**Unusual Temperature Dependence of Zero-Field 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**

194 Ferromagnetic Resonance – Theory and Applications

Physics 2006; 99 103904-1-103904-4.

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[54] Alvarez G., Montiel H., Barron J.F., Gutiérrez M.P., Zamorano R. Yafet-Kittel-type magnetic ordering in Ni0.35Zn0.65Fe2O4 ferrite detected by magnetosensitive microwave absorption measurements. Journal of Magnetism and Magnetic Materials 2009; 322(3)

[55] Montiel H, Alvarez G, Betancourt I, Zamorano R. and Valenzuela R. Correlation between low-field microwave absorption and magnetoimpedance in Co-based

[56] Barandiarán M., García-Arribas, A., and de Cos, D. Transition from quasistatic to ferromagnetic resonance regime in giant magnetoimpedance. Journal of Applied

[57] Knobel M, Kraus L, and Vázquez M. Magnetoimpedance. Handbook of Magnetic

[58] Valenzuela R., Ammar S., Herbst F., Ortega-Zempoalteca R. Low field microwave absorption in Ni-Zn ferrite nanoparticles in different aggregation states. Nanoscience

[59] Valenzuela R., Montiel H., Alvarez G., and Zamorano R. Low-field non-resonant microwave absorption in glass-coated Co-rich microwires. Physica Status Solidi A 2009;

[61] Broese van Groenou A., Schulkes J.A., and Annis D.A. Magnetic anisotropy in some

[62] Yafet Y. and Kittel, C. Antiferromagnetic arrangements in ferrites. Physical Review

[63] Alvarez G., Montiel H., Barrón J.F., Gutiérrez M.P. Zamorano R. Yafet-Kittel-type magnetic ordering on NiZnFe2O4 ferrite detected by magnetosensitive microwave absorption measurements. Journal of Magnetism and Magnetic Materials 2010; 322(3)

[64] Alvarez G., Font R., Portelles J., Zamorano R., and Valenzuela R. Microwave power absorption as a function of temperature and magnetic field in the ferroelectromagnet Pb(Fe1/2Nb )O3. Journal of the Physics and Chemistry of Solids 2007; 68(7) 1436-1442.

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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 report on ε-Fe2O3.

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 wireless communication using higher frequency millimeter waves.

© 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, and reproduction in any medium, provided the original work is properly cited.

In this chapter, we first introduce the synthesis, crystal structure, magnetic properties, and the formation mechanism of the original ε-Fe2O3 [8–10]. Then we report the physical properties of Al-substituted ε-Fe2O3, mainly focusing on its millimeter wave absorption properties due to zero-field ferromagnetic resonance. The resonance frequency was widely controlled from 112–182 GHz by changing the aluminum substitution ratio [23]. Furthermore, from a scientific point of view, temperature dependence of zero-field ferromagnetic resonance was investigated and was found to show an anomalous behavior caused by the spin reorientation phenomenon [28].

Unusual Temperature Dependence of Zero-Field

Ferromagnetic Resonance in Millimeter Wave Region on Al-Substituted ε-Fe2O3 197

With this synthesis method, rod-shaped ε-Fe2O3 is obtained due to the effect of Ba2+ ions, which adsorb on particular planes of ε-Fe2O3, inducing growth towards one direction. Spherical ε-Fe2O3 nanoparticles can also be synthesized by a different method without Ba2+ ions, which is an impregnation method using mesoporous silica nanoparticles [17,29,30]. Methanol and water solution containing Fe(NO3)3 is immersed into mesoporous silica and

The crystal structure of ε-Fe2O3 is shown in Figure 2a. It has an orthorhombic crystal structure (space group *Pna*21) with four non-equivalent Fe sites, A, B, C, and D sites. A, B, and C sites are six-coordinated octahedral sites, and D site is a four-coordinated tetrahedral site. ε-Fe2O3 exhibits spontaneous magnetization at a Curie temperature (*T*C) of 500 K. Figure 2b presents magnetization versus external magnetic field curve at 300 K, which shows a huge *H*c value of 20 kOe. Before this finding, the largest *H*c value among metal oxide was 6 kOe of barium ferrite, BaFe12O19 [31], which indicates that the *H*c of ε-Fe2O3 is over three times larger. The magnetic structure has been investigated using molecular field theory, which indicated that B and C sites have positive sublattice magnetizations, and A and D sites have negative sublattice magnetizations [32]. This result was consistent with the experimental results from neutron diffraction measurements, Mössbauer spectroscopy measurements, etc. [13,14], and was also consistent with first-principles calculation results

**Figure 2.** (a) Crystal structure of ε-Fe2O3. Dark blue, purple, light blue, and pink polyhedrons indicate A, B, C, and D sites, respectively. (b) Magnetization versus external magnetic field curve of ε-Fe2O3 at

Here we discuss the formation mechanism of ε-Fe2O3 from the viewpoint of phase transformation. By changing the sintering temperature in the present synthesis, a phase transformation of γ-Fe2O3 → ε-Fe2O3 → α-Fe2O3 was observed accompanied by an increase of particle size. γ- and α-Fe2O3 are very common phases of Fe2O3, and it has been well known that γ-Fe2O3 transforms directly into α-Fe2O3 in a bulk form. In the present case, it is considered that ε-Fe2O3 appeared as a stable phase at an intermediate size region due to the

300 K. Inset is a schematic illustration of the sublattice magnetizations of each site.

**2.2. Formation mechanism of ε-Fe2O3**

heated in air at 1200°C for 4 hours. The etching process is the same as above.

[33].

### **2. ε-Fe2O3**

This section introduces the synthesis, crystal structure, magnetic properties, and the formation mechanism of ε-Fe2O3. ε-Fe2O3 had only been known as impurity in iron oxide materials, and its properties were clarified for the first time after our success in the synthesis of single-phase ε-Fe2O3 in 2004 [8].

### **2.1. Synthesis, crystal structure, and magnetic properties of ε-Fe2O3**

Single-phase ε-Fe2O3 nanoparticles are synthesized by a chemical method, combining reversemicelle and sol-gel techniques (Figure 1) [8−10,16]. In the reverse-micelle step, two reversemicelle systems, A and B, are formed by cetyl trimethyl ammonium bromide (CTAB) and 1 butanol in *n*-octane. Reverse-micelle A contains aqueous solution of Fe(NO3)3 and Ba(NO3)2, and reverse-micelle B contains NH3 aqueous solution. These two microemulsion systems are mixed under rapid stirring. Tetraethoxysilane (C2H5O)4Si is then added to this solution, which forms SiO2 matrix around the Fe(OH)3 nanoparticles through 20 hours of stirring. The precipitation is separated by centrifugation and sintered in air at 1000C for 4 hours. The SiO2 matrix is removed by stirring in NaOH solution at 60C for 24 hours.

**Figure 1.** Synthetic procedure of ε-Fe2O3 nanomagnets using a combination method of reverse-micelle and sol-gel techniques. The inset is a transmission electron microscopy image of ε-Fe2O3 nanorods.

With this synthesis method, rod-shaped ε-Fe2O3 is obtained due to the effect of Ba2+ ions, which adsorb on particular planes of ε-Fe2O3, inducing growth towards one direction. Spherical ε-Fe2O3 nanoparticles can also be synthesized by a different method without Ba2+ ions, which is an impregnation method using mesoporous silica nanoparticles [17,29,30]. Methanol and water solution containing Fe(NO3)3 is immersed into mesoporous silica and heated in air at 1200°C for 4 hours. The etching process is the same as above.

The crystal structure of ε-Fe2O3 is shown in Figure 2a. It has an orthorhombic crystal structure (space group *Pna*21) with four non-equivalent Fe sites, A, B, C, and D sites. A, B, and C sites are six-coordinated octahedral sites, and D site is a four-coordinated tetrahedral site. ε-Fe2O3 exhibits spontaneous magnetization at a Curie temperature (*T*C) of 500 K. Figure 2b presents magnetization versus external magnetic field curve at 300 K, which shows a huge *H*c value of 20 kOe. Before this finding, the largest *H*c value among metal oxide was 6 kOe of barium ferrite, BaFe12O19 [31], which indicates that the *H*c of ε-Fe2O3 is over three times larger. The magnetic structure has been investigated using molecular field theory, which indicated that B and C sites have positive sublattice magnetizations, and A and D sites have negative sublattice magnetizations [32]. This result was consistent with the experimental results from neutron diffraction measurements, Mössbauer spectroscopy measurements, etc. [13,14], and was also consistent with first-principles calculation results [33].

**Figure 2.** (a) Crystal structure of ε-Fe2O3. Dark blue, purple, light blue, and pink polyhedrons indicate A, B, C, and D sites, respectively. (b) Magnetization versus external magnetic field curve of ε-Fe2O3 at 300 K. Inset is a schematic illustration of the sublattice magnetizations of each site.

### **2.2. Formation mechanism of ε-Fe2O3**

196 Ferromagnetic Resonance – Theory and Applications

of single-phase ε-Fe2O3 in 2004 [8].

**2. ε-Fe2O3** 

caused by the spin reorientation phenomenon [28].

In this chapter, we first introduce the synthesis, crystal structure, magnetic properties, and the formation mechanism of the original ε-Fe2O3 [8–10]. Then we report the physical properties of Al-substituted ε-Fe2O3, mainly focusing on its millimeter wave absorption properties due to zero-field ferromagnetic resonance. The resonance frequency was widely controlled from 112–182 GHz by changing the aluminum substitution ratio [23]. Furthermore, from a scientific point of view, temperature dependence of zero-field ferromagnetic resonance was investigated and was found to show an anomalous behavior

This section introduces the synthesis, crystal structure, magnetic properties, and the formation mechanism of ε-Fe2O3. ε-Fe2O3 had only been known as impurity in iron oxide materials, and its properties were clarified for the first time after our success in the synthesis

Single-phase ε-Fe2O3 nanoparticles are synthesized by a chemical method, combining reversemicelle and sol-gel techniques (Figure 1) [8−10,16]. In the reverse-micelle step, two reversemicelle systems, A and B, are formed by cetyl trimethyl ammonium bromide (CTAB) and 1 butanol in *n*-octane. Reverse-micelle A contains aqueous solution of Fe(NO3)3 and Ba(NO3)2, and reverse-micelle B contains NH3 aqueous solution. These two microemulsion systems are mixed under rapid stirring. Tetraethoxysilane (C2H5O)4Si is then added to this solution, which forms SiO2 matrix around the Fe(OH)3 nanoparticles through 20 hours of stirring. The precipitation is separated by centrifugation and sintered in air at 1000C for 4 hours. The SiO2

**Figure 1.** Synthetic procedure of ε-Fe2O3 nanomagnets using a combination method of reverse-micelle and sol-gel techniques. The inset is a transmission electron microscopy image of ε-Fe2O3 nanorods.

**2.1. Synthesis, crystal structure, and magnetic properties of ε-Fe2O3**

matrix is removed by stirring in NaOH solution at 60C for 24 hours.

Here we discuss the formation mechanism of ε-Fe2O3 from the viewpoint of phase transformation. By changing the sintering temperature in the present synthesis, a phase transformation of γ-Fe2O3 → ε-Fe2O3 → α-Fe2O3 was observed accompanied by an increase of particle size. γ- and α-Fe2O3 are very common phases of Fe2O3, and it has been well known that γ-Fe2O3 transforms directly into α-Fe2O3 in a bulk form. In the present case, it is considered that ε-Fe2O3 appeared as a stable phase at an intermediate size region due to the large surface energy effect. Free energy of each *i*-phase (*Gi*, *i* = γ, ε, or α) is expressed as a sum of chemical potential (*μi*) and surface energy (*Aiσi*):

$$G\_i = \mu\_i + A\_i \sigma\_{i\prime} \tag{1}$$

Unusual Temperature Dependence of Zero-Field

Ferromagnetic Resonance in Millimeter Wave Region on Al-Substituted ε-Fe2O3 199

In this section, synthesis, crystal structure, and various physical properties of Al-substituted ε-Fe2O3, ε-Al*x*Fe2–*x*O3, is discussed. Especially, the millimeter wave absorption property by

ε-Al*x*Fe2–*x*O3 samples (*x* = 0.06, 0.09, 0.21, 0.30, 0.40) were synthesized by the same method as the original ε-Fe2O3, using the combination of reverse-micelle and sol-gel techniques. Reverse-micelle A contained aqueous solution of Fe(NO3)3 and Al(NO3)3, and the mixing ratio was adjusted to obtain the different samples, *x* = 0.06, 0.09, 0.21, 0.30, and 0.40. The sintering temperature was 1050°C for *x* = 0.06, 0.09, 0.30, and 0.40, and 1025°C for *x* = 0.21. Only the sample for *x* = 0 was prepared by an impregnation method using mesoporous silica nanoparticles. The SiO2 matrices for all samples were etched by NaOH solution. The morphology and size of the obtained samples were examined using transmission electron microscopy (TEM), which showed spherical nanoparticles with an average particle size

**Figure 4.** Transmission electron microscopy images of ε-Al*x*Fe2–*x*O3 samples. The black bars indicate the

X-ray diffraction (XRD) patterns indicated the samples to have the same orthorhombic crystal structure as the original ε-Fe2O3. The Rietveld analyses of the XRD patterns showed a constant decrease in the lattice constants with the degree of Al-substitution. The analysis results also indicated that the Al3+ ions introduced in the samples have site selectivity in the substitution. For example, in the *x* = 0.21 sample, the Al3+ substitution ratio of each Fe site was 0%, 3%, 8%, and 30% for A, B, C, and D site, respectively. This tendency for the Al3+ ion to prefer D site was consistent with all of the Al-substituted samples (Figure 5). This site

**3.2. Al-substitution effect in crystal structure and magnetic properties** 

**3. Al-substituted ε-Fe2O3** 

between 20-50 nm (Figure 4).

scale.

zero-field ferromagnetic resonance is focused.

**3.1. Synthesis of Al-substituted ε-Fe2O3** 

where *Ai* is molar surface area and *σi* is surface free energy of a particle. Since, *Ai* is equal to 6*V*m,*<sup>i</sup>*/*d*, where *V*m,*<sup>i</sup>* and *d* represent the molar volume and particle diameter, respectively, the free energy per molar volume is expressed as

$$\mathcal{G}\_i \wr V\_{m,i} = \mu\_i \wr V\_{m,i} + \mathfrak{G}\sigma\_i \wr d. \tag{2}$$

This equation indicates that the contribution of the surface energy increases with the decrease of particle diameter. When the parameters satisfy the following three conditions, , , and ( )/( ) ( )/( ) , the free energy curve for each phase, *G*γ/*V*m,γ, *G*ε/*V*m,ε, and *G*α/*V*m,α intersect to form ε-Fe2O3 as the most stable phase at an intermediate *d* value (Figure 3). Such nanosize effect has also been reported for other metal oxide materials, e.g. Al2O3 [34,35] and Ti3O5 [36].

**Figure 3.** Representation of free energy per volume (*G*/*V*) versus particle diameter (*d*) for the three phases of Fe2O3. Green, blue, and red lines indicate the *G*/*V* curves for γ-Fe2O3, ε-Fe2O3, and α-Fe2O3, respectively. Below the graph are the crystal structures of γ-Fe2O3, ε-Fe2O3, and α-Fe2O3 from the left to right.

### **3. Al-substituted ε-Fe2O3**

198 Ferromagnetic Resonance – Theory and Applications

free energy per molar volume is expressed as

right.

 ,

   

 

sum of chemical potential (*μi*) and surface energy (*Aiσi*):

large surface energy effect. Free energy of each *i*-phase (*Gi*, *i* = γ, ε, or α) is expressed as a

, *G A i i ii* 

where *Ai* is molar surface area and *σi* is surface free energy of a particle. Since, *Ai* is equal to 6*V*m,*<sup>i</sup>*/*d*, where *V*m,*<sup>i</sup>* and *d* represent the molar volume and particle diameter, respectively, the

> , , / / 6 /. *GV V d i mi i mi i*

This equation indicates that the contribution of the surface energy increases with the decrease of particle diameter. When the parameters satisfy the following three conditions,

> 

curve for each phase, *G*γ/*V*m,γ, *G*ε/*V*m,ε, and *G*α/*V*m,α intersect to form ε-Fe2O3 as the most stable phase at an intermediate *d* value (Figure 3). Such nanosize effect has also been reported for

 

**Figure 3.** Representation of free energy per volume (*G*/*V*) versus particle diameter (*d*) for the three phases of Fe2O3. Green, blue, and red lines indicate the *G*/*V* curves for γ-Fe2O3, ε-Fe2O3, and α-Fe2O3, respectively. Below the graph are the crystal structures of γ-Fe2O3, ε-Fe2O3, and α-Fe2O3 from the left to

 

other metal oxide materials, e.g. Al2O3 [34,35] and Ti3O5 [36].

, and ( )/( ) ( )/( )

 

 

 

 

 

(1)

(2)

, the free energy

In this section, synthesis, crystal structure, and various physical properties of Al-substituted ε-Fe2O3, ε-Al*x*Fe2–*x*O3, is discussed. Especially, the millimeter wave absorption property by zero-field ferromagnetic resonance is focused.

### **3.1. Synthesis of Al-substituted ε-Fe2O3**

ε-Al*x*Fe2–*x*O3 samples (*x* = 0.06, 0.09, 0.21, 0.30, 0.40) were synthesized by the same method as the original ε-Fe2O3, using the combination of reverse-micelle and sol-gel techniques. Reverse-micelle A contained aqueous solution of Fe(NO3)3 and Al(NO3)3, and the mixing ratio was adjusted to obtain the different samples, *x* = 0.06, 0.09, 0.21, 0.30, and 0.40. The sintering temperature was 1050°C for *x* = 0.06, 0.09, 0.30, and 0.40, and 1025°C for *x* = 0.21. Only the sample for *x* = 0 was prepared by an impregnation method using mesoporous silica nanoparticles. The SiO2 matrices for all samples were etched by NaOH solution. The morphology and size of the obtained samples were examined using transmission electron microscopy (TEM), which showed spherical nanoparticles with an average particle size between 20-50 nm (Figure 4).

**Figure 4.** Transmission electron microscopy images of ε-Al*x*Fe2–*x*O3 samples. The black bars indicate the scale.

### **3.2. Al-substitution effect in crystal structure and magnetic properties**

X-ray diffraction (XRD) patterns indicated the samples to have the same orthorhombic crystal structure as the original ε-Fe2O3. The Rietveld analyses of the XRD patterns showed a constant decrease in the lattice constants with the degree of Al-substitution. The analysis results also indicated that the Al3+ ions introduced in the samples have site selectivity in the substitution. For example, in the *x* = 0.21 sample, the Al3+ substitution ratio of each Fe site was 0%, 3%, 8%, and 30% for A, B, C, and D site, respectively. This tendency for the Al3+ ion to prefer D site was consistent with all of the Al-substituted samples (Figure 5). This site

selectivity can be understood by the smaller ion radius of Al3+ (0.535 Å) compared to Fe3+ (0.645 Å) [37]. The Al3+ ions prefer to occupy the smaller tetrahedral D site than the octahedral A, B, and C sites.

Unusual Temperature Dependence of Zero-Field

*H* (3)

Ferromagnetic Resonance in Millimeter Wave Region on Al-Substituted ε-Fe2O3 201

**Figure 6.** Magnetization versus external magnetic field curve at 300 K for the samples *x* = 0, 0.21, and 0.40 (left), Curie temperature (*T*C) versus *x* value plot (upper right), and coercive field (*H*c) versus *x*

**3.3. Electromagnetic wave absorption of Al-substituted ε-Fe2O3 by zero-field** 

 

where *ν* is the gyromagnetic ratio. If the sample is consisted of randomly oriented particles with uniaxial magnetic anisotropy, the *H*a value is proportional to *H*c. Therefore, electromagnetic wave absorption at high frequencies is expected with insulating materials

With the general electromagnetic wave absorption measurement using free space absorption measurement system, the absorption frequencies of the present ε-Al*x*Fe2-*x*O3 samples exceeded the measurement range, where the maximum is 110 GHz. Therefore, the absorption measurements were conducted using terahertz time domain spectroscopy (THz-TDS) at room temperature. The THz-TDS measurement system is shown in Figure 8. A

magnetocrystalline anisotropy (*H*a) and can be expressed as

exhibiting large coercivity, which is the case for ε-Fe2O3 based magnets.

( /2 ) , *r a f*

Zero-field ferromagnetic resonance is a resonance phenomenon caused by the gyromagnetic effect induced by an electromagnetic wave irradiation under no magnetic field (Figure 7). This phenomenon is observed in ferromagnetic materials with magnetic anisotropy. When the magnetization is tilted away from the easy-axis by the magnetic component of the electromagnetic wave, precession of the magnetization occurs around the easy-axis due to gyromagnetic effect. Resonance is observed when this precession frequency coincides with the electromagnetic wave frequency, resulting in electromagnetic wave absorption at the particular frequency [38]. This resonance frequency (*f*r) is proportional to the

value plot (lower right).

**ferromagnetic resonance**

**Figure 5.** Al3+ occupancy ratio for A, B, C, and D site. Square, diamond, circle, and triangle plots represent A, B, C, and D site, respectively. Inset is the crystal structure of ε-Fe2O3.

The magnetic properties of the samples are shown in Table 1. The field-cooled magnetization curves under an external magnetic field of 10 Oe showed that the *T*C value decreased from 500 K to 448 K with the increase of Al-substitution (Figure 6, upper right). From the magnetization versus external magnetic field measurements, gradual change of the hysteresis loops was also observed. The obtained hysteresis loops of *x* = 0, 0.21, and 0.40 samples are shown in Figure 6. With Al-substitution, the *H*c value decreased from 22.5 kOe to 10.2 kOe, and saturation magnetization (*M*s) value increased. These changes in the magnetic properties can be explained by the metal replacement of Fe3+ magnetic ions (3d5, *S* = 5/2) by non-magnetic Al3+ ions (3d0, *S* = 0). As mentioned previously, ε-Fe2O3 is a ferrimagnet with positive sublattice magnetizations at B and C sites and negative sublattice magnetizations at A and D sites. With the substitution of D site Fe3+ ions with non-magnetic Al3+, the total magnetization increases, leading to the increase of *M*s value. In addition, the non-magnetic Al3+ ions reduce the superexchange interaction between the magnetic sites, resulting in a decrease of *T*C [32]. In this way, the magnetic properties can be widely controlled by Al-substitution.


**Table 1.** Magnetic properties of ε-Al*x*Fe2–*x*O3.

Unusual Temperature Dependence of Zero-Field Ferromagnetic Resonance in Millimeter Wave Region on Al-Substituted ε-Fe2O3 201

200 Ferromagnetic Resonance – Theory and Applications

octahedral A, B, and C sites.

controlled by Al-substitution.

**Table 1.** Magnetic properties of ε-Al*x*Fe2–*x*O3.

selectivity can be understood by the smaller ion radius of Al3+ (0.535 Å) compared to Fe3+ (0.645 Å) [37]. The Al3+ ions prefer to occupy the smaller tetrahedral D site than the

**Figure 5.** Al3+ occupancy ratio for A, B, C, and D site. Square, diamond, circle, and triangle plots

The magnetic properties of the samples are shown in Table 1. The field-cooled magnetization curves under an external magnetic field of 10 Oe showed that the *T*C value decreased from 500 K to 448 K with the increase of Al-substitution (Figure 6, upper right). From the magnetization versus external magnetic field measurements, gradual change of the hysteresis loops was also observed. The obtained hysteresis loops of *x* = 0, 0.21, and 0.40 samples are shown in Figure 6. With Al-substitution, the *H*c value decreased from 22.5 kOe to 10.2 kOe, and saturation magnetization (*M*s) value increased. These changes in the magnetic properties can be explained by the metal replacement of Fe3+ magnetic ions (3d5, *S* = 5/2) by non-magnetic Al3+ ions (3d0, *S* = 0). As mentioned previously, ε-Fe2O3 is a ferrimagnet with positive sublattice magnetizations at B and C sites and negative sublattice magnetizations at A and D sites. With the substitution of D site Fe3+ ions with non-magnetic Al3+, the total magnetization increases, leading to the increase of *M*s value. In addition, the non-magnetic Al3+ ions reduce the superexchange interaction between the magnetic sites, resulting in a decrease of *T*C [32]. In this way, the magnetic properties can be widely

> *x T*C (K) *H*c (kOe) *Ms* (emu/g) 0 500 22.5 14.9 0.06 496 19.1 15.1 0.09 490 17.5 14.6 0.21 480 14.9 17.0 0.30 466 13.8 20.3 0.40 448 10.2 19.7

represent A, B, C, and D site, respectively. Inset is the crystal structure of ε-Fe2O3.

**Figure 6.** Magnetization versus external magnetic field curve at 300 K for the samples *x* = 0, 0.21, and 0.40 (left), Curie temperature (*T*C) versus *x* value plot (upper right), and coercive field (*H*c) versus *x* value plot (lower right).

### **3.3. Electromagnetic wave absorption of Al-substituted ε-Fe2O3 by zero-field ferromagnetic resonance**

Zero-field ferromagnetic resonance is a resonance phenomenon caused by the gyromagnetic effect induced by an electromagnetic wave irradiation under no magnetic field (Figure 7). This phenomenon is observed in ferromagnetic materials with magnetic anisotropy. When the magnetization is tilted away from the easy-axis by the magnetic component of the electromagnetic wave, precession of the magnetization occurs around the easy-axis due to gyromagnetic effect. Resonance is observed when this precession frequency coincides with the electromagnetic wave frequency, resulting in electromagnetic wave absorption at the particular frequency [38]. This resonance frequency (*f*r) is proportional to the magnetocrystalline anisotropy (*H*a) and can be expressed as

$$f\_r = (\nu / 2\pi) \mathcal{H}\_{a'} \tag{3}$$

where *ν* is the gyromagnetic ratio. If the sample is consisted of randomly oriented particles with uniaxial magnetic anisotropy, the *H*a value is proportional to *H*c. Therefore, electromagnetic wave absorption at high frequencies is expected with insulating materials exhibiting large coercivity, which is the case for ε-Fe2O3 based magnets.

With the general electromagnetic wave absorption measurement using free space absorption measurement system, the absorption frequencies of the present ε-Al*x*Fe2-*x*O3 samples exceeded the measurement range, where the maximum is 110 GHz. Therefore, the absorption measurements were conducted using terahertz time domain spectroscopy (THz-TDS) at room temperature. The THz-TDS measurement system is shown in Figure 8. A

Unusual Temperature Dependence of Zero-Field

Ferromagnetic Resonance in Millimeter Wave Region on Al-Substituted ε-Fe2O3 203

**Figure 8.** A schematic diagram of the terahertz time domain spectroscopy measurement system.

**Figure 9.** Electromagnetic wave absorption spectra of ε-Al*x*Fe2–*x*O3 (left). Red, orange, yellow, green, light blue, and blue lines are the absorption spectra for *x* = 0.40, 0.30, 0.21, 0.09, 0.06, and 0, respectively.

Right is the zero-field ferromagnetic resonance frequency (*f*r) versus *x* value plot.

**Figure 7.** (a) A schematic illustration of zero-field ferromagnetic resonance (natural resonance), resulting in electromagnetic wave absorption due to the precession of magnetization around the easyaxis. The *M* and *E* of the electromagnetic wave indicate the magnetic and electric components, respectively. (b) An illustration indicating that larger coercive field (*H*c) results in higher resonance frequency (*f*r).

mode-locked Ti:sapphire femtosecond pulse laser with a time duration of 20 fs at a repetition rate of 76 MHz was used. The output was divided into a pump and probe beam for the time-domain system. For THz wave emitter and detector, dipole type and bowtie type low-temperature-grown GaAs photoconductive antennas were used, respectively. The sample was set on a sample holder, which was inserted between a set of paraboloidal mirrors concentrating the THz wave at the location of the sample. The temporal waveforms of the electric component of the transmitted THz pulse waves were obtained by changing the delay time between the pump and probe pulses. The temporal waves were Fourier transferred to obtain the frequency dependence, and the absorption spectra were calculated using the following equation:

$$(Absorption) = -10 \log \left| t(o) \right|^2 \text{ (dB)},\tag{4}$$

where *t*( ) is the complex amplitude transmittance. An absorption of 20 dB indicates 99% absorption.

The electromagnetic wave absorption spectra are shown in Figure 9. Absorption peaks were observed at 112 GHz (*x* = 0.40), 125 GHz (*x* = 0.30), 145 GHz (*x* = 0.21), 162 GHz (*x* = 0.09), 172 GHz (*x* = 0.06), and 182 GHz (*x* = 0). The *f*r value decreased with Al-substitution, consistent with the behavior of the *H*c value (Figure 6, lower right). The observed electromagnetic wave absorption due to zero-field ferromagnetic resonance at exceptional high frequencies was achieved by the large *H*a value of this series with large coercivity.

frequency (*f*r).

where *t*( ) 

absorption.

using the following equation:

**Figure 7.** (a) A schematic illustration of zero-field ferromagnetic resonance (natural resonance), resulting in electromagnetic wave absorption due to the precession of magnetization around the easyaxis. The *M* and *E* of the electromagnetic wave indicate the magnetic and electric components, respectively. (b) An illustration indicating that larger coercive field (*H*c) results in higher resonance

mode-locked Ti:sapphire femtosecond pulse laser with a time duration of 20 fs at a repetition rate of 76 MHz was used. The output was divided into a pump and probe beam for the time-domain system. For THz wave emitter and detector, dipole type and bowtie type low-temperature-grown GaAs photoconductive antennas were used, respectively. The sample was set on a sample holder, which was inserted between a set of paraboloidal mirrors concentrating the THz wave at the location of the sample. The temporal waveforms of the electric component of the transmitted THz pulse waves were obtained by changing the delay time between the pump and probe pulses. The temporal waves were Fourier transferred to obtain the frequency dependence, and the absorption spectra were calculated

<sup>2</sup> ( ) 10log ( ) (dB), *Absorption t*

The electromagnetic wave absorption spectra are shown in Figure 9. Absorption peaks were observed at 112 GHz (*x* = 0.40), 125 GHz (*x* = 0.30), 145 GHz (*x* = 0.21), 162 GHz (*x* = 0.09), 172 GHz (*x* = 0.06), and 182 GHz (*x* = 0). The *f*r value decreased with Al-substitution, consistent with the behavior of the *H*c value (Figure 6, lower right). The observed electromagnetic wave absorption due to zero-field ferromagnetic resonance at exceptional high frequencies

was achieved by the large *H*a value of this series with large coercivity.

is the complex amplitude transmittance. An absorption of 20 dB indicates 99%

(4)

**Figure 8.** A schematic diagram of the terahertz time domain spectroscopy measurement system.

**Figure 9.** Electromagnetic wave absorption spectra of ε-Al*x*Fe2–*x*O3 (left). Red, orange, yellow, green, light blue, and blue lines are the absorption spectra for *x* = 0.40, 0.30, 0.21, 0.09, 0.06, and 0, respectively. Right is the zero-field ferromagnetic resonance frequency (*f*r) versus *x* value plot.

### **3.4. Temperature dependence of zero-field ferromagnetic resonance in Alsubstituted ε-Fe2O3**

Among the ε-Al*x*Fe2–*x*O3 samples discussed in the previous section, we focused on the *x* = 0.06 sample and measured the temperature dependence of zero-field ferromagnetic resonance. The Al3+ substitution ratios of each Fe site in ε-Al0.06Fe1.94O3 are 3%, 0%, 0%, and 11% for A, B, C, and D site, respectively. The magnetic properties of ε-Al0.06Fe1.94O3 are shown in Figure 10. The field-cooled magnetization curve under 10 Oe external magnetic field showed a *T*C value of 496 K and a cusp at 131 K (= *T*p). The cusp in the magnetization is due to the spin reorientation phenomenon, which is known to occur in this temperature region [11,12]. The magnetization versus external magnetic field curve exhibited an *H*c value of 19.1 kOe at 300 K.

Unusual Temperature Dependence of Zero-Field

Ferromagnetic Resonance in Millimeter Wave Region on Al-Substituted ε-Fe2O3 205

increased from 19.4 kOe at 300 K to 22.6 kOe at 200 K, and then decreased to 4.5 kOe at 70 K with the decrease of temperature. The *H*c versus temperature plot indicates a sigmoid decrease in a wide temperature range of 200–60 K centered at *T*p (i.e., ±70 K from the center temperature, *T*p = 131 K) (Figure 12a). In other words, the beginning and ending temperatures of the spin reorientation are about 200 K and 60 K, respectively, with decreasing temperature. The temperature region of the sigmoid decrease of *f*r almost corresponds to the temperature range of the spin reorientation. The sigmoid increase of Δ*f*

**Figure 11.** (a) Electromagnetic wave absorption spectra of ε-Al0.06Fe1.94O3 at different temperatures. Black lines and red lines indicate the observed spectra and fitted Lorentz function. (b) Temperature dependence of zero-field ferromagnetic resonance frequency (*f*r). (c) Temperature dependence of full

width at half maximum (Δ*f*) of the absorption spectra. Dotted lines are to guide the eye.

**Figure 10.** Magnetic properties of ε-Al0.06Fe1.94O3. (a) Field-cooled magnetization curve under an external magnetic field of 10 Oe. (b) Magnetization versus external magnetic field curve at 300 K.

For the THz-TDS measurement, ε-Al0.06Fe1.94O3 powder sample was pressed into a pelletform. The absorption spectra at different temperatures are shown in Figure 11a. These absorption spectra versus frequency were obtained by calibration of the background noise. They were also fitted by Lorentz function. At 301 K, the *f*r value was 172 GHz, consistent with the result in the previous section. With the decrease of temperature, the *f*r value gradually increased to 186 GHz at 204 K, and turned to an abrupt decrease down to 147 GHz at 77 K. The *f*r value continued to decrease with lowering the temperature, and at 21 K, the *f*r value was 133 GHz (Figure 11b). Temperature dependence was also observed in the linewidth of the absorption spectra. The full width at half maximum (Δ*f*) value increased from 5 GHz at 301 K to 19 GHz at 77 K with decreasing temperature, and then, decreased to 16 GHz at 21 K (Figure 11c).

Temperature dependencies of magnetic hysteresis loop and ac magnetic susceptibility was studied in order to understand the anomalous temperature dependencies of *f*r and Δ*f*. As mentioned in Figure 10a, the field-cooled magnetization curve shows an increase below *T*C, but a cusp appears at *T*p = 131 K, where the magnetization turns to a decrease. The *H*c value increased from 19.4 kOe at 300 K to 22.6 kOe at 200 K, and then decreased to 4.5 kOe at 70 K with the decrease of temperature. The *H*c versus temperature plot indicates a sigmoid decrease in a wide temperature range of 200–60 K centered at *T*p (i.e., ±70 K from the center temperature, *T*p = 131 K) (Figure 12a). In other words, the beginning and ending temperatures of the spin reorientation are about 200 K and 60 K, respectively, with decreasing temperature. The temperature region of the sigmoid decrease of *f*r almost corresponds to the temperature range of the spin reorientation. The sigmoid increase of Δ*f*

204 Ferromagnetic Resonance – Theory and Applications

**substituted ε-Fe2O3** 

of 19.1 kOe at 300 K.

16 GHz at 21 K (Figure 11c).

**3.4. Temperature dependence of zero-field ferromagnetic resonance in Al-**

**Figure 10.** Magnetic properties of ε-Al0.06Fe1.94O3. (a) Field-cooled magnetization curve under an external magnetic field of 10 Oe. (b) Magnetization versus external magnetic field curve at 300 K.

For the THz-TDS measurement, ε-Al0.06Fe1.94O3 powder sample was pressed into a pelletform. The absorption spectra at different temperatures are shown in Figure 11a. These absorption spectra versus frequency were obtained by calibration of the background noise. They were also fitted by Lorentz function. At 301 K, the *f*r value was 172 GHz, consistent with the result in the previous section. With the decrease of temperature, the *f*r value gradually increased to 186 GHz at 204 K, and turned to an abrupt decrease down to 147 GHz at 77 K. The *f*r value continued to decrease with lowering the temperature, and at 21 K, the *f*r value was 133 GHz (Figure 11b). Temperature dependence was also observed in the linewidth of the absorption spectra. The full width at half maximum (Δ*f*) value increased from 5 GHz at 301 K to 19 GHz at 77 K with decreasing temperature, and then, decreased to

Temperature dependencies of magnetic hysteresis loop and ac magnetic susceptibility was studied in order to understand the anomalous temperature dependencies of *f*r and Δ*f*. As mentioned in Figure 10a, the field-cooled magnetization curve shows an increase below *T*C, but a cusp appears at *T*p = 131 K, where the magnetization turns to a decrease. The *H*c value

Among the ε-Al*x*Fe2–*x*O3 samples discussed in the previous section, we focused on the *x* = 0.06 sample and measured the temperature dependence of zero-field ferromagnetic resonance. The Al3+ substitution ratios of each Fe site in ε-Al0.06Fe1.94O3 are 3%, 0%, 0%, and 11% for A, B, C, and D site, respectively. The magnetic properties of ε-Al0.06Fe1.94O3 are shown in Figure 10. The field-cooled magnetization curve under 10 Oe external magnetic field showed a *T*C value of 496 K and a cusp at 131 K (= *T*p). The cusp in the magnetization is due to the spin reorientation phenomenon, which is known to occur in this temperature region [11,12]. The magnetization versus external magnetic field curve exhibited an *H*c value

**Figure 11.** (a) Electromagnetic wave absorption spectra of ε-Al0.06Fe1.94O3 at different temperatures. Black lines and red lines indicate the observed spectra and fitted Lorentz function. (b) Temperature dependence of zero-field ferromagnetic resonance frequency (*f*r). (c) Temperature dependence of full width at half maximum (Δ*f*) of the absorption spectra. Dotted lines are to guide the eye.

was also observed in the spin reorientation temperature region. Figure 12b is the ac magnetic susceptibility versus temperature with frequency of 10 Hz under field amplitude of 1 Oe. As the temperature decreased, the real part of the ac magnetic susceptibility (*χ*') gradually increased to a maximum value of 3.8 × 10–4 emu/g·Oe at 60 K and then decreased. The imaginary part (*χ*'') showed similar temperature dependence with a maximum around 70 K. These temperature dependencies of ac magnetic susceptibility correspond to that of Δ*f* [39,40].

Unusual Temperature Dependence of Zero-Field

Ferromagnetic Resonance in Millimeter Wave Region on Al-Substituted ε-Fe2O3 207

advantage in the viewpoint of industrial applications, such as electromagnetic wave absorbers in the near future, where high-frequency millimeter waves are likely to be used in

The present research was supported partly by the Core Research for Evolutional Science and Technology (CREST) program of the Japan Science and Technology Agency (JST), a Grantin-Aid for Young Scientists (S) from Japan Society for the Promotion of Science (JSPS), DOWA Technofund, the Asahi Glass Foundation, Funding Program for Next Generation World-Leading Researchers from JSPS, a Grant for the Global COE Program "Chemistry Innovation through Cooperation of Science and Engineering", Advanced Photon Science Alliance (APSA) from the Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT), the Cryogenic Research Center, The University of Tokyo, and the Center for Nano Lithography & Analysis, The University of Tokyo, supported by MEXT Japan. M. Y. is grateful to Advanced Leading Graduate Course for Photon Science (ALPS) and JSPS Research Fellowships for Young Scientists. A. N. is grateful to JSPS KAKENHI Grant Number 24850004 and Office for Gender Equality, The University of Tokyo. We are grateful to Dr. S. Sakurai of The University of Tokyo. We also thank Prof. M. Nakajima and Prof. T. Suemoto for support in THz-TDS measurements, Mr. Y. Kakegawa and Mr. H. Tsunakawa for collecting the TEM images, and Mr. K. Matsumoto, Mr. M. Goto, Mr. S. Sasaki, Mr. T. Miyazaki, and Mr. T. Yoshida of DOWA Electronics Materials Co., Ltd. for the valuable

order to transport heavy data at high speed.

Marie Yoshikiyo, Asuka Namai and Shin-ichi Ohkoshi

*CREST, JST, K's Gobancho, 7 Gobancho, Chiyoda-ku, Tokyo, Japan* 

[1] Zhou ZG. Magnetic Ferrite Materials. Beijing: Science Press; 1981.

[2] Yoshida S, Sato M, Sugawara E, Shimada Y. Permeability and Electromagnetic-Interference Characteristics of Fe-Si-Al Alloy Flakes-Polymer Composite. J. Appl. Phys.

[3] Naito Y, Suetake K. Application of Ferrite to Electromagnetic Wave Absorber and its

[4] Yusoff AN, Abdullah MH, Ahmed SH, Jusoh SF, Mansor AA, Hamid SAA. Electromagnetic and Absorption Properties of Some Microwave Absorbers. J. Appl.

Characteristics. IEEE Trans. Microwave Theory Tech. 1971;MT19(1) 65–72.

*Department of Chemistry, School of Science, The University of Tokyo, Tokyo, Japan* 

**Author details** 

Shin-ichi Ohkoshi

discussions.

**5. References** 

1999;85(8) 4636–4638.

Phys. 2002;92 (2) 876–882.

**Acknowledgement** 

**Figure 12.** (a) Temperature dependence of coercive field (*H*c). Dotted line is to guide the eye. (b) Temperature dependence of ac magnetic susceptibility (real part *χ*' and imaginary part *χ*'') measured at 10 Hz and 1 Oe field amplitude.

As mentioned previously, the *f*r value is proportional to the *H*a value, and in this case with randomly oriented samples, *f*r is also related to the *H*c value. Therefore, the observed anomalous temperature dependence of *f*r in ε-Al0.06Fe1.94O3 was understood by the temperature dependence of *H*c. The sigmoid decrease centered at *T*p originates from the disappearance of magnetic anisotropy due to the spin reorientation phenomenon [11–13].

### **4. Conclusion**

In this chapter, a rare phase of diiron trioxide, ε-Fe2O3, and its Al-substituted series were introduced. The synthesis, crystal structure, and its exceptional physical properties were discussed, especially its huge magnetic anisotropy exhibiting a gigantic coercive field, which enables electromagnetic wave absorption due to zero-field ferromagnetic resonance at high frequencies in the millimeter wave region. Al-substitution effect was observed in the ε-Al*x*Fe2–*x*O3 series, widely controlling the magnetic properties and the zero-field ferromagnetic resonance frequency: ε-Al*x*Fe2–*x*O3 absorbed millimeter waves from 112–182 GHz at room temperature. Temperature dependence of zero-field ferromagnetic resonance was also investigated for ε-Al0.06Fe1.94O3 sample, and an anomalous behavior was observed due to spin reorientation phenomenon.

Since ε-Al*x*Fe2–*x*O3 is composed of very common and low costing elements, it is friendly to the environment and can be economically produced. Its chemical stability is also an advantage in the viewpoint of industrial applications, such as electromagnetic wave absorbers in the near future, where high-frequency millimeter waves are likely to be used in order to transport heavy data at high speed.

### **Author details**

206 Ferromagnetic Resonance – Theory and Applications

10 Hz and 1 Oe field amplitude.

due to spin reorientation phenomenon.

**4. Conclusion** 

[39,40].

was also observed in the spin reorientation temperature region. Figure 12b is the ac magnetic susceptibility versus temperature with frequency of 10 Hz under field amplitude of 1 Oe. As the temperature decreased, the real part of the ac magnetic susceptibility (*χ*') gradually increased to a maximum value of 3.8 × 10–4 emu/g·Oe at 60 K and then decreased. The imaginary part (*χ*'') showed similar temperature dependence with a maximum around 70 K. These temperature dependencies of ac magnetic susceptibility correspond to that of Δ*f*

**Figure 12.** (a) Temperature dependence of coercive field (*H*c). Dotted line is to guide the eye. (b) Temperature dependence of ac magnetic susceptibility (real part *χ*' and imaginary part *χ*'') measured at

As mentioned previously, the *f*r value is proportional to the *H*a value, and in this case with randomly oriented samples, *f*r is also related to the *H*c value. Therefore, the observed anomalous temperature dependence of *f*r in ε-Al0.06Fe1.94O3 was understood by the temperature dependence of *H*c. The sigmoid decrease centered at *T*p originates from the disappearance of magnetic anisotropy due to the spin reorientation phenomenon [11–13].

In this chapter, a rare phase of diiron trioxide, ε-Fe2O3, and its Al-substituted series were introduced. The synthesis, crystal structure, and its exceptional physical properties were discussed, especially its huge magnetic anisotropy exhibiting a gigantic coercive field, which enables electromagnetic wave absorption due to zero-field ferromagnetic resonance at high frequencies in the millimeter wave region. Al-substitution effect was observed in the ε-Al*x*Fe2–*x*O3 series, widely controlling the magnetic properties and the zero-field ferromagnetic resonance frequency: ε-Al*x*Fe2–*x*O3 absorbed millimeter waves from 112–182 GHz at room temperature. Temperature dependence of zero-field ferromagnetic resonance was also investigated for ε-Al0.06Fe1.94O3 sample, and an anomalous behavior was observed

Since ε-Al*x*Fe2–*x*O3 is composed of very common and low costing elements, it is friendly to the environment and can be economically produced. Its chemical stability is also an Marie Yoshikiyo, Asuka Namai and Shin-ichi Ohkoshi *Department of Chemistry, School of Science, The University of Tokyo, Tokyo, Japan* 

Shin-ichi Ohkoshi *CREST, JST, K's Gobancho, 7 Gobancho, Chiyoda-ku, Tokyo, Japan* 

### **Acknowledgement**

The present research was supported partly by the Core Research for Evolutional Science and Technology (CREST) program of the Japan Science and Technology Agency (JST), a Grantin-Aid for Young Scientists (S) from Japan Society for the Promotion of Science (JSPS), DOWA Technofund, the Asahi Glass Foundation, Funding Program for Next Generation World-Leading Researchers from JSPS, a Grant for the Global COE Program "Chemistry Innovation through Cooperation of Science and Engineering", Advanced Photon Science Alliance (APSA) from the Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT), the Cryogenic Research Center, The University of Tokyo, and the Center for Nano Lithography & Analysis, The University of Tokyo, supported by MEXT Japan. M. Y. is grateful to Advanced Leading Graduate Course for Photon Science (ALPS) and JSPS Research Fellowships for Young Scientists. A. N. is grateful to JSPS KAKENHI Grant Number 24850004 and Office for Gender Equality, The University of Tokyo. We are grateful to Dr. S. Sakurai of The University of Tokyo. We also thank Prof. M. Nakajima and Prof. T. Suemoto for support in THz-TDS measurements, Mr. Y. Kakegawa and Mr. H. Tsunakawa for collecting the TEM images, and Mr. K. Matsumoto, Mr. M. Goto, Mr. S. Sasaki, Mr. T. Miyazaki, and Mr. T. Yoshida of DOWA Electronics Materials Co., Ltd. for the valuable discussions.

### **5. References**


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Unusual Temperature Dependence of Zero-Field

Ferromagnetic Resonance in Millimeter Wave Region on Al-Substituted ε-Fe2O3 209

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[28] Yoshikiyo M, Namai A, Nakajima M, Suemoto T, Ohkoshi S. Anomalous Behavior of High-Frequency Zero-Field Ferromagnetic Resonance in Aluminum-Substituted ε-

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**Chapter 9** 

© 2013Fu and Wang, 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.

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,

**Optical Properties of Antiferromagnetic/Ion-**

Increasing attention has been paid to magnetic photonic crystals (MPCs) because the properties of the MPCs can be modulated not only with the change of their structure (including components, layer thickness or thickness ratio) but also with the external magnetic field. MPCs are capable of acting as tunable filters [1] at different frequencies, and that controllable gigantic Faraday rotation angles [2-6] are simultaneously obtained. The nonmagnetic media in MPCs generally are ordinary dielectrics, so the electromagnetic wave modes are just magnetic polaritons. The effect of magnetic permeability and dielectric permittivity of two component materials in MPCs on the photonic band groups were discussed, where the permeability and permittivity were considered as scalar quantities [7]. Recently, our group investigated the optical properties of antiferromagnetic/ ion-crystal (AF/IC) PCs [8-11]. It is well known that the two resonant frequencies of AFs, such as, FeF2 and MnF2, fall into the millimeter or far infrared frequencies regions and some ionic semiconductors possess a very low phonon-resonant frequency range like the AFs. Especially, these frequency regions also are situated the working frequency range of THz technology, so the AF/IC PCs may be available to make the new elements in the field of THz technology. Note that in ICs, including ionic semiconductors, when the frequencies of the phonon and the transverse optical (TO) phonon modes of ICs are close, the dispersion curves of phonon and TO phonon modes will be changed and a kind of coupled mode called phonon polariton will be formed. Therefore, in the AF/IC PCs, the TO phonon modes of ICs can directly couple with the electric field in an electromagnetic wave and this coupling generates the phonon polaritons, however, the magnetization's motion in magnets can directly couple with the magnetic field, which is the origin of magnetic polaritons. Thus in such an AF/IC PCs, we refer to collective polaritons as the magneto-phonon polaritons (MPPs). In the presence of external magnetic field and damping, MPPs spectra display two

**Crystalic Photonic Crystals** 

Additional information is available at the end of the chapter

Shu-Fang Fu and Xuan-Zhang Wang

http://dx.doi.org/10.5772/56397

**1. Introduction** 


## **Optical Properties of Antiferromagnetic/Ion-Crystalic Photonic Crystals**

Shu-Fang Fu and Xuan-Zhang Wang

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/56397

### **1. Introduction**

210 Ferromagnetic Resonance – Theory and Applications

Nat. Chem. 2010;2(7) 539–545.

1988;49(2) 213–222.

[36] Ohkoshi S, Tsunobuchi Y, Matsuda T, Hashimoto K, Namai A, Hakoe F, Tokoro H. Synthesis of a Metal Oxide with a Room-Temperature Photoreversible Phase Transition.

[37] Shannon RD. Revised Effective Ionic Radii and Systematic Studies of Interatomic Distances in Halides and Chalcogenides. Acta Cryst. 1976;A32(Sep1) 751–767. [38] Chikazumi S. Physics of Ferromagnetism. New York: Oxford University Press; 1997. [39] Algarabel PA, Moral A, Ibarra MR, Arnaudas JI. Spin Reorientation in ReCo5 Compounds: A.C. Susceptibility and Thermal Expansion. J. Phys. Chem. Solids

[40] Malik SK, Adroja DT, Ma BM, Boltich EB, Sohn JG, Sankar SG, Wallace WE. Spin Reorientation Phenomenon in Nd0.5Er1.5Fe14–*x*M*x*B (M = Al and Co), as Determined by

AC Susceptibility Measurements. J. Appl. Phys. 1990;67(9) 4589–4591.

Increasing attention has been paid to magnetic photonic crystals (MPCs) because the properties of the MPCs can be modulated not only with the change of their structure (including components, layer thickness or thickness ratio) but also with the external magnetic field. MPCs are capable of acting as tunable filters [1] at different frequencies, and that controllable gigantic Faraday rotation angles [2-6] are simultaneously obtained. The nonmagnetic media in MPCs generally are ordinary dielectrics, so the electromagnetic wave modes are just magnetic polaritons. The effect of magnetic permeability and dielectric permittivity of two component materials in MPCs on the photonic band groups were discussed, where the permeability and permittivity were considered as scalar quantities [7].

Recently, our group investigated the optical properties of antiferromagnetic/ ion-crystal (AF/IC) PCs [8-11]. It is well known that the two resonant frequencies of AFs, such as, FeF2 and MnF2, fall into the millimeter or far infrared frequencies regions and some ionic semiconductors possess a very low phonon-resonant frequency range like the AFs. Especially, these frequency regions also are situated the working frequency range of THz technology, so the AF/IC PCs may be available to make the new elements in the field of THz technology. Note that in ICs, including ionic semiconductors, when the frequencies of the phonon and the transverse optical (TO) phonon modes of ICs are close, the dispersion curves of phonon and TO phonon modes will be changed and a kind of coupled mode called phonon polariton will be formed. Therefore, in the AF/IC PCs, the TO phonon modes of ICs can directly couple with the electric field in an electromagnetic wave and this coupling generates the phonon polaritons, however, the magnetization's motion in magnets can directly couple with the magnetic field, which is the origin of magnetic polaritons. Thus in such an AF/IC PCs, we refer to collective polaritons as the magneto-phonon polaritons (MPPs). In the presence of external magnetic field and damping, MPPs spectra display two

© 2013Fu and Wang, 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.

petty bulk mode bands with negative group velocity. It is worthy of mentioning that many surface modes emerge in the vicinity of two petty bulk mode bands, and that some surface modes bear nonreciprocality [11]. The optical properties of the AF/IC PCs can be modulated by an external magnetic field.

Optical Properties of Antiferromagnetic/Ion-Crystalic Photonic Crystals 213

will much longer than the period of AF/IC PCs. With this

2 21 2 2 1

2 21 2 2 1

 

 *H* , and 1 2 [ (2 )] 

*r aea* , where *M*0 is

the magnetic damping

(1)

 *i* (2)

the gyromagnetic ratio, and

as the dielectric constant of the AF. Subsequently, we present the

  , the

regarding the polaritons[30,32-34]. Actually, most materials are dispersive and absorbing.

An interesting configuration in experiment is the Voigt geometry as illustrated in Fig.1, where the polariton wave propagates in the *x-y* plane and the magnetic field of an electromagnetic wave is parallel to this plane, but the wave electric field aligns the *z* direction. We concentrate our attention on the case where the external magnetic field and AF anisotropy axis both are along the *z* axis and parallel to layers. The *y* axis is perpendicular to layers in the structure. The semi-space ( *y* 0 ) is of vacuum, where *<sup>a</sup> d* and *<sup>i</sup> d* are thicknesses of AF and IC layers, respectively. For the far infrared wave, the order of the wavelength is about 100*μm*. Thus, as long as the thicknesses of AF and IC layers are less

condition, the AF/IC PCs will become a uniform film by means of an effective-medium

We first present the permeability of the AF film. In the external magnetic field *H*<sup>0</sup>

xx yy <sup>0</sup> <sup>0</sup> 1 {[ ( i ) ] [ ( i ) ] }, *ma r <sup>r</sup>*

xy yx <sup>0</sup> <sup>0</sup> {[ ( i ) ] [ ( i ) ] }. *ma r <sup>r</sup>*

the sublattice magnetization, *Ha* represents the anisotropy field, and *H*e the exchange field.

magnetic permeability is well-known, with its nonzero elements [33, 37]

Therefore, it is also necessary to consider the effect of damping.

**2.1. MPPs in one-dimension AF/IC PCs** 

than 10*μm*, the wavelength

**Figure 1.** Illustration and coordinate system.

constant. We use *<sup>a</sup>*

with 

 

> 

 0 0 *H* , <sup>0</sup> 4 

*<sup>r</sup>* is the AF resonant frequency,

dielectric function of the IC [38],

*2.1.1. EMM for one-dimensional AF/IC PCs* 

 

> 

 *<sup>m</sup> M* ,

 

> *a a H* ,*e e*

method (EMM).

In addition, we have concluded that there is a material match of an AF and an IC, for which a common frequency range is found, in which the AF has a negative magnetic permeability and the IC has negative dielectric permittivity [10]. Consequently, the AF/IC structures are thought to be of the left-handed materials (LHMs) which have attracted much attention from the research community in recent years because of their completely different properties from right-handed materials (RHMs). In a LHM, the electric field, magnetic field and wave vector of a plane electromagnetic wave form a left-handed triplet, the energy flow of the plane wave is opposite in direction to that of the wave vector [12-17]. LHMs have to be constructed artificially since there is no natural LHM. Several variations of the design have been studied through experiments [18-20]. Up to now, scientists have found some LHMs available in infrared and visible ranges [21-25], but each design has a rather complicated structure. We noticed a work that discussed the left-handed properties of a superlattice composed of alternately semiconductor and antiferromagnetic (AF) layers, where the interaction between AF polaritons and semiconductor plasmons lead to the lefthandedness of the superlattice [26]. However the plasmon resonant frequency sensitively depends on the free charge carrier's density, or impurity concentration in semiconductor layers, so if one wants to see a plasmon resonant frequency near to AF resonant frequencies, the density must be very low since AF resonant frequencies are distributed in the millimeter to far infrared range. In the case of such a low density, the effect of the charge carriers on the electromagnetic properties may be very weak [27] so that there is not the left-handedness of the superlattice. According the discussion above, we propose a simple structure of multilayer which consists of AF and IC layers. An analytical condition under which both left-handeness and negative refraction phenomenon appear in the film is established by calculating the angle between the energy flow and wave vector of a plane electromagnetic wave in AF/IC PCs and its refraction angle.

### **2. Magneto-phonon polaritons (MPPs) in AF/IC PCs**

Polaritons in solids are a kind of electromagnetic modes determining optical or electromagnetic properties of the solids. Natures of various polaritons, including the surface and bulk polaritons, were very clearly discussed in Ref. [28]. Recent years, based on magnetic multilayers or superlattices, where nonmagnetic layers are of ordinary dielectric and their dielectric function is a constant, the polaritons in these structures called the MPCs were discussed [29-34]. On the other hand, ones were interested in the phonon polaritons [35-36], where the surface polariton modes could be focused by a simple way and probably possess new applications. In this part, the collective polaritons, MPPs in a superlattice structure comprised of alternating AF and IC layers, will be discussed. In the past, for simplicity, the damping was generally ignored in the discussion of dispersion properties regarding the polaritons[30,32-34]. Actually, most materials are dispersive and absorbing. Therefore, it is also necessary to consider the effect of damping.

### **2.1. MPPs in one-dimension AF/IC PCs**

212 Ferromagnetic Resonance – Theory and Applications

wave in AF/IC PCs and its refraction angle.

**2. Magneto-phonon polaritons (MPPs) in AF/IC PCs** 

Polaritons in solids are a kind of electromagnetic modes determining optical or electromagnetic properties of the solids. Natures of various polaritons, including the surface and bulk polaritons, were very clearly discussed in Ref. [28]. Recent years, based on magnetic multilayers or superlattices, where nonmagnetic layers are of ordinary dielectric and their dielectric function is a constant, the polaritons in these structures called the MPCs were discussed [29-34]. On the other hand, ones were interested in the phonon polaritons [35-36], where the surface polariton modes could be focused by a simple way and probably possess new applications. In this part, the collective polaritons, MPPs in a superlattice structure comprised of alternating AF and IC layers, will be discussed. In the past, for simplicity, the damping was generally ignored in the discussion of dispersion properties

by an external magnetic field.

petty bulk mode bands with negative group velocity. It is worthy of mentioning that many surface modes emerge in the vicinity of two petty bulk mode bands, and that some surface modes bear nonreciprocality [11]. The optical properties of the AF/IC PCs can be modulated

In addition, we have concluded that there is a material match of an AF and an IC, for which a common frequency range is found, in which the AF has a negative magnetic permeability and the IC has negative dielectric permittivity [10]. Consequently, the AF/IC structures are thought to be of the left-handed materials (LHMs) which have attracted much attention from the research community in recent years because of their completely different properties from right-handed materials (RHMs). In a LHM, the electric field, magnetic field and wave vector of a plane electromagnetic wave form a left-handed triplet, the energy flow of the plane wave is opposite in direction to that of the wave vector [12-17]. LHMs have to be constructed artificially since there is no natural LHM. Several variations of the design have been studied through experiments [18-20]. Up to now, scientists have found some LHMs available in infrared and visible ranges [21-25], but each design has a rather complicated structure. We noticed a work that discussed the left-handed properties of a superlattice composed of alternately semiconductor and antiferromagnetic (AF) layers, where the interaction between AF polaritons and semiconductor plasmons lead to the lefthandedness of the superlattice [26]. However the plasmon resonant frequency sensitively depends on the free charge carrier's density, or impurity concentration in semiconductor layers, so if one wants to see a plasmon resonant frequency near to AF resonant frequencies, the density must be very low since AF resonant frequencies are distributed in the millimeter to far infrared range. In the case of such a low density, the effect of the charge carriers on the electromagnetic properties may be very weak [27] so that there is not the left-handedness of the superlattice. According the discussion above, we propose a simple structure of multilayer which consists of AF and IC layers. An analytical condition under which both left-handeness and negative refraction phenomenon appear in the film is established by calculating the angle between the energy flow and wave vector of a plane electromagnetic

An interesting configuration in experiment is the Voigt geometry as illustrated in Fig.1, where the polariton wave propagates in the *x-y* plane and the magnetic field of an electromagnetic wave is parallel to this plane, but the wave electric field aligns the *z* direction. We concentrate our attention on the case where the external magnetic field and AF anisotropy axis both are along the *z* axis and parallel to layers. The *y* axis is perpendicular to layers in the structure. The semi-space ( *y* 0 ) is of vacuum, where *<sup>a</sup> d* and *<sup>i</sup> d* are thicknesses of AF and IC layers, respectively. For the far infrared wave, the order of the wavelength is about 100*μm*. Thus, as long as the thicknesses of AF and IC layers are less than 10*μm*, the wavelength will much longer than the period of AF/IC PCs. With this condition, the AF/IC PCs will become a uniform film by means of an effective-medium method (EMM).

**Figure 1.** Illustration and coordinate system.

#### *2.1.1. EMM for one-dimensional AF/IC PCs*

We first present the permeability of the AF film. In the external magnetic field *H*<sup>0</sup> , the magnetic permeability is well-known, with its nonzero elements [33, 37]

$$\mu\_{\infty} = \mu\_{\text{yy}} = \mu = 1 + \alpha\_m \alpha\_a \left\{ \left[ \alpha\_r^2 - \left( \alpha\_0 - \alpha - \text{i}\tau \right)^2 \right]^{-1} + \left[ \alpha\_r^2 - \left( \alpha\_0 + \alpha + \text{i}\tau \right)^2 \right]^{-1} \right\}. \tag{1}$$

$$\mu\_{\rm xy} = -\mu\_{\rm yx} = \mu\_{\perp} = \mathrm{i}o\_m o\_a \{ \left[ \mathrm{o}\_r^2 - (\mathrm{o}\_0 - \alpha \mathrm{-i}\,\mathrm{\tau})^2 \right]^{-1} - \left[ \mathrm{o}\_r^2 - (\mathrm{o}\_0 + \alpha + \mathrm{i}\,\mathrm{\tau})^2 \right]^{-1} \}. \tag{2}$$

with 0 0 *H* , <sup>0</sup> 4 *<sup>m</sup> M* ,*a a H* ,*e e H* , and 1 2 [ (2 )] *r aea* , where *M*0 is the sublattice magnetization, *Ha* represents the anisotropy field, and *H*e the exchange field. *<sup>r</sup>* is the AF resonant frequency, the gyromagnetic ratio, and the magnetic damping constant. We use *<sup>a</sup>* as the dielectric constant of the AF. Subsequently, we present the dielectric function of the IC [38],

$$\varepsilon\_{i} = \varepsilon\_{h} + \frac{(\varepsilon\_{l} - \varepsilon\_{h})\alpha\_{\Gamma}^{2}}{\alpha\_{\Gamma}^{2} - \alpha^{2} - \mathrm{i}\eta\alpha'} \,\tag{3}$$

Optical Properties of Antiferromagnetic/Ion-Crystalic Photonic Crystals 215

and

0. (11)

*k c* (12)

*k c* (13)

(14)

2

(17)

 

 are the decay

. The wave equation

(9)

(10)

<sup>z</sup> E ( )exp( )e , *y y A B E e E e ikx i t*

<sup>z</sup> E ( )exp( )e , *y y C D E e E e ikx i t*

coefficients when they are real, otherwise they correspond to the *y* wave-vector components. *<sup>j</sup> E* (*j=A, B, C or D*) denotes the amplitudes of the electric fields. Additionally, the

> 2 22 <sup>0</sup> ( ) *E E cE*

> > 2 2 22 *a v*

2 2 22 . *<sup>i</sup>*

 

Employing the well-known TMM, together withthe boundary conditions of *<sup>z</sup> E* and *Hx* continuous at the interfaces, we can find a matrix relation between wave amplitudes in any two adjacent bi-layers, or the relation between amplitudes in the *n*th and

> 1 1

*an an bn bn*

(1 ) [( )cosh( ) (1 )sinh( )], sinh( ), *<sup>a</sup> <sup>a</sup> d d*

( 1) sinh( ), [( )cosh( ) ( 1)sinh( )], *a a d d*

(15)

*i i i*

*i i i*

 *k* . The Bloch's theorem

(16)

 

 

*E E T E E* 

*e e <sup>T</sup> <sup>d</sup> d T <sup>d</sup>*

*e e <sup>T</sup> d T <sup>d</sup> <sup>d</sup>*

 

*E E e E E*

 

. *an iQL an bn bn*

 

1 1

Based on matrix relations (14) and (17), we obtain the polariton dispersion equation

     

 

 

corresponding magneticfields can be found with the relation *E iB*

and *c* is the light velocity in vacuum, and

where *T* is the transfer matrix expected and its components are

 

11 12

*<sup>k</sup>* and 2 2 ( ) / [ ( )]

respectively. *k* is the wave-vector component along x axis.

resulting from the Maxwell equations is

We see from the wave equation that

with 2 2 v 

*n+*1th bi-layers

 ( ) 

with 2 2 ( ) / [ ( )] 

implies another relation

2

21 22

 

Where *<sup>h</sup>* and *<sup>l</sup>* are the high- and low-frequency dielectric constants, but *<sup>T</sup>* is the TO resonant frequency of *k* 0 and is the phonon damping coefficient. The IC is nonmagnetic, so its magnetic permeability is taken as 1 *<sup>i</sup>* .

We assume that there are an effective relation *eff B H* between effective magnetic induction and magnetic field, and an effective relation *D E eff* between effective electric field and displacement, where these fields are considered as the wave fields in the structures. But *b h* and *d e* in any layer, where is given in Eqs.(1) for AF layer and <sup>1</sup> for IC layers. These fields are local fields in the layers. For the components of magnetic induction and field continuous at the interface, one assumes

$$H\_x = h\_{1x} = h\_{2x'} \\ H\_z = h\_{1z} = h\_{2z'} \\ B\_y = b\_{1y} = b\_{2y'} \tag{4}$$

And for those components discontinuous at the interface, one assumes

$$\mathbf{B}\_x = f\_a \mathbf{b}\_{1x} + f\_i \mathbf{b}\_{2x'} \mathbf{B}\_z = f\_a \mathbf{b}\_{1z} + f\_i \mathbf{b}\_{2z'} H\_y = f\_a \mathbf{h}\_{1y} + f\_i \mathbf{h}\_{2y}.\tag{5}$$

where the AF volume fraction / *a a f dL* and the IC volume fraction / *i i f dL* with the period *a i Ld d* .Thus the effective magnetic permeability is achieved from Eqs. (4),(5) and it is definite by *eff B H* ,

$$
\vec{\mu}\_{\rm eff} = \begin{pmatrix}
\mu\_{xx}^{\epsilon} & \mathrm{i}\mu\_{xy}^{\epsilon} & 0 \\
0 & 0 & 1
\end{pmatrix} \tag{6}
$$

with the elements

$$
\mu\_{\rm xx}^{\rm e} = \mathbf{f}\_{\rm a} \mu + \mathbf{f}\_{\rm i} \cdot (\mathbf{f}\_{\rm a} \mathbf{f}\_{\rm i} \mu\_{\perp}^2) \\
\Big/ (\mu \mathbf{f}\_{\rm i} + \mathbf{f}\_{\rm a}) \,, \\
\mu\_{\rm yy}^{\rm e} = \mu \Big/ (\mu \mathbf{f}\_{\rm i} + \mathbf{f}\_{\rm a}) \,, \\
\mu\_{\rm xy}^{\rm e} = \mu\_{\rm yx}^{\rm e} = \mathbf{f}\_{\rm a} \mu\_{\perp} \Big/ (\mu \mathbf{f}\_{\rm i} + \mathbf{f}\_{\rm a}) \,, \tag{7}
$$

On the similar principle, we can find that the effective dielectric permittivity tensor is diagonal and its elements are

$$
\boldsymbol{\varepsilon}\_{xx}^{\varepsilon} = \boldsymbol{\varepsilon}\_{zz}^{\varepsilon} = f\_a \boldsymbol{\varepsilon}\_a + f\_i \boldsymbol{\varepsilon}\_{i'}, \\
\boldsymbol{\varepsilon}\_{yy}^{\varepsilon} = \boldsymbol{\varepsilon}\_a \boldsymbol{\varepsilon}\_i / (f\_a \boldsymbol{\varepsilon}\_a + f\_i \boldsymbol{\varepsilon}\_i). \tag{8}
$$

On the base of these effective permeability and permittivity, one can consider the AF/IC PCs as homogeneous and anisotropical AF films or bulk media. The similar discussions can be found in the Chapter 3 of the book "Propagation of Electromagnetic Waves in Complex Matter" edited by Ahmed Kishk [39].

#### *2.1.2. Dispersion relations of surface and bulk MPP with transfer matrix method (TMM)*

The wave electric fields in an AF layer and IC layer are written as

Optical Properties of Antiferromagnetic/Ion-Crystalic Photonic Crystals 215

$$\vec{\mathbf{E}} = (E\_A e^{\alpha y} + E\_B e^{-\alpha y}) \exp(i k x - i \alpha t) \vec{\mathbf{e}}\_{x'} \tag{9}$$

$$\vec{\mathbf{E}} = (E\_{\mathbb{C}}e^{\beta y} + E\_{D}e^{-\beta y}) \exp(i\mathbf{k}\mathbf{x} - \mathbf{i}ot)\vec{\mathbf{e}}\_{x'} \tag{10}$$

respectively. *k* is the wave-vector component along x axis. and are the decay coefficients when they are real, otherwise they correspond to the *y* wave-vector components. *<sup>j</sup> E* (*j=A, B, C or D*) denotes the amplitudes of the electric fields. Additionally, the corresponding magneticfields can be found with the relation *E iB* . The wave equation resulting from the Maxwell equations is

$$
\nabla(\nabla \cdot \vec{E}) - \nabla^2 \vec{E} - \mu\_0 \, o^2 \not{\not{c}}^2 \vec{\varepsilon} \cdot \vec{E} = 0. \tag{11}
$$

We see from the wave equation that

214 Ferromagnetic Resonance – Theory and Applications

resonant frequency of *k* 0 and

where the AF volume fraction / *a a*

,

it is definite by *eff B H*

with the elements

diagonal and its elements are

Matter" edited by Ahmed Kishk [39].

The wave electric fields in an AF layer and IC layer are written as

 

and *d e*

Where *<sup>h</sup>* and *<sup>l</sup>* 

structures. But *b h*

2

.

12 12 12 , , , *H h hH h hB b b x x x z z zy y y* (4)

12 12 1 2 , , . *x ax i x z az i z y ay i y B fb fb B fb fb H f h fh* (5)

*f dL* and the IC volume fraction / *i i*

0 0 ,

(6)

(μf f ), μ μ f ( f f ),

 *ff ff* (8)

(7)

   

(3)

between effective magnetic

between effective electric

is given in Eqs.(1) for AF layer and

is the phonon damping coefficient. The IC is

*<sup>T</sup>* is the TO

*f dL* with the

( ) , *l hT*

*<sup>T</sup> i*

 

are the high- and low-frequency dielectric constants, but

field and displacement, where these fields are considered as the wave fields in the

<sup>1</sup> for IC layers. These fields are local fields in the layers. For the components of magnetic

period *a i Ld d* .Thus the effective magnetic permeability is achieved from Eqs. (4),(5) and

*e e xx xy e e eff xy yy*

*i* 

 

<sup>e</sup> 2 e e e <sup>μ</sup>xx a i a i i a yy i a xy yx a i a =f <sup>μ</sup>+f -(f f<sup>μ</sup> ) (μf +f ), <sup>μ</sup> 

On the similar principle, we can find that the effective dielectric permittivity tensor is

, / ( ). *e e <sup>e</sup> xx zz a a i i yy a i a a i i*

On the base of these effective permeability and permittivity, one can consider the AF/IC PCs as homogeneous and anisotropical AF films or bulk media. The similar discussions can be found in the Chapter 3 of the book "Propagation of Electromagnetic Waves in Complex

*2.1.2. Dispersion relations of surface and bulk MPP with transfer matrix method (TMM)* 

 

*i*

 

 

0 01

in any layer, where

i 2 2

 

*h*

 

nonmagnetic, so its magnetic permeability is taken as 1 *<sup>i</sup>*

We assume that there are an effective relation *eff B H*

induction and magnetic field, and an effective relation *D E eff*

And for those components discontinuous at the interface, one assumes

induction and field continuous at the interface, one assumes

$$\alpha^2 = k^2 - \varepsilon\_a \mu\_v \, o^2 \Big/ c^2 \tag{12}$$

with 2 2 v ( ) and *c* is the light velocity in vacuum, and

$$
\mathcal{B}^2 = k^2 - \varepsilon\_i \,\alpha^2 \Big/c^2\,. \tag{13}
$$

Employing the well-known TMM, together withthe boundary conditions of *<sup>z</sup> E* and *Hx* continuous at the interfaces, we can find a matrix relation between wave amplitudes in any two adjacent bi-layers, or the relation between amplitudes in the *n*th and *n+*1th bi-layers

$$
\begin{pmatrix} E\_{an+1} \\ E\_{bn+1} \end{pmatrix} = T \begin{pmatrix} E\_{an} \\ E\_{bn} \end{pmatrix} \tag{14}
$$

where *T* is the transfer matrix expected and its components are

$$T\_{11} = \frac{e^{ad\_s}}{\Delta - \Delta'} [ (\Delta - \Delta') \cosh(\beta d\_i) + (1 - \Delta \Delta') \sinh(\beta d\_i) ], \\ T\_{12} = \frac{e^{-ad\_s}(1 - \Delta'^2)}{\Delta - \Delta'} \sinh(\beta d\_i), \tag{15}$$

$$T\_{21} = \frac{e^{ad\_s}(\Lambda^2 - 1)}{\Lambda - \Lambda'} \sinh(\beta d\_i),\\ T\_{22} = \frac{e^{-ad\_s}}{\Lambda - \Lambda'} [(\Lambda - \Lambda') \cosh(\beta d\_i) + (\Lambda \Lambda' - 1) \sinh(\beta d\_i)],\tag{16}$$

with 2 2 ( ) / [ ( )] *<sup>k</sup>* and 2 2 ( ) / [ ( )] *k* . The Bloch's theorem implies another relation

$$
\begin{pmatrix} E\_{an+1} \\ E\_{bn+1} \end{pmatrix} = e^{iQL} \begin{pmatrix} E\_{an} \\ E\_{bn} \end{pmatrix}. \tag{17}
$$

Based on matrix relations (14) and (17), we obtain the polariton dispersion equation

$$\cos(QL) = \cosh(ad\_a)\cosh(\beta d\_i) + (\frac{\beta \mu\_v}{2a} + \frac{k^2 - \varepsilon\_a \mu \, o^2/c^2}{2a\beta \mu})\sinh(ad\_a)\sinh(\beta d\_i). \tag{18}$$

*Q* is the Bloch wave number for an infinite structure and is a real number, and then equation (18) describes the bulk polariton modes.

For a semi-infinite structure, it is interesting in physics that *Q i* is an imaginary number. Thus equation (18) can be used to determine the surface modes traveling along the *x* axis. We need the electromagnetic boundary conditions at the surface of this structure to find another necessary equation for the surface polariton modes. This equation is just

$$T\_{11} + \varphi T\_{12} = \exp(-\rho L) = \rho^{-1} T\_{21} + T\_{22} \,\prime \tag{19}$$

Optical Properties of Antiferromagnetic/Ion-Crystalic Photonic Crystals 217

**Figure 3.** Effective magnetic permeability and dielectric function for / 4:1 *a i d d* . (a) The magnetic

The MPP spectra are displayed in Fig.2, 3, and 5. In these spectra for dimensionless reduced

determined by Eq. (18) or (19) with *Q* 0 and *k* 0 , respectively. Meanwhile, the surface modes are obtained by Eqs. (18), (19) and (21). The thick curves with the serially numbered

as L1 and L2.Fig.2 shows the bulk bands and surface modes for the ratio / 4:1 *a i d d* <sup>a</sup> ( 0.8) *f* , which are obtained by the TMM. Four bulk continua appear including two minibands (see Figs. 2(b) and 2(c)). The top and bottom bands (see Fig.2(a)) correspond to the positive real parts of the effective magnetic permeability and dielectric function, so the effective refraction index is positive in the two bands. However, the two mini bands (see Figs.2(b) and 2(c)) are related to the negative real parts of the permeability and dielectric function, which can be found from Fig.3. It means that the effective refraction index is negative in the mini bands. Ref.14 proved that the negative refraction and left-handedness exist in the mini bands, where the transmission and refraction of the same structure were

Fig.4 displays the bulk bands and surface modes for ratio / 1:1 *a i d d* . The top bulk band is distinctly ascended to the high frequency direction, together with surface mode 1and 2. The bottom bulk band is widened conspicuously. Contrary to the previous situation, the two mini bands get significantly narrower. Compared with Fig.2, the slopes of surface modes 7 and 12 diminish appreciably, meaning their group velocity dwindles as the AF volume

<sup>r</sup> versus *k*, the shaded regions stand for the bulk bands whose boundaries are

in vacuum are labeled

permeability and (b) the dielectric function. After Wang & Ta, 2010.

examined in the absence of the external magnetic field.

sign S denote the surface polaritons. The photonic lines 2 2 *k c* ( /)

frequency

fraction ( <sup>a</sup>

*f* ) decreases.

 

where 0 0 ( )/( ) with 2 2 22 0 *k* c . <sup>0</sup> is the decay coefficient in vacuum and must be positive. It needs to be emphasized that the existence of surface modes requires Re( ) >0 .Eqs. (18) and (19) will be applied to determine the bulk polariton bands and surface polaritons.

Numerical simulations based on FeF2/TlBr will be performed with TMM. The reason is that their resonant frequencies lie in the far infrared range and are close to each other. The physical parameters here applied are *He* 533kG , <sup>0</sup> 4 7.04kG *M* , *Ha* 197kG , , 4 5 10 , and*<sup>r</sup>* 498.8kG ( <sup>1</sup> 52.45*cm* ) for AF layers; 30.4 *<sup>l</sup>* , 5.34 *<sup>h</sup>* , <sup>1</sup> <sup>48</sup> *<sup>T</sup> cm* , 3 8 10 for IC layers. The external field 0 *H T* 3.0 [10,33,34]. 5.5 *<sup>a</sup>* 

The MPP spectra are displayed in Fig.2, 3, and 5. In these spectra for dimensionless reduced f

**Figure 2.** Bulk polariton bands (shaded areas) and surface polaritons (thick solid curves) for / 4:1 *a i d d* via TMM: (a) a whole dispersion pattern; (b) and (c) are the partially enlarged figures. After Wang & Ta, 2010.

where 0 0 

surface polaritons.

4

3

After Wang & Ta, 2010.

5 10 , and

Re( 

  equation (18) describes the bulk polariton modes.

 

 ( )/( ) with 2 2 22

 

For a semi-infinite structure, it is interesting in physics that *Q i*

physical parameters here applied are *He* 533kG , <sup>0</sup> 4 7.04kG

8 10 for IC layers. The external field 0 *H T* 3.0 [10,33,34].

0 

2 22

(18)

<sup>0</sup> is the decay coefficient in vacuum

*M* , *Ha* 197kG , ,

 , <sup>1</sup> <sup>48</sup> *<sup>T</sup>* 

*cm* ,

5.5 *<sup>a</sup>* 

 , 5.34 *<sup>h</sup>* 

exp( ) , (19)

 

is an imaginary number.

*a a i a i*

 

1

 

<sup>v</sup> cos( ) cosh( )cosh( ) ( )sinh( )sinh( ). 2 2

*Q* is the Bloch wave number for an infinite structure and is a real number, and then

Thus equation (18) can be used to determine the surface modes traveling along the *x* axis. We need the electromagnetic boundary conditions at the surface of this structure to find

11 12 21 22 *T T L TT*

 *k* c .

and must be positive. It needs to be emphasized that the existence of surface modes requires

Numerical simulations based on FeF2/TlBr will be performed with TMM. The reason is that their resonant frequencies lie in the far infrared range and are close to each other. The

*<sup>r</sup>* 498.8kG ( <sup>1</sup> 52.45*cm* ) for AF layers; 30.4 *<sup>l</sup>*

**Figure 2.** Bulk polariton bands (shaded areas) and surface polaritons (thick solid curves) for

/ 4:1 *a i d d* via TMM: (a) a whole dispersion pattern; (b) and (c) are the partially enlarged figures.

The MPP spectra are displayed in Fig.2, 3, and 5. In these spectra for dimensionless reduced f

) >0 .Eqs. (18) and (19) will be applied to determine the bulk polariton bands and

 

another necessary equation for the surface polariton modes. This equation is just

*k c QL d d d d* 

**Figure 3.** Effective magnetic permeability and dielectric function for / 4:1 *a i d d* . (a) The magnetic permeability and (b) the dielectric function. After Wang & Ta, 2010.

The MPP spectra are displayed in Fig.2, 3, and 5. In these spectra for dimensionless reduced frequency <sup>r</sup> versus *k*, the shaded regions stand for the bulk bands whose boundaries are determined by Eq. (18) or (19) with *Q* 0 and *k* 0 , respectively. Meanwhile, the surface modes are obtained by Eqs. (18), (19) and (21). The thick curves with the serially numbered sign S denote the surface polaritons. The photonic lines 2 2 *k c* ( /) in vacuum are labeled as L1 and L2.Fig.2 shows the bulk bands and surface modes for the ratio / 4:1 *a i d d* <sup>a</sup> ( 0.8) *f* , which are obtained by the TMM. Four bulk continua appear including two minibands (see Figs. 2(b) and 2(c)). The top and bottom bands (see Fig.2(a)) correspond to the positive real parts of the effective magnetic permeability and dielectric function, so the effective refraction index is positive in the two bands. However, the two mini bands (see Figs.2(b) and 2(c)) are related to the negative real parts of the permeability and dielectric function, which can be found from Fig.3. It means that the effective refraction index is negative in the mini bands. Ref.14 proved that the negative refraction and left-handedness exist in the mini bands, where the transmission and refraction of the same structure were examined in the absence of the external magnetic field.

Fig.4 displays the bulk bands and surface modes for ratio / 1:1 *a i d d* . The top bulk band is distinctly ascended to the high frequency direction, together with surface mode 1and 2. The bottom bulk band is widened conspicuously. Contrary to the previous situation, the two mini bands get significantly narrower. Compared with Fig.2, the slopes of surface modes 7 and 12 diminish appreciably, meaning their group velocity dwindles as the AF volume fraction ( <sup>a</sup> *f* ) decreases.

Optical Properties of Antiferromagnetic/Ion-Crystalic Photonic Crystals 219

0 ), which is due to the combined contributions of

. Our aim is to determine general

(22)

*f* 0.8 ), which are

Fig.5 shows the bulk bands and surface modes for the ratio / 4:1 *a i d d* ( <sup>a</sup>

negative group velocities ( *d dk*

ratio, 2 2 1 / *<sup>a</sup> f RL* 

condition of

magnetic damping and phonon damping.

**2.2. MPPs in two-dimension AF/IC PCs** 

*2.2.1. EMM for the two-dimensional AF/IC PCs* 

, and the IC filling ratio, 2 2 / *<sup>i</sup>*

before establishing an EMM, a TMM should be introduced. This matrix is

*T*

*L* (λ is the polariton wavelength).

obtained by the EMM. For the bulk bands, we see that the results obtained within the two methods almost are equal. However, the surface modes obtained by the EMM start from the photonic lines and are continuous, but those achieved by the TMM do not. It is because the interfacial effects are efficiently considered within the TMM, but the EMM neglects these. For the surface polaritons, their many main features attained by the two methods are still analogous. 12 surface mode branches are seen from Fig.2 within the TMM, but 11 surface modes from Fig.5 within the EMM. 10 surface mode branches arise in the common vicinities of two AF resonant frequencies and TO phonon frequency.Except branches 1 and 2, all surface modes are nonreciprocal and their non-reciprocity results from the magnetic contribution in AF layers. Surface polaritons 1 and 2 should be called the quasi-phonon polaritons since the contribution of the magnetic response to the polaritons is very weak in their frequency range. Another interesting feature is that many surface modes possess

In this part, we consider such an AF/IC PCs constructed by periodically embedding cylinders of ionic crystal into an AF, as shown in Fig.6. We focus our attention on the situation where the external magnetic field and the AF anisotropy axis both are along the cylinder axis, or the *z*-axis. The surface of the MPC is parallel to the *x-z* plane. L and R indicate the lattice constant and cylindrical radius, respectively. We introduce the AF filling

characteristics of the surface and bulk polaritons with an effective-medium method under the

When the AF/IC PCs cell size is much shorter than the wavelength of electromagnetic wave, an EMM can be established for one to obtain the effective permeability and permittivity of the AF/IC PCs. The principle of this method is in a cell, an electromagnetic-field component continuous at the interface is assumed to be equal in the two media and equal to the corresponding effective-field component in the MPC, but one component discontinuous at the interface is averaged in the two media into another corresponding effective-field component [30,33,40-41]. Because the interface between the two media is of cylinder-style,

> cos sin 0 sin cos 0 , 0 01

 

 

Thus, we find the expression of the permeability in the cylinder coordinate system

*f RL* 

**Figure 4.** Bulk polariton bands and surface modes with the parameters and explanations as the same as those in Fig.2, except for / 1:1 *a i d d* . After Wang & Ta, 2010.

**Figure 5.** Bulk polariton bands and surface polaritons for / 4:1 *a i d d* . The interpretations of (a), (b) and (c) are the same as those of Fig.2. After Wang & Ta, 2010.

#### *2.1.3. Limiting case of small period (EMM)*

To examine the limiting case of small period or long wavelength is meaningful in physics. We let 0 *<sup>a</sup> d* and 0 *<sup>i</sup> d* in Eqs.(18) ,(19) and then find

$$\text{Ca}^{\text{2}} + (\mu\_{\text{ox}}^{\text{e}} \,/ \, \mu\_{\text{yy}}^{\text{e}}) \text{k}^{2} - \varepsilon\_{zz}^{\text{e}} \mu\_{\text{\lambda}}^{\text{e}} \text{(} \text{o} \text{/} \text{c} \text{)}^{2} = \text{0},\tag{20}$$

for the bulk modes with <sup>2</sup> ( ) / *e ee e e xx yy xy yy* , and

$$
\rho + (\mu\_{\rm xy}^{\rm e} / \mu\_{\rm yy}^{\rm e}) \mathbf{k} + \alpha\_0 \mu\_{\lambda}^{\rm e} = \mathbf{0},\\
\rho^2 = (\mu\_{\rm xx}^{\rm e} / \mu\_{\rm yy}^{\rm e}) \mathbf{k}^2 - \varepsilon\_{zz}^{\rm e} \mu\_{\lambda}^{\rm e} (\boldsymbol{\alpha} / \boldsymbol{\varepsilon})^2 = \mathbf{0},\tag{21}$$

for the surface polaritons. If the external magnetic field implicitly included in Eqs.(20) and (21) is equal to zero, the dispersion relations can be reduced to those in our earlier paper [10].Hence equations (20) and (21) also can be considered as the results achieved by the EMM.

Fig.5 shows the bulk bands and surface modes for the ratio / 4:1 *a i d d* ( <sup>a</sup> *f* 0.8 ), which are obtained by the EMM. For the bulk bands, we see that the results obtained within the two methods almost are equal. However, the surface modes obtained by the EMM start from the photonic lines and are continuous, but those achieved by the TMM do not. It is because the interfacial effects are efficiently considered within the TMM, but the EMM neglects these. For the surface polaritons, their many main features attained by the two methods are still analogous. 12 surface mode branches are seen from Fig.2 within the TMM, but 11 surface modes from Fig.5 within the EMM. 10 surface mode branches arise in the common vicinities of two AF resonant frequencies and TO phonon frequency.Except branches 1 and 2, all surface modes are nonreciprocal and their non-reciprocity results from the magnetic contribution in AF layers. Surface polaritons 1 and 2 should be called the quasi-phonon polaritons since the contribution of the magnetic response to the polaritons is very weak in their frequency range. Another interesting feature is that many surface modes possess negative group velocities ( *d dk* 0 ), which is due to the combined contributions of magnetic damping and phonon damping.

#### **2.2. MPPs in two-dimension AF/IC PCs**

218 Ferromagnetic Resonance – Theory and Applications

those in Fig.2, except for / 1:1 *a i d d* . After Wang & Ta, 2010.

and (c) are the same as those of Fig.2. After Wang & Ta, 2010.

We let 0 *<sup>a</sup> d* and 0 *<sup>i</sup> d* in Eqs.(18) ,(19) and then find

 

> 

 

*xx yy xy yy*

  

*2.1.3. Limiting case of small period (EMM)* 

for the bulk modes with <sup>2</sup> ( ) / *e ee e e* 

> 

**Figure 4.** Bulk polariton bands and surface modes with the parameters and explanations as the same as

**Figure 5.** Bulk polariton bands and surface polaritons for / 4:1 *a i d d* . The interpretations of (a), (b)

To examine the limiting case of small period or long wavelength is meaningful in physics.

2 e e2 e 2 xx yy <sup>λ</sup> ( / )k ( / ) 0, *<sup>e</sup> Q c zz*

 , and

> 

equations (20) and (21) also can be considered as the results achieved by the EMM.

 

e e e 2 e e2 e 2 xy yy 0 xx yy <sup>λ</sup> ( / )k 0, ( / )k ( / ) 0, *<sup>e</sup>*

for the surface polaritons. If the external magnetic field implicitly included in Eqs.(20) and (21) is equal to zero, the dispersion relations can be reduced to those in our earlier paper [10].Hence

  (20)

*zz c*

 

(21)

In this part, we consider such an AF/IC PCs constructed by periodically embedding cylinders of ionic crystal into an AF, as shown in Fig.6. We focus our attention on the situation where the external magnetic field and the AF anisotropy axis both are along the cylinder axis, or the *z*-axis. The surface of the MPC is parallel to the *x-z* plane. L and R indicate the lattice constant and cylindrical radius, respectively. We introduce the AF filling ratio, 2 2 1 / *<sup>a</sup> f RL* , and the IC filling ratio, 2 2 / *<sup>i</sup> f RL* . Our aim is to determine general characteristics of the surface and bulk polaritons with an effective-medium method under the condition of *L* (λ is the polariton wavelength).

#### *2.2.1. EMM for the two-dimensional AF/IC PCs*

When the AF/IC PCs cell size is much shorter than the wavelength of electromagnetic wave, an EMM can be established for one to obtain the effective permeability and permittivity of the AF/IC PCs. The principle of this method is in a cell, an electromagnetic-field component continuous at the interface is assumed to be equal in the two media and equal to the corresponding effective-field component in the MPC, but one component discontinuous at the interface is averaged in the two media into another corresponding effective-field component [30,33,40-41]. Because the interface between the two media is of cylinder-style, before establishing an EMM, a TMM should be introduced. This matrix is

$$T = \begin{pmatrix} \cos \theta & \sin \theta & 0 \\ -\sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix} \tag{22}$$

Thus, we find the expression of the permeability in the cylinder coordinate system

Optical Properties of Antiferromagnetic/Ion-Crystalic Photonic Crystals 221

(29)

*f f* (30)

(32)

and one

system.

0 0 ,

0 01

*e e rr r e ee r*

( ) *<sup>e</sup> rr rr a rr i*

<sup>2</sup> ( ) *<sup>e</sup> a rr i a i r a rr i f f ff f f*

( ). *e e r r a a rr i r ff f*

When we discuss the surface polaritons, the AF/IC PCs will be considered as a semi-infinite system with a single plane surface, so the corresponding permeability in the rectangular coordinate system (orthe xyz system) will be used. Applying the transformation matrix (22)

 

 

cos sin ( )sin cos 0 ( )sin cos sin cos 0 .

 

*e e e e e rr rr r*

 

( )2 0

 

symmetry, as shown by Fig.1, which leads to the *xx* and *yy* elements of the final

By a similarprocedure, the effective dielectric permittivity can be easily found. According to the principle of EMM, we present the equations for the electric-field and electric-

*e e e rr r e e ee a r rr* 

0 01

If one applies directly this form into the Maxwell equations, the resulting wave equation will be very difficult to solve. Thus, a further approximation is necessary. We think that if the wavelength of an electromagnetic wave is much longer than the cell size, then the wave will feel very slightly the structure information of the AF/IC PCs. Here, the averages of some

0 0 1

(33)

> 

( )2 0.

. In physics, this AF/IC PCs should be gyromagnetic and be of C4-

(34)

 

 (31)

> 

2 2

is averaged with respect to angle

 

 

 

 

 

 

Formula (29) is the expression of the effective magnetic permeability in the *r z*

2 2

 

determines the averaged effective magnetic permeability,

 

 

physics quantities are important. Hence, *<sup>e</sup>*

This means *eee*

*xx yy* 

displacement components as follows,

 

*e ee ee e rr r rr* 

 

permeability should be equal and its xy element equal to -*yx* element.

 

with

again, we find

**Figure 6.** Geometry configuration and coordinate system.

with *rr xx* and *r xy* . The theoretical processes of obtaining effective magnetic permeability, *<sup>e</sup>* , and electric permittivity, *<sup>e</sup>* , are presented as follows. According to the principle,we can introduce the following equations:

$$\mathbf{H}\_x = \mathbf{h}\_{az} = \mathbf{h}\_{iz}, \; \mathbf{H}\_0 = \mathbf{h}\_{a0} = \mathbf{h}\_{i0} \,\prime \tag{24}$$

, *r ar ir Bb b* (25)

$$H\_r = f\_a h\_{ar} + f\_i h\_{ir} \,\prime \tag{26}$$

$$B\_o = f\_a b\_{a0} + f\_i b\_{i0}, \; B\_x = f\_a b\_{ax} + f\_i b\_{ix'} \tag{27}$$

where the field components on the left side of Eqs. (24)-(27) are defined as the effective components in the AF/IC PCs and those on the right side are the field components in the AF and IC media within the cell. In the AF, the relation between *b* and *h* is determined by (23) in the *r z* system, but in the IC, the relation is

$$
\overrightarrow{b} = \overrightarrow{h}.\tag{28}
$$

After defining the relation between the effective fields in the AF/IC PCs, *<sup>e</sup> B H* , the effective permeabilityresulting from (24)-(27) is

Optical Properties of Antiferromagnetic/Ion-Crystalic Photonic Crystals 221

$$
\mu^e = \begin{pmatrix}
\mu\_{rr}^e & \mu\_{r\theta}^e & 0 \\
0 & 0 & 1
\end{pmatrix},
\tag{29}
$$

with

220 Ferromagnetic Resonance – Theory and Applications

1

*aa r T T*

 

**Figure 6.** Geometry configuration and coordinate system.

 and *r xy* 

principle,we can introduce the following equations:

, and electric permittivity, *<sup>e</sup>*

system, but in the IC, the relation is

effective permeabilityresulting from (24)-(27) is

 

<sup>z</sup> , , *H h hH h h az iz a i* 

z z , , *a a i i a az i i B fb fb B fb fb*

where the field components on the left side of Eqs. (24)-(27) are defined as the effective components in the AF/IC PCs and those on the right side are the field components in the AF and IC media within the cell. In the AF, the relation between *b* and *h* is determined by (23)

*b h* .

After defining the relation between the effective fields in the AF/IC PCs, *<sup>e</sup> B H*

 

with *rr*

in the *r z* 

 

permeability, *<sup>e</sup>*

*xx*

 

0 0 .

(23)

. The theoretical processes of obtaining effective magnetic

(24)

, *r ar ir Bb b* (25)

, *H fh fh r a ar i ir* (26)

(28)

, the

(27)

, are presented as follows. According to the

0 01

 

*rr r*

 

 

 

$$
\mu\_{rr}^{\epsilon} = \mu\_{rr} \left| (f\_a + \mu\_{rr} f\_i) \right. \tag{30}
$$

$$
\mu\_{00}^{\epsilon} = f\_a \mu\_{rr} + f\_{\ i\_l} + f\_a f\_i \mu\_{r0}^2 \left[ (f\_a + \mu\_{rr} f\_i) \right] \tag{31}
$$

$$
\mu\_{r0}^{\epsilon} = \mu\_{r0} f\_a \left| (f\_a + \mu\_{rr} f\_i) = -\mu\_{0r}^{\epsilon} \right. \tag{32}
$$

Formula (29) is the expression of the effective magnetic permeability in the *r z* system. When we discuss the surface polaritons, the AF/IC PCs will be considered as a semi-infinite system with a single plane surface, so the corresponding permeability in the rectangular coordinate system (orthe xyz system) will be used. Applying the transformation matrix (22) again, we find

$$
\bar{\mu}\_{\perp}^{\varepsilon} = \begin{bmatrix}
\mu\_{rr}^{\varepsilon}\cos^{2}\theta + \mu\_{\theta\theta}^{\varepsilon}\sin^{2}\theta & (\mu\_{rr}^{\varepsilon} - \mu\_{\theta\theta}^{\varepsilon})\sin\theta\cos\theta + \mu\_{r\theta}^{\varepsilon} & 0 \\
(\mu\_{rr}^{\varepsilon} - \mu\_{\theta\theta}^{\varepsilon})\sin\theta\cos\theta - \mu\_{r\theta}^{\varepsilon} & \mu\_{rr}^{\varepsilon}\sin^{2}\theta + \mu\_{\theta\theta}^{\varepsilon}\cos^{2}\theta & 0 \\
0 & 0 & 1
\end{bmatrix}.
\tag{33}
$$

If one applies directly this form into the Maxwell equations, the resulting wave equation will be very difficult to solve. Thus, a further approximation is necessary. We think that if the wavelength of an electromagnetic wave is much longer than the cell size, then the wave will feel very slightly the structure information of the AF/IC PCs. Here, the averages of some physics quantities are important. Hence, *<sup>e</sup>* is averaged with respect to angle and one determines the averaged effective magnetic permeability,

$$
\bar{\mu}^{\varepsilon}\_{\perp a} = \begin{pmatrix}
(\mu^{\varepsilon}\_{rr} + \mu^{\varepsilon}\_{\theta\theta}) / 2 & \mu^{\varepsilon}\_{r\theta} & 0 \\
& -\mu^{\varepsilon}\_{r\theta} & (\mu^{\varepsilon}\_{rr} + \mu^{\varepsilon}\_{\theta\theta}) / 2 & 0 \\
& 0 & 0 & 1
\end{pmatrix}.
\tag{34}
$$

This means *eee xx yy* . In physics, this AF/IC PCs should be gyromagnetic and be of C4 symmetry, as shown by Fig.1, which leads to the *xx* and *yy* elements of the final permeability should be equal and its xy element equal to -*yx* element.

By a similarprocedure, the effective dielectric permittivity can be easily found. According to the principle of EMM, we present the equations for the electric-field and electricdisplacement components as follows,

$$E\_{\rm az} = \mathbf{e}\_{\rm az} = e\_{\rm iz}, \; E\_0 = \mathbf{e}\_{\rm a} = e\_{i0}, \; D\_r = d\_{ar} = d\_{ir'} \tag{35}$$

Optical Properties of Antiferromagnetic/Ion-Crystalic Photonic Crystals 223

(43)

(44)

0 and 0

*c* (47)

 and <sup>0</sup> are

. From the Maxwell equations,

(46)

 

<sup>r</sup> in figures.

0 . Of course,

0 0 <sup>z</sup> *E E y ikx i t y* [ exp( )exp( )]e , ( 0)

1 z *E E y ikx i t y* [ exp( )exp( )]e , ( 0)

the decay coefficients and are positive for the surface polariton. The corresponding

2 2 *E cE y* ( / ) 0, ( 0)

2 2 ( / ) 0, ( 0) *e e zz v E c Ey*

2 2 22 2 e 2 0 v k ( / ), k ( / ) *<sup>e</sup>*

conditions of the field components, *<sup>z</sup> E* and *Hx* , continuous at the surface. Through a simple

 

it is very easy to find the dispersion relation of bulk polaritons. For the infinite AF/IC PCs,

2e 2 <sup>v</sup> k ( /) *<sup>e</sup> zz* 

The bulk polariton bands are just such regions determined by (46). One can calculate

FeF2 and TlBr are utilized as constituent materials in the AF/IC PCs, which the parameters have been introduced in the last section. We place the AF/IC PCs into an external field of

Surface mode curves are plotted against wave vector k along the x-axis. Bulk modes form

For comparison, we first present the polariton dispersion figures in the AF FeF2 and IC TlBr, as indicated in Fig. 7, respectively. For the AF, there exist three bulk bands and two surface modes. The surface modes appear in a nonreciprocal way and have a positive group

The surface modes are of reciprocity. Comparing Fig. 7(b) with the previous results without phonon damping [42], it is different that the two surface modes bear bended-back and end

0 ). For the IC, its surface modes and bulk bands are depicted in Fig. 7(b).

<sup>0</sup> k , *ee e r v i*

*zz*

. The dispersion relation can be found from the boundary

 *c c* (45)

 

 

 

 

directly the dispersion curves of the surface polariton from (45).

some continuous regions shown with shaded areas.

on the vacuum light line, due to the phonon damping.

<sup>0</sup> are determined by (45) with the conditions,

<sup>0</sup> *H T* 5.0 along the z-axis and employ a dimensionless reduced frequency

we find from the wave equation (44) that the dispersion relation in the *x-y* plane is

 

 

(41)

(42)

in the AF/IC PCs, where *k* is the wave-vector component along the *x*-axis, but

magneticfield can be obtained with the relation *E iB*

we confirm the electric field obeys the wave equations

 

which lead to two relations

with 2 2 [( ) ( ) )] / *ee e e v r*

 

where and 

velocity ( *d dk* 

mathematical process, we obtain this relation

$$D\_{\mathbf{Z}} = f\_a \mathbf{d}\_{\mathbf{a}\mathbf{z}} + f\_l \mathbf{d}\_{\mathbf{i}\mathbf{z}\prime} \; D\_0 = f\_a \mathbf{d}\_{\mathbf{a}0} + f\_l \mathbf{d}\_{i0\prime\prime} \; E\_{\mathbf{r}} = f\_a \mathbf{e}\_{ar} + f\_l \mathbf{e}\_{ir\prime} \tag{36}$$

with *ai ai* () () *d e* in the AF or IC. After using the definition, *<sup>e</sup> D E* , the effective dielectric permittivity of the AF/IC PCs in the *r z* system is determined as

$$
\bar{\mathcal{E}}^{\epsilon} = \begin{pmatrix}
\mathcal{E}\_{rr}^{\epsilon} & 0 & 0 \\
0 & \mathcal{E}\_{\theta\theta}^{\epsilon} & 0 \\
0 & 0 & \mathcal{E}\_{zz}^{\epsilon}
\end{pmatrix},
\tag{37}
$$

With

$$\mathcal{L}\_{rr}^{\varepsilon} = \varepsilon\_a \varepsilon\_i \Big/ (f\_{\;\;a} \varepsilon\_i + f\_i \varepsilon\_a), \; \varepsilon\_{00}^{\varepsilon} = f\_a \varepsilon\_a + f\_i \varepsilon\_{i\prime}, \; \varepsilon\_{zz}^{\varepsilon} = f\_a \varepsilon\_a + f\_i \varepsilon\_i \tag{38}$$

Transforming (37) into the form for the *xyz* system, we see

$$
\bar{\varepsilon}\_{\perp}^{\varepsilon} = \begin{pmatrix}
\varepsilon\_{rr}^{\varepsilon} \cos^2 \theta + \varepsilon\_{\theta\theta}^{\varepsilon} \sin^2 \theta & \varepsilon\_{rr}^{\varepsilon} \cos \theta \sin \theta - \varepsilon\_{\theta\theta}^{\varepsilon} \sin \theta \cos \theta & 0 \\
\varepsilon\_{rr}^{\varepsilon} \cos \theta \sin \theta - \varepsilon\_{\theta\theta}^{\varepsilon} \sin \theta \cos \theta & \varepsilon\_{rr}^{\varepsilon} \sin^2 \theta + \varepsilon\_{\theta\theta}^{\varepsilon} \cos^2 \theta & 0 \\
0 & 0 & \varepsilon\_{zz}^{\varepsilon}
\end{pmatrix}.
\tag{39}
$$

Then, its average value with respect to angle is

$$
\bar{\varepsilon}\_{\perp a}^{\varepsilon} = \begin{pmatrix}
(\varepsilon\_{rr}^{\varepsilon} + \varepsilon\_{\theta\theta}^{\varepsilon}) / 2 & 0 & 0 \\
0 & (\varepsilon\_{rr}^{\varepsilon} + \varepsilon\_{\theta\theta}^{\varepsilon}) / 2 & 0 \\
0 & 0 & \varepsilon\_{zz}^{\varepsilon}
\end{pmatrix} . \tag{40}
$$

We see *eee xx yy* . Now, this AF/IC PCs can be considered as an effective medium with effective electric permittivity <sup>e</sup> *a* and magnetic permeability *<sup>e</sup> a* . If the AF medium is taken as FeF2 with its resonant frequency about <sup>1</sup> / 2 52.45 *<sup>r</sup> c cm* , proper wavelengths should be between 170 and 190*m*. When the cell size is taken as *m* order of magnitude, such as 5*m* , the effective-medium theory is available and expressions (34) and (40) are reasonable.

#### *2.2.2. Dispersion equations of surface and bulk MPP*

The effective permittivity (40) and permeability (34) are applied to determine the dispersion equations of surface and bulk MPP in the AF/IC PCs. In the geometry of Fig.6, if the magnetic field of a plane electromagnetic wave is along the *z*-axis, the sublattice magnetizations in the AF do not couple with it, so the AF plays a role of an ordinary dielectric. Thus, we propose the electric fields of polariton waves in the AF/IC PCs are along the *z*-axis and the magnetic field is in the *x-y* plane. For a surface polariton, its electric field decaying with distance from the surface can be written as

Optical Properties of Antiferromagnetic/Ion-Crystalic Photonic Crystals 223

$$\vec{E} = [E\_0 \exp(-\alpha\_0 y) \exp(ikx - i\alpha t)] \vec{\mathbf{e}}\_{x'} (y > 0) \tag{41}$$

$$\vec{E} = [E\_1 \exp(\alpha y) \exp(i \mathbf{k} \mathbf{x} - i \alpha t)] \vec{\mathbf{e}}\_{\mathbf{z}'} \ (y < 0) \tag{42}$$

in the AF/IC PCs, where *k* is the wave-vector component along the *x*-axis, but and <sup>0</sup> are the decay coefficients and are positive for the surface polariton. The corresponding magneticfield can be obtained with the relation *E iB* . From the Maxwell equations, we confirm the electric field obeys the wave equations

$$
\nabla^2 \vec{E} + (\alpha / \ c)^2 \vec{E} = 0,\\
\text{(} y > 0\text{)}\tag{43}$$

$$
\nabla^2 \vec{E} + \text{(} \text{or} \text{/} \text{c)}^2 \varepsilon\_{zz}^{\epsilon} \mu\_v^{\epsilon} \vec{E} = 0, \text{(} y < 0 \text{)}\tag{44}
$$

which lead to two relations

222 Ferromagnetic Resonance – Theory and Applications

permittivity of the AF/IC PCs in the *r z*

Then, its average value with respect to angle

 

 

We see *eee xx yy* 

be between 170 and 190

5

effective electric permittivity <sup>e</sup>

Transforming (37) into the form for the *xyz* system, we see

 

*2.2.2. Dispersion equations of surface and bulk MPP* 

decaying with distance from the surface can be written as

*a* 

as FeF2 with its resonant frequency about <sup>1</sup> / 2 52.45 *<sup>r</sup>*

 

2 2

*rr rr e e e e e rr rr*

with *ai ai* () () *d e* 

With

z a <sup>a</sup> e ,e , , *z iz i r ar ir E eE eD d d*

(35)

(37)

 

2 2

*e zz*

*a* 

(40)

 

 

> 

 

, the effective dielectric

(38)

*e zz*

. If the AF medium is taken

*m* order of magnitude, such as

*c cm* , proper wavelengths should

(36)

 

Z a a r , ,e , *D fd fd D fd fd E f fe a z i iz a i i a ar i ir*

0 0

 

*rr a i i i a a a i i zz a a i i <sup>a</sup> f f ff ff* 

> 

( )2 0 0

 

0 ( )2 0 .

. Now, this AF/IC PCs can be considered as an effective medium with

0 0

is

0 0

and magnetic permeability *<sup>e</sup>*

*m* , the effective-medium theory is available and expressions (34) and (40) are reasonable.

The effective permittivity (40) and permeability (34) are applied to determine the dispersion equations of surface and bulk MPP in the AF/IC PCs. In the geometry of Fig.6, if the magnetic field of a plane electromagnetic wave is along the *z*-axis, the sublattice magnetizations in the AF do not couple with it, so the AF plays a role of an ordinary dielectric. Thus, we propose the electric fields of polariton waves in the AF/IC PCs are along the *z*-axis and the magnetic field is in the *x-y* plane. For a surface polariton, its electric field

 

*m*. When the cell size is taken as

 

cos sin cos sin sin cos 0 cos sin sin cos sin cos 0 .

   

(39)

*ee e e*

*e rr e e*

( ), , *e ee*

 

> 

 

*e e rr*

 

*e e e a rr*

 

system is determined as

*e zz*

> 

0 0 0 0,

 

in the AF or IC. After using the definition, *<sup>e</sup> D E*

$$\alpha\_0^2 = \mathbf{k}^2 - (\alpha / \mathbf{c})^2,\\ \alpha^2 = \mathbf{k}^2 - \varepsilon\_{zz}^e \mu\_\mathbf{v}^e (\alpha / \mathbf{c})^2 \tag{45}$$

with 2 2 [( ) ( ) )] / *ee e e v r* . The dispersion relation can be found from the boundary conditions of the field components, *<sup>z</sup> E* and *Hx* , continuous at the surface. Through a simple mathematical process, we obtain this relation

$$
\alpha = -\mathbf{i}\,\mu\_{r0}^{\epsilon}\mathbf{k}\Big/\mu^{\epsilon} - \alpha\_0\mu\_{v\prime}^{\epsilon} \tag{46}
$$

where and <sup>0</sup> are determined by (45) with the conditions, 0 and 0 0 . Of course, it is very easy to find the dispersion relation of bulk polaritons. For the infinite AF/IC PCs, we find from the wave equation (44) that the dispersion relation in the *x-y* plane is

$$\mathbf{k}^2 = \varepsilon\_{zz}^e \mu\_\mathbf{v}^e (o\nu/c)^2 \tag{47}$$

The bulk polariton bands are just such regions determined by (46). One can calculate directly the dispersion curves of the surface polariton from (45).

FeF2 and TlBr are utilized as constituent materials in the AF/IC PCs, which the parameters have been introduced in the last section. We place the AF/IC PCs into an external field of <sup>0</sup> *H T* 5.0 along the z-axis and employ a dimensionless reduced frequency <sup>r</sup> in figures. Surface mode curves are plotted against wave vector k along the x-axis. Bulk modes form some continuous regions shown with shaded areas.

For comparison, we first present the polariton dispersion figures in the AF FeF2 and IC TlBr, as indicated in Fig. 7, respectively. For the AF, there exist three bulk bands and two surface modes. The surface modes appear in a nonreciprocal way and have a positive group velocity ( *d dk* 0 ). For the IC, its surface modes and bulk bands are depicted in Fig. 7(b). The surface modes are of reciprocity. Comparing Fig. 7(b) with the previous results without phonon damping [42], it is different that the two surface modes bear bended-back and end on the vacuum light line, due to the phonon damping.

Optical Properties of Antiferromagnetic/Ion-Crystalic Photonic Crystals 225

, is negative in a large frequency range. The two

(see Eq. (1-30b)). The negative refraction and

, is negative in

*f* 0.8 to display the relevant effective permeability and permittivity.

The two mini bulk bands possess a special interest, corresponding to the negative effective magnetic permeability and negative effective dielectric permittivity of the AF/IC PCs. We

the vicinities of the AF resonant frequencies. Thus, for electromagnetic waves traveling in the *x-y* plane, the refraction index is negative and the left-handedness can exist in the two mini bulk bands. When electromagnetic waves propagate along the *z*-axis, there is no coupling between AF magnetizations and electromagnetic fields, so the electromagnetic

left-handedness were predicted in a one-dimension structure composed of identical

**Figure 9.** Bulk polariton bands and surface polaritons of the MPC for . The explanations of (a), (b) and

**Figure 10.** Effective permittivity and permeability in the MPC for : (a) the effective permittivity and (b)

*zz* 

AF resonant frequencies lie in this range and the magnetic permeability, *<sup>e</sup>*

present Fig. 10 for a

materials [10].

One can see that dielectric permittivity, *<sup>e</sup>*

waves cannot enter this range where 0 *<sup>e</sup>*

(c) are identical to those of Fig.8. After Wang & Ta, 2012.

the effective permeability. After Wang & Ta, 2012.

**Figure 7.** (a) Bulk polariton bands (shaded regions) and surface polaritons of FeF2. (b) Bulk polariton bands and surface modes of TlBr, with two vacuum light lines (thin lines). After Wang & Ta, 2012.

**Figure 8.** Bulk polariton bands (shaded regions) and surface polaritons of the MPC for 0.9 *<sup>a</sup> f* : (a) a whole dispersion pattern;(b) and (c) are the partially amplified figures. After Wang & Ta, 2012.

For the AF/IC PCs with <sup>a</sup> *f* 0.9 , Fig. 8 illustrates the dispersion features of magneto-phonon polaritons. Four bulk bands and 13 surface mode branches are found, where the surface modes are nonreciprocal (meaning the surface modes are changed when reversing their propagation directions). Two mini bulk bands and 11 surface mode branches exist in the vicinities of two AF resonant frequencies, where they are neither similar to those of the AF nor to those of the IC. Due to the combined contributions of the magnetic damping and phononic damping, the surface-mode group velocities become negative in some frequency ranges. For frequencies near the higher AF resonant frequency, the top bulk band bears a resemblance in nature to the top one of the AF, but the bottom band is analogous to the bottom one of the IC for frequencies close to the IC resonant frequency.

Figure 9 shows the bulk bands and surface modes for a *f* 0.8 . The bulk bands in this figure are characteristically similar to those in Fig. 8, but the two mini bands are narrowed and the top one risesstrikingly. For the surface modes, mode 6 in Fig. 8 splits into two surface modes in Fig. 9. Modes 5 and 11 in Fig. 8 disappear from the field of view. The surface modes are still nonreciprocal.

The two mini bulk bands possess a special interest, corresponding to the negative effective magnetic permeability and negative effective dielectric permittivity of the AF/IC PCs. We present Fig. 10 for a *f* 0.8 to display the relevant effective permeability and permittivity. One can see that dielectric permittivity, *<sup>e</sup> zz* , is negative in a large frequency range. The two

224 Ferromagnetic Resonance – Theory and Applications

For the AF/IC PCs with <sup>a</sup>

still nonreciprocal.

**Figure 7.** (a) Bulk polariton bands (shaded regions) and surface polaritons of FeF2. (b) Bulk polariton bands and surface modes of TlBr, with two vacuum light lines (thin lines). After Wang & Ta, 2012.

**Figure 8.** Bulk polariton bands (shaded regions) and surface polaritons of the MPC for 0.9 *<sup>a</sup>*

whole dispersion pattern;(b) and (c) are the partially amplified figures. After Wang & Ta, 2012.

bottom one of the IC for frequencies close to the IC resonant frequency.

Figure 9 shows the bulk bands and surface modes for a

polaritons. Four bulk bands and 13 surface mode branches are found, where the surface modes are nonreciprocal (meaning the surface modes are changed when reversing their propagation directions). Two mini bulk bands and 11 surface mode branches exist in the vicinities of two AF resonant frequencies, where they are neither similar to those of the AF nor to those of the IC. Due to the combined contributions of the magnetic damping and phononic damping, the surface-mode group velocities become negative in some frequency ranges. For frequencies near the higher AF resonant frequency, the top bulk band bears a resemblance in nature to the top one of the AF, but the bottom band is analogous to the

are characteristically similar to those in Fig. 8, but the two mini bands are narrowed and the top one risesstrikingly. For the surface modes, mode 6 in Fig. 8 splits into two surface modes in Fig. 9. Modes 5 and 11 in Fig. 8 disappear from the field of view. The surface modes are

*f* 0.9 , Fig. 8 illustrates the dispersion features of magneto-phonon

*f* : (a) a

*f* 0.8 . The bulk bands in this figure

AF resonant frequencies lie in this range and the magnetic permeability, *<sup>e</sup>* , is negative in the vicinities of the AF resonant frequencies. Thus, for electromagnetic waves traveling in the *x-y* plane, the refraction index is negative and the left-handedness can exist in the two mini bulk bands. When electromagnetic waves propagate along the *z*-axis, there is no coupling between AF magnetizations and electromagnetic fields, so the electromagnetic waves cannot enter this range where 0 *<sup>e</sup>* (see Eq. (1-30b)). The negative refraction and left-handedness were predicted in a one-dimension structure composed of identical materials [10].

**Figure 9.** Bulk polariton bands and surface polaritons of the MPC for . The explanations of (a), (b) and (c) are identical to those of Fig.8. After Wang & Ta, 2012.

**Figure 10.** Effective permittivity and permeability in the MPC for : (a) the effective permittivity and (b) the effective permeability. After Wang & Ta, 2012.

### **3. Presence of left-handedness and negative refraction of AF/IC PCs**

In the previous section, we have discussed MPPs in AF/IC PCs with the TMM and EMM for one- and two-dimension. Based on FeF2/TIBr, there are a number of surface and bulk polaritons in which the negative refraction and left-handedness can appear. In order to investigate the formation mechanism of LHM in AF/IC PCs, the external magnetic field and magnetic damping is set to be zero. In this case, according Eqs.(7) and (8), the effective permeability *<sup>e</sup>* and dielectric permittivity *<sup>e</sup>* will be described as

$$
\mu\_{xx}^{\epsilon} = (f\_a \mu + f\_i)\_{\prime} \,\mu\_{yy}^{\epsilon} = \mu \, / (f\_a + f\_i \mu)\_{\prime} \,\mu\_{zz}^{\epsilon} = 1,\tag{48}
$$

Optical Properties of Antiferromagnetic/Ion-Crystalic Photonic Crystals 227

 *f f* .The wave electric

**H** (56)

(55)

can occur simultaneously only when AF layers are thicker than IC layers, which

0 0 [ exp( ) exp( )]exp( ), *z y yx E A ik y B ik y ik x i t*

<sup>1</sup> . *z z e e xx yy*

*iyx*

**Figure 11.** Draft for incidence, reflection, refraction and transmission rays. After Wang & Song, 2009.

2 2

 

*<sup>A</sup>* is equal to

0 0 Re ( ) , Re ( ) . 2 2 *y y x x e e e e yy xx yy xx*

 

*<sup>B</sup>* and larger than

**<sup>A</sup> x yB x y S e eS e e** (57)

. Thus the angle between energy flow and wave vector can be

*I* **K Sj j** (58)

 

*e A zz I A* 

, or

/ 2 in the range (52).

*A B k k k k*

The inner product between a wave vector ( [ , ,0] *x y k k* **KA** or [ , ,0] *x y k k* **KB** ) and its corresponding energy flow is given by expression <sup>2</sup> Re( ) / 2 <sup>0</sup>

arccos[ / )], ( *j j*

The radiation in the film consists of two parts, one is the forward light (refraction light) related to amplitude *A*0 and the other is the backward light (reflection light) related to amplitude <sup>0</sup> *B* . Here, *<sup>y</sup> k* is defined as a negative number, otherwise the refraction wave corresponds to amplitude <sup>0</sup> *B* . These two situations are equivalent in essence. According to the definition of energy flow density of electromagnetic wave \* **S EH** Re( ) / 2 , the flow

 **<sup>x</sup> <sup>y</sup> <sup>e</sup> <sup>e</sup>**

*E E*

   

 

corresponds to spectral domain <sup>2</sup> 1/2 <sup>2</sup> 1/2 (2 ) (2 ) *r iam r aam* 

> 

and the corresponding magnetic field can be given by

densities of the two lights can be given by

<sup>2</sup> Re( ) / 2 <sup>0</sup> *e B zz I B* 

where *j*=*A* or *B*. It is obvious that

expressed as

field in the film can be written at

$$
\varepsilon\_{xx}^{\varepsilon} = \varepsilon\_{yy}^{\varepsilon} = \varepsilon\_a \varepsilon\_i \mid (f\_a \varepsilon\_i + f\_i \varepsilon\_a), \ \varepsilon\_{zz}^{\varepsilon} = (f\_a \varepsilon\_a + f\_i \varepsilon\_i), \tag{49}
$$

where / *a a f dD* is the AF filling ratio, and / *i i f dD* is IC filling ratio with *Dd d i a* as a bi-layer thickness.

Let us consider an incident plane electromagnetic wave propagating in the x-y plane as shown in Fig.11. Such a wave can be divided into two polarizations, a TE mode with its electric field parallel to axis z and a TM mode with its magnetic field parallel to axis z. According to Maxwell's equations, these wave vectors and frequencies of the two modes inside the film satisfy the following expressions

$$\frac{k\_y^2}{\varepsilon\_{zz}^\varepsilon \mu\_{xx}^\varepsilon} + \frac{k\_x^2}{\varepsilon\_{zz}^\varepsilon \mu\_{yy}^\varepsilon} = (o \,/\, c)^2 \quad \left(\text{for TE mode}\right). \tag{50}$$

$$k\_y^2 + k\_x^2 = \left(\alpha/c\right)^2 \varepsilon\_{xx}^\varepsilon \mu\_{zz}^\varepsilon \left(for \text{ TE mode}\right),\tag{51}$$

Since 1 *<sup>e</sup> zz* is a positive real number, relation (51) corresponds to the case of an ordinary optical (when 0 *<sup>e</sup> xx* ) or an opaque (when 0 *<sup>e</sup> xx* , contributing to an imaginary *<sup>y</sup> k* ) film, and so, we no longer consider the TM case, but deal with only the case of TE incident mode, and found the left-handed feature and negative refraction behavior. For the TE mode, the presence of left-handed feature (or negative refraction) needs the satisfaction of the prerequisite condition that 0 *<sup>e</sup> zz* and *<sup>e</sup> xx* and *<sup>e</sup> yy* can not be simultaneously positive [i.e., at least one of *<sup>e</sup> xx* and *<sup>e</sup> yy* is negative, see (50)]. According to expressions (48) and (49),

$$
\rho\_{\Gamma} \le \alpha \le \alpha\_{\Gamma} \sqrt{1 + f\_i(\varepsilon\_0 - \varepsilon\_a) / (f\_a \varepsilon\_a + f\_i \varepsilon\_a)} \text{ (for } \varepsilon\_{zz}^{\varepsilon} < 0\text{)}.\tag{52}
$$

$$
\Delta o\_r \le o \le \sqrt{o\_r^2 + 2f\_a o\_a o\_m} \text{ (for } \mu\_{\text{xx}}^\epsilon < 0\text{)}\,\tag{53}
$$

$$\sqrt{\alpha\_r^2 + 2f\_i \alpha\_a \alpha\_m} \le \alpha \le \sqrt{\alpha\_r^2 + 2\alpha\_a \alpha\_m} \text{ (for } \mu\_{yy}^\varepsilon < 0\text{)}\,\tag{54}$$

where *a a H* and 0 4 *<sup>m</sup> M* . Frequency region (52) is very large and covers regions (53) and (54) for the selected physical parameters. It is noted that both 0 *<sup>e</sup> xx* and 0 *<sup>e</sup> yy* 

can occur simultaneously only when AF layers are thicker than IC layers, which corresponds to spectral domain <sup>2</sup> 1/2 <sup>2</sup> 1/2 (2 ) (2 ) *r iam r aam f f* .The wave electric field in the film can be written at

$$E\_z = \left[ A\_0 \exp(ik\_y y) + B\_0 \exp(-ik\_y y) \right] \exp(ik\_x x - i\alpha t),\tag{55}$$

**H** (56)

and the corresponding magnetic field can be given by

226 Ferromagnetic Resonance – Theory and Applications

permeability *<sup>e</sup>*

where / *a a*

Since 1 *<sup>e</sup> zz* 

optical (when 0 *<sup>e</sup>*

at least one of *<sup>e</sup>*

where

*a a* 

*xx* 

prerequisite condition that 0 *<sup>e</sup>*

*xx* and *<sup>e</sup>*

bi-layer thickness.

and dielectric permittivity *<sup>e</sup>*

 

*f dD* is the AF filling ratio, and / *i i*

 

2 2

*k k*

*ee ee zz xx zz yy*

) or an opaque (when 0 *<sup>e</sup>*

and *<sup>e</sup>*

 

> 

(53) and (54) for the selected physical parameters. It is noted that both 0 *<sup>e</sup>*

*zz* 

> 

   

*yy* 

 *H* and 0 4 

   

*y x xx zz k k* 

inside the film satisfy the following expressions

**3. Presence of left-handedness and negative refraction of AF/IC PCs** 

In the previous section, we have discussed MPPs in AF/IC PCs with the TMM and EMM for one- and two-dimension. Based on FeF2/TIBr, there are a number of surface and bulk polaritons in which the negative refraction and left-handedness can appear. In order to investigate the formation mechanism of LHM in AF/IC PCs, the external magnetic field and magnetic damping is set to be zero. In this case, according Eqs.(7) and (8), the effective

 

 

( ), / ( ), 1, *ee e xx a i yy a i zz*

 

/ ( ), ( ), *e e <sup>e</sup> xx yy a i a i i a zz a a i i*

Let us consider an incident plane electromagnetic wave propagating in the x-y plane as shown in Fig.11. Such a wave can be divided into two polarizations, a TE mode with its electric field parallel to axis z and a TM mode with its magnetic field parallel to axis z. According to Maxwell's equations, these wave vectors and frequencies of the two modes

<sup>2</sup> ( /) , *<sup>y</sup> <sup>x</sup>*

22 2 ( ) / , *e e*

*xx* 

and so, we no longer consider the TM case, but deal with only the case of TE incident mode, and found the left-handed feature and negative refraction behavior. For the TE mode, the presence of left-handed feature (or negative refraction) needs the satisfaction of the

is a positive real number, relation (51) corresponds to the case of an ordinary

*yy* 

> 

<sup>0</sup> 1 ( )/( ) ( , 0) *<sup>e</sup>*

<sup>2</sup> 2 ( 0), *<sup>e</sup>*

 

is negative, see (50)]. According to expressions (48) and (49),

 

 

*f f f for* (52)

 

*f for* (53)

 *f for* (54)

*<sup>m</sup> M* . Frequency region (52) is very large and covers regions

 

*xx* and *<sup>e</sup>*

*T Ti aa i zz*

*r r a a m xx*

 

2 2 2 2 ( 0), *<sup>e</sup> r i a m r a m yy*

 

will be described as

 *f f ff* (48)

> 

(50)

*c for TE m do e* (51)

, contributing to an imaginary *<sup>y</sup> k* ) film,

can not be simultaneously positive [i.e.,

*xx* 

and 0 *<sup>e</sup>*

*yy* 

*for T*

*c E mode*

 *ff ff* (49)

*f dD* is IC filling ratio with *Dd d i a* as a

<sup>1</sup> . *z z e e xx yy*

*iyx*

 **<sup>x</sup> <sup>y</sup> <sup>e</sup> <sup>e</sup>**

  *E E*

 

**Figure 11.** Draft for incidence, reflection, refraction and transmission rays. After Wang & Song, 2009.

The radiation in the film consists of two parts, one is the forward light (refraction light) related to amplitude *A*0 and the other is the backward light (reflection light) related to amplitude <sup>0</sup> *B* . Here, *<sup>y</sup> k* is defined as a negative number, otherwise the refraction wave corresponds to amplitude <sup>0</sup> *B* . These two situations are equivalent in essence. According to the definition of energy flow density of electromagnetic wave \* **S EH** Re( ) / 2 , the flow densities of the two lights can be given by

$$\mathbf{S}\_{\mathbf{A}} = \text{Re}\left[\frac{\left|A\_{0}\right|^{2}}{2\alpha\nu}(\frac{k\_{\text{x}}}{\mu\_{yy}^{\varepsilon}}\mathbf{e}\_{\mathbf{x}} + \frac{k\_{\text{y}}}{\mu\_{\text{xx}}^{\varepsilon}}\mathbf{e}\_{\mathbf{y}})\right], \mathbf{S}\_{\mathbf{B}} = \text{Re}\left[\frac{\left|B\_{0}\right|^{2}}{2\alpha\nu}(\frac{k\_{\text{x}}}{\mu\_{yy}^{\varepsilon}}\mathbf{e}\_{\mathbf{x}} - \frac{k\_{\text{y}}}{\mu\_{\text{xx}}^{\varepsilon}}\mathbf{e}\_{\mathbf{y}})\right].\tag{57}$$

The inner product between a wave vector ( [ , ,0] *x y k k* **KA** or [ , ,0] *x y k k* **KB** ) and its corresponding energy flow is given by expression <sup>2</sup> Re( ) / 2 <sup>0</sup> *e A zz I A* , or <sup>2</sup> Re( ) / 2 <sup>0</sup> *e B zz I B* . Thus the angle between energy flow and wave vector can be expressed as

$$\alpha\_{j} = \arccos[I\_{j}/(\left|\mathbf{K}\_{j}\right| \left|\mathbf{S}\_{j}\right|)].\tag{58}$$

where *j*=*A* or *B*. It is obvious that *<sup>A</sup>* is equal to *<sup>B</sup>* and larger than / 2 in the range (52). It can be seen from the expression (57) of **AS** that its *x* component is negative and *y*  component is positive when both 0 *<sup>e</sup> xx* and 0 *<sup>e</sup> yy* , since *<sup>x</sup> k* is positive and identical to the equivalent component of the incident wave vector. Thus an important condition is found for the existence of negative refraction, or AF layers must be thicker than IC layers. The refraction angle can be expressed as

$$\theta' = \arctan(k\_x \mu\_{xx}^\varepsilon / k\_y \mu\_{yy}^\varepsilon). \tag{59}$$

Optical Properties of Antiferromagnetic/Ion-Crystalic Photonic Crystals 229

[43]. The matrix

. According to the boundary conditions of *<sup>z</sup> <sup>E</sup>*

(61)

*kd i kd* (62)

*k d* (63)

*k d* (64)

**Figure 13.** Refraction angle versus frequency for various incident angles, and for filling ratios (a)

**4. Transmission, refraction and absorption properties of AF/IC PCs** 

In this section, we shall examine transmission, refraction and absorption of AF/IC PCs, where the condition of the period much smaller than the wavelength is not necessary. The transmission spectra based on FeF2/TIBr PCs reveal that there exist two intriguing guided modes in a wide stop band [11]. Additionally, FeF2/TIBr PCs possess either the negative refraction or the quasi left-handedness, or even simultaneously hold them at certain frequencies of two guided modes, which require both negative magnetic permeability of AF layers and negative permittivity of IC layers. The handedness and refraction properties of the system can be manipulated by modifying the external magnetic field which will

The geometry is shown in Fig. 1. We assume the electric field solutions in AF and IC layers

[ exp( ) exp( )]exp( ), *j zj j j j x E e A ik y B ik y ik x i t*

where *j a i*, signify AF or IC layers, respectively. The corresponding magnetic field

and *Hx* continuous at interfaces, the relation between wave amplitudes in the two same

<sup>11</sup> [( )cos( ) (1 )sin( )]/ ( ), *<sup>a</sup> i i i i T*

<sup>12</sup> (1 )sin( ) / ( ), *<sup>a</sup> i i T i*

<sup>21</sup> ( 1)sin( ) / ( ), *<sup>a</sup> i i T i*

1 2

2

layers of the two adjacent periods can be shown as a transfer matrix T

*f* . After Wang & Song, 2009.

determine the frequency regimes of the guided modes.

solutions are also achieved via *j j E iB*

elements are expressed by the following equations:

0.8 *<sup>a</sup>*

as

*f* and (b) 0.6 *<sup>a</sup>*

FeF2 and TlBr are used as constituent materials where the AF resonant frequency *<sup>r</sup>* is closer to the phonon resonant frequency *<sup>T</sup>* and located in the far infrared regime. Fig.12 shows this angle as a function of frequency for 0.8 *<sup>a</sup> f* and 0.6. It can be seen from Fig.12 that for most of the frequency range occupied by the curves, angle is at least bigger than 160*<sup>o</sup>* for various incident angles. So the wave vector, electric and magnetic fields form an approximate left-handed triplet. The operation frequency range becomes narrow as *<sup>a</sup> f* decreases, as shown in Fig.12b.

As shown in Fig.13(a) for 0.8 *<sup>a</sup> f* , the refraction angle is positive on the left side and negative on the right side of the intersection point of the curves. This corresponds to the following critical frequency obtained under the condition of 0 *<sup>e</sup> xx* and 0 *<sup>e</sup> yy* :

$$
\alpha o\_c = \sqrt{o\_r^2 + 2f\_i o\_m o\_a} \,. \tag{60}
$$

It can be seen from Fig.13(b) in comparing with Fig.13(a) that the frequency region of negative refraction is obviously narrower and the negative refraction angle becomes smaller. Numerical simulations also show both positive and negative refraction angles are in the spectral range of approximate left-handed feature shown in Fig.12.

**Figure 12.** Angle beteen refraction energy flow and corresponding wave vector versus frequency for different incident angles and for filling ratios (a) 0.8 *<sup>a</sup> f* and (b) 0.6 *<sup>a</sup> f* . After Wang & Song, 2009.

component is positive when both 0 *<sup>e</sup>*

closer to the phonon resonant frequency

decreases, as shown in Fig.12b.

**Figure 12.** Angle

different incident angles and for filling ratios (a) 0.8 *<sup>a</sup>*

As shown in Fig.13(a) for 0.8 *<sup>a</sup>*

refraction angle can be expressed as

It can be seen from the expression (57) of **AS** that its *x* component is negative and *y* 

the equivalent component of the incident wave vector. Thus an important condition is found for the existence of negative refraction, or AF layers must be thicker than IC layers. The

> 

160*<sup>o</sup>* for various incident angles. So the wave vector, electric and magnetic fields form an approximate left-handed triplet. The operation frequency range becomes narrow as *<sup>a</sup>*

negative on the right side of the intersection point of the curves. This corresponds to the

<sup>2</sup> 2 . *c r ima*

It can be seen from Fig.13(b) in comparing with Fig.13(a) that the frequency region of negative refraction is obviously narrower and the negative refraction angle becomes smaller. Numerical simulations also show both positive and negative refraction angles are in the

 

beteen refraction energy flow and corresponding wave vector versus frequency for

*f* and (b) 0.6 *<sup>a</sup>*

FeF2 and TlBr are used as constituent materials where the AF resonant frequency

*yy* 

*x xx y yy*

, since *<sup>x</sup> k* is positive and identical to

*<sup>r</sup>* is

*f*

is at least bigger than

*k k* (59)

*f* , the refraction angle is positive on the left side and

*xx* 

*<sup>T</sup>* and located in the far infrared regime. Fig.12

and 0 *<sup>e</sup>*

*f* (60)

*yy* :

*f* . After Wang & Song, 2009.

*f* and 0.6. It can be seen from Fig.12

and 0 *<sup>e</sup>*

*xx* 

arctan( / ). *e e*

shows this angle as a function of frequency for 0.8 *<sup>a</sup>*

that for most of the frequency range occupied by the curves, angle

following critical frequency obtained under the condition of 0 *<sup>e</sup>*

spectral range of approximate left-handed feature shown in Fig.12.

 

**Figure 13.** Refraction angle versus frequency for various incident angles, and for filling ratios (a) 0.8 *<sup>a</sup> f* and (b) 0.6 *<sup>a</sup> f* . After Wang & Song, 2009.

### **4. Transmission, refraction and absorption properties of AF/IC PCs**

In this section, we shall examine transmission, refraction and absorption of AF/IC PCs, where the condition of the period much smaller than the wavelength is not necessary. The transmission spectra based on FeF2/TIBr PCs reveal that there exist two intriguing guided modes in a wide stop band [11]. Additionally, FeF2/TIBr PCs possess either the negative refraction or the quasi left-handedness, or even simultaneously hold them at certain frequencies of two guided modes, which require both negative magnetic permeability of AF layers and negative permittivity of IC layers. The handedness and refraction properties of the system can be manipulated by modifying the external magnetic field which will determine the frequency regimes of the guided modes.

The geometry is shown in Fig. 1. We assume the electric field solutions in AF and IC layers as

$$\vec{E}\_{j} = \vec{e}\_{z} \{ A\_{j} \exp(ik\_{j}y) + B\_{j} \exp(-ik\_{j}y) \} \exp(ik\_{x}x - i\alpha t),\tag{61}$$

where *j a i*, signify AF or IC layers, respectively. The corresponding magnetic field solutions are also achieved via *j j E iB* . According to the boundary conditions of *<sup>z</sup> <sup>E</sup>* and *Hx* continuous at interfaces, the relation between wave amplitudes in the two same layers of the two adjacent periods can be shown as a transfer matrix T [43]. The matrix elements are expressed by the following equations:

$$T\_{11} = \delta\_a \left[ (\Delta - \Delta') \cos(k\_i d\_j) + i(1 - \Delta \Delta') \sin(k\_i d\_j) \right] / (\Delta - \Delta'),\tag{62}$$

$$T\_{12} = \mathrm{i}\,\delta\_a^{-1} (1 - \Delta'^2) \sin(k\_i d\_i) / (\Delta - \Delta'),\tag{63}$$

$$T\_{21} = \mathrm{i}\mathcal{S}\_{\underline{a}}(\Delta^2 - 1)\sin(k\_{\underline{i}}d\_{\underline{i}})/(\Delta - \Delta'),\tag{64}$$

$$T\_{22} = \delta\_a^{-1} [ (\Delta - \Delta') \cos(k\_i d\_i) + i (\Delta \Delta' - 1) \sin(k\_i d\_i) ] / (\Delta - \Delta') \tag{65}$$

Optical Properties of Antiferromagnetic/Ion-Crystalic Photonic Crystals 231

(74)

, where volume fractions are

. The stacking number is *N* 9 . Figure 14 shows the

*arctg f S f S f S f S* (75)

(72)

*r*

*r* to 1.2

(69)

(70)

*ik y ik x i t*

1

 

1

 

> 

\* \*\* Re 2 Re <sup>2</sup> , ,. *j jj jy jz x jz jx y S E H H Ee EH e i <sup>j</sup> <sup>a</sup>* (73)

(71)

view to achieving refraction properties. Based on wave solutions of the electric field and Maxwell equations, the magnetic field components of the forward-going wave in AF layers

> ( /) exp( ). *a x ax a ax k ik <sup>H</sup> A ik y ik x i t*

> (/ ) exp( ). *a x ay a ax ik k <sup>H</sup> A ik y ik x i t*

exp( ), exp( ). *i i x i*

1 2

(1 ) (1 ) (1 ) (1 ) *i a aa i a a a A A B B* 

1 2

According to boundary conditions, the electric and magnetic fields of every layer are acquired when the incident wave is known. Then the expressions of refraction energy flow

What needs to be emphasized is that we here concentrate only on the refraction, so only the forward-going wave corresponding to the first term in Eq.(61) is considered and the backward-going wave is ignored. Owing to refraction angles being different in various layers, the refraction angle of the AF/IC PCs should be effective one. The angle between the energy flow and wave vector, and the refraction angle of the AF/IC PCs are defined as

arccos[( ) / ( )], *KS KS*

12 12 ' [( ) / ( )], *ax ix ay iy*

Numerical calculations based on FeF2/TlBr PCs. We take the AF layer thickness 4 *<sup>a</sup> d m*

transmission spectra with specific angles of the incidence in an external magnetic field

0

0

 

*k A k A <sup>H</sup> ik y ik x i t H*

<sup>0</sup> *H T* 3 . As illustrated in Fig. 14(a), the forbidden band ranges from 0.9

<sup>1</sup> *aai f ddd* and 2 1 *f f* 1 , respectively.

and the thickness of IC layers 1 *<sup>i</sup> d m*

and 1 2 *a i S fS fS*

in all layers are written as

with *K ke fk fk e xx a i y* 1 2

The amplitudes of two neighboring layers satisfy

 

 

0 0

The magnetic field components of the forward-going wave in the adjacent IC layers are

*ix i x iy i x*

 

are shown as

with / *x ai ik k k* and / *x ai ik k k* . In AF layers, there are the relation 2 22 2 *av x* <sup>a</sup> *k ck* with 2 2 *v* , 2 22 2 sin *<sup>x</sup> k c* and *<sup>a</sup>* the dielectric constant. The magnetic permeability tensor components of AF layers are represented by

$$\mu = 1 + \alpha\_m \alpha\_a \{ 1/\left[ \alpha\_r^2 - (\alpha\_0 - \alpha - i\sigma)^2 \right] + 1/\left[ \alpha\_r^2 - (\alpha\_0 + \alpha + i\sigma)^2 \right] \}.\tag{66}$$

$$\mu\_{\perp} = \alpha\_m \alpha\_a \{ 1/\left[ \alpha\_r^2 - (\alpha\_0 - \alpha - i\sigma)^2 \right] - 1/\left[ \alpha\_r^2 - (\alpha\_0 + \alpha + i\sigma)^2 \right] \}. \tag{67}$$

with 0 4 *<sup>m</sup> M* , *a a H* , 1 2 [ (2 )] *r aea* ,*e e H* and 0 0 *H* . *M*<sup>0</sup> represents the sublattice magnetization, and *H*<sup>0</sup> , *Ha* and *H*<sup>e</sup> indicate the external magnetic field, anisotropy and the exchange fields, respectively. *<sup>r</sup>* denotes the zero-field resonant frequency, and are the gyromagnetic ratio and the magnetic damping. In IC layers, we have the relation 2 22 2 *ii x k ck* with the dielectric function 2 22 ( ) *i h l hT T i* , where *<sup>h</sup>* and *<sup>l</sup>* are the high- and low-frequency dielectric constants, but *T* indicates the frequency of the TO vibrating mode in the longwavelength limition and denotes the phonon damping. The magnetic permeability of IC layers is considered as 1 *<sup>i</sup>* . We assume here that the stacking number included in the magnetic superlattices is *N*. Then transmission and reflection coefficients of the AISL can be written as

$$
\begin{pmatrix} 1 \\ r \end{pmatrix} = \frac{1}{2} \begin{pmatrix} 1+\Lambda\_1 & 1+\Lambda\_1' \\ 1-\Lambda\_1 & 1-\Lambda\_1' \end{pmatrix} \overleftrightarrow{T}^{N-1} \begin{pmatrix} k\_i \delta\_a \delta\_i \left(1+\Lambda\right)/\left(k\_1+k\_i\right) & k\_i \delta\_a^{-1} \delta\_i \left(1+\Lambda'\right)/\left(k\_1+k\_i\right) \\ k\_i \delta\_a \delta\_i^{-1} \left(1-\Lambda\right)/\left(k\_i-k\_1\right) & k\_i \delta\_a^{-1} \delta\_i^{-1} \left(1-\Lambda'\right)/\left(k\_i-k\_1\right) \end{pmatrix} \begin{pmatrix} t \\ t \end{pmatrix}. \tag{68}
$$

Note that incident wave amplitude is taken as 1. Therefore, the transmission and reflection coefficients can be determined with Eq. (68), and then the transmission ratio is <sup>2</sup> *t* and reflection ratio is <sup>2</sup> *<sup>r</sup>* . Additionally, absorption ratio is represented with 2 2 *<sup>A</sup>* <sup>1</sup> *r t* . Other quantities in Eq. (68) are 1 1 / *x a ik k k* , 1 1 / *x a ik k k* , exp( ) *i ii ik d* , 2 22 2 <sup>1</sup>*k c* cos .

As described in Ref. [8], magnetic superlattices possess two mini-bands with negative group velocity. When the incident wave is located in the frequency regions corresponding to the two mini-bands, what are the optical properties of the AF/IC PCs? In the preceding section, the expressions of transmission and absorption to be used have been derived. To grasp handedness and refraction properties of the AF/IC PCs, the refraction angle and propagation direction need to be determined. Therefore, subsequently we give the expression of the refraction angle. However, this structure possibly possesses a negative refraction, and generally the directions of the energy flow of electromagnetic wave and the wave vector misalign. We start with the definition of energy flow ( \* *S EH* Re[ ] / 2 ) with a view to achieving refraction properties. Based on wave solutions of the electric field and Maxwell equations, the magnetic field components of the forward-going wave in AF layers are shown as

$$H\_{ax} = \frac{\left(k\_a + ik\_x \mu\_\perp \mid \mu\right)}{o\mu\_0 \mu\_\nu} A\_a \exp(ik\_a y + ik\_x \mathbf{x} - i\alpha t). \tag{69}$$

$$H\_{ay} = \frac{(ik\_a \mu\_\perp / \mu - k\_x)}{\alpha \mu\_0 \mu\_\nu} A\_a \exp(ik\_a y + ik\_x x - i\alpha t). \tag{70}$$

The magnetic field components of the forward-going wave in the adjacent IC layers are

$$\frac{k\_i H\_i}{\alpha \mu\_0} = \frac{k\_i A\_i}{\alpha \mu\_0} \exp(ik\_i y + ik\_x x - i\alpha t),\\ H\_{ij} = -\frac{k\_x A\_i}{\alpha \mu\_0} \exp(ik\_i y + ik\_x x - i\alpha t). \tag{71}$$

The amplitudes of two neighboring layers satisfy

230 Ferromagnetic Resonance – Theory and Applications

*av x* <sup>a</sup> *k ck* 

 

2 22 ( ) *i h l hT T*

*a a* 

have the relation 2 22 2

 

anisotropy and the exchange fields, respectively.

*T*

Other quantities in Eq. (68) are 1 1 / *x a ik k k*

.

 

with / *x ai ik k k*

relation 2 22 2

with 0 4 

frequency,

written as

 *<sup>m</sup> M* ,

> and

dielectric constants, but

wavelength limition and

reflection ratio is <sup>2</sup>

*ik d* , 2 22 2

<sup>1</sup>*k c* cos

exp( ) *i ii*

layers is considered as 1 *<sup>i</sup>*

  1

> 

 

 and / *x ai ik k k*

<sup>22</sup> [( )cos( ) ( 1)sin( )] / ( ) *<sup>a</sup> i i i i T*

0 0 1 {1 / [ ( ) ] 1 / [ ( ) ]}, *ma r <sup>r</sup>*

the sublattice magnetization, and *H*<sup>0</sup> , *Ha* and *H*<sup>e</sup> indicate the external magnetic field,

 and *<sup>l</sup>* 

magnetic superlattices is *N*. Then transmission and reflection coefficients of the AISL can be

1 1 1 1 1

*r k kk k k k t*

Note that incident wave amplitude is taken as 1. Therefore, the transmission and reflection

As described in Ref. [8], magnetic superlattices possess two mini-bands with negative group velocity. When the incident wave is located in the frequency regions corresponding to the two mini-bands, what are the optical properties of the AF/IC PCs? In the preceding section, the expressions of transmission and absorption to be used have been derived. To grasp handedness and refraction properties of the AF/IC PCs, the refraction angle and propagation direction need to be determined. Therefore, subsequently we give the expression of the refraction angle. However, this structure possibly possesses a negative refraction, and generally the directions of the energy flow of electromagnetic wave and the wave vector misalign. We start with the definition of energy flow ( \* *S EH* Re[ ] / 2 ) with a

coefficients can be determined with Eq. (68), and then the transmission ratio is <sup>2</sup>

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

*N iai i ia i i iai i ia i i*

*<sup>r</sup>* . Additionally, absorption ratio is represented with 2 2 *<sup>A</sup>* <sup>1</sup> *r t* .

 , 2 22 2 sin *<sup>x</sup> k c*

2 22 2

2 22 2 0 0 {1 / [ ( ) ] 1 / [ ( ) ]}. *ma r <sup>r</sup>*

 

 

*e e H* and

*i i* (67)

are the gyromagnetic ratio and the magnetic damping. In IC layers, we

*T* indicates the frequency of the TO vibrating mode in the long-

. We assume here that the stacking number included in the

denotes the phonon damping. The magnetic permeability of IC

 

(68)

 

*k kk k k k t*

 

1

 , 1 1 / *x a ik k k*

*i i* (66)

constant. The magnetic permeability tensor components of AF layers are represented by

 with 2 2 *v* 

*H* , 1 2 [ (2 )]

*ii x k ck* 

> 

 

*i* , where *<sup>h</sup>*

*r aea* ,

*kd i k d* (65)

. In AF layers, there are the

 and *<sup>a</sup>* 

 

with the dielectric function

*<sup>r</sup>* denotes the zero-field resonant

are the high- and low-frequency

 ,

the dielectric

0 0 *H* . *M*<sup>0</sup> represents

*t* and

 

 

$$
\begin{pmatrix} A\_i \\ B\_i \end{pmatrix} = \begin{pmatrix} (1 + \Delta\_1)\delta\_a & (1 - \Delta\_2)\delta\_a^{-1} \\ (1 - \Delta\_1)\delta\_a & (1 + \Delta\_2)\delta\_a^{-1} \end{pmatrix} \begin{pmatrix} A\_a \\ B\_a \end{pmatrix} \tag{72}
$$

According to boundary conditions, the electric and magnetic fields of every layer are acquired when the incident wave is known. Then the expressions of refraction energy flow in all layers are written as

$$\vec{S}\_{j} = \text{Re}\left(\vec{E}\_{j} \times \vec{H}\_{j}^{\ast}\right) \Big/ \mathbf{2} = \text{Re}\left(-H\_{\dagger y}^{\ast} E\_{\dagger z} \vec{e}\_{x} + E\_{\dagger z} H\_{\dagger x}^{\ast} \vec{e}\_{y}\right) \Big/ \mathbf{2} \,, \tag{73} \tag{73}$$

What needs to be emphasized is that we here concentrate only on the refraction, so only the forward-going wave corresponding to the first term in Eq.(61) is considered and the backward-going wave is ignored. Owing to refraction angles being different in various layers, the refraction angle of the AF/IC PCs should be effective one. The angle between the energy flow and wave vector, and the refraction angle of the AF/IC PCs are defined as

$$\alpha = \arccos[\left(\bar{K} \cdot \bar{S}\right) / \left(\left|\bar{K}\right| \left|\bar{S}\right|\right)].\tag{74}$$

$$\theta' = \operatorname{arctg} [(f\_1 S\_{ax} + f\_2 S\_{ix}) / (f\_1 S\_{ay} + f\_2 S\_{iy})] \,\text{.}\tag{75}$$

with *K ke fk fk e xx a i y* 1 2 and 1 2 *a i S fS fS* , where volume fractions are <sup>1</sup> *aai f ddd* and 2 1 *f f* 1 , respectively.

Numerical calculations based on FeF2/TlBr PCs. We take the AF layer thickness 4 *<sup>a</sup> d m* and the thickness of IC layers 1 *<sup>i</sup> d m* . The stacking number is *N* 9 . Figure 14 shows the transmission spectra with specific angles of the incidence in an external magnetic field <sup>0</sup> *H T* 3 . As illustrated in Fig. 14(a), the forbidden band ranges from 0.9*r* to 1.2*r*

Optical Properties of Antiferromagnetic/Ion-Crystalic Photonic Crystals 233

*<sup>r</sup>* , the absorption is obviously

*f* 0.8 , *N* 9 , 0 *H T* 3 and <sup>0</sup> *H T* 1 . (a)

maximum transmission. In the absorbing band at 1.024

strengthened with enhancing the incident angle.

**Figure 15.** Partially enlarged absorption spectra with 1

AF/IC PCs possesses a quasi left-handedness in these frequency regions.

16. We find the refraction angles are negative in the neighborhood of 0.943

handedness in Fig.17. However, the result is opposite in the vicinity of 1.064

Therefore, we here reckon the left-handedness is not always accompanied by negative refraction. FeF2/TlBrsuperlattices have the natures of either negative refraction or quasi lefthandedness, or even simultaneously bear them at the certain frequencies of two guided

To have a deeper understanding of the negative refraction and quasi- left-handedness of the

*<sup>r</sup>* . The two frequency ranges for negative angle do not completely coincide with those of the quasi-left-handedness in Fig. 17. Namely, the frequency regime of negative

 

> 

AF/IC PCs, subsequently the expressions of the dielectric function

corresponding to Fig. 15(d). After Wang & Ta, 2012.

frequency for 0 *H T* <sup>3</sup> and <sup>o</sup>

Figure 17 shows the refraction angle

angle 

modes.

0.943

1.064

 

refraction near to 0.943

 *<sup>r</sup>* and 1.064 

corresponding to Fig. 14(b); (b) corresponding to Fig.14(d); (c) corresponding to Fig. 15(b); (d)

To capture the handedness and refraction behaviors of the AF/IC PCs, the angle of refraction and the angle between the energy flow and wave vector are illustrated. Fig. 16 shows the

between the energy flow and wave vector of forward-going wave varies with

45 . As illustrated in Fig. 16, the angles in the vicinities of

' versus frequency under the same condition as Fig.

 *<sup>r</sup>* and

> *<sup>r</sup>* .

*<sup>r</sup>* are greater than <sup>o</sup> 90 , but less than 180 . It indicates that the

*<sup>r</sup>* is strikingly greater than that occupied by the quasi-left-

**Figure 14.** Transmission spectra for the fraction 1 *f* 0.8 , external magnetic field 0 *H T* 3 and stacking number *N* 9 . (a) the incident angle <sup>0</sup> 0 ; (b) a zoomed view of guided modes in (a); (c) the incident angle <sup>0</sup> 45 ; (d) a zoomed view of guided modes in (c). After Wang & Ta, 2012.

corresponding to the band gap of magneto-phonon polariton in Ref. [8]. Here the most interesting may be that guided modes arise in the forbidden band. The two guided modes lie in the proximity of 0.943 *<sup>r</sup>* and 1.064 *<sup>r</sup>* , corresponding to the mini-bands with negative group velocity in Ref. [8]. At the same time, the magnetic permeability of AF layers and the dielectric function for IC layers are both negative. To distinctly observe two guided modes, the partially enlarged Fig. 14(b) corresponding to Fig. 14(a) is exhibited. Seen from Fig. 14(b), the maximum transmission of the guided mode with lower-frequency is 40% and that of higher mode is 28.4%. As is well known, the optical thicknesses of films are determined by the frequency-dependent magnetic permeability and the dielectric function. Then the optical thicknesses of thin films are varied with the frequency of incident wave. The optical path of wave in media can also be altered by changing the incident angle. Fig. 14(c) shows the transmission spectrum with incident angle <sup>o</sup> 45 and other parameters are the same as those in 14(a). The partially enlarged Fig. 14(d) corresponding to Fig. 14(c) is given. Compared with the normal incidence case, for 0 the forbidden band becomes wide and their maximum transmissions are reduced to 27.1% and 21.9%, but two guided modes keep their frequency positions unaltered.

As already noted, the damping is included and then the absorption appears. We are more interest in the two guided modes, so only the absorption corresponding to two guided modes will be considered in Fig. 15 (a) and (c) display the absorption spectra in the case of right incidence, but (b) and (d) illustrate the absorption spectra for incident angle <sup>o</sup> 45 . We see that the absorption has a great influence on the transmission spectra. In the absorbing band at 0.9426 *<sup>r</sup>* , relative tiny absorption corresponds exactly to the maximum transmission. In the absorbing band at 1.024 *<sup>r</sup>* , the absorption is obviously strengthened with enhancing the incident angle.

232 Ferromagnetic Resonance – Theory and Applications

**Figure 14.** Transmission spectra for the fraction 1

the incident angle <sup>0</sup>

lie in the proximity of 0.943

stacking number *N* 9 . (a) the incident angle <sup>0</sup>

given. Compared with the normal incidence case, for

modes keep their frequency positions unaltered.

 absorbing band at 0.9426

*<sup>r</sup>* and 1.064 

45 ; (d) a zoomed view of guided modes in (c). After Wang & Ta, 2012.

*<sup>r</sup>* , relative tiny absorption corresponds exactly to the

corresponding to the band gap of magneto-phonon polariton in Ref. [8]. Here the most interesting may be that guided modes arise in the forbidden band. The two guided modes

negative group velocity in Ref. [8]. At the same time, the magnetic permeability of AF layers and the dielectric function for IC layers are both negative. To distinctly observe two guided modes, the partially enlarged Fig. 14(b) corresponding to Fig. 14(a) is exhibited. Seen from Fig. 14(b), the maximum transmission of the guided mode with lower-frequency is 40% and that of higher mode is 28.4%. As is well known, the optical thicknesses of films are determined by the frequency-dependent magnetic permeability and the dielectric function. Then the optical thicknesses of thin films are varied with the frequency of incident wave. The optical path of wave in media can also be altered by changing the incident angle. Fig. 14(c) shows the transmission spectrum with incident angle <sup>o</sup> 45 and other parameters are the same as those in 14(a). The partially enlarged Fig. 14(d) corresponding to Fig. 14(c) is

wide and their maximum transmissions are reduced to 27.1% and 21.9%, but two guided

As already noted, the damping is included and then the absorption appears. We are more interest in the two guided modes, so only the absorption corresponding to two guided modes will be considered in Fig. 15 (a) and (c) display the absorption spectra in the case of right incidence, but (b) and (d) illustrate the absorption spectra for incident angle <sup>o</sup>

We see that the absorption has a great influence on the transmission spectra. In the

*f* 0.8 , external magnetic field 0 *H T* 3 and

0 ; (b) a zoomed view of guided modes in (a); (c)

*<sup>r</sup>* , corresponding to the mini-bands with

0 the forbidden band becomes

45 .

**Figure 15.** Partially enlarged absorption spectra with 1 *f* 0.8 , *N* 9 , 0 *H T* 3 and <sup>0</sup> *H T* 1 . (a) corresponding to Fig. 14(b); (b) corresponding to Fig.14(d); (c) corresponding to Fig. 15(b); (d) corresponding to Fig. 15(d). After Wang & Ta, 2012.

To capture the handedness and refraction behaviors of the AF/IC PCs, the angle of refraction and the angle between the energy flow and wave vector are illustrated. Fig. 16 shows the angle between the energy flow and wave vector of forward-going wave varies with frequency for 0 *H T* <sup>3</sup> and <sup>o</sup> 45 . As illustrated in Fig. 16, the angles in the vicinities of 0.943 *<sup>r</sup>* and 1.064 *<sup>r</sup>* are greater than <sup>o</sup> 90 , but less than 180 . It indicates that the AF/IC PCs possesses a quasi left-handedness in these frequency regions.

Figure 17 shows the refraction angle ' versus frequency under the same condition as Fig. 16. We find the refraction angles are negative in the neighborhood of 0.943 *<sup>r</sup>* and 1.064 *<sup>r</sup>* . The two frequency ranges for negative angle do not completely coincide with those of the quasi-left-handedness in Fig. 17. Namely, the frequency regime of negative refraction near to 0.943 *<sup>r</sup>* is strikingly greater than that occupied by the quasi-lefthandedness in Fig.17. However, the result is opposite in the vicinity of 1.064 *<sup>r</sup>* . Therefore, we here reckon the left-handedness is not always accompanied by negative refraction. FeF2/TlBrsuperlattices have the natures of either negative refraction or quasi lefthandedness, or even simultaneously bear them at the certain frequencies of two guided modes.

To have a deeper understanding of the negative refraction and quasi- left-handedness of the AF/IC PCs, subsequently the expressions of the dielectric function

**Figure 16.** Angle between energy flow and wave vector of down going wave versus the change of frequencies at 0 *H T* <sup>3</sup> , <sup>0</sup> 45 and 1 *f* 0.8 . After Wang & Ta, 2012.

**Figure 17.** Refraction angle ' versus the alteration of frequencies at 0 *H T* <sup>3</sup> , <sup>0</sup> 45 and 1 *f* 0.8 . After Wang & Ta, 2012.

 2 22 ( ) *i h l hT T i* of IC layers and the magnetic permeability of AF layers are analyzed. We find that when frequency lies in

$$
\alpha \alpha\_{\Gamma} < \alpha < \left(\varepsilon \sqrt{\varepsilon\_h \beta\_h}\right)^{1/2} \alpha\_{\Gamma} \tag{76}
$$

Optical Properties of Antiferromagnetic/Ion-Crystalic Photonic Crystals 235

versus the different frequencies at 0 *H T* <sup>3</sup> , <sup>0</sup>

45 and

refraction in the limit of long wavelength. Regarding the results arising from the two methods mentioned above, we conclude that the necessary condition of negative refraction

are both negative in this PCs.

or left-handedness is that *<sup>v</sup>*

**Figure 18.** Voigt magnetic permeability *<sup>v</sup>*

Shu-Fang Fu and Xuan-Zhang Wang

1

2012.

**5. Summary** 

**Author details** 

**Acknowledgement** 

**6. References** 

Province, with no. ZD200913.

 and *<sup>i</sup>* 

*f* 0.8 (solid line denotes real parts and broken line indicates imaginary parts). After Wang & Ta,

This chapter aims to discover optical properties of AF/IC PCs in the presence of external static magnetic field. First, within the effective-medium theory, we investigated dispersion properties of MPPs in one- and two-dimension AF/IC PCs. The ATR (attenuated total reflection) technique should be powerful in probing these MPPs. Second, there is a frequency region where the negative refraction and the quasi left-handedness appear when the AF/IC PCs period is much shorter than the incident wavelength. Finally, an external

magnetic field can be used to modulate the optical properties of the AF/IC PCs.

*Physics and Electronic Engineering, Harbin Normal University, Harbin, China* 

*Key Laboratory for Photonic and Electronic Bandgap Materials, Ministry of Education, School of* 

This work was financially supported by the National Natural Science Foundation of China with Grant no.11084061, 11104050, and the Natural Science Foundation of Heilongjiang

[1] T. Goto, A .V. Baryshev, M. Inoue, A. V. Dorofeenko, A. M. Merzlikin, A. P. Vinogradov, A. A. Lisyansky, A. B. Granovsky, Tailoring surfaces of one-dimensional

the dielectric function *<sup>i</sup>* is negative, namely the range 0.9152 2.1835 *r r* . This completely covers the frequency range of AF resonance, so the dielectric function must be negative in the region of negative magnetic permeability *<sup>v</sup>* . It is found from Fig.18 that the magnetic permeability *<sup>v</sup>* is negative in certain regions, where the dielectric function *<sup>i</sup>* is also negative. In our previous work [8], utilizing the effective medium theory, we verified the effective dielectric function and magnetic permeability are both negative in the long wavelength limit when *<sup>v</sup>* and *<sup>i</sup>* are negative. In other words, the AF/IC PCs is of negative refraction in the limit of long wavelength. Regarding the results arising from the two methods mentioned above, we conclude that the necessary condition of negative refraction or left-handedness is that *<sup>v</sup>* and *<sup>i</sup>* are both negative in this PCs.

**Figure 18.** Voigt magnetic permeability *<sup>v</sup>* versus the different frequencies at 0 *H T* <sup>3</sup> , <sup>0</sup> 45 and 1 *f* 0.8 (solid line denotes real parts and broken line indicates imaginary parts). After Wang & Ta, 2012.

### **5. Summary**

234 Ferromagnetic Resonance – Theory and Applications

**Figure 16.** Angle

layers are analyzed. We find that when frequency lies in

negative in the region of negative magnetic permeability *<sup>v</sup>*

 and *<sup>i</sup>* 

 

45 and 1

frequencies at 0 *H T* <sup>3</sup> , <sup>0</sup>

**Figure 17.** Refraction angle

the dielectric function *<sup>i</sup>*

magnetic permeability *<sup>v</sup>*

wavelength limit when *<sup>v</sup>*

2 22 ( ) *i h l hT T*

 

After Wang & Ta, 2012.

 

between energy flow and wave vector of down going wave versus the change of

*f* 0.8 . After Wang & Ta, 2012.

' versus the alteration of frequencies at 0 *H T* <sup>3</sup> , <sup>0</sup>

1 2 ,

 

is negative, namely the range 0.9152 2.1835

is negative in certain regions, where the dielectric function *<sup>i</sup>*

are negative. In other words, the AF/IC PCs is of negative

*T lh T*

completely covers the frequency range of AF resonance, so the dielectric function must be

also negative. In our previous work [8], utilizing the effective medium theory, we verified the effective dielectric function and magnetic permeability are both negative in the long

*i* of IC layers and the magnetic permeability of AF

(76)

. It is found from Fig.18 that the

*r r* 

 . This

45 and 1

*f* 0.8 .

is This chapter aims to discover optical properties of AF/IC PCs in the presence of external static magnetic field. First, within the effective-medium theory, we investigated dispersion properties of MPPs in one- and two-dimension AF/IC PCs. The ATR (attenuated total reflection) technique should be powerful in probing these MPPs. Second, there is a frequency region where the negative refraction and the quasi left-handedness appear when the AF/IC PCs period is much shorter than the incident wavelength. Finally, an external magnetic field can be used to modulate the optical properties of the AF/IC PCs.

### **Author details**

Shu-Fang Fu and Xuan-Zhang Wang

*Key Laboratory for Photonic and Electronic Bandgap Materials, Ministry of Education, School of Physics and Electronic Engineering, Harbin Normal University, Harbin, China* 

### **Acknowledgement**

This work was financially supported by the National Natural Science Foundation of China with Grant no.11084061, 11104050, and the Natural Science Foundation of Heilongjiang Province, with no. ZD200913.

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## *Edited by Orhan Yalçın*

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 technical activity. The modern technical 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.

Ferromagnetic Resonance - Theory and Applications

Ferromagnetic Resonance

Theory and Applications

*Edited by Orhan Yalçın*

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