anisotropic contributions and a narrow one, presumably corresponding to the surface isotropic contributions. This leads to a characteristic FMR shape for nanoparticulated systems. The general features include a broad component (becoming wider and shifting to lower fields upon cooling, see figure 2), a narrow component, and a large broadening and shifting as the particle size decreases. Figure 2 shows the FMR signal evolution with temperature variation from room temperature down to 100 K for a well diluted magnetite suspension [26]. A double component spectrum is well observed at high temperatures, while its two components seem to overlap in a single broad signal as temperature goes down. This can be attributed to an important decrease on isotropic

**2. Ferromagnetic resonance (FMR) in ferrite nanoparticles** 

reviewed.

included here, except for some references.

*Temperature and size dependences* 

on this dependence [24].

must be assumed.

### *Synthesis of MNPs*

The most common method to synthesize MNPs are based on coprecipitation and microemulsion [11]. The coprecipitation method produces NPs by a pH change in a solution containing the desired metals in the form of nitrates or chlorides. Average size and size distribution, as well as shape depend on the pH and the ionic strength of the precipitating solution. In the microemulsion method, an aqueous metal solution phase is dispersed (entrapped) as microdroplets in a continuous oil phase within a micellar assembly of stabilizing surfactants. The advantage is that the microdroplets provide a confined space which limits the growth and agglomeration of NPs.

An emerging method for preparation of uniform NPs is the polyol technique, where metallic salts (acetates, oxalates), dissolved in an alcohol (such as diethylenglycol) are directly precipitated by high temperature decomposition [12]. This method can produce metals; by addition of a controlled amount of water, it can lead to oxide MNPs.

Spray and laser pyrolysis, with great commercial scale-up potential have been reported [13]. In spray pyrolysis, a solution of a ferric salt (and a reducing agent) is sprayed through a reactor to produce evaporation of the solvent within each droplet. In laser pyrolysis, the laser energy is used to heat a flowing mixture of gases leading to a chemical reaction. Under the appropriate conditions, homogeneous nucleation occurs and NPs are produced.

A different method utilizes high-energy ultrasound waves to create acoustic cavitations resulting in extremely hot spots. The sound waves produced by these cavities can lead to particle size reduction and hence the formation of NPs [14]. Other methods are based on electrochemical deposition of metal in a cathode [11], and also the use of magnetotactic bacteria [15].

### *Microwave absorption in MNPs*

In this brief review, some of the recent developments in microwave absorption in MNPs. The response of ferromagnetic resonance (FMR) of MNPs (under different conditions) is first reviewed. FMR results on consolidated materials by spark plasma sintering (SPS) techniques are included, as this method allows the preparation of nanostructured ferrites (grains under 100 nm in size) with high densities. The behavior of electron paramagnetic resonance (EPR) of some relevant magnetically disordered MNPs systems is also presented. We devote a part of this review to the emerging low field microwave absorption technique (LFMA), which is a non-resonant method providing valuable information on magnetically-ordered materials. Results on MNPs in different aggregation states, as well as SPS-sintered materials are briefly reviewed.

As FMR [16] and EPR [17] techniques are well known, so no additional treatment of them is included here, except for some references.

### **2. Ferromagnetic resonance (FMR) in ferrite nanoparticles**

### *Temperature and size dependences*

170 Ferromagnetic Resonance – Theory and Applications

*Synthesis of MNPs* 

bacteria [15].

*Microwave absorption in MNPs* 

for a ~100 nm nanoparticle, surface atom fraction is about 6% of the total NP atoms, while for a ~5 nm NP this fraction can attain 78% [10]. Surface layer of materials exhibits different properties simply because these atoms have a very different structure than the core's. Surface atoms, for instance, have a reduced coordination number (unsatisfied bonding), crystal defects and modified crystal planes ("broken symmetry"). In the case of magnetically ordered materials (ferro, ferri, antiferromagnetic phases), additionally, several magnetic properties critically change at the nanometric scale. These properties are, for instance, the change from multidomain to single domain magnetic structure, domain wall thickness, the decrease in anisotropy energy giving rise to superparamagnetic phenomena. MNPs can thus exhibit many property changes with the reduction in size. Last (but no least), MNPs can show important macroscopic effects of interparticle magnetic interactions, which can

The most common method to synthesize MNPs are based on coprecipitation and microemulsion [11]. The coprecipitation method produces NPs by a pH change in a solution containing the desired metals in the form of nitrates or chlorides. Average size and size distribution, as well as shape depend on the pH and the ionic strength of the precipitating solution. In the microemulsion method, an aqueous metal solution phase is dispersed (entrapped) as microdroplets in a continuous oil phase within a micellar assembly of stabilizing surfactants. The advantage is that the microdroplets provide a confined space

An emerging method for preparation of uniform NPs is the polyol technique, where metallic salts (acetates, oxalates), dissolved in an alcohol (such as diethylenglycol) are directly precipitated by high temperature decomposition [12]. This method can produce metals; by

Spray and laser pyrolysis, with great commercial scale-up potential have been reported [13]. In spray pyrolysis, a solution of a ferric salt (and a reducing agent) is sprayed through a reactor to produce evaporation of the solvent within each droplet. In laser pyrolysis, the laser energy is used to heat a flowing mixture of gases leading to a chemical reaction. Under

A different method utilizes high-energy ultrasound waves to create acoustic cavitations resulting in extremely hot spots. The sound waves produced by these cavities can lead to particle size reduction and hence the formation of NPs [14]. Other methods are based on electrochemical deposition of metal in a cathode [11], and also the use of magnetotactic

In this brief review, some of the recent developments in microwave absorption in MNPs. The response of ferromagnetic resonance (FMR) of MNPs (under different conditions) is first reviewed. FMR results on consolidated materials by spark plasma sintering (SPS) techniques are included, as this method allows the preparation of nanostructured ferrites (grains under

the appropriate conditions, homogeneous nucleation occurs and NPs are produced.

involve additive forces (exchange), or attraction/repulsion (dipole).

addition of a controlled amount of water, it can lead to oxide MNPs.

which limits the growth and agglomeration of NPs.

One of the most studied phenomena on MNP's is the change of their FMR spectra as a function of temperature. The main parameters describing FMR signal are plotted versus temperature and eventually compared with the bulk counterpart, as an attempt to characterize the changes associated with the nanometric scale. Such parameters are generally peak-to-peak resonance linewidth, ∆*H*pp, resonance field, *H*res, and the intensity or height of the resonant absorption signal. Often, a simple linear dependence with *T* is observed [18-23]. Magnetic and structural phase transitions appear as a discontinuous event on this dependence [24].

The resonance field behavior with temperature for bulk and MNP's decreases with decreasing temperature, as a consequence of the enhancement of the contributions to the internal field associated with magnetic ordering (mainly exchange and anisotropy). This effect is stronger for small particles (figure 1) [25], revealing additional contributions to the internal field at low temperatures as a consequence of the size decreasing. These contributions can be assumed as an extra unidirectional internal field arising from surface disorder, where the magnetization processes are presumably to be isotropic and causes an extra shift of the resonance field. Therefore, as the surface area increases with decreasing the particle size, the isotropic effects on the magnetic resonance behavior are more pronounced and an additional distribution of energy barriers, promoted by surface isotropic disorder, must be assumed.

As observed in many works [23,25-27], the FMR spectra for MNPs at intermediate temperatures results to be a mixing of two lines: a broad component corresponding to typical anisotropic contributions and a narrow one, presumably corresponding to the surface isotropic contributions. This leads to a characteristic FMR shape for nanoparticulated systems. The general features include a broad component (becoming wider and shifting to lower fields upon cooling, see figure 2), a narrow component, and a large broadening and shifting as the particle size decreases. Figure 2 shows the FMR signal evolution with temperature variation from room temperature down to 100 K for a well diluted magnetite suspension [26]. A double component spectrum is well observed at high temperatures, while its two components seem to overlap in a single broad signal as temperature goes down. This can be attributed to an important decrease on isotropic

contributions which causes the narrow component to disappear at low temperatures. In addition, decreasing temperature makes the broad component to widen, becoming more symmetric, and shifting to lower fields, thus revealing a random distribution of the anisotropy axis enhanced at low temperatures. Inter-particle interactions are negligible and do not contribute to this broadening since the high dilution of the studied suspension promotes isolation between particles.

Microwave Absorption in Nanostructured Spinel Ferrites 173

Figure 3 shows the absorption signals for different particle distribution sizes of noninteracting maghemite (γ-Fe2O3) nanoparticle ferrofluid samples at room temperature [27]. The absorption signal changes drastically with changes on the particle size. Two limit cases are observed. The upper limit, for the largest size particles (sample 2, *d* = 10 nm), corresponds to a wide line shifted to a lower field in comparison with the reference (ω/γ = 3.3 kOe, where γ is the gyromagnetic ratio for free electrons). This line seems to be characteristic for anisotropic contributions. On the contrary, for the smaller size particles (sample 6, *d* = 10 nm) the spectrum shows a much narrower line, apparently characteristic for isotropic contributions and not shifted from the reference. The FMR signals for intermediate cases result in a more complex double-feature shape spectrum (samples 3, 4 and 5). Such behavior may be explained in terms of a mixing of two overlapped signals, the broad anisotropic shifted line and the narrower one, isotropic and not shifted. The coexistence of these two different contributions reveals the presence of a core-shell structure

**Figure 3.** Influence of the particle size on X-band FMR spectra measured at room temperature for ZFC non-interacting maghemite (γ-Fe2O3) nanoparticle ferrofluid samples. The particle size decreases from sample 2 to 6; sample 2 has the largest particle size distribution (*d* = 10 nm) and sample 6 the smaller

for the studied nanoparticles.

particle size distribution (*d* = 4.8 nm) [27].

**Figure 1.** (a) Isotropic shift from the reference field (*H*res=3.3 kOe) as a function of temperature for ZFC non-interacting maghemite (γ-Fe2O3) nanoparticle ferrofluid samples: (■ ) *d* = 4.8 nm (▲ ) *d* = 10 nm. (b) Peak to peak linewidth *ΔH*pp as a function of temperature for ZFC non-interacting maghemite (γ-Fe2O3) nanoparticle ferrofluid samples: (□) *d* = 4.8 nm (Δ) *d* = 10 nm [25].

**Figure 2.** FMR signal evolution with temperature for a 0.1 wt% polymer Fe3O4 suspension [26].

Figure 3 shows the absorption signals for different particle distribution sizes of noninteracting maghemite (γ-Fe2O3) nanoparticle ferrofluid samples at room temperature [27]. The absorption signal changes drastically with changes on the particle size. Two limit cases are observed. The upper limit, for the largest size particles (sample 2, *d* = 10 nm), corresponds to a wide line shifted to a lower field in comparison with the reference (ω/γ = 3.3 kOe, where γ is the gyromagnetic ratio for free electrons). This line seems to be characteristic for anisotropic contributions. On the contrary, for the smaller size particles (sample 6, *d* = 10 nm) the spectrum shows a much narrower line, apparently characteristic for isotropic contributions and not shifted from the reference. The FMR signals for intermediate cases result in a more complex double-feature shape spectrum (samples 3, 4 and 5). Such behavior may be explained in terms of a mixing of two overlapped signals, the broad anisotropic shifted line and the narrower one, isotropic and not shifted. The coexistence of these two different contributions reveals the presence of a core-shell structure for the studied nanoparticles.

172 Ferromagnetic Resonance – Theory and Applications

promotes isolation between particles.

contributions which causes the narrow component to disappear at low temperatures. In addition, decreasing temperature makes the broad component to widen, becoming more symmetric, and shifting to lower fields, thus revealing a random distribution of the anisotropy axis enhanced at low temperatures. Inter-particle interactions are negligible and do not contribute to this broadening since the high dilution of the studied suspension

**Figure 1.** (a) Isotropic shift from the reference field (*H*res=3.3 kOe) as a function of temperature for ZFC non-interacting maghemite (γ-Fe2O3) nanoparticle ferrofluid samples: (■ ) *d* = 4.8 nm (▲ ) *d* = 10 nm. (b) Peak to peak linewidth *ΔH*pp as a function of temperature for ZFC non-interacting maghemite (γ-Fe2O3)

**Figure 2.** FMR signal evolution with temperature for a 0.1 wt% polymer Fe3O4 suspension [26].

nanoparticle ferrofluid samples: (□) *d* = 4.8 nm (Δ) *d* = 10 nm [25].

**Figure 3.** Influence of the particle size on X-band FMR spectra measured at room temperature for ZFC non-interacting maghemite (γ-Fe2O3) nanoparticle ferrofluid samples. The particle size decreases from sample 2 to 6; sample 2 has the largest particle size distribution (*d* = 10 nm) and sample 6 the smaller particle size distribution (*d* = 4.8 nm) [27].

### *Particle concentration dependence*

Double component FMR spectra have also been reported for solid and liquid Fe3O4 suspensions [26]. Important changes were observed on the absorption signal depending on the particle concentration. Figure 4 shows the decrease of the narrow component as the concentration increases. Both components exhibit also a broadening as the concentration increases. This important dependence of the signal linewidth with concentration is due to an increase of particle dipolar interactions at mean distances and a consequence of aggregation [28-30].

Microwave Absorption in Nanostructured Spinel Ferrites 175

**Figure 5.** Frequency and shape orientation dependence of the applied field, at which FMR occurs, for commercial NiFe2O4 nanoparticles, dispersed in KBr and then compressed with different packing volume fractions. The inset shows the evolution of the resonance field as a function of the packing

The anisotropy distributions and its FMR signal effects on MNP's can be carried out by measuring the angle variation between the cooling field of field-cooled (FC) samples. By comparing the obtained spectra for different orientations of FC samples, it is possible to determine the contribution of the anisotropy distribution within each particle to the FMR signal. The corresponding resonance lines often lead to a double component signal. It is then possible to separate the contributions which have a strong dependence with the angle from the weakly dependent counterpart. The narrow component can be attributed to surface anisotropy, while the the broad component should be associated with internal anisotropy of the NPs. Recent works have demonstrated the isotropic/anisotropic nature of most common MNP's, as described above. Figure 6 shows differences between FMR spectra for parallel and perpendicular configurations of FC samples. The samples consisted of maghemite nanoparticles with a particle diameter reported as 4.8 nm [27]. At the high temperature (top in figure 6), no variation as a function of θ is observed for the narrow component, while the broad component, on the contrary, shifts to higher resonance fields as the angle θ increases

(bottom in figure 6). The narrow component is not affected by angular variations.

The resonance field of the anisotropic component generally is plotted against the angle θ, the dependences with θ use to be fitted, in good agreement, with sin2 θ or cos2 θ functions [24-32]. Its well established the axial symmetry nature for this kind of functions, as magnetization processes perform the higher energy absorption (energy improved by the resonance field) when the magnetizing field is perpendicular to the axial orientation, the

volume fraction [31].

*Angular dependence* 

**Figure 4.** FMR signals for solid (a) and liquid (b) Fe3O4 suspensions at different concentrations [26].

Particle interactions always play an important role on the magnetic resonance absorption phenomena. FMR at different frequencies on dispersed KBr and then compressed NiFe2O4 commercial nanoparticles has recently been reported [31]. The resulting pellets showed strong shape anisotropy, as in-plane and out-of-plane analyzed measurements diverge with the increase of nanoparticle packing fraction (figure 5).

The resonance field decreases with increasing volume for in-plane measurements. In contrast, for out-of-plane measurements it increases with volume fraction. Both cases (in and out of plane), showed a resonance field shifted to a lower field in comparison with an ideal bulk as reported. As the packing fraction decreases, the in-plane and the out-of-plane curves converge to the same field value (≈ 0.04 T). This value can be identified as the average effective anisotropy field of the particles. A useful and powerful tool to estimate particle anisotropy can be based on these measurements.

**Figure 5.** Frequency and shape orientation dependence of the applied field, at which FMR occurs, for commercial NiFe2O4 nanoparticles, dispersed in KBr and then compressed with different packing volume fractions. The inset shows the evolution of the resonance field as a function of the packing volume fraction [31].

#### *Angular dependence*

174 Ferromagnetic Resonance – Theory and Applications

Double component FMR spectra have also been reported for solid and liquid Fe3O4 suspensions [26]. Important changes were observed on the absorption signal depending on the particle concentration. Figure 4 shows the decrease of the narrow component as the concentration increases. Both components exhibit also a broadening as the concentration increases. This important dependence of the signal linewidth with concentration is due to an increase of particle dipolar interactions at mean distances and a consequence of aggregation

**Figure 4.** FMR signals for solid (a) and liquid (b) Fe3O4 suspensions at different concentrations [26].

the increase of nanoparticle packing fraction (figure 5).

anisotropy can be based on these measurements.

Particle interactions always play an important role on the magnetic resonance absorption phenomena. FMR at different frequencies on dispersed KBr and then compressed NiFe2O4 commercial nanoparticles has recently been reported [31]. The resulting pellets showed strong shape anisotropy, as in-plane and out-of-plane analyzed measurements diverge with

The resonance field decreases with increasing volume for in-plane measurements. In contrast, for out-of-plane measurements it increases with volume fraction. Both cases (in and out of plane), showed a resonance field shifted to a lower field in comparison with an ideal bulk as reported. As the packing fraction decreases, the in-plane and the out-of-plane curves converge to the same field value (≈ 0.04 T). This value can be identified as the average effective anisotropy field of the particles. A useful and powerful tool to estimate particle

*Particle concentration dependence* 

[28-30].

The anisotropy distributions and its FMR signal effects on MNP's can be carried out by measuring the angle variation between the cooling field of field-cooled (FC) samples. By comparing the obtained spectra for different orientations of FC samples, it is possible to determine the contribution of the anisotropy distribution within each particle to the FMR signal. The corresponding resonance lines often lead to a double component signal. It is then possible to separate the contributions which have a strong dependence with the angle from the weakly dependent counterpart. The narrow component can be attributed to surface anisotropy, while the the broad component should be associated with internal anisotropy of the NPs. Recent works have demonstrated the isotropic/anisotropic nature of most common MNP's, as described above. Figure 6 shows differences between FMR spectra for parallel and perpendicular configurations of FC samples. The samples consisted of maghemite nanoparticles with a particle diameter reported as 4.8 nm [27]. At the high temperature (top in figure 6), no variation as a function of θ is observed for the narrow component, while the broad component, on the contrary, shifts to higher resonance fields as the angle θ increases (bottom in figure 6). The narrow component is not affected by angular variations.

The resonance field of the anisotropic component generally is plotted against the angle θ, the dependences with θ use to be fitted, in good agreement, with sin2 θ or cos2 θ functions [24-32]. Its well established the axial symmetry nature for this kind of functions, as magnetization processes perform the higher energy absorption (energy improved by the resonance field) when the magnetizing field is perpendicular to the axial orientation, the

resonance field also increases and finds a maximum at this position. The differences between the resonance field at θ = 0° and θ = 90°, are related to the anisotropy field, promoted by the respective axial anisotropy within the core of the nanoparticles revealing a mono-domain configuration on this region (figure 7).

Microwave Absorption in Nanostructured Spinel Ferrites 177

where *Hr* is the resonance field, *Ha* = 4|*K*|/*M* is the anisotropy field (with |*K*| the absolute value of the anisotropy constant and *M* the magnetization), and *Ho* measures the *g* value. The *g* value is a constant describing the relation between the energy of the microwave radiation and the *dc* magnetic field. For the parallel configuration the resonance field is:

 *Hr* = *Ho* – 4|*K*|/*M* (2)

**Figure 8.** Reduced magnetization dependence with temperature for a FC hematite NPs [24].

**Figure 9.** Resonance field dependence with temperature for hematite sample of Fig. 8 [24].

field. The parallel configuration can therefore be used to investigate phase transitions.

Magnetic phase transitions produce significant changes in |*K*|/*M* and hence on the resonance

**Figure 6.** X-band FMR spectra of maghemite FC nanoparticles dispersed on glycerol, with a 10 kOe cooling field. Solid and dashed lines correspond respectively, to configurations with the scanning field parallel (θ = 0°) and perpendicular (θ = 90°) to the cooling field [27].

**Figure 7.** Dependence of the resonance field with θ for field-cooled (FC) hematite NPs at 300 and 155 K (lower curve) [24].

The fitting of the angular resonance field dependence is useful to determine anisotropy parameters. The dependence can be given by [32]:

$$H\_r = H\_o - H\_a (1/2)(3\cos^2\theta - 1)\tag{1}$$

where *Hr* is the resonance field, *Ha* = 4|*K*|/*M* is the anisotropy field (with |*K*| the absolute value of the anisotropy constant and *M* the magnetization), and *Ho* measures the *g* value. The *g* value is a constant describing the relation between the energy of the microwave radiation and the *dc* magnetic field. For the parallel configuration the resonance field is:

176 Ferromagnetic Resonance – Theory and Applications

mono-domain configuration on this region (figure 7).

resonance field also increases and finds a maximum at this position. The differences between the resonance field at θ = 0° and θ = 90°, are related to the anisotropy field, promoted by the respective axial anisotropy within the core of the nanoparticles revealing a

**Figure 6.** X-band FMR spectra of maghemite FC nanoparticles dispersed on glycerol, with a 10 kOe cooling field. Solid and dashed lines correspond respectively, to configurations with the scanning field

**Figure 7.** Dependence of the resonance field with θ for field-cooled (FC) hematite NPs at 300 and 155 K

The fitting of the angular resonance field dependence is useful to determine anisotropy

�� ���� � ���1 2� ���������1� (1)

parallel (θ = 0°) and perpendicular (θ = 90°) to the cooling field [27].

parameters. The dependence can be given by [32]:

(lower curve) [24].

$$Hr = H\rho - 4\,\mathrm{l}\,\mathrm{K}\,\mathrm{l}\,\mathrm{/M}\tag{2}$$

**Figure 8.** Reduced magnetization dependence with temperature for a FC hematite NPs [24].

**Figure 9.** Resonance field dependence with temperature for hematite sample of Fig. 8 [24].

Magnetic phase transitions produce significant changes in |*K*|/*M* and hence on the resonance field. The parallel configuration can therefore be used to investigate phase transitions.

Figure 8 shows the temperature dependence of magnetization of FC hematite NPs, parallel to the cooling field [24]. A change in the slope is observed in good agreement with observed transitions from a disordered phase to a magnetically ordered phase. In contrast with bulk hematite (antiferromagnetic above 260 K), these hematite NPs exhibited a superparamagnetic behavior, which can be explained in terms of a weakly ferromagnet below ~ 200 K, as shown in figure 8. The reported transition is also observed in plots of the resonance field against temperature (figure 9) and other FMR spectrum parameters, which suggest the FMR technique as a useful tool to investigate phase transitions on MNP's.

Microwave Absorption in Nanostructured Spinel Ferrites 179

**Figure 11.** FMR spectra of Ni-Zn ferrites at 300 K, in monodisperse state (Sample A), and clusters

Ferrite NPs have to be consolidated as a high density solid for many applications (in electronic devices, for instance), where a powder is unstable. Typical sintering processes needing high temperatures are difficult to apply, as NPs tend to grow very rapidly at temperatures above 500°C, losing the nanometric size range and thus the different properties associated with this size range. A particularly well suited method to consolidate NPs into a high density nanostructured solid preserving grains within the nanometric range is Spark Plasma Sintering (SPS for short) [33]. Also known as pulsed electric current sintering (PECS), in this technique the sample (typically in the form of a powder) is placed in a graphite die and pressed by two punches at pressures in the 200 MPa range, while a strong electric current goes through the system; the die is shown in Fig. 12. SPS therefore consolidates powders under the simultaneous action of pressure and electric current pulses

The electric current pulses result in a very rapid heating of the sample, at rates as high as 1000°C/min. If the sample is a good conductor, current goes through it and the heating is even more efficient. A significant point is that the electric current has a significant impact on the atomic diffusion during the process [34]. The sintering process can then reach high densities at very low temperatures and extremely short times [35]. Obviously, SPS can also

SPS has been used to consolidate spinel [37], garnet [38], and hexagonal ferrites [39]. In the case of spinel Ni0.5Zn0.5Fe2O4 ferrites, samples prepared in the form of 6-8 nm nanoparticles by the forced hydrolysis in a polyol method [12], were consolidated by SPS at temperatures in the 350-500°C range by times as short as 5 min. Just for comparison, the typical conditions for sintering in the classic solid state reaction are 1200°C for at least 4 hours. NPs growth

**3. Ferromagnetic resonance in nanostructured ferrites** 

be utilized for reactive sintering involving a chemical reaction [36].

(typically of a few milliseconds in duration [33]).

(sample B) [18].

#### *Effects of the aggregation state*

Ferrite NPs can show the effects of two extreme aggregation states, namely monodisperse NPs, and clusters of a few hundreds of NPs. Samples of composition Zn0.5Ni0.5Fe2O4 were prepared by the polyol method, and designated as "A" for monodisperse state, and "B" for clusters. FMR spectra exhibited significant differences, as shown in Fig. 10. The decrease in the effects of surface for sample B appear in the form of a lower resonance field and an increase in the linewidth, as compared with sample A. The former can be understood in terms of a larger internal field in the cluster sample as a consequence of the aggregation of NPs; simply, the NPs inside the cluster tend to behave as grains in a bulk material. The effects of surface (crystal defects, unsatisfied bonds, etc.) are relieved by the presence of other NPs. Magnetic exchange interactions among neighboring NPs can also be assumed, which increases the internal field and thereby decreases the applied magnetic field needed to fulfill the Larmor equation conditions. On the other hand, these interactions increase the linewidth, especially the random distribution of anisotropy axes.

**Figure 10.** FMR spectra of Ni-Zn ferrites at 77 K, in monodisperse state (Sample A), and clusters (sample B) [18].

At room temperature, as shown in Fig. 11, isolated NPs show a superparamagnetic phase and again a larger resonance field with a significant reduction in linewidth as the factors just mentioned, associated with an ordered magnetic structure, are absent. The cluster sample exhibits a reduced linewidth, but always larger than the superparamagnetic state.

**Figure 11.** FMR spectra of Ni-Zn ferrites at 300 K, in monodisperse state (Sample A), and clusters (sample B) [18].

### **3. Ferromagnetic resonance in nanostructured ferrites**

178 Ferromagnetic Resonance – Theory and Applications

*Effects of the aggregation state* 

(sample B) [18].

Figure 8 shows the temperature dependence of magnetization of FC hematite NPs, parallel to the cooling field [24]. A change in the slope is observed in good agreement with observed transitions from a disordered phase to a magnetically ordered phase. In contrast with bulk hematite (antiferromagnetic above 260 K), these hematite NPs exhibited a superparamagnetic behavior, which can be explained in terms of a weakly ferromagnet below ~ 200 K, as shown in figure 8. The reported transition is also observed in plots of the resonance field against temperature (figure 9) and other FMR spectrum parameters, which suggest the FMR technique as a useful tool to investigate phase transitions on MNP's.

Ferrite NPs can show the effects of two extreme aggregation states, namely monodisperse NPs, and clusters of a few hundreds of NPs. Samples of composition Zn0.5Ni0.5Fe2O4 were prepared by the polyol method, and designated as "A" for monodisperse state, and "B" for clusters. FMR spectra exhibited significant differences, as shown in Fig. 10. The decrease in the effects of surface for sample B appear in the form of a lower resonance field and an increase in the linewidth, as compared with sample A. The former can be understood in terms of a larger internal field in the cluster sample as a consequence of the aggregation of NPs; simply, the NPs inside the cluster tend to behave as grains in a bulk material. The effects of surface (crystal defects, unsatisfied bonds, etc.) are relieved by the presence of other NPs. Magnetic exchange interactions among neighboring NPs can also be assumed, which increases the internal field and thereby decreases the applied magnetic field needed to fulfill the Larmor equation conditions. On the other hand, these interactions increase the

linewidth, especially the random distribution of anisotropy axes.

**Figure 10.** FMR spectra of Ni-Zn ferrites at 77 K, in monodisperse state (Sample A), and clusters

exhibits a reduced linewidth, but always larger than the superparamagnetic state.

At room temperature, as shown in Fig. 11, isolated NPs show a superparamagnetic phase and again a larger resonance field with a significant reduction in linewidth as the factors just mentioned, associated with an ordered magnetic structure, are absent. The cluster sample Ferrite NPs have to be consolidated as a high density solid for many applications (in electronic devices, for instance), where a powder is unstable. Typical sintering processes needing high temperatures are difficult to apply, as NPs tend to grow very rapidly at temperatures above 500°C, losing the nanometric size range and thus the different properties associated with this size range. A particularly well suited method to consolidate NPs into a high density nanostructured solid preserving grains within the nanometric range is Spark Plasma Sintering (SPS for short) [33]. Also known as pulsed electric current sintering (PECS), in this technique the sample (typically in the form of a powder) is placed in a graphite die and pressed by two punches at pressures in the 200 MPa range, while a strong electric current goes through the system; the die is shown in Fig. 12. SPS therefore consolidates powders under the simultaneous action of pressure and electric current pulses (typically of a few milliseconds in duration [33]).

The electric current pulses result in a very rapid heating of the sample, at rates as high as 1000°C/min. If the sample is a good conductor, current goes through it and the heating is even more efficient. A significant point is that the electric current has a significant impact on the atomic diffusion during the process [34]. The sintering process can then reach high densities at very low temperatures and extremely short times [35]. Obviously, SPS can also be utilized for reactive sintering involving a chemical reaction [36].

SPS has been used to consolidate spinel [37], garnet [38], and hexagonal ferrites [39]. In the case of spinel Ni0.5Zn0.5Fe2O4 ferrites, samples prepared in the form of 6-8 nm nanoparticles by the forced hydrolysis in a polyol method [12], were consolidated by SPS at temperatures in the 350-500°C range by times as short as 5 min. Just for comparison, the typical conditions for sintering in the classic solid state reaction are 1200°C for at least 4 hours. NPs growth

was controlled, as the final grain size in the consolidated ferrite was about 60 nm, even for the highest (500°C) SPS temperature [40].The densities reached values as high as 94% of the theoretical value. FMR spectra obtained at 77 and 300 K are shown in Fig 13, together with the FMR signal corresponding to the original NPs. The resonance field exhibited a decrease as the SPS temperature increased, which can be explained in terms of the components of the total field in the Larmor expression, = *H*. The main components of the total field are *H* = *H*DC + *H*X + *H*a + …, where *H*DC is the applied field, *H*X is the exchange field responsible for magnetic ordering, and *H*a is the anisotropy field. As magnetic ordering sets in, *H*X and *H*<sup>a</sup> increase and therefore the applied field is decreased in order to fulfill the Larmor resonance conditions. As sintering progresses, the surface effects decrease and the material has the tendency to behave as a bulk ferrite.

Microwave Absorption in Nanostructured Spinel Ferrites 181

All the samples exhibited a large broadening in the linewidth, generally interpreted by considering a random distribution of the anisotropy axis in single domain NPs [41]. The broadening decreases as consolidation increases, as the surface effects are diminished by formation of grain boundaries. At room temperature the linewidth decreases since all the samples approach the paramagnetic state where the internal field is eliminated and only the applied field is involved in the Larmor expression. The NPs signal exhibited the lowest linewidth as at room temperature these NPs are superparamagnetic; all the consolidated samples showed a ferrimagnetic behavior at 300 K as compared with the as-produced

Zinc ferrite is a very good material to investigate the site occupancy by cations. In spite of a relatively large cation radius, Zn+2 has a tendency to occupy tetrahedral sites, while Fe3+ fill octahedral sites [1], in other words, a "normal" spinel. This arrangement leads to B-O-B superexchange interactions between iron cations and therefore to an antiferromagnetic structure with a Néel temperature about 9 K. In the case of nanosized Zn ferrite, however, the cation distribution can be significantly different; some degree of inversion occurs [1,42], with a fraction of Zn2+ on octahedral sites, and hence Fe3+ on both sites. The spinel becomes

The variations in linewidth, *B*PP, and *g*-factor, *g*eff in these ferrites depend mainly on the interparticle magnetic dipole–dipole interactions and intraparticle super-exchange interactions. On the other hand, the interparticle superexchange between magnetic ions (through oxygen) can reduce the value of linewidth. The magnitude of this interaction is determined by the relative position of metallic and oxygen ions. When the distance between the metallic and oxygen ions is short, the metal cations have an exactly half-filled orbital, and the angle between these two bonds is close to 180*◦*, the superexchange interaction is the

Several factors can produce a distribution of local field, such as unresolved hyperfine structure, g-value anisotropy, strain distribution, crystal defects. The field strength on a particular spin is then modulated by local field distribution and leads to an additional

The strain distribution produced by a small average particle size can cause a small resonance signal [1]. As the particle size is increased, such small signal disappears and only the broad signal remains. This suggests that the particle size has a threshold value above

In order to gain some insight into the relationship between internal structure and EPR spectra, it can be useful to compare the properties of several iron-based oxide NPs embedded in a polytethylene matrix, prepared by the same method: Fe2O3, BaFe2O4, and BaFe12O19 [46]. The experimental EPR spectra of the samples are presented in Figs. (14-20). At room temperature the EPR spectra of all the samples show a ''two-line pattern'' (Fig. 14)

sample which has a blocking temperature about 90 K.

ferromagnetic, with a Curie temperature well above 9 K.

strongest [43].

linewidth broadening [44,45].

which the strains are relieved.

**4. Paramagnetic resonance in ferrite nanoparticles** 

**Figure 12.** Schematic of spark plasma sintering apparatus [33].

**Figure 13.** Ferromagnetic resonance of Ni-Zn ferrites prepared by a chimie douce method followed by SPS at temperatures in the 350-500°C range for 5 min. The FMR response of the original NPs is also shown for comparison. (Adapted from [40]).

All the samples exhibited a large broadening in the linewidth, generally interpreted by considering a random distribution of the anisotropy axis in single domain NPs [41]. The broadening decreases as consolidation increases, as the surface effects are diminished by formation of grain boundaries. At room temperature the linewidth decreases since all the samples approach the paramagnetic state where the internal field is eliminated and only the applied field is involved in the Larmor expression. The NPs signal exhibited the lowest linewidth as at room temperature these NPs are superparamagnetic; all the consolidated samples showed a ferrimagnetic behavior at 300 K as compared with the as-produced sample which has a blocking temperature about 90 K.

### **4. Paramagnetic resonance in ferrite nanoparticles**

180 Ferromagnetic Resonance – Theory and Applications

tendency to behave as a bulk ferrite.

**Figure 12.** Schematic of spark plasma sintering apparatus [33].

shown for comparison. (Adapted from [40]).

**Figure 13.** Ferromagnetic resonance of Ni-Zn ferrites prepared by a chimie douce method followed by SPS at temperatures in the 350-500°C range for 5 min. The FMR response of the original NPs is also

was controlled, as the final grain size in the consolidated ferrite was about 60 nm, even for the highest (500°C) SPS temperature [40].The densities reached values as high as 94% of the theoretical value. FMR spectra obtained at 77 and 300 K are shown in Fig 13, together with the FMR signal corresponding to the original NPs. The resonance field exhibited a decrease as the SPS temperature increased, which can be explained in terms of the components of the total field in the Larmor expression, = *H*. The main components of the total field are *H* = *H*DC + *H*X + *H*a + …, where *H*DC is the applied field, *H*X is the exchange field responsible for magnetic ordering, and *H*a is the anisotropy field. As magnetic ordering sets in, *H*X and *H*<sup>a</sup> increase and therefore the applied field is decreased in order to fulfill the Larmor resonance conditions. As sintering progresses, the surface effects decrease and the material has the

> Zinc ferrite is a very good material to investigate the site occupancy by cations. In spite of a relatively large cation radius, Zn+2 has a tendency to occupy tetrahedral sites, while Fe3+ fill octahedral sites [1], in other words, a "normal" spinel. This arrangement leads to B-O-B superexchange interactions between iron cations and therefore to an antiferromagnetic structure with a Néel temperature about 9 K. In the case of nanosized Zn ferrite, however, the cation distribution can be significantly different; some degree of inversion occurs [1,42], with a fraction of Zn2+ on octahedral sites, and hence Fe3+ on both sites. The spinel becomes ferromagnetic, with a Curie temperature well above 9 K.

> The variations in linewidth, *B*PP, and *g*-factor, *g*eff in these ferrites depend mainly on the interparticle magnetic dipole–dipole interactions and intraparticle super-exchange interactions. On the other hand, the interparticle superexchange between magnetic ions (through oxygen) can reduce the value of linewidth. The magnitude of this interaction is determined by the relative position of metallic and oxygen ions. When the distance between the metallic and oxygen ions is short, the metal cations have an exactly half-filled orbital, and the angle between these two bonds is close to 180*◦*, the superexchange interaction is the strongest [43].

> Several factors can produce a distribution of local field, such as unresolved hyperfine structure, g-value anisotropy, strain distribution, crystal defects. The field strength on a particular spin is then modulated by local field distribution and leads to an additional linewidth broadening [44,45].

> The strain distribution produced by a small average particle size can cause a small resonance signal [1]. As the particle size is increased, such small signal disappears and only the broad signal remains. This suggests that the particle size has a threshold value above which the strains are relieved.

> In order to gain some insight into the relationship between internal structure and EPR spectra, it can be useful to compare the properties of several iron-based oxide NPs embedded in a polytethylene matrix, prepared by the same method: Fe2O3, BaFe2O4, and BaFe12O19 [46]. The experimental EPR spectra of the samples are presented in Figs. (14-20). At room temperature the EPR spectra of all the samples show a ''two-line pattern'' (Fig. 14)

which is typical of superparamagnetic resonance (SPR) spectra. The relative intensity of these lines depends on the particle size and shape distribution function [47]. For the Fe2O3 and BaFe2O4 samples, the broad line predominates in the room temperature spectra; the opposite is observed for the BaFe12O19 sample, where the narrow line is more pronounced.

Microwave Absorption in Nanostructured Spinel Ferrites 183

At room temperature, the spectra of all samples show a "two-line pattern" (Figure 15) which is typical of superparamagnetic resonance. These spectra can be considered as a broader line superimposed on a narrow line. The relative intensity of these lines depends on the particle size and shape distribution function, as well as on the magnitude of the magnetic anisotropy. For Fe2O3 and BaFe2O4 samples, the broad line predominates in the RT spectra. This line is characterized by a peak-to-peak linewidth of *H* ~ 850 Oe and an effective *g*value of 2.07. In the RT spectrum of BaFe12O19 sample, in contrast, the narrow line is more

At low temperatures, the EPR spectra of Fe2O3 change significantly (Fig.16). On cooling below 100 K, the broad line S1 shows a monotonous increase of the linewidth ∆*H* and a decrease of the amplitude *A*. However, below about 50 K new resonances, S2 and S3, appear in the spectra of Fe2O3 (Figs. 16 and 17). It is interesting that S2 behaves like a typical paramagnetic resonance signal, namely, the amplitude increases and the linewidth decreases as the temperature is diminished. The EPR spectra suggest that Fe2O3 nanoparticles contain both ferromagnetic and antiferromagnetic phases. The weak EPR signal of the rhombic symmetry (g ≈ 4.3) that appears at low temperatures may be attributable to an -Fe2O3 phase

visible, with *H* ~ 120 Oe and *g* ~ 2.0.

that undergoes an antiferromagnetic-like transition near 6 K. [46]

**Figure 16.** EPR spectra of Fe2O3 nanoparticles at different temperatures. [46]

temperatures.

EPR spectra of the BaFe2O4 nanoparticles at different temperatures are shown in Fig. 18. At all temperatures the spectra are broad and rather asymmetric. A significant shift of the line position to low magnetic fields and a marked spectrum broadening are observed at low

On the other hand, the thermal variations of EPR spectra of the BaFe12O19 sample is typical of superparamagnetic resonance. The relatively narrow line that dominates at room

**Figure 14.** EPR spectra of zinc ferrite sintered at different temperatures.

**Figure 15.** Room temperature EPR spectra of nanoparticles: Fe2O3 (curve1), BaFe2O4 (curve 2), and BaFe12O19 (curve3) [46].

At room temperature, the spectra of all samples show a "two-line pattern" (Figure 15) which is typical of superparamagnetic resonance. These spectra can be considered as a broader line superimposed on a narrow line. The relative intensity of these lines depends on the particle size and shape distribution function, as well as on the magnitude of the magnetic anisotropy. For Fe2O3 and BaFe2O4 samples, the broad line predominates in the RT spectra. This line is characterized by a peak-to-peak linewidth of *H* ~ 850 Oe and an effective *g*value of 2.07. In the RT spectrum of BaFe12O19 sample, in contrast, the narrow line is more visible, with *H* ~ 120 Oe and *g* ~ 2.0.

182 Ferromagnetic Resonance – Theory and Applications

Relative Absorbance (

 )

**Figure 14.** EPR spectra of zinc ferrite sintered at different temperatures.

Magnetic Field

BaFe12O19 (curve3) [46].

**Figure 15.** Room temperature EPR spectra of nanoparticles: Fe2O3 (curve1), BaFe2O4 (curve 2), and

which is typical of superparamagnetic resonance (SPR) spectra. The relative intensity of these lines depends on the particle size and shape distribution function [47]. For the Fe2O3 and BaFe2O4 samples, the broad line predominates in the room temperature spectra; the opposite is observed for the BaFe12O19 sample, where the narrow line is more pronounced.

> At low temperatures, the EPR spectra of Fe2O3 change significantly (Fig.16). On cooling below 100 K, the broad line S1 shows a monotonous increase of the linewidth ∆*H* and a decrease of the amplitude *A*. However, below about 50 K new resonances, S2 and S3, appear in the spectra of Fe2O3 (Figs. 16 and 17). It is interesting that S2 behaves like a typical paramagnetic resonance signal, namely, the amplitude increases and the linewidth decreases as the temperature is diminished. The EPR spectra suggest that Fe2O3 nanoparticles contain both ferromagnetic and antiferromagnetic phases. The weak EPR signal of the rhombic symmetry (g ≈ 4.3) that appears at low temperatures may be attributable to an -Fe2O3 phase that undergoes an antiferromagnetic-like transition near 6 K. [46]

**Figure 16.** EPR spectra of Fe2O3 nanoparticles at different temperatures. [46]

EPR spectra of the BaFe2O4 nanoparticles at different temperatures are shown in Fig. 18. At all temperatures the spectra are broad and rather asymmetric. A significant shift of the line position to low magnetic fields and a marked spectrum broadening are observed at low temperatures.

On the other hand, the thermal variations of EPR spectra of the BaFe12O19 sample is typical of superparamagnetic resonance. The relatively narrow line that dominates at room

temperature disappears as temperature decreases (Figs. 19 and 20). The BaFe12O19 nanoparticles reveal an EPR signal that is signicantly narrowed at high temperatures by superparamagnetic uctuations. This is evidence of the reduced magnetic anisotropy energy that may be due to the particle's nanosize (effective diameter <10 nm).

Microwave Absorption in Nanostructured Spinel Ferrites 185

**Figure 19.** Low-temperature EPR spectra of BaFe12O19 nanoparticles. [46]

**Figure 20.** EPR spectra of BaFe12O19 nanoparticles at different temperatures. [46]

Low field microwave absorption (LFMA for short) refers to the non-resonant, hysteretical losses of a material subjected to a high frequency electromagnetic field. Recently, it has become a useful method to investigate magnetization processes [48], magnetoelastic effects [49], phase transitions [50], non-aligned ferromagnetic resonance [51,52], spin arrangements [53,54]. LFMA is similar to giant magnetoimpedance (GMI) [55], but physically different to

**5. Low Field Microwave Absorption (LFMA)** 

**Figure 17.** EPR spectra of Fe2O3 nanoparticles at low temperatures. [46]

**Figure 18.** EPR spectra of the BaFe2O4 nanoparticles at different temperatures. [46]

These results show that EPR spectra at low temperatures are desirable for the correct identification of NPs and a comparison with high temperature experiments allows a better understanding of phenomena related with variations associated with nanosized materials.

**Figure 19.** Low-temperature EPR spectra of BaFe12O19 nanoparticles. [46]

temperature disappears as temperature decreases (Figs. 19 and 20). The BaFe12O19 nanoparticles reveal an EPR signal that is signicantly narrowed at high temperatures by superparamagnetic uctuations. This is evidence of the reduced magnetic anisotropy energy

that may be due to the particle's nanosize (effective diameter <10 nm).

**Figure 17.** EPR spectra of Fe2O3 nanoparticles at low temperatures. [46]

**Figure 18.** EPR spectra of the BaFe2O4 nanoparticles at different temperatures. [46]

These results show that EPR spectra at low temperatures are desirable for the correct identification of NPs and a comparison with high temperature experiments allows a better understanding of phenomena related with variations associated with nanosized materials.

**Figure 20.** EPR spectra of BaFe12O19 nanoparticles at different temperatures. [46]

### **5. Low Field Microwave Absorption (LFMA)**

Low field microwave absorption (LFMA for short) refers to the non-resonant, hysteretical losses of a material subjected to a high frequency electromagnetic field. Recently, it has become a useful method to investigate magnetization processes [48], magnetoelastic effects [49], phase transitions [50], non-aligned ferromagnetic resonance [51,52], spin arrangements [53,54]. LFMA is similar to giant magnetoimpedance (GMI) [55], but physically different to

ferromagnetic resonance (FMR) [56]). GMI, generally defined as the variations of impedance of a magnetic conductor carrying an alternate electrical current when subjected to an external magnetic field [57], extends into a very wide frequency range. Clearly, GMI is a non-resonant phenomenon as confirmed by two facts: GMI does not fulfill the resonant Larmor conditions, and exhibits magnetic hysteresis.

Microwave Absorption in Nanostructured Spinel Ferrites 187

the 150 K-240K temperature range [63]. The LFMA sign changed from negative at ~ 154 K to

Zn0.65Ni0.35Fe2

O4

100 150 200 250 300 350 400 450

T (K)

**Figure 22.** Fig. 21. Correlation between the field on LFMA critical points (divided by 2) and a direct calculation of magnetocrystalline anisotropy in Zn0.65Ni0.35Fe2O4 bulk ferrites [60]. *K*1 anisotropy constant

**Figure 23.** LFMA from a Co-rich CoFeBSi glass-covered amorphous microwire [59]. This is an example of a "negative" LFMA signal, mostly observed in conductor materials (compare with Fig. W above).

LFMA can provide useful insights into the structure of NPs. When two different aggregation states are compared, i.e., monodisperse and clustered NPs with the same composition and NP diameter, clearly different spectra are obtained. By varying the synthesis conditions in

calculated

experimental

0

data for the calculation was obtained from ref [61].

50

100

150

HK (Oe)

200

250

300

positive at *T* ≥ 240 K.

*LFMA in NPs* 

**Figure 21.** Typical "positive" LFMA signal from a Ni-Zn bulk ferrite (Adapted from [59]).

LFMA is associated with magnetization processes in magnetically ordered materials, in the process from the unmagnetized state to the magnetic saturation. In bulk ferro and ferrimagnetic materials, LFMA exhibits a flat response in the paramagnetic phase. To measure experimentally LFMA in a typical FMR/EPR facility, the applied field has to be cycled; usually between -1 kOe and +1kOe is enough. Also, a device to compensate for the remanent magnetization is needed in most electromagnets.

An important parameter is, of course, the total anisotropy field of the particular sample. In most cases, LFMA exhibits a critical behavior at the total anisotropy field in the form of a maximum and a minimum, leading to a characteristic signal as shown in Fig. 21 [59]. In bulk Ni-Zn ferrite, a correlation exists between the magnetocrystalline anisotropy and the halfpeak-to-peak, measurement of LFMA. Figure 22 shows a comparison between the peak-topeak LFMA field (divided by 2) [60] and a calculation from magnetocrystalline constant, *K*1,published results [61]. The small differences can be attributed to shape anisotropy, as the calculation is based solely on *K*1.

LFMA seems to be associated with spin structure. By convention, this signal can be assigned as "positive", simply because it has the same shape than the FMR/EPR signal, i.e., a maximum and a minimum when observed from left to right. A positive LFMA sign has been observed in most insulator and semiconductor materials, while a "negative" signal appears for most metallic conductors, as shown in Fig. 23, for a Co-rich CoFeBSi amorphous microwire [59]. An interesting result was found in bulk Ni-Zn ferrites showing the Yafet-Kittel triangular arrangement [62], by measuring the LFMA as a function of temperature in the 150 K-240K temperature range [63]. The LFMA sign changed from negative at ~ 154 K to positive at *T* ≥ 240 K.

**Figure 22.** Fig. 21. Correlation between the field on LFMA critical points (divided by 2) and a direct calculation of magnetocrystalline anisotropy in Zn0.65Ni0.35Fe2O4 bulk ferrites [60]. *K*1 anisotropy constant data for the calculation was obtained from ref [61].

**Figure 23.** LFMA from a Co-rich CoFeBSi glass-covered amorphous microwire [59]. This is an example of a "negative" LFMA signal, mostly observed in conductor materials (compare with Fig. W above).

#### *LFMA in NPs*

186 Ferromagnetic Resonance – Theory and Applications

dP/d

calculation is based solely on *K*1.

H (a.u.)

Larmor conditions, and exhibits magnetic hysteresis.

Ni0.36Zn0.64Fe2

remanent magnetization is needed in most electromagnets.

O4

ferromagnetic resonance (FMR) [56]). GMI, generally defined as the variations of impedance of a magnetic conductor carrying an alternate electrical current when subjected to an external magnetic field [57], extends into a very wide frequency range. Clearly, GMI is a non-resonant phenomenon as confirmed by two facts: GMI does not fulfill the resonant


LFMA is associated with magnetization processes in magnetically ordered materials, in the process from the unmagnetized state to the magnetic saturation. In bulk ferro and ferrimagnetic materials, LFMA exhibits a flat response in the paramagnetic phase. To measure experimentally LFMA in a typical FMR/EPR facility, the applied field has to be cycled; usually between -1 kOe and +1kOe is enough. Also, a device to compensate for the

An important parameter is, of course, the total anisotropy field of the particular sample. In most cases, LFMA exhibits a critical behavior at the total anisotropy field in the form of a maximum and a minimum, leading to a characteristic signal as shown in Fig. 21 [59]. In bulk Ni-Zn ferrite, a correlation exists between the magnetocrystalline anisotropy and the halfpeak-to-peak, measurement of LFMA. Figure 22 shows a comparison between the peak-topeak LFMA field (divided by 2) [60] and a calculation from magnetocrystalline constant, *K*1,published results [61]. The small differences can be attributed to shape anisotropy, as the

LFMA seems to be associated with spin structure. By convention, this signal can be assigned as "positive", simply because it has the same shape than the FMR/EPR signal, i.e., a maximum and a minimum when observed from left to right. A positive LFMA sign has been observed in most insulator and semiconductor materials, while a "negative" signal appears for most metallic conductors, as shown in Fig. 23, for a Co-rich CoFeBSi amorphous microwire [59]. An interesting result was found in bulk Ni-Zn ferrites showing the Yafet-Kittel triangular arrangement [62], by measuring the LFMA as a function of temperature in

T= 362 K

H(Oe)

**Figure 21.** Typical "positive" LFMA signal from a Ni-Zn bulk ferrite (Adapted from [59]).

LFMA can provide useful insights into the structure of NPs. When two different aggregation states are compared, i.e., monodisperse and clustered NPs with the same composition and NP diameter, clearly different spectra are obtained. By varying the synthesis conditions in

the forced hydrolysis in a polyol method [12], Ni-Zn ferrites can be obtained as monodisperse, well crystallized ~ 6 nm NPs on one hand, and labeled as sample "A"; on the other, clusters about ~ 100 NPs constituted by NPs with the same composition and diameter [58], labeled as sample "B". It is interesting to mention that high resolution transmission electron microscopy (HRTEM) showed some epitaxial arrangements within the clusters of sample B.

Microwave Absorption in Nanostructured Spinel Ferrites 189

LFMA signal at low temperatures towards a positive signal at temperatures close to room temperature [63]. Inspection of Figs. 24-26 reveals that in the case of sample A at 77 K (monodisperse NPs), no YK arrangement is manifested. Some evolution is found in sample B (clusters) at low T, but no full inversion of LFMA signal is observed. A preliminary explanation of these findings is that the surface effects inhibit the formation of the YK structure. Another possibility is based on the fact that cation distribution in NPs as synthesized by the polyol method is often different [42] as compared with bulk ferrites (prepared by the solid state reaction at high temperatures). A different cation distribution (in the present case a non-negligible concentration of Zn on B sites, for instance) could hence

**Figure 25.** LFMA signal from epitaxial clusters of Zn0.5Ni0.5Fe2O4 ferrites, formed by a few hundreds of

Sample B

decreasing

T= 77 K

> HDC

increasing


HDC (Oe)

**Figure 26.** LFMA signal of epitaxial clusters of Zn0.5Ni0.5Fe2O4 ferrites, formed by a few hundreds of ~ 6

HDC

lead to different conditions for the formation of the YK structure.

~ 6 nm NPs, at 300 K.

nm NPs, at 77 K.

dP/dH (a.u.)

Sample A showed a superparamagnetic behavior at room temperature, and a blocking temperature about 50 K [58]. Sample B, in contrast, exhibited a ferromagnetic behavior up to 300 K (blocking temperature > 300 K), in spite of being constituted by NPs of the same composition and NP diameter. Clusters effectively decrease the effects of surface producing samples with a general behavior between that of NPs and bulk materials.

Figure 24 shows the LFMA signal from sample A at 77 and 300 K. AT the low temperature, a positive LFMA behavior is observed, corresponding to a non-conductor material. At 300 K, however, a flat response appears, associated with the superparamagnetic phase of these monodisperse NPs. Clearly, the superparamagnetic phase behaves like a paramagnetic signal. This flat signal with a small slope is associated with the microwave absorption of non-interacting dipoles [64].

**Figure 24.** LFMA signal from monodisperse, ~ 6 nm diameter, Zn0.5Ni0.5Fe2O4 NPs, at 77 and 300 K (adapted from [58]).

Clusters of the same NPs, in contrast, showed a clear positive LFMA signal at room temperature, see Fig. 254. This is consistent, as this sample B behaved as a ferromagnetic phase close to the bulk properties.

For low temperatures, however, sample B exhibited a very different signal as shown in Fig. 26, with an important hysteresis and no similarity to either positive or negative character. As mentioned above, bulk Ni-Zn ferrites with Zn content *x* > 0.5 (in Zn*x*Ni1-*x*Fe2O4) show a triangular Yafet-Kittel spin arrangement, which appears as an evolution from negative LFMA signal at low temperatures towards a positive signal at temperatures close to room temperature [63]. Inspection of Figs. 24-26 reveals that in the case of sample A at 77 K (monodisperse NPs), no YK arrangement is manifested. Some evolution is found in sample B (clusters) at low T, but no full inversion of LFMA signal is observed. A preliminary explanation of these findings is that the surface effects inhibit the formation of the YK structure. Another possibility is based on the fact that cation distribution in NPs as synthesized by the polyol method is often different [42] as compared with bulk ferrites (prepared by the solid state reaction at high temperatures). A different cation distribution (in the present case a non-negligible concentration of Zn on B sites, for instance) could hence lead to different conditions for the formation of the YK structure.

188 Ferromagnetic Resonance – Theory and Applications

sample B.

non-interacting dipoles [64].

dP/dH (a.u.)

phase close to the bulk properties.

(adapted from [58]).

the forced hydrolysis in a polyol method [12], Ni-Zn ferrites can be obtained as monodisperse, well crystallized ~ 6 nm NPs on one hand, and labeled as sample "A"; on the other, clusters about ~ 100 NPs constituted by NPs with the same composition and diameter [58], labeled as sample "B". It is interesting to mention that high resolution transmission electron microscopy (HRTEM) showed some epitaxial arrangements within the clusters of

Sample A showed a superparamagnetic behavior at room temperature, and a blocking temperature about 50 K [58]. Sample B, in contrast, exhibited a ferromagnetic behavior up to 300 K (blocking temperature > 300 K), in spite of being constituted by NPs of the same composition and NP diameter. Clusters effectively decrease the effects of surface producing

Figure 24 shows the LFMA signal from sample A at 77 and 300 K. AT the low temperature, a positive LFMA behavior is observed, corresponding to a non-conductor material. At 300 K, however, a flat response appears, associated with the superparamagnetic phase of these monodisperse NPs. Clearly, the superparamagnetic phase behaves like a paramagnetic signal. This flat signal with a small slope is associated with the microwave absorption of


HDC (Oe)

**Figure 24.** LFMA signal from monodisperse, ~ 6 nm diameter, Zn0.5Ni0.5Fe2O4 NPs, at 77 and 300 K

Clusters of the same NPs, in contrast, showed a clear positive LFMA signal at room temperature, see Fig. 254. This is consistent, as this sample B behaved as a ferromagnetic

For low temperatures, however, sample B exhibited a very different signal as shown in Fig. 26, with an important hysteresis and no similarity to either positive or negative character. As mentioned above, bulk Ni-Zn ferrites with Zn content *x* > 0.5 (in Zn*x*Ni1-*x*Fe2O4) show a triangular Yafet-Kittel spin arrangement, which appears as an evolution from negative

77 K

300 K

samples with a general behavior between that of NPs and bulk materials.

Sample A

**Figure 25.** LFMA signal from epitaxial clusters of Zn0.5Ni0.5Fe2O4 ferrites, formed by a few hundreds of ~ 6 nm NPs, at 300 K.

**Figure 26.** LFMA signal of epitaxial clusters of Zn0.5Ni0.5Fe2O4 ferrites, formed by a few hundreds of ~ 6 nm NPs, at 77 K.

### **6. Conclusions**

Microwave absorption (MA) is a very sensitive phenomenon and has become an extremely powerful characterization tool. MA accurately depends on all the factors surrounding unpaired electrons; it can play a significant role in the characterization of the complex and fascinating development of magnetic nanoparticles. In this brief review, recent results on the characterization of magnetic nanoparticles and consolidated spinel ferrites by means of ferromagnetic resonance, paramagnetic resonance, and low field microwave absorption have been presented.

Microwave Absorption in Nanostructured Spinel Ferrites 191

[9] Hua M, Zhang S, Pan B, Zang W, Lv L, Zhang Q. Heavy metal removal from water/wastewater by nanosized metal oxides. Journal of Hazardous Materials 2012; 211-

[10] Hosokawa M, Nogi K, Naito M, Yokoyama T. Nanoparticle Technology Handbook

[11] Tartaj P, Morales MP, Gonzalez-Carreño V, Veintemillas-Verdaguer S, Serna CJ. Advances in magnetic nanoparticles for biotechnology applications. Journal of

[12] Beji Z., Ben Chaabane T., Smiri L.S., Ammar S., Fiévet F., Jouini N., and Grenèche J.M. Synthesis of nickel-zinc ferrite nanoparticles in polyol: morphological, structural and

[13] Tartaj P, Morales MP, Gonzalez-Carreño V, Veintemillas-Verdaguer S, Serna CJ. The preparation of magnetic nanoparticles for applications in biomedicine. Journal of

[14] Dang F, Enomoto N, Hojo J, Enpuku K. A novel method to synthesize monodispersed

[15] Matsunaga T, Okamura Y, Tanaka T. Biotechnological application of nano-scale engineered bacterial magnetic particles. Journal of Materials Chemistry 2004; 14(14)

[16] Guimaraes AP. Magnetism and magnetic resonance in solids. Wiley VCH (1998). (ISBN:

[17] Well JA, Bolton JR. Electron paramagnetic resonance: elementary theory and practical applications 2nd edition. John Wiley and Sons, 2007. (ISBN: 047175496X); Brustolon MR. Electron paramagnetic resonance: a practitioner toolkit, 1st edition. John Wiley and Sons

[18] Valenzuela R, Herbst F, Ammar S. Ferromagnetic resonance in Ni–Zn ferrite nanoparticles in different aggregation states. Journal of Magnetism and Magnetic

[19] Guskos N, Zolnierkiewicz G ,Typek J, Guskos A, Czech Z. FMR study of γ-Fe2O3 agglomerated nanoparticles dispersed in glues. Reviews on Advanced Materials

[20] Thirupathi G, Singh R. Magnetic Properties of Zinc Ferrite Nanoparticles. Institute of Electrical and Electronics Engineers Transactions on Magnetics 2012; 48 3630-3633. [21] De Biasi E, Lima E, Ramos C A, Butera A, Zysler R D. Effect of thermal fluctuations in FMR experiments in uniaxial magnetic nanoparticles: Blocked vs superparamagnetic

regimes . Journal of Magnetism and Magnetic Materials 2013; 326(1) 138-146. [22] Sobón M, Lipinski I E, Typek J, Guskos A. FMR Study of Carbon Coated Cobalt Nanoparticles Dispersed in a Paraffin Matrix. Solid State Phenomena 2007; 128 193-198. [23] Shames A I, Rozenberg E, Sominski E, Gedanken A. Nanometer size effects on magnetic order in La12xCaxMnO3 (x = 0.5 and 0.6) manganites, probed by ferromagnetic resonance. Journal of Applied Physics 2012; 111 07D701-1-07D701-2.

212 317-331.

2099-2105.

0471197742).

(2009) (ISBN: 0470258829).

Science 2007;14 57-60.

Materials 2012; 324(21) 3398-3401.

(Elsevier, Amsterdam) 2007.

Magnetism and Magnetic Materials 2005; 290(1) 28-34.

Physics D: Applied Physics 2003; 36 R182.

magnetic properties. physica status solidi (a) 2006; 203(3) 504-512.

magnetite nanoparticles. Chemical Letters 2008; 37(5) 530-531.

### **Author details**

Gabriela Vázquez-Victorio, Ulises Acevedo-Salas and Raúl Valenzuela *Institute for Materials Research, National Autonomous University of México, México* 

## **Acknowledgement**

Authors acknowledge partial support for this work from ANR-CONACyT grant 139292, as well as PAPIIT-UNAM grant IN141012.

### **7. References**


[9] Hua M, Zhang S, Pan B, Zang W, Lv L, Zhang Q. Heavy metal removal from water/wastewater by nanosized metal oxides. Journal of Hazardous Materials 2012; 211- 212 317-331.

190 Ferromagnetic Resonance – Theory and Applications

Microwave absorption (MA) is a very sensitive phenomenon and has become an extremely powerful characterization tool. MA accurately depends on all the factors surrounding unpaired electrons; it can play a significant role in the characterization of the complex and fascinating development of magnetic nanoparticles. In this brief review, recent results on the characterization of magnetic nanoparticles and consolidated spinel ferrites by means of ferromagnetic resonance, paramagnetic resonance, and low field microwave absorption

Authors acknowledge partial support for this work from ANR-CONACyT grant 139292, as

[1] Valenzuela R. Magnetic Ceramics. Cambridge University Press (2005) (ISBN: 0-521-

[2] Amiri S, Shokrollahi H. The role of cobalt ferrite magnetic nanoparticles in medical

[3] Mahmoudi M, Sant S, Wang B, Laurent S, Sen T. Superparamagnetic iron oxide nanoparticles (SPIONs): development, surface modification and applications.

[4] Kashevsky BE, Agabekov BE, Kashevsky SB, Kekalo KA, Manina EY, Prokhorov IV, Ulashchik VS. Study of cobalt ferrite nanosuspensions for low frequency ferromagnetic

[5] Kliza E, Strijkers GJ, Nicolay K. Multifunctional magnetic resonance imaging probes.

[6] Xu Y, Wang E. Electrochemical biosensors based on magnetic micro/nanoparticles.

[7] Dahawan SK, Singh K, Bakhshi AK, Ohlan A. Conducting polymer embedded with nanoferrite and titanium dioxide nanoparticles for microwave absorption. Synthetic

[8] Liu R, Lal R. Nanoenhanced materials for reclamation of mine lands and other degrade soils: a review. Journal of Nanotechnology 2012; 461468-1-461468-18. doi:

Gabriela Vázquez-Victorio, Ulises Acevedo-Salas and Raúl Valenzuela

science. Materials Science and engineering C 2013: 33(1) 1-8.

Advanced Drug Delivery Reviews 2011: 63(1) 24-46-

Recent Results in Cancer Research 2013; 187 151-190.

hyperthermia. Particuology 2008; 6(5) 322-333.

Electrochimica Acta 2012; 84(1) 62-73.

Metals 2009, 159(21-22) 2259-2262.

10.1155/2012/461468.

*Institute for Materials Research, National Autonomous University of México, México* 

**6. Conclusions** 

have been presented.

**Author details** 

**Acknowledgement** 

**7. References** 

01843-9).

well as PAPIIT-UNAM grant IN141012.

	- [24] Owens F. Ferromagnetic resonance observation of a phase transition in magnetic field aligned Fe2O3 nanoparticles. Journal of Magnetism and Magnetic Materials 2009; 321(15) 2386-2391.

Microwave Absorption in Nanostructured Spinel Ferrites 193

[39] Nakamura T, Okano Y, Tabuchi M, Takeuchi T. Synthesis of hexagonal ferrite via spark plasma sintering technique. Journal of the Japan Society of Powder and Powder

[40] Valenzuela R, Beji Z, Herbst F, and Ammar S. Ferromagnetic resonance behavior of spark plasma sintered Ni-Zn ferrite nanoparticles produced by a chemical route.

[41] Sukhov A, Usadel KD, and Nowak U. Ferromagnetic resonance in an ensemble of nanoparticles with randomly distributed anisotropy axes. Journal of Magnetism and

[42] Ammar S, Jouini N, Fièvet F, Beji Z, Smiri L, Moliné P, Danot M, Grenèche JM. Magnetic properties of zin-ferrite nanoparticles synthesized by hydrolysis in a polyol

[43] Sui Y, Xu DP, Zheng FL and Su WH. Electrons spin resonance study of NiFe2O4 nanosolids compacted under high pressure. Journal of Applied Physics 1996; 80 719-

[44] A. Abragam and B. Bleaney, *Electron Paramagnetic Resonance of Transition Ions* 

[45] C. P. Poole and H. A. Farach, *Relaxation in Magnetic Resonance* (Academic Press, London,

[46] Koksharov Yu. A, Pankratov D A., Gubin SP, Kosobudsky ID, Beltran M, Khodorkovsky Y and Tishin AM. Electron Paramagnetic Resonance of Ferrite

[47] Kliava J and Berger R. Size and Shape Distribution of Magnetic Nanoparticles in Disordered Systems: Computer Simulations of Superparamagnetic Resonance Spectra.

[48] Valenzuela R, Alvarez G, Montiel H, Gutiérrez MP, Mata-Zamora ME, Barrón F., Sanchez AY, Betancourt I, Zamorano R. Characterization of magnetic materials by lowfield microwave absorption techniques. Journal of Magnetism and Magnetic Materials

[49] Montiel H, Alvarez G, Gutiérrez MP, Zamorano R, and Valenzuela R. The effect of metal-to-glass ratio on the low field microwave absorption at 9.4 GHz of glass coated

[51] Prinz G.A., Rado G.T. and Krebs J.J. Magnetic properties of single-crystal {110} iron films grown on GaAs by molecular beam epitaxy. Journal of Applied Physics 1982; 53

[52] Gerhardter F, Li Yi, and Baberschke K. Temperature-dependent ferromagneticresonance study in ultrahigh vacuum: magnetic anisotropies of thin iron films. Physical

[53] Valenzuela R. The temperature behavior of resonant and non-resonant microwave absorption in Ni-n ferrites. In *Electromagnetic Waves* / Book 1, InTech Open Access

CoFeBSi microwires. IEEE Transactions on Magnetics 2006; 42(10) 3380-3382. [50] Montiel H., Alvarez G., Gutiérrez M.P., Zamorano R., and Valenzuela R. Microwave absorption in Ni-Zn ferrites through the Curie transition. Journal of Alloys and

medium. Journal of Physics: Condensed Matter 2006; 18(39), 9055-9069.

Journal of Applied Physics 2011; 109(\*) 07A329-1-07A329-3.

Nanoparticles. Journal of Applied Physics 2001;89 2293-2298.

Journal of Magnetism and Magnetic Materials 1999; 205 328-342.

Metallurgy 2001; 48(2) 166-169.

Magnetic Materials 2008; 320(1) 31-33.

(Clarendon Press, Oxford,1970).

2008; 320(14) 1961-1965.

(3) 2087-2091.

Compounds 2004; 369(1) 141-143.

Review B 1993; 47(17) 11204-11210.

723.

1971).


[39] Nakamura T, Okano Y, Tabuchi M, Takeuchi T. Synthesis of hexagonal ferrite via spark plasma sintering technique. Journal of the Japan Society of Powder and Powder Metallurgy 2001; 48(2) 166-169.

192 Ferromagnetic Resonance – Theory and Applications

321(15) 2386-2391.

181-184.

[24] Owens F. Ferromagnetic resonance observation of a phase transition in magnetic field aligned Fe2O3 nanoparticles. Journal of Magnetism and Magnetic Materials 2009;

[25] Gazeau F, Bacri J C, Gendron F, Perzynski R, Raikher Yu L, Stepanov V I, Dubois E. Magnetic resonance of ferrite nanoparticles: Evidence of surface effects. Journal of

[26] Noginov M, Noginova N, Amponsah O, Bah R, Rakhimov R, Atsarkin V A. Magnetic resonance in iron oxide nanoparticles: Quantum features and effect of size. Journal of

[27] Gazeau F, Bacri J C, Gendron F, Perzynski R, Raikher Yu L, Stepanov V I, Dubois E. Magnetic resonance of nanoparticles in a ferrofluid: Evidence of thermofluctuational

[28] Edelman I, Petrakovskaja E, Petrov D, Zharkov S, Khaibullin R, Nuzhdin V, Stepanov A. FMR and TEM studies of Co and Ni nanoparticles implanted in the SiO2 matrix.

[29] Martinez B, Obradors X, Balcells Ll, Rouanet A, Monty C. Low temperature surface spin-glass transition in gamma-Fe2O3 nanoparticles. Physical Review Letters 1998; 80

[30] Winkler E, Zysler R D, Fiorani D. Surface and magnetic interaction effects in Mn3O4

[31] Song H, Mulley S, Coussens N, Dhagat P, Jander V. Effect of packing fraction on ferromagnetic resonance in NiFe2O4 nanocomposites. Journal of Applied Physics 2012;

[32] De Biasi R, Devezas T. Anisotropy field of small magnetic particles as measured by

[33] Munir Z.A., Quach D.V. and Ohyanagi M. Electric current activation of sintering: a review of the pulsed electric current sintering process. Journal of the American Ceramic

[34] Anselmi-Tamburini U, Garay JE and Munir ZA. Fundamental investigation on the spark plasma sintering/synthesis process III. Current effect on reactivity. Materilas

[35] Mizuguchi T, Guo S and Kagawa Y. Transmission electron microscopy characterization of spark plasma sintered ZrB2 ceramicc. Ceramics International 2010; 36(3) 943-946. [36] Regaieg Y, Delaizir G, Herbst F, Sicard L, Monnier J, Montero D, Villeroy B, Ammar-Merah S, Cheikhrouhou A, Godart C, Koubaa M. Rapid solid state synthesis by spark plasma sintering and magnetic properties of LaMnO3 perovskite manganite. Materials

[37] Valenzuela R, Ammar S, Nowak S, and Vázquez G. Low field microwave absorption in nanostructured ferrite ceramics consolidated by spark plasma sintering. Journal of

[38] Gaudisson T, Acevedo U, Nowak S, Yaacoub N, Greneche JM, Ammar S, Valenzuela R. Combining soft chemistry and Spark Plasma Sintering to produce highly dense and

finely grained soft ferrimagnetic Y3Fe5O12 (YIG) ceramics. (to be published).

Superconductivity and Novel Magnetism 2012; 25(7) 2389-2393.

effects. Journal of Magnetism and Magnetic Materials 1999; 202(2-3) 535-546.

Magnetism and Magnetic Materials 1998; 186(2) 175-187.

Magnetism and Magnetic Materials 2008; 320(18) 2228-2232.

nanoparticles. Physical Review B 2004; 70 174406-1-174406-5.

resonance. Journal of Applied Physics 1978; 49 2466-2470.

Applied Magnetic Resonance 2011; 40 363-375.

111 (07E348) 07E348\_1 - 07E348\_3.

Science and Engineering A 2005; 407 24-30.

Society 2011; 94(1) 1-19.

Letters 2012; 80(1) 195-198.


Publisher, edited by Vitaliy Zhurbenko, pp. 387 - 402 (2011), ISBN: 978 – 953 – 307 – 304 - 0. Online June 24, 2011 at: http://www.intechopen.com/articles/show/title/thetemperature-behavior-of-resonant-and-non-resonant-microwave-absorption-in-ni-znferrites.

**Chapter 8** 

© 2013 Ohkoshi et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**Unusual Temperature Dependence of Zero-Field** 

Insulating magnetic materials absorb electromagnetic waves. This absorption property is one of the important functions of magnetic materials, which is widely applied in our daily life as electromagnetic wave absorbers to avoid electromagnetic interference problems [1-5]. For example, spinel ferrites are used as absorbers for the present Wi-Fi communication, which uses 2.4 GHz and 5 GHz frequency waves. With the development of information technology, the demand is rising for sending heavy data such as high-resolution images at high speed. Recently, high-frequency electromagnetic waves in the frequency range of 30– 300 GHz, called millimeter waves, are drawing attention as a promising carrier for the next generation wireless communication. For example, 76 GHz is an important frequency, which is beginning to be used for vehicle radars. There are also new audio products coming to use, applying millimeter wave communication in the 60 GHz region [6,7]. However, there had been no magnetic material that could absorb millimeter waves above 80 GHz before our

Well-known forms of Fe2O3 are α-Fe2O3 and γ-Fe2O3, commonly called as hematite and maghemite, respectively. However, our research group first succeeded in preparing a pure phase of ε-Fe2O3, which is a rare phase of iron oxide Fe2O3 that is scarcely found in nature [8–10]. Since then, its physical properties have been actively studied, and one of the representative properties is the gigantic coercive field (*H*c) of 20 kilo-oersted (kOe) at room temperature [11–18]. We have also reported metal-substituted ε-Fe2O3 (ε-*Mx*Fe2–*x*O3, *M* = In, Ga, Al, and Rh), and showed that this series absorb millimeter waves from 35–209 GHz at room temperature due to zero-field ferromagnetic resonance (so called natural resonance) [19-29]. ε-Fe2O3 based magnet is expected to be a leading absorbing material for the future

and reproduction in any medium, provided the original work is properly cited.

wireless communication using higher frequency millimeter waves.

**Ferromagnetic Resonance in Millimeter Wave** 

**Region on Al-Substituted ε-Fe2O3**

Marie Yoshikiyo, Asuka Namai and Shin-ichi Ohkoshi

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/55779

**1. Introduction** 

report on ε-Fe2O3.

