Preface

Chapter 8 **Giant Magnetoimpedance Effect and AC Magnetic**

**Co/garnet Heterostructures 195**

Leszek Malkinski and Rahmatollah Eskandari

Chapter 11 **Magnetic Properties of Gadolinium-Doped ZnO Films and**

Iman S. Roqan, S. Assa Aravindh and Singaravelu Venkatesh

**FeCoNbBSiCu 181**

**VI** Contents

Andrés Rosales-Rivera

Andrzej Stupakiewicz

**Nanostructures 249**

Chapter 10 **Magnetic Micro-Origami 223**

**Susceptibility in Amorphous Alloys System of**

Chapter 9 **Magnetization Statics and Ultrafast Photoinduced Dynamics in**

Zulia Isabel Caamaño De Ávila, Amilkar José Orozco Galán and

This book provides up-to-date information on recent trends and developments in material technology, characterization techniques, and theory and applications of magnetic materials with novel contributions from the renowned scientists in the field of material science and magnetism. The book addresses diverse groups of readers, including students, engineers, and researchers, from different fields—physics, chemistry, engineering, electronics, and ma‐ terials science—who wish to enhance their knowledge and research capabilities in magnet‐ ism and magnetic materials.

The book contains eleven chapters discussing scaling in magnetic materials, characterization of cylindrical magnetic nanowires, magnetic dynamics–induced charge and spin transport on the surface of topological insulator with magnetism, metamaterial properties of 2D ferro‐ magnetic nanostructures, molecular magnetism modeling with their application in spincrossover compounds, proteresis of core-shell nanocrystals, radiation and propagation of waves in magnetic materials with helicoidal magnetic structure, giant magneto-impedance effect and AC magnetic susceptibility in amorphous alloys systems, magnetization statics and ultrafast photo-induced dynamics in Co/garnet heterostructures, magnetic micro-origa‐ mi, and magnetic properties of rare earth and transition metals. The topics in the book are brought to an appropriate level covering a wide range of magnetic materials. At the end of each chapter, proper references are included, which can lead the readers to the best sources in the literature and help them to go into more depth in the field of magnetism.

I am grateful to the InTech's publishing team for making this project possible and to all the authors who have contributed to this book. I am also thankful to the Publishing Process Manager Ms. Romina Rovan for her cooperative attitude during the reviewing and publish‐ ing processes. I hope that this book will help the readers to learn more about magnetic mate‐ rials and will provide an opportunity to strengthen their knowledge in the field of magnetism and magnetic materials.

> **Dr. Maaz Khan** Physics Division PINSTECH, Islamabad Pakistan

## **Chapter 1**

## **Scaling in Magnetic Materials**

Krzysztof Z. Sokalski, Barbara Ślusarek and Jan Szczygłowski

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/63285

#### **Abstract**

The chapter presents applications of the scaling in several problems of magnetic materials. Soft magnetic materials (SMMs) and soft magnetic composites (SMCs) are considered. Application of scaling in investigations of problems, such as power losses, losses separation, data collapse of the losses characteristics and modelling of the magnetic hysteresis, is presented. The symmetry group generated by scaling and gauge transformations enables us to introduce the classification of the hysteresis loops with respect to the equivalence classes. SMC materials require special treatment in the production process. Therefore, algorithms for optimization of the power losses are created. The algorithm for optimization processes is based on the scaling and the notion of the pseudo-equation of state. The scaling makes modelling and calculations easy; however, the data must obey the scaling. Checking procedure of statistical data to this respect is presented.

**Keywords:** magnetic materials, hysteresis loop, power losses, losses separation, scal‐ ing, gauge

#### **1. Introduction**

The notion of scaling describes invariance of various phenomena and their mathematical models with respect to a change of scale. Let us take into account the simplest mathematical model revealing such behaviour:

$$\mathbf{y} = A\mathbf{x}^{\alpha}.\tag{1}$$

© 2016 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.

where α and *A* are the constants of model. Such functions appear in mathematical modelling in physics, mathematics, biology, economics and engineering. In this section, we consider data of classical gas. By simple calculation, we will prove that these data are self-similar. Let us change the scale of the both variables *x* and y with respect to *λ* > 0 multiplier.

$$
\dot{\mathbf{x}}' = \mathcal{X}^{\theta} \mathbf{x}, \dot{\mathbf{y}}' = \mathcal{X}^{\prime} \mathbf{y}. \tag{2}
$$

Substituting (2)–(1) we obtain:

$$\mathbf{y}' = \mathbf{A}'\mathbf{x}'^{a},\tag{3}$$

where β and γ are scaling exponents and *A*′ = *λγ*−*αβ A* is the model constant in new scale. (1)– (3) reveal that the phenomenon described by the model (1) is self-similar. This means that the phenomenon reproduces itself on different scales. In achieving the property of self-similarity, an important role plays dimensional analysis. Its idea is very simple: physical laws cannot depend on an arbitrary choice of basic units of measurement [1]. Set of all transformations (2) and multiplication constitutes =(ℝ<sup>+</sup> , ·) group. In the next section, we will consider selfsimilar model of hysteresis loop. Extension of (2) to the two parameters group is necessary to this respect. Let us extend (1) to non-homogenous form:

$$\mathbf{y} = A\mathbf{x}^a + \mathbf{c},\tag{4}$$

The full symmetry of (4) consists of the two transformations, *λ* scaling and *χ* gauge transfor‐ mation:

$$\mathbf{y}' = A(\lambda \mathbf{x})^a + \mathbf{c} + \mathbf{y}.\tag{5}$$

where *χ* is additive gauge operation which constitutes additive group =(ℝ, + ). Therefore, the full symmetry of (5) consists of the following direct product: [2]. However, the symmetry of the hysteresis loop will occur to be semi-direct product.

In this chapter, we will consider more advanced function and then the models (1) and (4). Therefore, we will need definition of homogenous function in general sense [3] which has played crucial role in the all achievements presented in this chapter. Let *f*(*x*1, *x*2, … *xn*) be a function of *n* variables. If ∃(*α*0, *<sup>α</sup>*1, *<sup>α</sup>*2, …*αn*)ℝ*n***+1** such that ∀*λ*ℝ<sup>+</sup> , the following relation holds:

$$
\mathcal{A}^{u\_0} f\left(\mathbf{x}\_1, \mathbf{x}\_2, \dots \mathbf{x}\_n\right) = f\left(\mathcal{A}^{u\_1} \mathbf{x}\_1, \mathcal{A}^{u\_2} \mathbf{x}\_2, \dots \mathcal{A}^{u\_s} \mathbf{x}\_n\right). \tag{6}
$$

Then, *f* (*x*1, *x*2, … *xn*) is homogenous function in general sense. Based on the measurement data of classical gas presented in **Table 1**, we present simple application of this notion. We assume that the phenomena are represented by measurement data which satisfy certain relation called law. Let us assume that searched on phenomenon have a form of the homogenous function in general sense:

$$
\lambda^a \rho(T, p) = \rho\left(\lambda^b T, \lambda^s p\right). \tag{7}
$$


**Table 1.** Measurement data of classical gas.

where α and *A* are the constants of model. Such functions appear in mathematical modelling in physics, mathematics, biology, economics and engineering. In this section, we consider data of classical gas. By simple calculation, we will prove that these data are self-similar. Let us

> g

(2)

¢ ¢¢ = (3)

= + (4)

+ (5)

(6)

 l

change the scale of the both variables *x* and y with respect to *λ* > 0 multiplier.

Substituting (2)–(1) we obtain:

2 Magnetic Materials

mation:

this respect. Let us extend (1) to non-homogenous form:

' ' *x xy y* , . b

> *y Ax* , a

*y Ax* c, a

*y Ax c* () . a ¢ = + l

symmetry of the hysteresis loop will occur to be semi-direct product.

a

l

The full symmetry of (4) consists of the two transformations, *λ* scaling and *χ* gauge transfor‐

where *χ* is additive gauge operation which constitutes additive group =(ℝ, + ). Therefore, the full symmetry of (5) consists of the following direct product: [2]. However, the

In this chapter, we will consider more advanced function and then the models (1) and (4). Therefore, we will need definition of homogenous function in general sense [3] which has played crucial role in the all achievements presented in this chapter. Let *f*(*x*1, *x*2, … *xn*) be a function of *n* variables. If ∃(*α*0, *<sup>α</sup>*1, *<sup>α</sup>*2, …*αn*)ℝ*n***+1** such that ∀*λ*ℝ<sup>+</sup> , the following relation holds:

> ( ) ( ) <sup>0</sup> 1 2 1 2 1 2 , , ,, . *<sup>n</sup> n n fxx x f x x x*

¼= ¼

a

ll

 a  a

 l

 c

where β and γ are scaling exponents and *A*′ = *λγ*−*αβ A* is the model constant in new scale. (1)– (3) reveal that the phenomenon described by the model (1) is self-similar. This means that the phenomenon reproduces itself on different scales. In achieving the property of self-similarity, an important role plays dimensional analysis. Its idea is very simple: physical laws cannot depend on an arbitrary choice of basic units of measurement [1]. Set of all transformations (2) and multiplication constitutes =(ℝ<sup>+</sup> , ·) group. In the next section, we will consider selfsimilar model of hysteresis loop. Extension of (2) to the two parameters group is necessary to

 = = l

> where *ρ, T* and *p* are gas density, temperature and pressure, respectively. Coefficients *a, b* and *c* are the scaling exponents. Since (7) holds for each value of *λ*, we are free to substitute the following expression *λ* = *T*−1/*<sup>b</sup>* and get the following relation:

$$T^{-a/b} \rho(T, p) = \rho\left(1, T^{-g/b} p\right). \tag{8}$$

Introducing new symbols for the scaling exponents: *<sup>α</sup>* <sup>=</sup> *<sup>a</sup> <sup>b</sup>* , *<sup>γ</sup>* <sup>=</sup> *<sup>g</sup> <sup>b</sup>* we derive the following equation of state:

$$T^{-\alpha}\,\rho = AT^{-\gamma}\,\mathbf{p}.\tag{9}$$

where the right-hand side of (8) was approximated by linear function, *A* is an expansion's coefficient. In the next sections of this chapter, we will use the Maclaurin expansion beyond the linear term as a way for creation of scaling function. Equation (9) depends on the one effective exponent *δ* = *γ*–*α*:

$$
\rho(T, p) = A \frac{p}{T^{\delta}}.\tag{10}
$$

where the model constants *A* and *δ* have to be determined from the experimental data of **Table 1**. Using formula (10) and **Table 1** we have created error function *Chi*<sup>2</sup> which was minimized by the SOLVER program of the Excel package. The obtained results are as follows: *A* = 0.121 (mol K J−1) and *δ* = 1.002 (−). Note that *A*−1 = 8.22 (mol−1 K−1J) reveals an approximation of the gas constant. Mentioned and illustrated above methodology for applications of the scaling and the gauge transformations will be applied to the following problems: self-similarity of hysteresis, self-similarity of total loss in SMM, multi-scaling of core losses in SMM, optimiza‐ tion of total loss in SMC and scaling conception of losses separation.

#### **2. Self-similar model of hysteresis loop**

The goal of the present section is to describe the self-similar mathematical model of hysteresis loop which enables us to express its self-similarity by the homogeneous function in general sense. Derivation of the model based on the well-known properties of tanh(⋅) suits for model of initial magnetization function [4]. It describes properly the saturation for both asymptotic values of the magnetic field *H* → ±∞, as well as and the behaviour of magnetization in the neighbourhood of origin. However, this is too rigid for scaling. We make tanh(⋅) to be a softer by making the base of the natural logarithm free parameters [5]:

$$
\tanh(x) - > \tan H(a, b, c, d; \boldsymbol{x}) = \frac{a^{\times} - b^{-\times}}{c^{\times} + d^{-\times}}.\tag{11}
$$

where the bases have to satisfy the following conditions: *a* > 1, *b >* 1, *c* > 1, *d* > 1. These conditions guarantee correctness of the model; however, a little deviations from the mentioned constrains are possible.

First, we write down the model expression for initial magnetization curve:

$$M\_{\,\,P}\left(X,\varepsilon\right) = M\_{\,\,0}P\left(X,\varepsilon\right);\, X\epsilon[0,X\_{\,\,max}].\tag{12}$$

where *M*<sup>0</sup> is magnetization corresponding to saturation expressed in tesla: [T], *<sup>X</sup>* <sup>=</sup> *<sup>H</sup> <sup>h</sup>* , where *H* is magnetic field, and *h* is a parameter of the magnetic field dimension (A m−1) to be determined. Function *P*(*X, ε*) is of the following form:

$$P\left(X,\varepsilon\right) = \frac{a^{X-\varepsilon} - b^{-X+\varepsilon}}{c^{X-\varepsilon} - d^{-X+\varepsilon}}\tag{13}$$

where ε is modelling parameter related to *θ*, where *<sup>ε</sup>* <sup>−</sup> *<sup>θ</sup>* <sup>2</sup> , <sup>+</sup> *<sup>θ</sup>* <sup>2</sup> . Let the upper and the lower branches of the hysteresis loop are of the following forms:

$$M\_F\left(X,\theta\right) = M\_0 F\left(X,\theta\right);\ M\_G\left(X,\theta\right) = M\_0 G\left(X,\theta\right). \tag{14}$$

where

the linear term as a way for creation of scaling function. Equation (9) depends on the one

d

where the model constants *A* and *δ* have to be determined from the experimental data of **Table**

by the SOLVER program of the Excel package. The obtained results are as follows: *A* = 0.121 (mol K J−1) and *δ* = 1.002 (−). Note that *A*−1 = 8.22 (mol−1 K−1J) reveals an approximation of the gas constant. Mentioned and illustrated above methodology for applications of the scaling and the gauge transformations will be applied to the following problems: self-similarity of hysteresis, self-similarity of total loss in SMM, multi-scaling of core losses in SMM, optimiza‐

The goal of the present section is to describe the self-similar mathematical model of hysteresis loop which enables us to express its self-similarity by the homogeneous function in general sense. Derivation of the model based on the well-known properties of tanh(⋅) suits for model of initial magnetization function [4]. It describes properly the saturation for both asymptotic values of the magnetic field *H* → ±∞, as well as and the behaviour of magnetization in the neighbourhood of origin. However, this is too rigid for scaling. We make tanh(⋅) to be a softer

where the bases have to satisfy the following conditions: *a* > 1, *b >* 1, *c* > 1, *d* > 1. These conditions guarantee correctness of the model; however, a little deviations from the mentioned constrains

= (10)

which was minimized

(11)

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

(, ) . *<sup>p</sup> Tp AT*

r

**1**. Using formula (10) and **Table 1** we have created error function *Chi*<sup>2</sup>

tion of total loss in SMC and scaling conception of losses separation.

by making the base of the natural logarithm free parameters [5]:

First, we write down the model expression for initial magnetization curve:

e

determined. Function *P*(*X, ε*) is of the following form:

*M X MP X X X <sup>P</sup>* ( , ) <sup>0</sup> ( , ; [0, ]. ) *max*

where *M*<sup>0</sup> is magnetization corresponding to saturation expressed in tesla: [T], *<sup>X</sup>* <sup>=</sup> *<sup>H</sup>*

=

 e

*H* is magnetic field, and *h* is a parameter of the magnetic field dimension (A m−1) to be

ò

(12)

**2. Self-similar model of hysteresis loop**

effective exponent *δ* = *γ*–*α*:

4 Magnetic Materials

are possible.

$$F(X,\theta) = \frac{a^{X\circ\theta} - b^{-X-\theta}}{c^{X\circ\theta} - d^{-X-\theta}} \quad G(X,\theta) = \frac{a^{X\circ\theta} - b^{-X\circ\theta}}{c^{X-\theta} - d^{-X\circ\theta}}.\tag{15}$$

Let us consider for illustration the following symmetric example: *a* = *b* = *c = d* = 4 and *θ* = 1.3, *ε* = 0, *M*0 = 1 (See **Figure 1**).

**Figure 1.** A model of nucleation-type hysteresis constructed with functions *F, P* and *G* according to (12)–(15).

Due to the asymptotic properties of magnetization, the functions *F***(***X, θ***)** and *G***(***X***, θ)** have to possess the same asymptotic properties. As we have mentioned, these components get equal values for *H* **→ ±∞**. However, due to the uncertainty of measured magnitudes, it is possible to accept the saturation points at *X* = *Xmin* and *X* = *Xmax* being the end points of the hysteresis loop. Therefore, the modelling process has to ensure that the initial function satisfies the following constrain:

$$\left| \left| F \left( X\_{\text{max}}, \theta \right) - G \left( X\_{\text{max}}, \theta \right) \right| \lessdot \left| \psi \right|. \tag{16}$$

where |*ψ*| =*su pX* | *MF* (*X* , *θ*) − *MG*(*X* , *θ*) | *<sup>M</sup>*<sup>0</sup> is dimensionless measure of uncertainty corresponding to |*MF* (*X* )−*MG*(*X* )|.

The scaling in space of loops is performed by scaling each loop's component (13), (14), (15) and (16). For simplicity of further investigations, we consider simplified symmetric model, where all the bases of tanH( ) are equal:

$$\frac{M\_F(X,\theta)}{M\_0} = \frac{(a)^{X+\theta} - (a)^{-X-\theta}}{(a)^{X+\theta} + (a)^{-X-\theta}}\tag{17}$$

$$\frac{M\_G(X,\theta)}{M\_0} = \frac{(a)^{X-\theta} - (a)^{-X+\theta}}{(a)^{X-\theta} + (a)^{-X+\theta}}\tag{18}$$

$$\frac{M\_F(X,\varepsilon)}{M\_0} = \frac{(a)^{X\circ\varepsilon} - (a)^{-X-\varepsilon}}{(a)^{X\circ\varepsilon} + (a)^{-X-\varepsilon}}\tag{19}$$

Let us perform scaling on (17) and (18). Since exponents are dimensionless, the scaling on this level cannot be supported by dimension analysis. However, we are able to scale the following magnitudes *a*, *MF*(*X*, θ), *MG*(*X*, θ), *MP*(*X*, θ), and to prove the following theorem:

For the symmetric model (17)–(19), the hysteresis loop is invariant with respect to scaling and gauge transformation [5]. Following definition of the homogeneous function in general sense (6), we write down the scaled form of the hysteresis loop:

$$\frac{M\_F(X,\theta)}{M\_0}\lambda^\nu = \frac{(\lambda^a a)^{X\ast\theta} a^\chi - (\lambda^a a)^{-X-\theta} a^{-\chi}}{(\lambda^a a)^{X\ast\theta} a^\chi + (\lambda^a a)^{-X-\theta} a^{-\chi}},\tag{20}$$

$$\frac{M\_G(X,\theta)}{M\_0}\lambda^\nu = \frac{(\lambda^a a)^{X-\theta} a^\chi - (\lambda^a a)^{-X \ast \theta} a^{-\chi}}{(\lambda^a a)^{X-\theta} a^\chi + (\lambda^a a)^{-X \ast \theta} a^{-\chi}},\tag{21}$$

where exponentials *a<sup>χ</sup>* and *a*<sup>−</sup>*<sup>χ</sup>* represent action of the gauge transformation. This formal trick guarantees proper order of actions: the first has to be performed scaling than after that gauge transformation. According to the assumption just above (6), we are free to make the following substitution:

$$
\mathcal{X}^a = a^{p-1},
\tag{22}
$$

where . Substituting (22) leads to the following forms of (20) and (21):

Therefore, the modelling process has to ensure that the initial function satisfies the following

The scaling in space of loops is performed by scaling each loop's component (13), (14), (15) and (16). For simplicity of further investigations, we consider simplified symmetric model, where

> () () *X X*

> () () *X X*

> () () *X X*

Let us perform scaling on (17) and (18). Since exponents are dimensionless, the scaling on this level cannot be supported by dimension analysis. However, we are able to scale the following

For the symmetric model (17)–(19), the hysteresis loop is invariant with respect to scaling and gauge transformation [5]. Following definition of the homogeneous function in general sense

( ,) ( ) ( ) , () ()

( ,) ( ) ( ) , () ()

where exponentials *a<sup>χ</sup>* and *a*<sup>−</sup>*<sup>χ</sup>* represent action of the gauge transformation. This formal trick guarantees proper order of actions: the first has to be performed scaling than after that gauge

*MX a a a a M aa a a*

*MX a a a a M aa a a*

a qc

a qc

a qc

a qc

l

l

n

n

l

ql

l

ql

*X X*

 l

 l

 a

+ -- - + -- -

 a

 a


 a  q c

 q c

 q c

 q c



*X X*

*X X*

 l

 l

*X X*

e

e

+ --

q

q


q

q

+ --

*X X*

*X X*

*X X*

+ --


+ --

 q

 q

 q

 q

 e

 e

( , ) () ()

( , ) () ()

( ,) () ()

*MX a a M aa*

magnitudes *a*, *MF*(*X*, θ), *MG*(*X*, θ), *MP*(*X*, θ), and to prove the following theorem:

*MX a a M aa*

*MX a a M aa*

 q

) - £ ( *max* ) *ψ* (16)




*<sup>M</sup>*<sup>0</sup> is dimensionless measure of uncertainty corresponding


q

0

0

0

q

q

e

*F*

*G*

*P*

(6), we write down the scaled form of the hysteresis loop:

0

0

*F*

*G*


constrain:

6 Magnetic Materials

where |*ψ*| =*su pX*

to |*MF* (*X* )−*MG*(*X* )|.

all the bases of tanH( ) are equal:

$$\frac{M\_F(X,\theta)}{M\_0}a^\* = \frac{(a)^{p(X+\theta)+\chi} - (a)^{p(-X-\theta)-\chi}}{(a)^{p(X+\theta)+\chi} + (a)^{p(-X-\theta)-\chi}}\tag{23}$$

$$\frac{M\_G(X,\theta)}{M\_0}a^{\
u} = \frac{(a)^{p(X-\theta)\cdot\chi} - (a)^{p(-X\circ\theta)\cdot\chi}}{(a)^{p(X-\theta)\cdot\chi} + (a)^{p(-X\circ\theta)\cdot\chi}}.\tag{24}$$

where for abbreviation, we introduce *<sup>n</sup>* <sup>=</sup> *<sup>ν</sup> <sup>α</sup>* (*p* −1). Introducing the following new variables:

$$M\_0' = a^{-\imath} M\_0,\\ X' = pX + \jmath, \theta' = p\,\theta,\tag{25}$$

**Figure 2.** Magnetic hysteresis family for *<sup>a</sup>* =4, p=1, n=1, *<sup>θ</sup>* =1.3, *<sup>ν</sup> <sup>α</sup>* =1.

we derive (17) and (18), which proves the considered thesis. The initial magnetization curve (19) is invariant with respect to scaling and gauge transformation as well. The proof goes the same way as for (17) and (18). Therefore, we can formulate conclusion that the presented model of hysteresis loop is self-similar. Below in **Figures 2**–**4**, we present some examples of the hysteresis loops which may suggest how to apply the scaling and gauge transformation for modelling of hysteresis phenomena.

**Figure 3.** Magnetic hysteresis family for *<sup>a</sup>* =4, <sup>χ</sup>=0, *<sup>n</sup>* =1, *<sup>θ</sup>* =1.3, *<sup>ν</sup> <sup>α</sup>* =1.

**Figure 4.** Magnetic hysteresis family for the following values of the model and scaling parameters: (*a*) *<sup>a</sup>* =4, <sup>θ</sup>=3, *<sup>p</sup>* =14, *<sup>v</sup> <sup>α</sup>* <sup>=</sup> <sup>−</sup>0.18, *<sup>χ</sup>* =0, (*b*) *<sup>a</sup>* =4, <sup>θ</sup> =2, *<sup>p</sup>* =13, *<sup>v</sup> <sup>α</sup>* = −0.18, *χ* =0, (*c*)*<sup>a</sup>* =4, <sup>θ</sup> =1.3, *<sup>χ</sup>* =0, (*b*) <sup>θ</sup> =2, *<sup>p</sup>* =13, *<sup>v</sup> <sup>α</sup>* = −0.18, *χ* =0.

**Figure 2** presents how the pure gauge transformations generate a displacement of transformed loops along the horizontal axis. **Figure 3** presents compression of loops along the vertical axis under the scaling. Finally, **Figure 4** presents the loop's family for large value of the scaling parameter *p*. Each element of this set resembles the Preisach hysteron [6, 7].

of hysteresis loop is self-similar. Below in **Figures 2**–**4**, we present some examples of the hysteresis loops which may suggest how to apply the scaling and gauge transformation for

*<sup>α</sup>* =1.

**Figure 4.** Magnetic hysteresis family for the following values of the model and scaling parameters:

*<sup>α</sup>* = −0.18, *χ* =0.

*<sup>α</sup>* = −0.18, *χ* =0,

*<sup>α</sup>* <sup>=</sup> <sup>−</sup>0.18, *<sup>χ</sup>* =0, (*b*) *<sup>a</sup>* =4, <sup>θ</sup> =2, *<sup>p</sup>* =13, *<sup>v</sup>*

modelling of hysteresis phenomena.

8 Magnetic Materials

**Figure 3.** Magnetic hysteresis family for *<sup>a</sup>* =4, <sup>χ</sup>=0, *<sup>n</sup>* =1, *<sup>θ</sup>* =1.3, *<sup>ν</sup>*

(*a*) *<sup>a</sup>* =4, <sup>θ</sup>=3, *<sup>p</sup>* =14, *<sup>v</sup>*

(*c*)*<sup>a</sup>* =4, <sup>θ</sup> =1.3, *<sup>χ</sup>* =0, (*b*) <sup>θ</sup> =2, *<sup>p</sup>* =13, *<sup>v</sup>*

**Multi-scaling of hysteresis loop**. The derived above mathematical model of the hysteresis loop is not full. There is need to extend considered model with respect to frequency, pick of induction and temperature. As we have shown the loop's model is self-similar with respect to scaling of the following magnitudes *X, M*0 and *θ*, these are as follows: dimensionless magnetic field, amplitude of magnetization and loop's closing parameter, respectively. By introducing new independent variables, we introduce new scales and corresponding new scaling param‐ eters [8]. We have shown in [8, 9] that it is always possible to introduce a new characteristic scale; however, one must investigate whether the considered system possesses corresponding symmetry. This can be known only from investigations of the measurement data. In the considered model, there are two places where the new variables can be implemented. These are base of the tanH() function and the loop's closing parameter *θ*. There are many possible combinations for configurations of the new variables in presented model which can be applied:

$$\mathbf{B} \cdot \left[ \mathbf{i} \right] B\_{n}^{\theta^{\prime}} a \left( \frac{f}{B\_{n}^{a^{\prime}}}, \frac{T}{B\_{n}^{\prime}} \right); B\_{n}^{\theta^{\prime}} \theta \left( \frac{f}{B\_{n}^{a^{\prime}}}, \frac{T}{B\_{n}^{\prime}} \right) \left[ ; \mathbf{i} \right] \left[ B\_{n}^{\theta} a \left( \frac{f}{B\_{n}^{a^{\prime}}}, \frac{T}{B\_{n}^{\prime}} \right); B\_{n}^{\theta^{\prime}} \theta \left( - \right) \right]; \mathbf{i} \text{(iii)} \left[ B\_{n}^{\theta^{\prime}} a \left( \frac{f}{B\_{n}^{a^{\prime}}}, \frac{T}{B\_{n}^{\prime}} \right); \theta \left( - \right) \right]. \tag{25a}$$

The list (25a) is not closed and can be extended as needed. We assume that *a*(*f, Bm*, *T*) and *θ* (*f, Bm*, *T*) are homogenous functions in general sense. To simplify considerations, we chose for illustration the temperature less model (*iii*), where *θ*(−) is a constant. For the model of the function *a*( *<sup>f</sup> Bm <sup>α</sup>* ′), we chose the square polynomial:

$$a = B\_{\
u}^{\rho^\*} \left( \Gamma\_{\text{,}}^{\cdot} \frac{f}{B\_{\
u}^{\alpha^\*}} + \Gamma\_{\text{,}}^{\cdot} \left( \frac{f}{B\_{\
u}^{\alpha^\*}} \right)^2 \right) \tag{25b}$$

where *α* ′ , *β* ′ , Γ′ 1, Γ′ 2 are model constants to be calculated from the measurement data, whereas the *Bm* pick of induction is correlated with *Xmin*and *Xmax*. Formulae (25b), (14) and (15) constitute frequency- and pick of induction-dependent loop model.

**Equivalence classes and partitioning [10]**. Transformation formulae (25) enable us to inves‐ tigate algebraic structure of all the transformations which are composed of scaling and gauge transformation. Therefore, the whole set of *p* and the multiplication constitute group. Moreover, ∀*a* > 0 and <sup>∀</sup> *<sup>v</sup> <sup>α</sup>* ≠0, the following expression *a* (1<sup>−</sup> *<sup>p</sup>*) *<sup>ν</sup> <sup>α</sup>* represents an infinite number of groups being isomorphic to . Gauge transformations *χ* ℝ and the addition constitute group. Each of revealed group possesses the own representation space. group operates in the spaces generated by *X, θ* and M0 *and* group operates in the space spanned by *X* variable. The considered model of loop reveals a combination of and which operates in space generated by (*X, θ*) pair. Therefore, total symmetry of the considered model can be presented in the following matrix representation (26):

$$
\begin{bmatrix} p & 0 & 0 \\ 0 & a^{(1-p)\frac{V}{a}} & 0 \\ 0 & 0 & p \end{bmatrix} \begin{bmatrix} X \\ M\_0 \\ \theta \end{bmatrix} + \begin{bmatrix} \mathcal{X} \\ \mathbf{0} \\ \mathbf{0} \end{bmatrix} = \begin{bmatrix} X' \\ M\_0' \\ \theta\_0' \end{bmatrix}. \tag{26}
$$

Therefore, the total symmetry of the considered group has got the structure of the following semi-direct product where *p* ′ =*a* (1<sup>−</sup> *<sup>p</sup>*) *<sup>ν</sup> <sup>α</sup>* represents isomorphic mapping . Let *Vh* be space of the all hysteresis' models represented by (14) and (15). Group element is automorphism in *Vh* if yields . Set of the all automorphisms G*<sup>A</sup> X*1

constitutes automorphism group, where Let us distinguish the loop *v*<sup>1</sup> = *M*<sup>01</sup> *θ*1 *∈Vh* , and

relate *v*1 with *v*2:

$$
\nu\_1 \mathcal{R} \nu\_2,\tag{26a}
$$

where ℛ means the binary relation given by (26).

#### *Definition:*

Equivalence binary relation on a set *V*(*h*) is a relation, which is reflexive, symmetric and transitive [11]. Due to the group structure of (26a) satisfies these conditions and ℛ is equivalence relation.

Let ℛ be an equivalence relation on *Vh*, then Ev1 ⊂ *Vh* containing all elements *v*2*∈Vh* satisfying (26a) is called the equivalence class of *v*1. The sets Evi are pairwise disjoint, that is Evi ∩Evj = ∅ if *vi* ≠ *vj*. Union of the all equivalence classes is the *Vh* space: ∪*vi Evi* =*Vh* . What practical use one may have from equivalence relations and partitioning. Let us assume the following *v*1ℛ*v*<sup>2</sup> relation, and then there exists a group element which relates both loops means that they are equivalent. However, in the opposite case, the two loops do not belong to a common class and then they are relevantly different from the algebra point of view.

## **3. Losses scaling in soft magnetic materials**

Density of the total power losses in magnetic materials under variable magnetic field is due to eddy currents generated in the material. These may be generated for various scales of dimen‐ sions: currents caused by Barkhausen jumps, what leads to a dependence *Phys* <sup>∝</sup>(*<sup>B</sup>* 1.6 *<sup>f</sup>* ) [11], currents around moving domain walls *Pexc* ∝(*Bf* ) 3 <sup>2</sup> [12] and currents in the whole material volume *Pclas* ∝(*Bf* ) <sup>2</sup> [11]. All of the aforementioned dependencies obey a power law, however, with a diverse value of exponents [13]. Therefore, one cannot talk about universality of the presented above formulae. However, certain universality of the power losses data of soft magnetic materials has been derived by applying the scaling. The proposed approach has been based on assumption that the density of the total power loss in soft magnetic materials was self-similar like intermittency of fully developed turbulent flow [1]. Moreover, using simple model of hysteresis loop (14), (15) and assuming semi-static conditions, we derive the following formula for the total power loss [14]:

spanned by *X* variable. The considered model of loop reveals a combination of and which operates in space generated by (*X, θ*) pair. Therefore, total symmetry of the considered model

0 0 0.

q

Therefore, the total symmetry of the considered group has got the structure of the following

Equivalence binary relation on a set *V*(*h*) is a relation, which is reflexive, symmetric and transitive [11]. Due to the group structure of (26a) satisfies these conditions and ℛ is

Let ℛ be an equivalence relation on *Vh*, then Ev1 ⊂ *Vh* containing all elements *v*2*∈Vh* satisfying

may have from equivalence relations and partitioning. Let us assume the following *v*1ℛ*v*<sup>2</sup> relation, and then there exists a group element which relates both loops means that they are equivalent. However, in the opposite case, the two loops do not belong to a common class and

Density of the total power losses in magnetic materials under variable magnetic field is due to eddy currents generated in the material. These may be generated for various scales of dimen‐

(26a) is called the equivalence class of *v*1. The sets Evi are pairwise disjoint, that is Evi

Union of the all equivalence classes is the *Vh* space: ∪*vi*

then they are relevantly different from the algebra point of view.

**3. Losses scaling in soft magnetic materials**

*<sup>p</sup> X X aM M*

é ùé ù éù é ù¢ ê ú ê úê ú êú ê ú + = ¢ ê ú ê ú ¢ ë û ëû ë û ê ú ë û

0 0

=*a* (1<sup>−</sup> *<sup>p</sup>*) *<sup>ν</sup>*

. Let *Vh* be space of the all hysteresis' models represented by (14) and (15). Group element is automorphism in *Vh* if yields . Set of the all automorphisms G*<sup>A</sup>*

c

0

(26)

*<sup>α</sup>* represents isomorphic mapping

1 2 *v v* R , (26a)

*Evi*

*X*1 *M*<sup>01</sup> *θ*1

*∈Vh* , and

∩Evj

=*Vh* . What practical use one

= ∅ if

 q

can be presented in the following matrix representation (26):

semi-direct product where *p* ′

where ℛ means the binary relation given by (26).

relate *v*1 with *v*2:

10 Magnetic Materials

*Definition:*

*vi* ≠ *vj*.

equivalence relation.

(1 )


*p*

0 0

n

a

00 0

constitutes automorphism group, where Let us distinguish the loop *v*<sup>1</sup> =

*p*

$$P\_{\rm tot} = f \mu\_0 M\_0 h \quad a^{-n} \ln \left( \frac{a^{z \cdot (p \theta + p \chi\_{\rm max + \chi})} + 1}{(a^{z \cdot (p \chi\_{\rm max + \chi})} + a^{2 p \theta})} \cdot \frac{\left(a^{z \cdot (p \chi\_{\rm min + \chi})} + a^{2 p \theta}\right)}{(a^{z \cdot \{p \theta + p \chi\_{\rm min + \chi}\}} + 1)}\right) \left(p \cdot \ln(a)\right)^{-1}.\tag{26b}$$

In case of the symmetric extrema of magnetic field *Xmin* = −*Xmax*, (26b) gets the following form:

$$P\_{\rm tot} = 4\theta f \,\mu\_0 M\_0 h a^{-\ast}.\tag{26c}$$

where *a*, *<sup>n</sup>* <sup>=</sup> *<sup>ν</sup> <sup>α</sup>* (*<sup>p</sup>* <sup>−</sup>1), *<sup>h</sup>* <sup>=</sup> *<sup>H</sup> <sup>X</sup>* , *θ* and *M*<sup>0</sup> are the hysteresis loop parameters and *μ*0 is the permea‐ bility of free space.

According to (26c), the formula for *Ptot* is monomial which is always self-similar mathematical expression. This theoretical result confirms experimental observations concerning homoge‐ neity of *Ptot* in soft magnetic materials.

Let us assume that the density of the total power losses is homogenous function in general sense (6). Let *Ptot* (*f, Bm*) be density of the total power losses, where *f* and *Bm*are frequency and the pick of magnetic induction of flux waveform. Applying (6) for the two independent variables, we derive the most general form for *Ptot*:

$$P\_m\left(f, B\_m\right) = B\_m^\beta F\left(B\_m^{-\alpha} f\right),\tag{27}$$

where *α* and *β* are scaling exponents and *F*(⋅) is an arbitrary function. These three unknown magnitudes have to be determined from experimental data. As the simplest approach to estimation of *F*(⋅), we have applied the Maclaurin expansion of (27):

$$P\_{\rm tot}\left(f, B\_{\rm m}\right) = B\_{\rm m}^{\beta} \left(\Gamma^{(1)}\left(B\_{\rm m}^{-\alpha}f\right) + \Gamma^{(2)}\left(B\_{\rm m}^{-\alpha}f\right)^{2}\right). \tag{28}$$

where Γ(*<sup>k</sup>* ) = 1 *<sup>k</sup>* ! <sup>Γ</sup>(*<sup>k</sup>* ) (0). Since the total losses vanish for *f =* 0 or for *Bm=* 0, the constant term of expansion (28) equals zero.

#### **3.1. Measurement data**

The measurement of density of total power losses was carried out following the IEC Standards (60404-2, 60404-6). During measurements, the shape factor of secondary voltage was equal to 1.111± per cent. Extended uncertainty of obtained measurements was equal to about 0.5%. The measurements covered the three following classes of soft magnetic materials:


Density of total power losses was measured as a function of the maximum induction *Bm*∈ [0.1] (*T*), 1.8(T)] at fixed values of *f* ≤ 400 (*Hz*). Samples of conventional crystalline materials were strips, whereas the remaining ones had the shape of cylinder.

#### **3.2. Estimation of expansion parameters (28) from measurement data**


**Table 2.** Values of scaling exponents *α, β* and values of amplitudes *Γ*(*k*) .

Firstly, the initial values of exponents *α, β* and amplitudes *Γ*(*k*) were assumed and differences between all measurement values of density of total power losses and values of density of total power losses obtained from expansion (28) were calculated. Next, the *Chi*<sup>2</sup> function was optimized. Constraint of normal distribution of error was applied. Results of optimization for exponents *α, β* and amplitudes *Γ*(*k*) for chosen magnetic materials are given in **Table 2**.

The results, that is scaled values of the measurement data and the values obtained from the mathematical model, for the chosen magnetic materials, are shown in **Figures 5**–**7**, in the *Ptot Bm <sup>β</sup>* , *<sup>f</sup> Bm <sup>α</sup>* coordinates system. Based on the all results concerning the density of total power losses, the universal relationship between the scaling exponents *α* and *β* was stated. This relation is of the following form [15]:

$$
\beta = 1.35x + 1.75,\tag{29}
$$

See also **Figure 8**. The origin of the relationship in (29) will be subject of further relations.

where Γ(*<sup>k</sup>* )

12 Magnetic Materials

= 1 *<sup>k</sup>* ! <sup>Γ</sup>(*<sup>k</sup>* )

**3.1. Measurement data**

**•** Crystalline materials;

**•** Nanocrystalline alloy.

**•** Amorphous alloys, Co-based and Fe-based; and

**Magnetic materials α(−) β(−)** *Γ* **(1)**

**Table 2.** Values of scaling exponents *α, β* and values of amplitudes *Γ*(*k*)

exponents *α, β* and amplitudes *Γ*(*k*)

relation is of the following form [15]:

*Ptot Bm <sup>β</sup>* , *<sup>f</sup> Bm*

strips, whereas the remaining ones had the shape of cylinder.

**3.2. Estimation of expansion parameters (28) from measurement data**

GO—3% Si-Fe −2.16 −1.19 12.78×10−3 37.68×10−6 Co71.5Fe2.5Mn2Mo1Si9B14 −1.55 −0.35 2.88×10−3 1.90×10−6 Fe73.5Cu1Nb3Si13.5B<sup>9</sup> −1.81 −0.70 0.17×10−3 0.71×10−6

expansion (28) equals zero.

(0). Since the total losses vanish for *f =* 0 or for *Bm=* 0, the constant term of

The measurement of density of total power losses was carried out following the IEC Standards (60404-2, 60404-6). During measurements, the shape factor of secondary voltage was equal to 1.111± per cent. Extended uncertainty of obtained measurements was equal to about 0.5%. The

Density of total power losses was measured as a function of the maximum induction *Bm*∈ [0.1] (*T*), 1.8(T)] at fixed values of *f* ≤ 400 (*Hz*). Samples of conventional crystalline materials were

**(***m* **<sup>2</sup>**

.

for chosen magnetic materials are given in **Table 2**.

Firstly, the initial values of exponents *α, β* and amplitudes *Γ*(*k*) were assumed and differences between all measurement values of density of total power losses and values of density of total power losses obtained from expansion (28) were calculated. Next, the *Chi*<sup>2</sup> function was optimized. Constraint of normal distribution of error was applied. Results of optimization for

The results, that is scaled values of the measurement data and the values obtained from the mathematical model, for the chosen magnetic materials, are shown in **Figures 5**–**7**, in the

losses, the universal relationship between the scaling exponents *α* and *β* was stated. This

*<sup>α</sup>* coordinates system. Based on the all results concerning the density of total power

**T(***α***−***β***) /** *s* **<sup>2</sup>**

**)** *Γ* **(2)**

**(***m* **<sup>2</sup>**

**T(2***α***−***β***) /** *s***)**

measurements covered the three following classes of soft magnetic materials:

**Figure 5.** A comparison of measurement data of total density of power losses *Ptot* (markers) and values obtained from the scaling theory (solid line) for Co-based amorphous alloy *Co*71.5*F e*2.5*M n*2*M o*1*Si* <sup>9</sup>*B*14.

**Figure 6.** A comparison of measurement data of total density of power losses *Ptot* (markers) and values obtained from the scaling theory (solid line) for nanocrystalline alloy *F e*73.5*Cu*1*N b*3*Si* 13.5*B*9.

**Figure 7.** A comparison of measurement data for total density of power losses *Ptot* (markers) and values obtained from the scaling theory (solid line) for grain-oriented silicon steel 3% Si-Fe.

**Figure 8.** The universal relationship of the scaling exponents *α* and *β*. Markers correspond to estimations from experi‐ mental data, and continuous line corresponds to (29).

The three achievements resulting from scaling should be emphasized. The first one is a satisfactory agreement between the measurement data and the theoretical description. The second one is relation (29), which establishes the universal linear relationship between the scaling exponents and decreases the number of free parameters. The third achievement consists in revealing the data collapse. **Figures 5**–**7** are drawn in *Ptot Bm <sup>β</sup>* , *<sup>f</sup> Bm <sup>α</sup>* coordinates system and present the continuous sets of the losses characteristics of different values of *Bm* collapsed just to a single curve. This effect is called "a single-sample data collapse" Reversible procedure ( *Ptot Bm <sup>β</sup>* , *<sup>f</sup> Bm <sup>α</sup>* )→(*Ptot*, *f* ) splits the collapsed curves to separate curves for different values of *Bm*. This effect will be demonstrated for more complicated case (see **Figure 14**). Therefore, the scaling can be also applied as method for a compression of data. All examples in **Figures 5**–**7** present the single-sample data collapses. Having data for different materials and introducing for each of them are the following dimensionless magnitudes:

$$
\tilde{P}\_{\text{tot}} = \frac{\Gamma^{(2)}}{\Gamma^{(1)^2}} \frac{P\_{\text{tot}}}{B\_{\text{in}}^{\beta}} \quad \tilde{f} = \frac{\Gamma^{(2)}}{\Gamma^{(1)}} \frac{f}{B\_{\text{in}}^{\alpha}},\tag{30}
$$

We obtain the multi-sample data collapse (see **Figure 9**). These results confirm the assumption of density of total power losses scaling. Applying these transformations to (28), we derive the dimensionless low for the density of power losses in soft magnetic materials:

$$
\tilde{P}\_{\text{tot}} = \tilde{\tilde{f}} + \tilde{\tilde{f}}^2. \tag{31}
$$

**Figure 9.** The multi-sample data collapse for total density of power losses *<sup>P</sup>*˜ *tot*(markers) and values obtained from the scaling theory (solid line).

#### **3.3. An application of the multi-sample data collapse**

**Figure 7.** A comparison of measurement data for total density of power losses *Ptot* (markers) and values obtained from

**Figure 8.** The universal relationship of the scaling exponents *α* and *β*. Markers correspond to estimations from experi‐

The three achievements resulting from scaling should be emphasized. The first one is a satisfactory agreement between the measurement data and the theoretical description. The second one is relation (29), which establishes the universal linear relationship between the scaling exponents and decreases the number of free parameters. The third achievement

and present the continuous sets of the losses characteristics of different values of *Bm* collapsed

*Ptot Bm <sup>β</sup>* , *<sup>f</sup> Bm*

*<sup>α</sup>* coordinates system

the scaling theory (solid line) for grain-oriented silicon steel 3% Si-Fe.

14 Magnetic Materials

mental data, and continuous line corresponds to (29).

consists in revealing the data collapse. **Figures 5**–**7** are drawn in

Mostly, the data collapse is applying as a tool for the detection of self-similarity. Here, we present a new application which solves problem of comparison measurements taken in different laboratories [16]. In 1995, the leading European Laboratories busy with measure‐ ments of the electrical steel magnetic properties were trying to compare the measurement results of the power loss in electrical sheet steel under the conditions of rotating and alternating flux [17]. Taking from [17] the idea of the inter-comparison of measurement data of the energy losses in soft magnetic materials, we perform such an inter-comparison with data taken in two laboratories [16, 18], however, under the conditions of axial and alternating flux. Self-similarity of the density power losses enables us to scale off the interference of the sample's geometry and the material type from the dependence of power losses versus the pick of induction and magnetizing frequencies. This property of SMM allows comparing different measurement sets. Successively, this fact allows introducing an absolute measure of uncertainty characterizing the given measurement set. Therefore, the way for assessing the uncertainty contributions would not interfere with the abovementioned data comparison. Formula (30) suits very well to the mentioned above phenomena and constitute background for solution of the presented problem.

For inter-comparison, we have selected two sets of power losses data. The first one belongs to Yuan [18] and consists of the following three sets of data: *S*<sup>1</sup> = *F e*76*M o*2*Si* <sup>2</sup>*P*10*C*7.5*B*2.5, *S*<sup>2</sup> = *F e*79.8*M o*2.1*Si* 2.1*P*8*C*6*B*2, *S*<sup>3</sup> = *F e*80*M o*1*Si* <sup>2</sup>*P*8*C*6*B*3. The samples were thin ribbons wound into toroids. For details concerning measurement methods, we refer readers to [18]. On the basis of measured data, the parameters' values of (28) have been estimated (see **Table 3**, after [18]).


**Table 3.** Scaling exponents and coefficients of (28).

The second set contains some of our results [15, 19] for the power losses in the following alloys: amorphous ribbon *P*<sup>1</sup> = *F e*7.8*Si* <sup>13</sup>*B*9, Co-based amorphous alloy *P*<sup>2</sup> =*Co*71.5*F e*2.5*Mn*2*Mo*1*Si*9*B*<sup>14</sup> and nanocrystalline alloy *P*<sup>3</sup> = *F e*73.5*Cu*1*N b*3*Si* 15.5*B*7. The corresponding scaling exponents *α, β* and the scaling coefficients Γ(1), Γ(2) are presented in **Table 3**.

Plotting *P*˜ *tot* versus *f* ˜ for the all considered samples (**Figure 10**), we confirm that the data collapse takes place for the selected samples.

**Figure 10.** The data collapse for total power losses of compared materials.

different laboratories [16]. In 1995, the leading European Laboratories busy with measure‐ ments of the electrical steel magnetic properties were trying to compare the measurement results of the power loss in electrical sheet steel under the conditions of rotating and alternating flux [17]. Taking from [17] the idea of the inter-comparison of measurement data of the energy losses in soft magnetic materials, we perform such an inter-comparison with data taken in two laboratories [16, 18], however, under the conditions of axial and alternating flux. Self-similarity of the density power losses enables us to scale off the interference of the sample's geometry and the material type from the dependence of power losses versus the pick of induction and magnetizing frequencies. This property of SMM allows comparing different measurement sets. Successively, this fact allows introducing an absolute measure of uncertainty characterizing the given measurement set. Therefore, the way for assessing the uncertainty contributions would not interfere with the abovementioned data comparison. Formula (30) suits very well to the mentioned above phenomena and constitute background for solution of the presented

For inter-comparison, we have selected two sets of power losses data. The first one belongs

toroids. For details concerning measurement methods, we refer readers to [18]. On the basis of measured data, the parameters' values of (28) have been estimated (see **Table 3**, after

The second set contains some of our results [15, 19] for the power losses in the following alloys:

**T(α−β)s−2)** *Γ***(2)(m<sup>2</sup>**

<sup>13</sup>*B*9, Co-based amorphous alloy *P*<sup>2</sup> =*Co*71.5*F e*2.5*Mn*2*Mo*1*Si*9*B*<sup>14</sup> and

˜ for the all considered samples (**Figure 10**), we confirm that the data

15.5*B*7. The corresponding scaling exponents *α, β* and

<sup>2</sup>*P*8*C*6*B*3. The samples were thin ribbons wound into

**T(2α−β)s−1)**

<sup>2</sup>*P*10*C*7.5*B*2.5,

to Yuan [18] and consists of the following three sets of data: *S*<sup>1</sup> = *F e*76*M o*2*Si*

*S1* −1.533 −0.319 6.744×10−3 1.322×10−6 *S*2 −0.364 1.259 1.412×10−2 1.917×10−6 *S*3 −0.504 1.069 9.11×10−3 3.389×10−6 *P*1 −2.945 −1.776 2.90×10−3 4.60×10−6 *P*2 −1.519 −0.375 2.53×10−3 6.79×10−6 *P*3 −3.231 −1.365 3.22×10−4 1.95×10−7

\*The data for *S*1, *S*2 and *S*3 have been kindly supplied by the authors of Yuan et al. [18].

2.1*P*8*C*6*B*2, *S*<sup>3</sup> = *F e*80*M o*1*Si*

**Sample α(−) β(−)** *Γ***(1)(m<sup>2</sup>**

**Table 3.** Scaling exponents and coefficients of (28).

nanocrystalline alloy *P*<sup>3</sup> = *F e*73.5*Cu*1*N b*3*Si*

collapse takes place for the selected samples.

the scaling coefficients Γ(1), Γ(2) are presented in **Table 3**.

amorphous ribbon *P*<sup>1</sup> = *F e*7.8*Si*

*tot* versus *f*

Plotting *P*˜

problem.

16 Magnetic Materials

[18]).

*S*<sup>2</sup> = *Fe*79.8*M o*2.1*Si*

Since all the magnitudes in (31) are dimensionless and the formula for *P*˜ *tot* is sample inde‐ pendent, we propose to introduce a measure of uncertainty characterizing the measurement set by the total distance of all empirical points from the scaling curve (31):

$$D = \sqrt{\frac{1}{N} \sum\_{i=1}^{N} \left( \tilde{P}\_{\text{tot}}^{\text{exp}}(\tilde{f}\_{i}) - \tilde{P}\_{\text{tot}}^{\text{th}}(\tilde{f}\_{i}) \right)^{2}},\tag{32}$$

where *P*˜ *tot exp* , *P*˜ *tot th* are the power losses, measured and calculated from formula (31), *f* ˜ *<sup>i</sup>* is dimensionless frequency, where index *i* is running through the whole series of experimental data and N denotes length of the measured series. We consider the two sets of experimental data S and ℙ corresponding to different LABs. Each set consists of the three series which correspond to different samples. To create effective measure of uncertainty characterizing measurement set of the given LAB, we calculate average measure *Dav*for the three samples belonging to either of two selected sets:

$$D\_{av}^2 = \sum\_{i=1}^3 \frac{N\_i}{N\_1 + N\_2 + N\_3} D\_i^2. \tag{33}$$


Comparisons of uncertainty measures are presented in **Table 4**.

**Table 4.** Comparisons of uncertainty measures.

Progress in modern technologies depends on the comprehensive knowledge of material properties under standard and non-standard conditions. However, an agreed standardized method does not exist, and the reproducibility of the different methods used in different laboratories is unknown. We are of the opinion that the reason of such situation is lacking of statistical method enabling the appropriate data's inter-comparison. In this section, we have proposed a solution of this problem. As we have shown, the data collapse supplies method that enables us to introduce universal measure of uncertainty, which compares different experimental sets even based on different measurement methods. Therefore, the introduced method also can serve as a tool to compare measurement data obtained in different laborato‐ ries. The measure (33) expresses the total uncertainty characterizing the data set for the chosen range of dimensionless frequency *f* ˜ . There are four contributions to *Dav*resulting from the following: (1) uncertainty characterizing the measurement method and construction of the measurement set, (2) uncertainty of measurements of elementary magnitudes, (3) errors resulting from the approximation (28) and uncertainty of estimations of *α, β, Γ*(1) and *Γ*(2). The derived method is universal and can be applied to investigations of any phenomenon satisfy‐ ing the self-similarity conditions.

## **4. Multi-scaling of core losses in soft magnetic materials**

The application of soft magnetic materials in electronic devices requires knowledge of the losses under different excitation conditions: sinusoidal and non-sinusoidal flux waveforms of different shapes, with and without DC bias. Scaling theory allows the total power losses density to be derived in the form of a general homogeneous function, which depends on the peak to peak of the magnetic inductance *ΔB*, frequency *f*, DC bias *HDC* and temperature *T*:

$$P\_{\rm tot} = F(f, \Delta B, H\_{\rm DC}, T). \tag{34}$$

The form of this function has been generated through the Maclaurin expansion with respect to scaled frequency. The parameters of the model consist of expansion coefficients, scaling exponents, parameters of DC bias mapping, parameters of temperature factor and tuning exponents. Values of these model parameters were estimated on the basis of measured data of total power density losses. However, influence of the DC bias on the self-similarity of measurement data was very relevant. In order to apply scaling to (34), the right-hand side has to be a homogeneous function in a general sense. This assumption has to be satisfied both by the experimental data and by the mathematical model. However, according to the results given in [20], Eq. (34) and measurement data are not uniform in the required sense when there is a DC bias. This problem has been solved by using the method invented by Van den Bossche and Valchev [21]. Their method consists in mapping of magnetic field into a pseudo-magnetization by using tanh(⋅):

Comparisons of uncertainty measures are presented in **Table 4**.

**2**

*S1* 48 1.64×10−5 8.11×10−3

*P1* 48 9.23×10−5 6.53×10−3

**(−)** *Dav*

Progress in modern technologies depends on the comprehensive knowledge of material properties under standard and non-standard conditions. However, an agreed standardized method does not exist, and the reproducibility of the different methods used in different laboratories is unknown. We are of the opinion that the reason of such situation is lacking of statistical method enabling the appropriate data's inter-comparison. In this section, we have proposed a solution of this problem. As we have shown, the data collapse supplies method that enables us to introduce universal measure of uncertainty, which compares different experimental sets even based on different measurement methods. Therefore, the introduced method also can serve as a tool to compare measurement data obtained in different laborato‐ ries. The measure (33) expresses the total uncertainty characterizing the data set for the chosen

following: (1) uncertainty characterizing the measurement method and construction of the measurement set, (2) uncertainty of measurements of elementary magnitudes, (3) errors resulting from the approximation (28) and uncertainty of estimations of *α, β, Γ*(1) and *Γ*(2). The derived method is universal and can be applied to investigations of any phenomenon satisfy‐

The application of soft magnetic materials in electronic devices requires knowledge of the losses under different excitation conditions: sinusoidal and non-sinusoidal flux waveforms of different shapes, with and without DC bias. Scaling theory allows the total power losses density to be derived in the form of a general homogeneous function, which depends on the peak to peak of the magnetic inductance *ΔB*, frequency *f*, DC bias *HDC* and temperature *T*:

**4. Multi-scaling of core losses in soft magnetic materials**

**<sup>2</sup> (−)**

˜ . There are four contributions to *Dav*resulting from the

( , , , ). *P F f BH T tot* = D *DC* (34)

**Sample** *Ni Di*

18 Magnetic Materials

*S2* 40 7.42×10−3 *S3* 32 1.29×10−4

*P2* 47 2.05×10−5 *P3* 49 1.53×10−5

**Table 4.** Comparisons of uncertainty measures.

range of dimensionless frequency *f*

ing the self-similarity conditions.

$$P\_{\rm tot} = F\left(f, \Delta B, \tanh(H\_{\rm DC} \cdot \mathbf{c}\_0), T\right). \tag{35}$$

Following Bossche and Valchev, we have applied series of the mappings as expansion coefficients for modelling *F*(⋅,⋅,⋅,⋅) function of (35):

$$H\_{DC} \to [M\_0, M\_1, M\_2, M\_3]. \tag{36}$$

where *Mi* =tanh(*HDCci* ), where *ci* are expansion coefficients, to be determined from measure‐ ment data.Therefore, applying definition for the homogeneous function in general sense, we have formulated the following scaling hypothesis: ∃{*a*, *<sup>b</sup>*, *<sup>c</sup>*, *<sup>d</sup>*, g}∈ℝ<sup>5</sup> :∀*<sup>λ</sup>* ∈ℝ<sup>+</sup> yields: *Ptot* (*<sup>λ</sup> <sup>a</sup> <sup>f</sup>* , *<sup>λ</sup> <sup>b</sup>*(*ΔB*), *<sup>λ</sup> <sup>c</sup> <sup>M</sup>*0, *<sup>M</sup>*1, *<sup>M</sup>*2, *<sup>M</sup>*<sup>3</sup> , *<sup>λ</sup> dT* ) <sup>=</sup>*<sup>λ</sup> gPtot*( *<sup>f</sup>* , *<sup>Δ</sup>B*, *<sup>M</sup>*0, *<sup>M</sup>*1, *<sup>M</sup>*2, *<sup>M</sup>*<sup>3</sup> , *<sup>T</sup>* ). Substituting the following *λ* = (ΔB)−1/*<sup>b</sup>* , we derive the most general form for *Ptot*which satisfies above hypothesis:

$$P\_{\rm tot} = \left(\Delta B\right)^{\rho} F \left(\frac{f}{\left(\Delta B\right)^{a}}, \frac{\left[M\_{0}, M\_{1}, M\_{2}, M\_{3}\right]}{\left(\Delta B\right)^{r}}, \frac{T}{\left(\Delta B\right)^{s}}\right). \tag{37}$$

where *<sup>α</sup>* <sup>=</sup> *<sup>a</sup> <sup>b</sup>* , *<sup>β</sup>* <sup>=</sup> <sup>g</sup> *<sup>b</sup>* , *<sup>γ</sup>* <sup>=</sup> *<sup>c</sup> <sup>b</sup>* , *<sup>δ</sup>* <sup>=</sup> *<sup>d</sup> <sup>b</sup>* are effective scaling exponents. *F*(⋅,⋅,⋅) is an arbitrary function of the three variables. Both the effective exponents and the F function have to be determined.

General formula (37) enables us to construct mathematical model which maps the fourdimensional space spanned by *f, ΔB, DC bias* and *T* into the one-dimensional *Ptot*space. In the first step, we separate the temperature factor Θ(⋅):

$$F\left(\frac{f}{\left(\Delta B\right)^{a}}, \frac{\left[M\_{0}, M\_{1}, M\_{2}, M\_{3}\right]}{\left(\Delta B\right)^{\circ}}, \frac{T}{\left(\Delta B\right)^{\circ}}\right) = \Phi\left(\frac{f}{\left(\Delta B\right)^{a}}, \frac{\left[M\_{0}, M\_{1}, M\_{2}, M\_{3}\right]}{\left(\Delta B\right)^{\circ}}\right) \Theta\left(\frac{T}{\left(\Delta B\right)^{\circ}}\right). \tag{38}$$

Let us assume that Φ(⋅,⋅) consists of two terms which need not be independent. However, the second one is *HDC* dependent in contrary to the first one. For both of them, we assume the Maclaurin expansions with respect to scaled frequency *f*/(Δ*B*) *<sup>α</sup>*, which is very much suited for the Bertotti decomposition. Moreover, the first term should describe losses for *HDC*→ 0, whereas the second term must vanish for this condition. The resulting expression takes the following form:

$$\Phi\left(\frac{f}{\left(\Delta B\right)^{a}}, H\_{DC}\right) = \Sigma\_{i=1}^{4}\Gamma\_{i}\left(\frac{f}{\left(\Delta B\right)^{a}}\right)^{i} + \Sigma\_{i=0}^{3}\Gamma\_{i\leftrightarrow 3}\left(\frac{f}{\left(\Delta B\right)^{a}}\right)\frac{\tanh(H\_{DC}\cdot\mathbf{c}\_{i})}{\left(\Delta B\right)^{\gamma}}.\tag{39}$$

Since (39) has been created by the Maclaurin expansion, all series exponents are integers. However, it may be so that the best error's minimum is obtained for fractional values of exponents. For this purposes, we introduce tuning exponents *x* and *y*:

$$\Phi\left(\frac{f}{\left(\Delta B\right)^{a^\*}}, H\_{DC}\right) = \Sigma\_{l=l}^4 \Gamma\_l \left(\frac{f}{\left(\Delta B\right)^{a^\*}}\right)^{l(l-x)} + \Sigma\_{l=0}^3 \Gamma\_{l \leftrightarrow 3} \left(\frac{f}{\left(\Delta B\right)^{a^\*}}\right)^{\left(l+\gamma\right)(l-x)} \frac{\tanh(H\_{DC} \cdot \mathbf{c}\_i)}{\left(\Delta B\right)^{\gamma}}.\tag{40}$$

On the basis of some numerical test simulations, we have selected the following Padé approx‐ imant for Θ(⋅):

$$\Theta = \left(\frac{\psi\_o + \theta(\psi\_1 + \theta\psi\_2)}{1 + \theta(\psi\_3 + \theta\psi\_4)}\right)^{1-z} \tag{41}$$

where θ<sup>=</sup> *<sup>T</sup>* <sup>+</sup> *<sup>τ</sup> <sup>Δ</sup><sup>B</sup> <sup>δ</sup>* is gauged and scaled temperature, *T* is measured temperature in °C, *z is* tuning parameter, and *ψ<sup>i</sup>* are Padé approximant coefficients. After all improvements of *F*(⋅,⋅,⋅), the final form is still homogenous function in general sense (6).

In order to perform core loss measurements, the *B-H* loop measurement has been evaluated as the most suitable. This technique enables rapid measurement while retaining a good ac‐ curacy. The measurement set works in the following way: two windings are placed around the core under test. Taking into account the number of secondary winding turns and the ef‐ fective core cross section, the secondary winding voltage *V* is integrated into the core flux density *B*. Next taking into account the number of primary winding turns and the effective magnetic path length of the core under test, the magnetic field strength *H* is calculated. Then, the total power losses per unit volume is the enclosed area of the *B-H* loop multiplied by the frequency *f*. The test system consists of a power stage, a power supply, an oscillo‐ scope and a heating chamber. It is controlled by a MATLAB program running on a PC com‐ puter under Microsoft Windows. The power stage is capable of a maximum input voltage of 450 V, output current of 25 A and a switching frequency of up to 200 kHz. The *B-H* loop measurements have been performed for SIFERRIT. The rectangular voltage shape across the core and DC bias has been applied, while the duty cycle was 50%.

*The tested core data were as follows:*


( )

20 Magnetic Materials

form:

[ ]

a

a

imant for Θ(⋅):

where θ<sup>=</sup> *<sup>T</sup>* <sup>+</sup> *<sup>τ</sup>*

parameter, and *ψ<sup>i</sup>*

=

agd

( ) ( ) ( )

Maclaurin expansions with respect to scaled frequency *f*/(Δ*B*)

<sup>0123</sup> <sup>0123</sup> ,,, ,,, ,, , . ΔΔΔ ΔΔ Δ *f Tf MMMM MMMM <sup>T</sup> <sup>F</sup> BBB BB B*

Let us assume that Φ(⋅,⋅) consists of two terms which need not be independent. However, the second one is *HDC* dependent in contrary to the first one. For both of them, we assume the

the Bertotti decomposition. Moreover, the first term should describe losses for *HDC*→ 0, whereas the second term must vanish for this condition. The resulting expression takes the following

*i i*

(1 ) ( )(1 )

1

*<sup>Δ</sup><sup>B</sup> <sup>δ</sup>* is gauged and scaled temperature, *T* is measured temperature in °C, *z is* tuning


*z*

are Padé approximant coefficients. After all improvements of *F*(⋅,⋅,⋅), the


ag

*ψ ψ* (41)

*i x iy x*

a

( ) ( ) ( ) ( ) 4 3 1 0 5 tanh( ) Φ , ΣΓ ΣΓ . Δ Δ ΔΔ

( ) ( ) ( ) ( )

*<sup>f</sup> <sup>f</sup> <sup>f</sup> H c <sup>H</sup> B B BB*

On the basis of some numerical test simulations, we have selected the following Padé approx‐

 q

 q

In order to perform core loss measurements, the *B-H* loop measurement has been evaluated as the most suitable. This technique enables rapid measurement while retaining a good ac‐ curacy. The measurement set works in the following way: two windings are placed around the core under test. Taking into account the number of secondary winding turns and the ef‐ fective core cross section, the secondary winding voltage *V* is integrated into the core flux density *B*. Next taking into account the number of primary winding turns and the effective magnetic path length of the core under test, the magnetic field strength *H* is calculated.

0 12 3 4 ( ) 1( )

q

æ ö + + Q = ç ÷ + + è ø *ψ ψψ*

q

æ ö æö æö <sup>×</sup> ç ÷ ç÷ ç÷ = +

a

*<sup>f</sup> <sup>f</sup> f Hc <sup>H</sup> B B BB*

= +

Since (39) has been created by the Maclaurin expansion, all series exponents are integers. However, it may be so that the best error's minimum is obtained for fractional values of

*DC i i i i*

è ø èø èø

exponents. For this purposes, we introduce tuning exponents *x* and *y*:

4 3 1 0 5 tanh( ) Φ , ΣΓ Σ Γ . Δ Δ ΔΔ

 a

= = + æ ö æö æö <sup>×</sup> ç ÷ ç÷ ç÷ = +

*DC i i i i*

è ø èø èø

final form is still homogenous function in general sense (6).

<sup>æ</sup> ö æ öæ ö <sup>ç</sup> ÷ ç <sup>=</sup> F Q÷ç ÷ <sup>ç</sup> ÷ ç ÷ç ÷ <sup>è</sup> ø è øè ø

[ ]

ag

( ) ( )

*DC i*

*DC i*

 g  d

*<sup>α</sup>*, which is very much suited for

(38)

(39)

(40)


*The following factors influence the accuracy of measurements:*






**Table 5.** The set of estimated model's parameters of (37)–(41).


**Table 6.** Selected 60 records of the measurement data of SIFERRIT N87.

**T ΔB f HDC Ptot T ΔB f HDC Ptot [°C] [T] [kHz] [A/m] [Wm−3] [°C] [T] [kHz] [A/m] [Wm−3]** 28.1 0.395 1 8.634 4064.3 28.1 0.391 1 20.146 4469.0 28.1 0.374 1 60.634 6332.4 28.3 0.351 1 86.651 6463.6 17.7 0.398 2 7.801 9452.1 17.8 0.398 2 20.555 10663.8 18.9 0.396 2 35.583 12745.8 18.5 0.377 2 89.240 16015.6 26.2 0.400 5 6.570 21131.3 26.4 0.4 5 17.820 23110.0 26.5 0.398 5 33.230 28057.3 27.1 0.386 5 89.400 35209.8 28.4 0.401 10 5.892 41549.0 28.6 0.401 10 17.477 45257.9 28.8 0.400 10 31.820 54650.9 29.7 0.393 10 73.960 63821.6 30.8 0.386 10 105.000 64632.1 28.4 0.49 1 11.694 6611.0 28.4 0.488 1 24.299 7196.0 28.4 0.451 1 78.390 8771.6 19.1 0.497 2 10.120 15234.1 19.2 0.496 2 23.718 16781.0 19.3 0.485 2 54.630 19235.9 19.8 0.475 2 76.860 20100.2 27.7 0.502 5 8.920 34634.8 27.4 0.503 5 15.020 36195.2 27.7 0.501 5 21.500 37496.6 28.6 0.496 5 47.500 41259.7 31.7 0.499 10 20.520 71226.8 32.2 0.494 10 45.040 76876.5 32.6 0.487 10 67.140 80858.2 28.5 0.588 1 14.420 10042.9 28.7 0.561 1 57.970 11239.6 28.7 0.541 1 78.080 11255.7 29.1 0.580 2 12.820 19689.9 28.7 0.576 2 54.360 22043.0 30.1 0.592 5 42.400 52126.7 31.1 0.599 10 10.290 92648.6 31.3 0.595 10 31.230 96446.4 28.9 0.684 1 22.050 14150.5 28.1 0.389 1 33.507 5358.8 28.4 0.346 1 91.066 6376.4 18.2 0.386 2 68.034 15049.1 18.7 0.367 2 110.590 16027.7 26.8 0.394 5 58.800 32614.3 27.5 0.386 5 97.779 35945.6 29.2 0.396 10 61.172 62814.4 30.2 0.387 10 99.190 64410.1 28.3 0.473 1 54.300 8296.5 28.5 0.443 1 85.100 8702.4 19.7 0.480 2 68.360 20073.3 20.2 0.469 2 87.440 20547.5 28.1 0.500 5 31.420 39530.2 31.5 0.499 10 7.570 65879.7 27.3 0.501 5 15.030 36194.3 28.5 0.58 1 36.010 10790.0 28.7 0.586 2 33.490 21002.2 30.2 0.616 5 36.050 54344.9 34.7 0.586 10 61.250 96583.3 29.0 0.669 1 41.330 14417.5

22 Magnetic Materials

**Table 6.** Selected 60 records of the measurement data of SIFERRIT N87.

**Figure 11.** Projection of the measurement points and the scaling theory points (38)–(41) in (( *<sup>f</sup>* <sup>Δ</sup>*<sup>B</sup> <sup>α</sup>* ) (1−*x*) , *Ptot* / <sup>Δ</sup>*<sup>B</sup> <sup>β</sup>* ) plane.

**Figure 12.** Projection of the measurement points and the scaling theory points (38)–(41) in ((tan *h*(c1*HDC*), *Ptot* / <sup>Δ</sup>*<sup>B</sup> <sup>β</sup>* ) plane.

Some comments concerning temperature change/stabilization have to be done. For details of the applied measurement method and the errors of the relevant factors, we refer to [22, 24]. The parameter values of (37)–(41) have been estimated by minimizing χ<sup>2</sup> of our experimental data and using the simplex method of Nelder and Mead [23]. The measurement series consists of 60 points (see **Table 5**). The standard deviation per point is equal to 15 Wm−3. Applying the formulae (37)–(41) and the estimated parameter values (**Table 6**), we have drawn the three scatter plots given in **Figures 11**–**13**, which compare estimated points those obtained through experimentation in the three projections, respectively. Note that in order to ensure numerical stability during the estimation process, the unit of frequency was set at 1 kHz, while other magnitudes were expressed in the SI unit system.

**Figure 13.** Projection of the measurement points and the scaling theory points (38)–(41) in ((*T* + τ) / *ΔB <sup>δ</sup>* , *Ptot* / <sup>Δ</sup>*<sup>B</sup> <sup>β</sup>* ) plane.

Scaled variables *Ptot*/(Δ*B*) *β* and *f*/(*ΔB*) *<sup>α</sup>* are very convenient for the model parameter estimations. By using these variables, the number of independent variables is reduced. Also the collapsed form of power losses characteristic is very compact and easy to implement. However, for the purpose of designing of magnetic circuits, it is necessary to have the split characteristics which describe the physical magnitude *Ptot*versus the physical ones: *T, f, HDC* and *ΔB*. Note that formula (37) is suitable just for this task. Let us assume the characteristics family for the following values of the independent variables: *T, f, HDC* and *<sup>T</sup>* =30°C, *HDC* =7 *<sup>A</sup> m* , *ΔB* = ∈{0.4−0.7}T, and *f* = ∈ {0.0 – 10.0}kHz. Using (37) and applying (38)–(41) as well as **Table 6**, we derive the characteristics presented in **Figure 14**.

The efficiency of scaling in solving problems concerning power losses in soft magnetic material has already been confirmed in recent papers [15, 25]. However, this paper is the first one which presents an application of scaling in modelling the temperature dependence of the core losses. The presented method is universal, which means that it works for a wide spectrum of excitations and different soft magnetic materials. Moreover, the presented model formulae (37)–(41) are not closed and can be adapted for a current problem by fitting the forms of both factors Φ and Θ. Ultimately, one must say that the degree of success achieved when applying the scaling depends on the property of the data. The data must obey the scaling.

**Figure 14.** Family of the power losses characteristics *Ptot* versus frequency *f* derived for SIFERRIT N87 material for *<sup>T</sup>* =30°C, *HDC* =7 *<sup>A</sup> m* .

## **5. Optimization of power losses in soft magnetic composites**

experimentation in the three projections, respectively. Note that in order to ensure numerical stability during the estimation process, the unit of frequency was set at 1 kHz, while other

**Figure 13.** Projection of the measurement points and the scaling theory points (38)–(41) in ((*T* + τ) / *ΔB <sup>δ</sup>*

By using these variables, the number of independent variables is reduced. Also the collapsed form of power losses characteristic is very compact and easy to implement. However, for the purpose of designing of magnetic circuits, it is necessary to have the split characteristics which describe the physical magnitude *Ptot*versus the physical ones: *T, f, HDC* and *ΔB*. Note that formula (37) is suitable just for this task. Let us assume the characteristics family for the following values of the independent variables: *T, f, HDC* and *<sup>T</sup>* =30°C, *HDC* =7 *<sup>A</sup>*

*ΔB* = ∈{0.4−0.7}T, and *f* = ∈ {0.0 – 10.0}kHz. Using (37) and applying (38)–(41) as well as **Table**

The efficiency of scaling in solving problems concerning power losses in soft magnetic material has already been confirmed in recent papers [15, 25]. However, this paper is the first one which presents an application of scaling in modelling the temperature dependence of the core losses. The presented method is universal, which means that it works for a wide spectrum of excitations and different soft magnetic materials. Moreover, the presented model formulae (37)–(41) are not closed and can be adapted for a current problem by fitting the forms of both factors Φ and Θ. Ultimately, one must say that the degree of success achieved when applying

the scaling depends on the property of the data. The data must obey the scaling.

, *Ptot* / <sup>Δ</sup>*<sup>B</sup> <sup>β</sup>*

*<sup>α</sup>* are very convenient for the model parameter estimations.

)

*m* ,

magnitudes were expressed in the SI unit system.

plane.

24 Magnetic Materials

Scaled variables *Ptot*/(Δ*B*)

*β*

**6**, we derive the characteristics presented in **Figure 14**.

and *f*/(*ΔB*)

Recently, novel concept of technological parameters' optimization has been applied in soft magnetic composites (SMCs) by Ślusarek et al. [26]. This concept is based on the assumption that power losses in SMC obey the scaling law. The efficiency of this approach has been confirmed in [9]. The scaling is very useful tool due of the three reasons:


Therefore, applying concept of the homogenous function in general sense, we apply the following expansion:

$$\frac{P\_{\rm tot}}{B\_{\rm m}^{\theta}} = \left(f \, \Big/ B\_{\rm m}^{a}\right) \cdot \left(\Gamma\_{1} + f \, \Big/ B\_{\rm m}^{a} \cdot \left(\Gamma\_{2} + f \, \Big/ B\_{\rm m}^{a} \cdot \left(\Gamma\_{3} + f \, \Big/ B\_{\rm m}^{a} \cdot \Gamma\_{4}\right)\right)\right). \tag{42}$$

Γ*n*, *α* and β parameters have been estimated for different values of pressure and temperature [9]. For the purpose of this paper, we take into account only one family of power losses characteristics which are presented in **Figures 15** and **16**. The corresponding estimated values of the model parameters are presented in **Table 7**. For all other details concerning SMC material and measurement data, we refer to [9]. Now we are ready to formulate the goals of this section. Main goal is to describe minimization of the power losses in SMC by using model density of power losses (42) and corresponding values of the model parameters. From the first row of **Table 7**, we can see that dimensions of the *Γn* coefficients depend on the values of *α* and *β* exponents. Therefore, the power losses characteristics presented in **Figures 15** and **16** are different dimensions. So, we have to answer the following question: Are we able to relate them in the optimization process which has been described in [9, 26]?

**Figure 15.** Selection of the power losses characteristics *Ptot*/(*Bm*) *<sup>α</sup>* versus *f*/(*Bm*) *<sup>α</sup>* calculated according to (42) and **Table 1** for Somaloy 500 [26], *T* = 500°C.

In this section, we show that if the considered characteristics obey the scaling, then the binary relation between them is invariant with respect to this transformation and comparison of two magnitudes of different dimensions has mathematical meaning. Reach measurement data of power losses in Somaloy 500 have been transformed into parameters of (42) versus hardening temperature and compaction pressure (**Table 7**) in [26]. Information contained in this table enables us to infer about topological structure of set of the power losses characteristics and finally to construct pseudo-state equation for SMC and derive new algorithm for the best values of technological parameters.

**Scaling of binary relations**. Let the power losses characteristic has the form determined by the scaling (27). It is important to remain that *α* and *β* are defined by initial exponents *a, b* and *c* (see after formula (27)):

$$
\alpha = \frac{a}{b}; \beta = \frac{c}{b}.\tag{43}
$$

Let us concentrate our attention at the point on the *<sup>f</sup> Bm <sup>α</sup>* axis of **Figures 15** or **16**:

#### Scaling in Magnetic Materials http://dx.doi.org/10.5772/63285 27

$$\frac{f}{B\_m^{\alpha}} = \frac{f\_1}{B\_{m1}^{\alpha 1}} = \frac{f\_2}{B\_{m2}^{\alpha 2}} = \frac{f\_3}{B\_{m3}^{\alpha 3}} = \frac{f\_4}{B\_{m4}^{\alpha 4}}.\tag{44}$$

**Figure 16.** Selection of the power losses characteristics *Ptot/*(*Bm*) *<sup>α</sup>* versus *f/*(*Bm*) *<sup>α</sup>* calculated according to (42) and **Table 1** for Somaloy 500 [26].

Let us take into account the two characteristics and let us assume that

$$\frac{P\_{\text{tot}1}}{B\_{m1}^{\beta 1}} > \frac{P\_{\text{tot}2}}{B\_{m2}^{\beta 2}}.\tag{45}$$


Values of scaling exponents and coefficients of (42) versus compaction pressure and hardening temperature, a selection from [26].

**Table 7.** Somaloy 500.

characteristics which are presented in **Figures 15** and **16**. The corresponding estimated values of the model parameters are presented in **Table 7**. For all other details concerning SMC material and measurement data, we refer to [9]. Now we are ready to formulate the goals of this section. Main goal is to describe minimization of the power losses in SMC by using model density of power losses (42) and corresponding values of the model parameters. From the first row of **Table 7**, we can see that dimensions of the *Γn* coefficients depend on the values of *α* and *β* exponents. Therefore, the power losses characteristics presented in **Figures 15** and **16** are different dimensions. So, we have to answer the following question: Are we able to relate them

*<sup>α</sup>* versus *f*/(*Bm*)

In this section, we show that if the considered characteristics obey the scaling, then the binary relation between them is invariant with respect to this transformation and comparison of two magnitudes of different dimensions has mathematical meaning. Reach measurement data of power losses in Somaloy 500 have been transformed into parameters of (42) versus hardening temperature and compaction pressure (**Table 7**) in [26]. Information contained in this table enables us to infer about topological structure of set of the power losses characteristics and finally to construct pseudo-state equation for SMC and derive new algorithm for the best

**Scaling of binary relations**. Let the power losses characteristic has the form determined by the scaling (27). It is important to remain that *α* and *β* are defined by initial exponents *a, b* and

> ; . *a c b b*

> > *Bm*

 b

a

Let us concentrate our attention at the point on the *<sup>f</sup>*

*<sup>α</sup>* calculated according to (42) and **Table 1**

= = (43)

*<sup>α</sup>* axis of **Figures 15** or **16**:

in the optimization process which has been described in [9, 26]?

**Figure 15.** Selection of the power losses characteristics *Ptot*/(*Bm*)

for Somaloy 500 [26], *T* = 500°C.

26 Magnetic Materials

values of technological parameters.

*c* (see after formula (27)):

Therefore, the considered binary relation is the strong inequality and corresponds to natural order presented in **Figures 15** and **16**. The most important question of this research is whether (45) is invariant with respect to scaling:

$$\frac{P\_{\text{not1}}^{\cdot}}{B\_{m1}^{\cdot,\beta\text{1}}} > \frac{P\_{\text{not2}}^{\cdot}}{B\_{m2}^{\cdot,\beta\text{2}}}.\tag{46}$$

Let *λ >* 0 be an arbitrary positive real number. Then, the scaling of (46) goes according to the following algorithm:

**•** Let us perform the scaling with respect to *λ* of all independent magnitudes and the de‐ pendent one:

$$\mathbb{L}f'\_i = \mathbb{X}^{a\_i} f'\_i; B'\_{mi} = \mathbb{X}^{b\_i} B\_{mi}; P'\_{\text{tot}} = \mathbb{X}^{c\_i} P\_{\text{tot}},\tag{47}$$

where *i* = 1, 2 …4 labels the considered characteristics.

**•** Substituting appropriate relations of (47) to (48), we derive:

$$\frac{P\_{\text{tot}1}}{B\_{m1}^{\beta\text{1}}} \mathcal{A}^{\epsilon\_1 - b\_1\beta\_1} > \frac{P\_{\text{tot}2}}{B\_{m2}^{\beta\text{2}}} \mathcal{A}^{\epsilon\_2 - b\_2\beta\_2}.\tag{48}$$

**•** Collecting all powers of λ on the left-hand side of (48) and taking into account (43) we derive that the resulting power has to be equal zero:

$$
\mathcal{X}^{c\_1 \cdots b\_l \beta\_l \cdots c\_2 \ast b\_2 \beta\_2} = \mathbf{l}.\tag{49}
$$

Therefore, (45) is invariant with respect to scaling. This binary relation has mathematical meaning and constitutes the total order in the set of characteristics.

**Binary equivalence relations**. The result derived in subsection **Scaling of binary relations** can be supplemented with the following binary equivalence relation. Let

$$X\_{i,j=}\left(\frac{f\_{i,j}}{B\_{mi,j}^{\alpha\_i}}, \frac{P\_{\text{not},j}}{B\_{mi,j}^{\beta\_i}}\right) \tag{50}$$

be the *j*th point of the *i*th characteristic. Two points, *j* and *k*, are related if they belong to the same *i*th characteristic:

$$X\_{i,f} \mathbf{R} X\_{i,k} \tag{51}$$

*Theorem:* **R** is equivalence relation. (The proof is trivial and can be done by checking out that the considered relation is reflexive, symmetric and transitive.) Therefore, **R** constitutes division of the positive-positive quarter of plane spanned by (50). The characteristics do not intersect each other except in the origin point which is excluded from the space. The result of this section implies that the power losses characteristics (27) and (42) are invariant with respect to scaling. Structure of derived here the set of all characteristics of which some examples are presented in **Figures 15** and **16** enable us to derive a formal pseudo-state equation of SMC. This equation constitutes a relation of the hardening temperature, the compaction pressure and a parameter characterizing the power losses characteristics corresponding to the values of these techno‐ logical parameters. Finally, the pseudo-state equation will improve the algorithm for designing the best values of technological parameters.

Therefore, the considered binary relation is the strong inequality and corresponds to natural order presented in **Figures 15** and **16**. The most important question of this research is whether

> b

Let *λ >* 0 be an arbitrary positive real number. Then, the scaling of (46) goes according to the

**•** Let us perform the scaling with respect to *λ* of all independent magnitudes and the de‐

; ,; *ii i ab c i i mi mi tot tot f fB B P P* ¢¢ ¢ == =

> 1 2 1 11 2 22 1 2 1 2 . *tot c b tot c b*

> > 1 11 2 2 2 1. *cb c b* b

 b

Therefore, (45) is invariant with respect to scaling. This binary relation has mathematical

**Binary equivalence relations**. The result derived in subsection **Scaling of binary relations** can

, ,

æ ö ç ÷ è ø

, , , *i i i j toti j*

be the *j*th point of the *i*th characteristic. Two points, *j* and *k*, are related if they belong to the

 b

*mi j mi j*

 b

**•** Collecting all powers of λ on the left-hand side of (48) and taking into account (43) we derive

 l

*m m P P B B* b

 l

 b

ll

where *i* = 1, 2 …4 labels the considered characteristics.

that the resulting power has to be equal zero:

**•** Substituting appropriate relations of (47) to (48), we derive:

b

l

l

meaning and constitutes the total order in the set of characteristics.

be supplemented with the following binary equivalence relation. Let

,

*i j*

*f P <sup>X</sup> B B* <sup>=</sup> a

> (46)

(47)

(50)



, , . *X X i j ik* **R** (51)

' ' 1 2 '1 ' 2 1 2 . *tot tot m m P P B B* b

(45) is invariant with respect to scaling:

following algorithm:

28 Magnetic Materials

pendent one:

same *i*th characteristic:

**Pseudo-equation of state for SMC**. Let C is set of all possible power losses characteristics in considered SMC. Each characteristic is smooth curve in [*f*/(*Bm*) *<sup>α</sup>*, *Ptot*/(*Bm*) *β* ] plane which corresponds to a point in [*T, p*] plane. In order to derive the pseudo-state equation, we transform each power losses characteristic into a number *V* corresponding to (*T, p*) point. By this way, we obtain a function of two variables:

$$(T, p) \to V.\tag{52}$$

This function must satisfy the following condition. Let us concentrate our attention at the two following points:

$$\frac{f\_1}{B\_{m1}^{\alpha\_1}} = \frac{f}{B\_m^{\alpha}} ; \frac{f\_2}{B\_{m2}^{\alpha\_2}} = \frac{f}{B\_m^{\alpha}} . \tag{53}$$

Let us consider the two characteristics *Ptot*<sup>1</sup> / (*Bm*1) *β*1 and *Ptot*<sup>2</sup> / (*Bm*2) *<sup>β</sup>*<sup>2</sup> of the two samples composed in *T*1, *p*1 and *T*2, *p*2 temperatures and pressures, respectively, while the other technological parameters such as powder compositions and volume fraction are constant. Let us assume that for (53), the following relation holds:

$$\frac{P\_{\text{tot}1}}{B\_{m1}^{\beta\text{1}}} > \frac{P\_{\text{tot}2}}{B\_{m2}^{\beta\text{2}}} \tag{54}$$

It results from the derived structure of that (54) holds for each value of (53).Therefore, we have to assume the following condition of sought *V*(*T, p*). If the relation (54) holds for given values of temperature and pressure *T*1, *p*1, *T*2, *p*2, then the following relation for *V*(*T, p*) has to be satisfied:

$$V\left(T\_1, p\_1\right) > V\left(T\_2, p\_2\right). \tag{55}$$

Moreover, *V* (*T, p*) has to indicate place of corresponding characteristic in the ordered set. The simplest choice satisfying these requirements is the following average:

$$V\left(T, p\right) = \frac{1}{\varphi\_{\text{max}} - \varphi\_{\text{min}}} \frac{P\_{\text{tot}}\left(\frac{f}{B\_m^{\alpha}}\right)}{B\_m^{\beta}} d\left(\frac{f}{B\_m^{\alpha}}\right),\tag{56}$$

The integration domain is common for all characteristics. We have selected the following common domain for the data presented in **Figures 15** and **16** *φmin* = 0, *φmax* = 4000 (*s*−1*T*<sup>−</sup>*<sup>α</sup>*).

Using (42), we transform (56) to the working formula for the *V* we measure:

$$V(T, p) = \frac{1}{\varphi\_{\max} - \varphi\_{\min}} \int\_{\varphi\_{\min}}^{\varphi\_{\max}} \mathbf{x} \left(\Gamma\_1 + \mathbf{x} \{\Gamma\_2 + \mathbf{x} \{\Gamma\_3 + \mathbf{x} \Gamma\_4\}\} \right) d\mathbf{x}.\tag{57}$$


**Table 8.** *V* measure versus hardening temperature and compaction pressure.

where *<sup>x</sup>* <sup>=</sup> *<sup>f</sup> Bm <sup>α</sup>* and *Γ<sup>i</sup>* are coefficients dependent on *T* and *p* (see **Table 7**). The values of *V* (*T, p*) are tabulated in **Table 8**. **Table 8** enables us to draw pseudo-isotherm. It is presented in **Figure 17**. However, in order to derive the complete pseudo-state equation, we must create a math‐ ematical model. On the basis of **Figure 17**, we start from the classical gas state equation as an initial approximation:

Moreover, *V* (*T, p*) has to indicate place of corresponding characteristic in the ordered set. The

<sup>1</sup> , , <sup>φ</sup> *max min*

ç ÷ æ ö è ø = ò ç ÷ - è ø

The integration domain is common for all characteristics. We have selected the following common domain for the data presented in **Figures 15** and **16** *φmin* = 0, *φmax* = 4000 (*s*−1*T*<sup>−</sup>*<sup>α</sup>*).

j

j

*<sup>B</sup> <sup>f</sup> VTp <sup>d</sup>*

*tot m m m*

*<sup>f</sup> <sup>P</sup>*

b

æ ö

*B B* a

are coefficients dependent on *T* and *p* (see **Table 7**). The values of *V* (*T, p*)

are tabulated in **Table 8**. **Table 8** enables us to draw pseudo-isotherm. It is presented in **Figure 17**. However, in order to derive the complete pseudo-state equation, we must create a math‐ ematical model. On the basis of **Figure 17**, we start from the classical gas state equation as an

 a (56)

(57)

simplest choice satisfying these requirements is the following average:

max min

Using (42), we transform (56) to the working formula for the *V* we measure:

j

( )

**T p V**

**[K] [MPa] [W kg−1T−β]** 723.15 800 40.60 773.15 900 43.75 773.15 700 47.25 673.15 800 50.30 773.15 600 57.12 823.15 800 81.50 773.15 500 89.28 742.15 764 492.3 753.15 780 509.2 804.15 764 528.5 711.15 764 547.0 873.15 800 720.0

**Table 8.** *V* measure versus hardening temperature and compaction pressure.

where *<sup>x</sup>* <sup>=</sup> *<sup>f</sup>*

30 Magnetic Materials

*Bm <sup>α</sup>* and *Γ<sup>i</sup>*

initial approximation:

**Figure 17.** Pseudo-isotherm *T* = 500*°*C of the low-losses phase, according to data of **Table 8**. for Somaloy 500 [1]. where *kB* is the pseudo-Boltzmann constant.

In order to extend (58) to a realistic equation, we apply again the scaling hypothesis (27):

$$V\left(\frac{T}{T\_c}, \frac{p}{p\_c}\right) = \left(\frac{p}{p\_c}\right)^{\gamma} \cdot \Phi\left(\frac{T}{\left(\frac{p}{p\_c}\right)^{\delta}}\right). \tag{59}$$

where Φ(*∙*) is an arbitrary function to be determined. Parameters *γ, δ* and *Tc*, *pc*are scaling exponents and scaling parameters, respectively, to be determined. For our conveniences, we introduce the following variables:

$$
\pi = \left(\frac{T}{T\_c}\right);\ \pi = \frac{p}{p\_c};\ X = \frac{\frac{T}{T\_c}}{\left(\frac{p}{p\_c}\right)^\delta} = \frac{\pi}{\pi^\delta}.\tag{60}
$$

In order to extend (58) to a full-state equation, we apply the Padé approximant by analogy to virial expansion derived by [27]:

$$V\left(\tau,\pi\right) = \pi^{\prime} \frac{G\_0 + X\left(G\_1 + X\left(G\_2 + X\left(G\_3 + XG\_4\right)\right)\right)}{1 + X\left(D\_1 + X\left(D\_2 + X\left(D\_3 + XD\_4\right)\right)\right)},\tag{61}$$

where *G*0,…, *G*4, *D*1,… *D*<sup>4</sup> are coefficients of the Padé approximant. All parameters have to be determined from the data presented in **Table 8**.

**Estimation of parameters for pseudo-equation of state**. At the beginning, we have to notice that the data collected in **Table 8** reveal sudden change of *V* between two points: [773, 15; 500, 0] and [742, 15; 764, 0]. This suggests existence of a crossover between two phases: low-losses phase and high-losses phase. We take this effect into account and we divide the data of **Table 8** into two subsets corresponding to these two phases, respectively. Since the crossover consists in changing of characteristic exponents for the given universality class, it is necessary to perform estimations of the model parameters for each phase separately. Minimizations of *χ*<sup>2</sup> for both phases have been performed by using MICROSOFT EXCEL 2010, where

$$\chi^2 = \sum\_{i=1}^{N} \left( V(\pi\_i, \pi\_i) - \pi\_i^\prime \frac{G\_0 + X\_i \left( G\_1 + X\_i \left( G\_2 + X\_i \left( G\_3 + X\_i G\_4 \right) \right) \right)}{1 + X\_i \left( D\_1 + X\_i \left( D\_2 + X\_i \left( D\_3 + X\_i D\_4 \right) \right) \right)} \right)^2. \tag{62}$$


Values of pseudo-state equation's parameters and the Padé approximant's coefficients of (61)

**Table 9.** Somaloy 500, low-losses phase.


Values of pseudo-state equation's parameters and the Padé approximant's coefficients of (61)

**Table 10.** Somaloy 500, high-losses phase.

;; . *<sup>c</sup>*

In order to extend (58) to a full-state equation, we apply the Padé approximant by analogy to

( ) ( ( ( )))

, , <sup>1</sup> *G X G X G X G XG*

*c*

( ( ( ))) 0 1 2 34

1 2 34

*X D X D X D XD*

where *G*0,…, *G*4, *D*1,… *D*<sup>4</sup> are coefficients of the Padé approximant. All parameters have to be

**Estimation of parameters for pseudo-equation of state**. At the beginning, we have to notice that the data collected in **Table 8** reveal sudden change of *V* between two points: [773, 15; 500, 0] and [742, 15; 764, 0]. This suggests existence of a crossover between two phases: low-losses phase and high-losses phase. We take this effect into account and we divide the data of **Table 8** into two subsets corresponding to these two phases, respectively. Since the crossover consists in changing of characteristic exponents for the given universality class, it is necessary to perform estimations of the model parameters for each phase separately. Minimizations of *χ*<sup>2</sup>

for both phases have been performed by using MICROSOFT EXCEL 2010, where

0 1 2 34 2

**γ δ Tc pc G0 G1 G2** 1.2812 0.1715 21.622 37.729 370315315 −47752251 1734952

*G3 G4 D1 D2 D3 D4* – −1.3764 −678.26 170.80 6243.8 386.96 −28.699 –

Values of pseudo-state equation's parameters and the Padé approximant's coefficients of (61)

g

 p

*N ii ii*

æ ö ++++ <sup>=</sup> ç ÷ - ++++ è ø

d d

++++ <sup>=</sup> ++++ (61)

( ( ( ))) ( ( ( )))

1 2 34

å (62)

*ii ii*

*X D X D X D XD*

*G X G X G X G XG*

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

2

(60)

t

p

*p*

ç ÷ è ø

*T*

*c c*

 p

t

g

 p

virial expansion derived by [27]:

32 Magnetic Materials

*V*

tp

determined from the data presented in **Table 8**.

1

c

**Table 9.** Somaloy 500, low-losses phase.

*V*

 tp

*ii i i*

=

*T pT <sup>X</sup> T p p*

æ ö = == = ç ÷ è ø æ ö

> where *N* = 7 and *N* = 5 for the low-losses and high-losses phases, respectively. **Tables 9** and **10** present estimated values of the model parameters for the low-losses and high-losses phases, respectively.

**Figure 18.** Phase diagram for Somaloy 500.

**Optimization of technological parameters**. Function *V* (*T, p*) serves a power loss measure versus the hardening temperature and compaction pressure. In order to explain how to optimize the technological parameters with the pseudo-state Eq. (61), we plot the phase diagram of considered SMC **Figure 17**. Note that all losses' characteristics collapsed to a one curve for the each phase. Taking into account the low-losses phase, we determine the lowest losses at *τ* ⋅ *π*<sup>−</sup>*<sup>δ</sup>* = 19, 75 (see **Figure 18**). This gives the following continuous subspace of the optimal points:

$$\frac{T}{\left(\frac{p}{p\_c}\right)^{\delta}} = 19,75. \tag{63}$$

Formula (63) represents the minimal iso-power loss curve. All points satisfying (63) are solutions of the optimization problem for technical parameters of SMC. By introducing the binary relations, we have revealed twofold. The power losses characteristics do not cross each other which makes the topology's set of these curves very useful and effectively that we can perform all calculations in the one-dimensional space spanned by the scaled frequency or here in the case of pseudo-state equation in the scaled temperature. For general knowledge concerning such a topology, we refer to the paper [28]. The obtained result is the continuous set of points satisfying (63). All solutions of these equations are equivalent for the optimization of the power losses. Therefore, the remaining degree of freedom can be used for optimizing the magnetic properties of the considered SMC.

#### **6. Scaling conception of losses separation**

In this section, we show how to expand losses into polynomial series. The distinction between different eddy current scales, that is a macroscale, covering the whole bulk material and a microscale covering the area of moving domain walls, introduced by Bertotti's theory, has led to the following relationship of the three terms:

$$P\_{\rm nu} = \mathcal{c}\_1 f \, B\_{\rm nu}^{\vartheta} + \mathcal{c}\_2 \, \sigma f^2 B\_{\rm nu}^2 + 8 \sqrt{(\sigma G S V\_0)} f^{1.5} B\_{\rm nu}^{1.5},\tag{64}$$

where *σ* is conductivity, *G* is the constant equal to 0.1356; *S* is sample cross section, whereas *V0* is a parameter dependent on flux density. In general case, (64) is not homogenous expres‐ sion; therefore, this can describe the self-similarity property only for *β* = 1. However, the Bertotti interpretation of each term is correct

$$P\_{\rm tot} = P\_{\rm hy} + P\_{\rm elas} + P\_{\rm env}.\tag{65}$$

where the presented in (65) components are hysteresis, classical and excess losses, respectively.

In this chapter, we have shown how the two-component formula for losses (28) can be transformed to dimensionless expression (30) and (31). This expression helps us to consider the data collapse. However, in the case of expansion of (27) over the square term, (31) does not apply.

Then, one can consider partial data collapses in the expansions up to necessary degree. Let us consider, for example, (42):

$$\frac{P\_{\rm tot}}{B\_{\rm m}^{\beta}} = \left(\Gamma\_{\rm 1} \cdot \left(f \not\slash B\_{\rm m}^{\alpha} + \cdot \Gamma\_{\rm 2} \cdot \left(f \not\slash B\_{\rm m}^{\alpha}\right)^{2} + \Gamma\_{\rm 3} \cdot \left(f \not\slash B\_{\rm m}^{\alpha}\right)^{3} + \Gamma\_{\rm 4} \cdot \left(f \not\slash B\_{\rm m}^{\alpha}\right)^{4}\right),$$

Formula (42) is the fourth-degree polynomial of the ( *f* / *Bm <sup>α</sup>*) scaled frequency. Let us span all possible binomial subspaces:

$$
\sum = \{S\_{12}, S\_{13}, S\_{14}, S\_{23}, S\_{24}, S\_{34}\},
$$

where *Si <sup>j</sup>* ={( *f* / (*Bm α*)) *i* , ( *f* / (*Bm <sup>α</sup>*)) *<sup>j</sup>* }. To consider partial data collapse in *S*12, we perform the following transformations:

$$P\_{\text{nor},2} = \frac{\Gamma\_2 P\_{\text{nor}}}{\Gamma\_1^2 B\_m^{\theta}}, f\_{1,2} = \frac{\Gamma\_2 f}{\Gamma\_1 B\_m^{\alpha}}.\tag{66}$$

Substituting (66) to (42), we get

19,75. *<sup>c</sup>*

Formula (63) represents the minimal iso-power loss curve. All points satisfying (63) are solutions of the optimization problem for technical parameters of SMC. By introducing the binary relations, we have revealed twofold. The power losses characteristics do not cross each other which makes the topology's set of these curves very useful and effectively that we can perform all calculations in the one-dimensional space spanned by the scaled frequency or here in the case of pseudo-state equation in the scaled temperature. For general knowledge concerning such a topology, we refer to the paper [28]. The obtained result is the continuous set of points satisfying (63). All solutions of these equations are equivalent for the optimization of the power losses. Therefore, the remaining degree of freedom can be used for optimizing

In this section, we show how to expand losses into polynomial series. The distinction between different eddy current scales, that is a macroscale, covering the whole bulk material and a microscale covering the area of moving domain walls, introduced by Bertotti's theory, has led

1 2 <sup>0</sup> 8( ) , *P c f B c f B GSV f B tot <sup>m</sup> <sup>m</sup> <sup>m</sup>*

b =+ + s

2 2 1.5 1.5

(64)

. *PPP P tot hys clas exc* =+ + (65)

 s

where *σ* is conductivity, *G* is the constant equal to 0.1356; *S* is sample cross section, whereas *V0* is a parameter dependent on flux density. In general case, (64) is not homogenous expres‐ sion; therefore, this can describe the self-similarity property only for *β* = 1. However, the Bertotti

where the presented in (65) components are hysteresis, classical and excess losses, respectively.

In this chapter, we have shown how the two-component formula for losses (28) can be transformed to dimensionless expression (30) and (31). This expression helps us to consider the data collapse. However, in the case of expansion of (27) over the square term, (31) does not

(63)

*c*

the magnetic properties of the considered SMC.

34 Magnetic Materials

**6. Scaling conception of losses separation**

to the following relationship of the three terms:

interpretation of each term is correct

apply.

*T T p p*

d <sup>=</sup> æ ö ç ÷ è ø

$$P\_{\text{tot }1,2} \{ f\_{1,2} \} = f\_{1,2} \{ 1 + f\_{1,2} \} + f\_{1,2}^3 \frac{\Gamma\_1}{\Gamma\_2^2} (\Gamma\_3 + f\_{1,2} \frac{\Gamma\_1 \Gamma\_4}{\Gamma\_2}). \tag{67}$$

Note that (67) is dimensionless full formula for the scaled loss. Moreover, expression *f*12(1 + *f*12) is sample independent. The linear and square terms describe the hysteresis and the classical losses, respectively. The cubic and the fourth-order terms correspond to the excess losses. Moreover, the square and linear terms describe the partial data collapse in *S*12 [29]. Using (67), one can compare losses data of different measurements projected on *S12* subspace. The analogical equations for the *S*34 subspace read:

$$P\_{\text{tot }3,4} = \frac{\Gamma\_4^3 P\_{\text{tot}}}{\Gamma\_3^4 B\_m^{\beta}}, f\_{3,4} = \frac{\Gamma\_4 f}{\Gamma\_3 B\_m^{\frac{\alpha}{\alpha}} m}, \tag{68}$$

$$P\_{\text{tot }3,4} \{ f\_{3,4} \} = f\_{3,4}^3 \{ 1 + f\_{3,4} \} + f\_{3,4} \frac{\Gamma}{\Gamma\_3^3} \{ \Gamma\_1 + f\_{3,4} \frac{\Gamma\_2 \Gamma\_3}{\Gamma\_4} \}. \tag{69}$$


**Table 11.** Scaling exponents and expansion coefficients.

Applying (68) and (69), one can complete the data comparison by considering partial data collapse using *f* <sup>3</sup>,<sup>4</sup> <sup>3</sup> (1 <sup>+</sup> *<sup>f</sup>* <sup>3</sup>,4) polynomial which is also sample independent. In the case of expansion (42), the comparisons performed in *S*12, *S*34 spaces are completed. To test the presented comparison formalism, we present the following measurement data: P1 amorphous alloy *F e*78*Si* <sup>13</sup>*B*9, P2—amorphous alloy *Co*71.5*F e*2.5*Mn*2*Mo*1*Si*9B14, P4—crystalline material-oriented electrotechnical steel shits 3% Si-Fe and P7—iron-nickel-alloy 79%Ni-Fe. Processed measurement data in the form of scaling exponents and expansion coefficients are presented in **Table 11**:

**Figures 19** and **20** present the completed partial collapses for the considered problem:

**Figure 19.** Partial data collapse in *S*12 space.

**Figure 20.** Partial data collapse in *S*34 space.

In order to make a numerical comparison of the measurement qualities taken from different samples, one can introduce analogically to (32) the measures of uncertainty for the both spaces *S*12 and *S*34. The comparisons must be done and interpreted independently for *S*12 and *S*34. Qualitative analysis on the basis of **Figures 19** and **20** shows that the uncertainty measure of *S*34 for the sample *P*4 is significantly high.

### **7. Summary**

**Sample α β Γ<sup>1</sup> Γ<sup>2</sup> Γ<sup>3</sup> Γ<sup>4</sup>** P1 −2.347 −1.407 2.25E-03 7.96E-06 −5.19E-09 1.76E-12 P2 −1.519 −0.375 2.53E-03 6.79E-06 −6.48E-09 2.78E-12 P4 −2.372 −1.295 1.80E-02 2.04E-05 7.68E-09 −1.37E-12 P7 −2.437 −1.401 2.28E-03 1.05E-05 3.08E-07 −8.38E-10

Applying (68) and (69), one can complete the data comparison by considering partial data

expansion (42), the comparisons performed in *S*12, *S*34 spaces are completed. To test the presented comparison formalism, we present the following measurement data: P1—

material-oriented electrotechnical steel shits 3% Si-Fe and P7—iron-nickel-alloy 79%Ni-Fe. Processed measurement data in the form of scaling exponents and expansion coefficients are

**Figures 19** and **20** present the completed partial collapses for the considered problem:

<sup>3</sup> (1 <sup>+</sup> *<sup>f</sup>* <sup>3</sup>,4) polynomial which is also sample independent. In the case of

<sup>13</sup>*B*9, P2—amorphous alloy *Co*71.5*F e*2.5*Mn*2*Mo*1*Si*9B14, P4—crystalline

**Table 11.** Scaling exponents and expansion coefficients.

collapse using *f* <sup>3</sup>,<sup>4</sup>

36 Magnetic Materials

amorphous alloy *F e*78*Si*

presented in **Table 11**:

**Figure 19.** Partial data collapse in *S*12 space.

We have presented many examples of the measurements of power losses in soft magnetic materials, including composites. Moreover, working conditions were determined by multidi‐ mensional parameter space: frequency, pick of induction, DC bias and temperature. On the basis of obtained results of experimental and theoretical considerations, we confirm that the total power loss in soft magnetic materials is self-similar. This is very important for practices, since the fundamental parameter used by technologists in the processes aimed at tailoring properties of magnetic materials as well as in design and work analysis of magnetic circuits is loss density. However, there is one important detail which has to be discussed at the end. In order to determine *F*(⋅) in (27), the Maclaurin expansion has been applied up to the secondorder term. Note that each two-term formula can be reduced to dimensionless form (28). Therefore, one could conclude that the data collapse is trivial. However, this is not so because relevance of the data collapse depends on measurement data. If data transformed by (30) get place on (31), then these data satisfy the axioms of homogeneity, they are invariant with respect to scaling as well as they are self-similar. What to do if the two-term expansion (28) is not sufficient? Then, one should extend (28) up to sufficient polynomial order. For an example, see (42). In general case, reduction of losses characteristic to dimensionless form is not possible. However, for comparison of different measurement data, it is possible always to perform transformation of data to dimensionless magnitudes partially in the two-dimensional subspa‐ ces (67), (69) and obtain full comparison by collecting the all independent comparisons in *Sij* subspaces.

## **Author details**

Krzysztof Z. Sokalski1\*, Barbara Ślusarek<sup>2</sup> and Jan Szczygłowski<sup>1</sup>

\*Address all correspondence to: ksokalski76@gmail.com

1 Czestochowa University of Technology, Czestochowa, Poland

2 Tele and Radio Research Institute, Warszawa, Poland

## **References**


equation of soft magnetic composites. Mater. Sci. Appl. 2014;5:1040–1047. doi:10.4236/ msa.2014.514107


see (42). In general case, reduction of losses characteristic to dimensionless form is not possible. However, for comparison of different measurement data, it is possible always to perform transformation of data to dimensionless magnitudes partially in the two-dimensional subspa‐ ces (67), (69) and obtain full comparison by collecting the all independent comparisons in *Sij*

and Jan Szczygłowski<sup>1</sup>

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**Author details**

**References**

Krzysztof Z. Sokalski1\*, Barbara Ślusarek<sup>2</sup>

\*Address all correspondence to: ksokalski76@gmail.com

2 Tele and Radio Research Institute, Warszawa, Poland

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1 Czestochowa University of Technology, Czestochowa, Poland


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Fanny Béron, Marcos V. Puydinger dos Santos, Peterson G. de Carvalho, Karoline O. Moura, Luis C.C. Arzuza and Kleber R. Pirota

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/63482

#### **Abstract**

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J. Chem. Phys. 1964;40:939–950. doi:10.1063/1.1725286

Cylindrical magnetic nanowires made through the help of nanoporous alumina templates are being fabricated and characterized since the beginning of 2000. They are still actively investigated nowadays, mainly due to their various promising applica‐ tions, ranging from high-density magnetic recording to high-frequency devices, passing by sensors, and biomedical applications. They also represent suitable systems in order to study the dimensionality effects on a given material. With time, the development in fabrication techniques allowed to increase the obtained nanowire complexity (control‐ led crystallinity, modulated composition and/or geometry, range of materials, etc.), while the improvements in nanomanipulation permitted to fabricate system based either on arrays or on single nanowires. On the other side, their increased complexity requires specific physical characterization methods, due to their particular features such as high anisotropy, small magnetic volume, dipolar interaction field between them, and interesting electronic properties. The aim of this chapter was to offer an ample overview of the magnetic, electric, and physical characterization techniques that are suitable for cylindrical magnetic nanowire investigation, of what is the specific care that one needs to take into account and which information will be extracted, with typical and varied examples.

**Keywords:** magnetic materials, nanotechnology, cylindrical magnetic nanowires, nanoporous alumina templates, characterization techniques

© 2016 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.

## **1. Introduction**

The continuing progress in fabrication techniques nowadays lead scientists to succeed in producing nanosystems that are always smaller, more complex, and/or fabricated with more control. This achievement is highly interesting for both fundamental studies and technologi‐ cal improvements, since it provides novel systems permitting to test phenomena at mankind frontiers of knowledge. However, the counterpart is that these new nanosystems also require improvements inthe characterizationtechnique field,toyieldmore efficient,more sensible, and sometimes up to revolutionary, characterization methods. Without the ability to adequately probe the fabricated system characteristics, they remain useless and their promised advances need to wait until the development of suitable characterization techniques.

**Figure 1.** Representation of nanowire arrays embedded in a nanoporous alumina template. Reprinted with permission from [11]. Copyright 2010 by InTech.

Cylindrical magnetic nanowires represent a good example of the kind of nanosystems that become enabled due to a new fabrication technique. After the finding regarding how to obtain controlled nanoporous alumina templates by Masuda et al. in 1995 [1], the ordered, parallel, and with high-aspect ratio pores were quickly identified as perfect mold for electrodeposited metallic nanowires (**Figure 1**) [2–4]. Using magnetic materials, those nanowires are still actively investigated nowadays, mainly due to their various promising applications, ranging from high-density magnetic recording [5] to high-frequency devices [6], passing by sensors [7], and biomedical applications [8]. They also represent suitable systems in order to study the dimensionality effects on a given material [9, 10]. With time, the development in fabrication techniques permitted to increase the obtained nanowire complexity (controlled crystallinity, modulated composition and/or geometry, range of materials, etc.), while the improvements in nanomanipulation permitted to fabricate systems based either on arrays or on single nano‐ wires.

However, characterizing cylindrical magnetic nanowires presents specific challenges, due to their particular features. Among them, the small volume of a nanowire, and therefore its low magnetic signal, complicates both the manipulation of individual nanowires for their subse‐ quent characterization and most of the single nanowire magnetic probing. It also limits the electrical current that a nanowire can support without melting, making them highly sensible to electrical discharge. Furthermore, nanowires present, by definition, a high-length/diameter ratio, typically larger than 50. This peculiar shape induces a mechanical fragility along the nanowire, thereby making its manipulation difficult and facilitating the apparition of longi‐ tudinal mechanical stress, which may interfere with their properties. From a magnetic point of view, this high-aspect ratio yields a large magnetic shape anisotropy that typically governs its magnetic behavior. Another important component influencing it is the large dipolar interaction field among nanowires when in array conformation, due to the small interdistance between them. However, this interaction field also needs to be taken into account when nanowires are free, like in solution. Finally, a last particular feature is the large nanowire surface (in comparison to its volume). It makes nanowires highly sensible to their environment, which may be an advantage, as in sensors, but can also be an issue while characterizing them (e.g., favoring the surface oxidation).

As previously stated, the magnetic nanowires' increased complexity requires specific charac‐ terizations, in order to efficiently understand their properties and behavior and take advantage of their novelty. The understanding of a magnetic system, such as magnetic nanowires, requires studying not only its magnetic behavior but also its electrical and physical properties. The emphasis on one or another aspect depends on the specific objective of the study or application. Therefore, the aim of this chapter is to offer an ample overview of the physical, electrical, and magnetic characterization techniques that are suitable for cylindrical magnetic nanowire investigation. For each method, the specific care that one needs to take into account and which information can be extracted are discussed, with typical and varied examples.

## **2. Physical characterization**

**1. Introduction**

42 Magnetic Materials

from [11]. Copyright 2010 by InTech.

The continuing progress in fabrication techniques nowadays lead scientists to succeed in producing nanosystems that are always smaller, more complex, and/or fabricated with more control. This achievement is highly interesting for both fundamental studies and technologi‐ cal improvements, since it provides novel systems permitting to test phenomena at mankind frontiers of knowledge. However, the counterpart is that these new nanosystems also require improvements inthe characterizationtechnique field,toyieldmore efficient,more sensible, and sometimes up to revolutionary, characterization methods. Without the ability to adequately probe the fabricated system characteristics, they remain useless and their promised advances

**Figure 1.** Representation of nanowire arrays embedded in a nanoporous alumina template. Reprinted with permission

Cylindrical magnetic nanowires represent a good example of the kind of nanosystems that become enabled due to a new fabrication technique. After the finding regarding how to obtain controlled nanoporous alumina templates by Masuda et al. in 1995 [1], the ordered, parallel, and with high-aspect ratio pores were quickly identified as perfect mold for electrodeposited metallic nanowires (**Figure 1**) [2–4]. Using magnetic materials, those nanowires are still actively

need to wait until the development of suitable characterization techniques.

Since magnetic properties are intrinsically linked to the system morphology, composition, and crystalline structure, they represent fundamental information while investigating magnetic nanowires. A common first step is to verify the obtained nanowire geometry by microscopy, either scanning electron microscopy (SEM) or transmission electron microscopy (TEM). Coupled to these microscopes, other spectroscopy techniques allow knowing the nanowire composition, such as energy-dispersive X-ray spectroscopy (EDS) and electron energy loss spectroscopy (EELS). Radiation diffraction techniques yield information about the crystalline or magnetic structure and can be performed on the complete array, through X-ray, neutron, or electron diffraction, or locally on a nanowire section, with the help of a high-resolution TEM (HRTEM). Finally, phase transitions are well probed through specific heat measurements.

## **2.1. Geometry/morphology**

As any nanostructure, nanowire fabrication often requires a visual inspection of the obtained product, before pursuing further characterization. Even if nanowires are basically long cylinders, their exact geometry (diameter, length, and interdistance in the array) directly influences their magnetic behavior. For example, their coercivity is highly sensible to their diameter, much more than to their length, while their interdistance controls the interaction field strength in the array, among others. Due to the fabrication technique used, nanowires grown in nanoporous alumina templates always exhibit a distribution, even very narrow, of their geometric parameter values. Furthermore, since nanowires are not restricted to homo‐ geneous cylinders, their morphology also needs to be probed.

Such information can be obtained by electron microscopy techniques, due to the length scale required (less than 5-nm resolution at minimum). Scanning electron microscopy (SEM) and transmission electron microscopy (TEM) are the two best equipments for nanowire imaging. While the resolution of the first is lower, it is compensated by the facts that it is a cheaper, more widespread and easier to operate microscope.

#### *2.1.1. Scanning electron microscopy (SEM)*

SEM is a technique that enables the inspection of nanostructures by means of an electron beam guided through magnetic and electrostatic lenses. The wealth of different information that can be obtained by SEM method is caused by the multitude of signals that arise when an electron interacts with the specimen. The different types of electron scattering (backscattered, secon‐ dary, and Auger electrons) are the basis of most electron microscopy methods [12]. The widespread use of SEM became possible after 1958, when researchers from Cambridge (UK) built the first commercial prototype [13]. The typical primary electron beam used in a SEM is of 1–30 kV, with a beam current of 1 pA to 20 nA that can be focused in about 2–100-nm spot size, depending on the emitter source [14, 15]. A detailed description of the SEM operation can be found in [16].

This technique is highly adequate for proper measurements of the nanowire (and array) dimensions. Array cross-sectional view (easily obtained by simply breaking the fragile alumina template) facilitates the evaluation of geometrical parameter distribution (**Figure 2a**). Free nanowires can also be investigated, after selectively dissolving the alumina template and dispersing them on a substrate, generally Si (**Figure 2b**). Despite the relatively low resolution of SEM, it commonly allows zooming on some specific nanowire regions, such as the dendritic region at the bottom, if voltage reduction protocol described in [17] is used to thin the alumina barrier layer and allows the pores filling by electrodeposition (**Figure 2c**). Finally, the evalua‐ tion of the pores-filling ratio can be completed by the top view of the array (**Figure 2d**). Additionally, as a non-destructive technique, one can inspect the nanowires along with electrical measurements, thermal annealing, or chemical reactions inside the SEM chamber [18]. However, electron beam exposure can induce hydrocarbon molecule deposition (from the vacuum chamber) on the scanned nanowire surface, leading to some contamination [19]. This can be avoided by reducing the beam current and exposure time over the surface [15].

**Figure 2.** (a–d) SEM images of Ni nanowires with modulated diameter fabricated through the three-step anodization technique [20]. (a) Cross-sectional view, showing the diameter modification region; (b) zoom on free nanowires upper segment; (c) zoom on the dendritic region, at the nanowire base; (d) array-top view, where the filled pores appear clearer; and (e) TEM image of a Ni nanowire region. Reprinted with permission from [21]. Copyright 2015 by IEEE.

#### *2.1.2. Transmission electron microscopy (TEM)*

composition, such as energy-dispersive X-ray spectroscopy (EDS) and electron energy loss spectroscopy (EELS). Radiation diffraction techniques yield information about the crystalline or magnetic structure and can be performed on the complete array, through X-ray, neutron, or electron diffraction, or locally on a nanowire section, with the help of a high-resolution TEM (HRTEM). Finally, phase transitions are well probed through specific heat measurements.

As any nanostructure, nanowire fabrication often requires a visual inspection of the obtained product, before pursuing further characterization. Even if nanowires are basically long cylinders, their exact geometry (diameter, length, and interdistance in the array) directly influences their magnetic behavior. For example, their coercivity is highly sensible to their diameter, much more than to their length, while their interdistance controls the interaction field strength in the array, among others. Due to the fabrication technique used, nanowires grown in nanoporous alumina templates always exhibit a distribution, even very narrow, of their geometric parameter values. Furthermore, since nanowires are not restricted to homo‐

Such information can be obtained by electron microscopy techniques, due to the length scale required (less than 5-nm resolution at minimum). Scanning electron microscopy (SEM) and transmission electron microscopy (TEM) are the two best equipments for nanowire imaging. While the resolution of the first is lower, it is compensated by the facts that it is a cheaper, more

SEM is a technique that enables the inspection of nanostructures by means of an electron beam guided through magnetic and electrostatic lenses. The wealth of different information that can be obtained by SEM method is caused by the multitude of signals that arise when an electron interacts with the specimen. The different types of electron scattering (backscattered, secon‐ dary, and Auger electrons) are the basis of most electron microscopy methods [12]. The widespread use of SEM became possible after 1958, when researchers from Cambridge (UK) built the first commercial prototype [13]. The typical primary electron beam used in a SEM is of 1–30 kV, with a beam current of 1 pA to 20 nA that can be focused in about 2–100-nm spot size, depending on the emitter source [14, 15]. A detailed description of the SEM operation can

This technique is highly adequate for proper measurements of the nanowire (and array) dimensions. Array cross-sectional view (easily obtained by simply breaking the fragile alumina template) facilitates the evaluation of geometrical parameter distribution (**Figure 2a**). Free nanowires can also be investigated, after selectively dissolving the alumina template and dispersing them on a substrate, generally Si (**Figure 2b**). Despite the relatively low resolution of SEM, it commonly allows zooming on some specific nanowire regions, such as the dendritic region at the bottom, if voltage reduction protocol described in [17] is used to thin the alumina barrier layer and allows the pores filling by electrodeposition (**Figure 2c**). Finally, the evalua‐

geneous cylinders, their morphology also needs to be probed.

widespread and easier to operate microscope.

*2.1.1. Scanning electron microscopy (SEM)*

be found in [16].

**2.1. Geometry/morphology**

44 Magnetic Materials

In order to observe the nanowires' geometrical aspects in more detail, higher resolution images can be obtained by using TEM. In a conventional TEM, a thin specimen is irradiated with an electron beam with uniform current density. Typically, the acceleration voltage is at least 100– 200 kV, while medium voltage instruments work at 200–500 kV to provide better transmission, and some equipments can reach up to 3 MV. The use of such a high voltage allows to produce a very small electron beam spot (typically <5 nm and, at best, <0.1 nm in diameter). On the other hand, strong electron/atom interactions through elastic and inelastic scattering can lead to some damages to the sample. This technique therefore requires a very thin specimen, typically of the order of 5–100 nm for 100-keV electrons, depending on the sample material [22, 23].

Fortunately, most nanowires are thin enough in order to be directly investigated by TEM, without further preparation than dispersing free nanowires on a carbon TEM grid. Interesting morphological information that can be obtained includes surface roughness, layer interface sharpness, or even atomic plane directions, to name a few, besides more precise geometrical characterization than with a SEM (**Figure 2e**). Generally, TEM nanowire characterization is not limited to geometry and morphology but specialized in local investigation. It usually takes advantage of the high-energy electron beam to perform local spectroscopy (Section 2.2). Moreover, given the small beam diameter, large electron diffraction occurs on the sample, which reveals useful crystallographic information that will be described more in detail in Section 2.3.2. Finally, even local magnetic information can be obtained, as will be discussed in Section 4.3.2. As expected, its operation presents much more difficulties compared to SEM.

A common challenge concerning the inspection by electron microscope of nanoporous alumina membrane is the charging effects from the electron beam on the scanned area (mainly insu‐ lating). This effect can be avoided by depositing a very thin layer (<50 nm) of carbon or metal on the top surface [12, 24], whereas the dimensions of the inspected structures are larger than the metal layer thickness. However, the SEM contrast of nonconductive specimens varies widely depending on the coating metal and thickness, which influences the measurement's accuracy on the scanned structures [25]. For TEM analysis, if the charge dissipation is insuffi‐ cient, as is the case for insulating materials, the sample becomes unstable under the beam and the analysis becomes impossible.

#### **2.2. Chemical composition**

Since chemical composition is not necessarily totally controlled during the nanowire fabrica‐ tion, either by electrodeposition or other alumina template-filling methods, probing their chemical composition is essential. This is especially important for alloyed or multi-element nanowires, as well as for multilayered ones. In the last case, a direct visualization of the chemical composition configuration is highly helpful. This is also true for local investigation, such as for surface oxidation, for example.

Therefore, it is highly advantageous to use some features of the electron microscopes described above to simultaneously perform a composition characterization of the nanowires. Mainly, both SEM and TEM possess a high-energy electron beam, which allows high sensibility, while their scanning possibilities permit to map the chemical composition, usually superimposed to their morphology image. The typical techniques used for nanowires are the energy-dispersive X-ray spectroscopy (EDS) and the electron energy loss spectroscopy (EELS). Generally speaking, while the first determines the presence or not of an element, the second can probe their chemical environment. Both techniques can lead to a mapping of local chemical properties with high spatial resolution when coupled to a TEM, whereas EDS can also be performed in a SEM chamber.

## *2.2.1. Energy dispersive X-ray spectroscopy (EDS)*

typically of the order of 5–100 nm for 100-keV electrons, depending on the sample material [22,

Fortunately, most nanowires are thin enough in order to be directly investigated by TEM, without further preparation than dispersing free nanowires on a carbon TEM grid. Interesting morphological information that can be obtained includes surface roughness, layer interface sharpness, or even atomic plane directions, to name a few, besides more precise geometrical characterization than with a SEM (**Figure 2e**). Generally, TEM nanowire characterization is not limited to geometry and morphology but specialized in local investigation. It usually takes advantage of the high-energy electron beam to perform local spectroscopy (Section 2.2). Moreover, given the small beam diameter, large electron diffraction occurs on the sample, which reveals useful crystallographic information that will be described more in detail in Section 2.3.2. Finally, even local magnetic information can be obtained, as will be discussed in Section 4.3.2. As expected, its operation presents much more difficulties compared to SEM.

A common challenge concerning the inspection by electron microscope of nanoporous alumina membrane is the charging effects from the electron beam on the scanned area (mainly insu‐ lating). This effect can be avoided by depositing a very thin layer (<50 nm) of carbon or metal on the top surface [12, 24], whereas the dimensions of the inspected structures are larger than the metal layer thickness. However, the SEM contrast of nonconductive specimens varies widely depending on the coating metal and thickness, which influences the measurement's accuracy on the scanned structures [25]. For TEM analysis, if the charge dissipation is insuffi‐ cient, as is the case for insulating materials, the sample becomes unstable under the beam and

Since chemical composition is not necessarily totally controlled during the nanowire fabrica‐ tion, either by electrodeposition or other alumina template-filling methods, probing their chemical composition is essential. This is especially important for alloyed or multi-element nanowires, as well as for multilayered ones. In the last case, a direct visualization of the chemical composition configuration is highly helpful. This is also true for local investigation,

Therefore, it is highly advantageous to use some features of the electron microscopes described above to simultaneously perform a composition characterization of the nanowires. Mainly, both SEM and TEM possess a high-energy electron beam, which allows high sensibility, while their scanning possibilities permit to map the chemical composition, usually superimposed to their morphology image. The typical techniques used for nanowires are the energy-dispersive X-ray spectroscopy (EDS) and the electron energy loss spectroscopy (EELS). Generally speaking, while the first determines the presence or not of an element, the second can probe their chemical environment. Both techniques can lead to a mapping of local chemical properties with high spatial resolution when coupled to a TEM, whereas EDS can also be performed in a

23].

46 Magnetic Materials

the analysis becomes impossible.

such as for surface oxidation, for example.

**2.2. Chemical composition**

SEM chamber.

EDS measures the energy and intensity distribution of the X-rays generated by the impact of the high-energy electron beam on the surface of the sample (**Figure 3a**). The principle is based on the inner-shell electrons that may be excited by the incident beam, thus leaving holes in the atom's electronic shells. When these holes are filled by higher energy electrons, the energy level difference creates a released X-ray. Since its energy depends on the electronic structure of the atom, the elemental composition within the probed area can be determined to a high degree of precision. More information about the EDS technique is available in [26].

**Figure 3.** (a) EDS spectrum of BiFeO3 nanowires made by sol–gel preparation into alumina template. Reprinted with permission from [10]. Copyright 2013 by Elsevier Ltd, (b) EDS composition mapping of Fe3Ga4 nanowires fabricated through the metallic-flux nanonucleation method. Reprinted with permission from [9]. Copyright 2016 by Nature Pub‐ lishing Group, and (c) EELS analysis for the composition mapping of Ni nanowire after dissolving the alumina tem‐ plate. Reprinted with permission from [28]. Copyright 2013 by Brazilian Microelectronics Society.

Specifically in nanowire characterization case, EDS is particularly useful to determine their chemical composition [27], their stoichiometry, and their impurity content. As mentioned before, since EDS is a local probe, it allows focusing on the nanowires, being either free or still in the alumina array (viewed in cross section). On counterpart, EDS data from nanowires require to be analyzed keeping in mind the low material thickness. Especially for light elements, the incident electron beam may not sufficiently interact with the atoms while passing through the nanowire diameter, thus preventing the emission of the characteristic X-ray spectrum of the material. In these cases, elemental percentage cannot be determined with precision, and EDS results are limited to probe the presence or not of the elements (in concen‐ tration above the detection limit). Care should also be taken about the EDS mapping spatial resolution (**Figure 3b**). Since it depends on the size of the interaction volume, which in turn is controlled by the accelerating voltage and the mean atomic number of the sample, the spatial resolution is better while performing EDS mapping in a TEM than in a SEM, on the order of 0.5 and 10 nm, respectively.

#### *2.2.2. Electron energy loss spectroscopy (EELS)*

In order to overcome the EDS disadvantages and obtain more precise information about the material chemical composition, one can use EELS technique instead. It allows obtaining chemical information with a better energy resolution, passing from tens of eV for EDS to around 1 eV for EELS. Moreover, the EELS spatial resolution is generally higher than the corresponding EDS experiment because the EDS data are affected by beam broadening, unless the sample is very thin [29]. However, as a main drawback, EELS represents a more difficult technique to operate.

During an EELS measurement, some incident electrons of known energy will be scattered through inelastic interactions. The detection of the energy and momentum of the scattered electrons provides information on the excitations in the sample. As in EDS, inner-shell ionizations allow to determine the material atomic composition. However, a careful analysis of the spectrum also gives access to data about the chemical environment, such as the chemical bonding and the valence-/conduction-band electronic properties, among others [30, 31]. Since EELS is based on energy losses, the spectrum identification is facilitated for sharp and welldefined excitation edges, such as exhibited by low atomic number elements. The reader interested by more information about EELS as surface analysis technique is referenced to [32].

Therefore, in addition to determine the atomic composition for nanowires including light elements, EELS nanowire characterization is commonly used to investigate their oxidation state. In this case, the EELS- and TEM-coupled results can show the nanowire surface oxidation after being released from the alumina template (**Figure 3c**), which is critical for good electrical contact, as will be explained in Section 3.2. Another EELS capacity especially useful for nanowires is the ability to locally determine the valence state of a given element in the nanowire.

#### **2.3. Structural characterization**

Nanowires are by definition highly anisotropic, due to their elongated shape. However, in addition to the induced large axial shape anisotropy, one may need to also consider the magnetocrystalline contribution to the effective anisotropy. Therefore, the crystalline texture is especially important to resolve for nanowires made of materials with non-negligible magnetocrystalline anisotropy, like Co in hexagonal-close package (hcp). In this case, the uniaxial magnetocrystalline anisotropy constant being of the same order of magnitude than the shape anisotropy of an infinite cylinder, it makes the effective anisotropy very sensitive to the nanowire texture.

Crystalline structure being, by definition, a repetitive pattern of a unit cell of atoms, the characterization techniques are based on the diffraction principle. X-ray diffraction (XRD) is the most common one and can be applied to nanowires. However, due to their small volume, the measurements require to be performed on several nanowires together, preventing to resolve local modifications of the crystalline structure along the nanowires. Substituting the incident beam from X-rays to high-energy electrons, precise local crystalline structure can be probed. Therefore, TEM chamber, through techniques such as selected area electron diffraction (SAED) and high-resolution TEM, represents an ideal environment for nanowire crystalline structure investigation.

#### *2.3.1. X-ray diffraction (XRD)*

chemical information with a better energy resolution, passing from tens of eV for EDS to around 1 eV for EELS. Moreover, the EELS spatial resolution is generally higher than the corresponding EDS experiment because the EDS data are affected by beam broadening, unless the sample is very thin [29]. However, as a main drawback, EELS represents a more difficult

During an EELS measurement, some incident electrons of known energy will be scattered through inelastic interactions. The detection of the energy and momentum of the scattered electrons provides information on the excitations in the sample. As in EDS, inner-shell ionizations allow to determine the material atomic composition. However, a careful analysis of the spectrum also gives access to data about the chemical environment, such as the chemical bonding and the valence-/conduction-band electronic properties, among others [30, 31]. Since EELS is based on energy losses, the spectrum identification is facilitated for sharp and welldefined excitation edges, such as exhibited by low atomic number elements. The reader interested by more information about EELS as surface analysis technique is referenced to [32].

Therefore, in addition to determine the atomic composition for nanowires including light elements, EELS nanowire characterization is commonly used to investigate their oxidation state. In this case, the EELS- and TEM-coupled results can show the nanowire surface oxidation after being released from the alumina template (**Figure 3c**), which is critical for good electrical contact, as will be explained in Section 3.2. Another EELS capacity especially useful for nanowires is the ability to locally determine the valence state of a given element in the

Nanowires are by definition highly anisotropic, due to their elongated shape. However, in addition to the induced large axial shape anisotropy, one may need to also consider the magnetocrystalline contribution to the effective anisotropy. Therefore, the crystalline texture is especially important to resolve for nanowires made of materials with non-negligible magnetocrystalline anisotropy, like Co in hexagonal-close package (hcp). In this case, the uniaxial magnetocrystalline anisotropy constant being of the same order of magnitude than the shape anisotropy of an infinite cylinder, it makes the effective anisotropy very sensitive to

Crystalline structure being, by definition, a repetitive pattern of a unit cell of atoms, the characterization techniques are based on the diffraction principle. X-ray diffraction (XRD) is the most common one and can be applied to nanowires. However, due to their small volume, the measurements require to be performed on several nanowires together, preventing to resolve local modifications of the crystalline structure along the nanowires. Substituting the incident beam from X-rays to high-energy electrons, precise local crystalline structure can be probed. Therefore, TEM chamber, through techniques such as selected area electron diffraction (SAED) and high-resolution TEM, represents an ideal environment for nanowire crystalline

technique to operate.

48 Magnetic Materials

nanowire.

**2.3. Structural characterization**

the nanowire texture.

structure investigation.

As mentioned above, monochromatic X-rays are used in XRD technique as incident beam that will be scattered through elastic interactions with the atom electrons. For some specific scattering angles that depend on the crystalline structure, a constructive interference is obtained and detected. Careful analysis of the diffraction pattern yields the identification of the crystal symmetry and unit cell, as well as allowing characterization of more subtle aspects, such as stress and disorder in the unit cell.

X-ray diffraction is commonly used to determine the basic composition, crystalline phase, and texture of electrodeposited nanowires. Due to geometry restrictions, the XRD experi‐ ment is usually performed in the *θ*–2*θ* mode with the scattering vector parallel to the nanowires inside the porous template, but it is also possible to remove the nanowires from the template and dispose them on a surface (such as a silicon wafer or glass) to set the scattering vector perpendicular to the nanowires. In the first geometry, only the crystalline planes perpendicular to the scattering vector contribute to the diffractogram peaks (**Figure 4a**). This particular situation yields that, besides the usual element and phase determina‐ tion through their position, their relative intensity gives information about the crystalline texture, that is, the preferential grain orientation in the nanowires [33]. The XRD rocking curve technique, where the detector angle is fixed and the sample slightly tilted around a given angle, is a more powerful tool for the nanowire crystalline texture determination [34] (**Figure 4a** inset). However, due to the small volume of the electrodeposited nanowires and because they often do not completely fill the pore length, the diffractogram count can be very low and additional diffraction peaks arising from the template/substrate complicate the data analysis. Therefore, synchrotron radiation is preferred, since it leads to a higher signal coming from the nanowires.

**Figure 4.** (a) XRD pattern of hcp/fcc bi-crystalline Co nanowires. Inset: rocking curves for two different hcp peaks. Re‐ printed with permission from [34]. Copyright 2015 by AIP Publishing LLC. (b–d) Structural characterization through electron diffraction of a polycrystalline Ni nanowire. Reprinted with permission from [21]. Copyright 2015 by IEEE, (b) SAED diffraction pattern, (c) dark-field TEM images, showing size distribution of planes under the same diffraction condition, and (d) HRTEM showing the crystallographic planes.

## *2.3.2. Electron diffraction*

Another way to improve the nanowire diffraction signal is to again take advantage of the incident high-energy electron beam present in a TEM. The diffraction principle remains similar as in XRD, with the difference that it is electrons that are elastically scattered, allowing a higher resolution. Also, the diffraction is probed in transmission, instead as in reflection, yielding a pattern of bright spots indicating the constructive interference conditions. Several probing techniques are available in a TEM chamber, but due to their morphology (thin but long) and nanoscale dimensions, nanowires represent an ideal system for those.

One can perform a selected area electron diffraction (SAED) in a region as small as of a few hundreds of nanometers, which can be a specific region along a nanowire. Indexing the obtained diffraction pattern is a powerful tool to study its crystalline structure (**Figure 4b**). The very short electron wavelength (of the pm order, but with relatively low energy when compared with X-rays) gives access to a precise description of the atom's position. Addition‐ ally, one can also select a diffraction spot of SAED and enhance the contrast of the volume that contributes to that spot (**Figure 4c**). This imaging technique, called dark-field TEM, is useful to investigate planar defects, stacking faults, and grain size along individual nanowires, quickly allowing identifying repetitive crystalline pattern. Additionally, direct atomic observation is possible through a high-resolution TEM (HRTEM), to image the crystalline planes (**Figure 4d**). Therefore, by directly measuring the interplanar distances, one can determine the phase and orientation of the grains, and also estimate their size. For a brief review of SAED and HRTEM applied to nanowires, one can refer to [35].

Finally, it is noteworthy to mention that specific heat, a technique which will be explained in the next section, can also be used to indirectly investigate nanowire crystalline structure [9]. The measurement does not require removing the nanowires from the template, which can represent a great advantage. On the counterpart, one requires a bulk sample of the same material, in a crystalline structure thought as similar as in the nanowires. Specific heat measurement yields the system Debye temperature, which is related to the phonon spectrum. By comparing the Debye temperatures, it is possible to see if both phonon spectra, and therefore crystalline structures, are similar or not.

#### **2.4. Phase transition**

In certain systems, it is advantageous to explore their physical properties by means of specific heat measurements. The specific heat of a material is one of the most important thermodynamic properties denoting its heat retention or loss of capacity. Therefore, its variation with temper‐ ature or magnetic field may indicate crystalline and/or magnetic phase transitions [36]. The specific heat acquisition can be performed in a small-mass calorimeter. In one of the various measurement protocols, it is placed into the sample chamber which controls the heat added to/removed from a sample while monitoring the resulting change in temperature. During a measurement, a given quantity of heat is first applied at constant power for a fixed time, before allowing the sample to cool down during an equivalent time. Unlike other techniques that monitor magnetic transition phase, such as magnetometry, specific heat measurements can provide important additional information on the electronic structure and crystal lattice. However, care to the extent and in the analysis of the results need to be made more cautiously.

Like bulk systems, nanowires can undergo phase transitions. Interestingly, the transition temperature and nature may be modified for nanowires, as consequences of their lower di‐ mensionality [9, 37] (**Figure 5**). However, performing specific heat measurements on nano‐ wires represents a challenging task, due to the requirements to remove the contributions to the signal arising from everything else than the nanowires with precision. For nanowire ar‐ rays, this means to also measure, in the same conditions, a similar (in mass and geometry) empty alumina template. Accuracy in results depends on the correct mass determination of each component, which needs to be estimated for the nanowires.

**Figure 5.** Specific heat divided by temperature as a function of temperature of GdIn3 nanowires (black curve) and its bulk correspondent (red curve), fabricated through the metallic-flux nanonucleation method. The sharp peaks show an antiferromagnetic transition, zoomed in the respective inset. Reprinted with permission from [37]. Copyright 2014 by Elsevier Ltd.

#### **3. Electrical characterization**

*2.3.2. Electron diffraction*

50 Magnetic Materials

Another way to improve the nanowire diffraction signal is to again take advantage of the incident high-energy electron beam present in a TEM. The diffraction principle remains similar as in XRD, with the difference that it is electrons that are elastically scattered, allowing a higher resolution. Also, the diffraction is probed in transmission, instead as in reflection, yielding a pattern of bright spots indicating the constructive interference conditions. Several probing techniques are available in a TEM chamber, but due to their morphology (thin but long) and

One can perform a selected area electron diffraction (SAED) in a region as small as of a few hundreds of nanometers, which can be a specific region along a nanowire. Indexing the obtained diffraction pattern is a powerful tool to study its crystalline structure (**Figure 4b**). The very short electron wavelength (of the pm order, but with relatively low energy when compared with X-rays) gives access to a precise description of the atom's position. Addition‐ ally, one can also select a diffraction spot of SAED and enhance the contrast of the volume that contributes to that spot (**Figure 4c**). This imaging technique, called dark-field TEM, is useful to investigate planar defects, stacking faults, and grain size along individual nanowires, quickly allowing identifying repetitive crystalline pattern. Additionally, direct atomic observation is possible through a high-resolution TEM (HRTEM), to image the crystalline planes (**Figure 4d**). Therefore, by directly measuring the interplanar distances, one can determine the phase and orientation of the grains, and also estimate their size. For a brief

Finally, it is noteworthy to mention that specific heat, a technique which will be explained in the next section, can also be used to indirectly investigate nanowire crystalline structure [9]. The measurement does not require removing the nanowires from the template, which can represent a great advantage. On the counterpart, one requires a bulk sample of the same material, in a crystalline structure thought as similar as in the nanowires. Specific heat measurement yields the system Debye temperature, which is related to the phonon spectrum. By comparing the Debye temperatures, it is possible to see if both phonon spectra, and

In certain systems, it is advantageous to explore their physical properties by means of specific heat measurements. The specific heat of a material is one of the most important thermodynamic properties denoting its heat retention or loss of capacity. Therefore, its variation with temper‐ ature or magnetic field may indicate crystalline and/or magnetic phase transitions [36]. The specific heat acquisition can be performed in a small-mass calorimeter. In one of the various measurement protocols, it is placed into the sample chamber which controls the heat added to/removed from a sample while monitoring the resulting change in temperature. During a measurement, a given quantity of heat is first applied at constant power for a fixed time, before allowing the sample to cool down during an equivalent time. Unlike other techniques that monitor magnetic transition phase, such as magnetometry, specific heat measurements can

nanoscale dimensions, nanowires represent an ideal system for those.

review of SAED and HRTEM applied to nanowires, one can refer to [35].

therefore crystalline structures, are similar or not.

**2.4. Phase transition**

The growing interest in magnetic nanowires is connected to the possibility of employing them for advanced applications in wide technological fields. For example, nanowires represent ideal candidates for sensor devices, since they present high sensitivity to their environment [38, 39]. Moreover, they are presently intensively investigated for a large range of spintronic devices, due to the nanowire dimensions comparable or smaller than scaling lengths in magnetism and spin-polarized transport. Transport and magnetic properties, such as anisotropic magnetoresistance, field-induced magnetization reversal in single nanowires, domain-wall magnetoresistance, quantized spin transport in nanoconstrictions, among others [40–43], represent essential characterization for nanowires intended to play a major role in tomorrow's high technology.

In this chapter, we focus on direct current (DC) electrical characterization. The reader inter‐ ested in radio-frequency (RF) nanowire measurements is referred to [6]. The main difficulty to perform electrical measurements on nanowires is to succeed in obtaining a good electrical contact. In this sense, several techniques have been implemented [38–40]. Here, we will restrain ourselves to the description of electrical characterization performed on single or few cylindrical nanowires. First, we describe a technique where the connected nanowires remain inside the nanowire array, thereby feeling the interaction field from their neighbors. In the second part, a method to measure free single nanowires is presented.

#### **3.1. Nanowires embedded in array**

Electrical characterization of nanowire arrays still embedded in porous alumina is an inter‐ esting technique because of their spatial ordering. In this case, the idea is to electrically connect nanowires both extremity, which is facilitated by the electrode already present at the nano‐ wires' bottom, used for their electrodeposition. Even if the barrier layer thinning method can be used to obtain an electrical contact [44], removing completely the barrier layer and closing the pores with a thick conductive film (usually Au) is preferred to lower the resistance. The top contact may be fabricated by filling the remaining pores length by a metallic material, like Cu [41, 43], or by mechanically polishing the top of the template before depositing an electrode by sputtering [40, 42].

Using lithography techniques, one can limit the template region electrically connected, measuring current transport over thousands of nanowires at the same time. The main advant‐ age of this technique is to improve signal-to-noise ratio. In counterpart, the electrical charac‐ teristic in response to an electrical excitation is averaged [41], which prevents spin-transfer experiments.

**Figure 6.** (a) Schematic illustration of the single nanowire contacting process on an array of nanowires electrodeposited in supported nanoporous alumina template and (b) magnetoresistance curve for Py/Cu/Py spin-valve nanowire with 80 nm of diameter and external magnetic field in-plane with the membrane. Reprinted with permission from [40]. Copyright 2007 by American Chemical Society.

However, one can also define electrical nanocontacts on a single electrodeposited nanowire inside the porous membrane. This can be done by the indentation of an ultrathin-insulating photoresist layer deposited on the top face of the thinned alumina template after the electro‐ deposition. A modified atomic force microscope designed for local resistance measurement is used as a nanoindenter, allowing an easy access point onto individual nanowires at the surface of the template [40, 42], as shown in **Figure 6a**. As a consequence, magnetic properties, such as magnetoresistance signal, can be extracted from one individual nanowire (**Figure 6b**).

Despite the presence of the interaction field on the measured nanowire, this technique presents several advantages. First, the template size and shape yield a convenient sample-handling method [41]. Second, since the probed nanowire remains inside the alumina template, it does not suffer surface oxidation due to the contact with alumina-etching solution and ambient atmosphere. The major drawback of this oxidation is that it increases the nanowire resistance. As a consequence, it makes spin-transfer experiments more difficult, leading to a small giant magnetoresistance signal. Moreover, it favors the apparition of heating problem due to Joule effect, limiting the current density that one can inject in the nanowire before melting it.

#### **3.2. Free nanowires**

[40–43], represent essential characterization for nanowires intended to play a major role in

In this chapter, we focus on direct current (DC) electrical characterization. The reader inter‐ ested in radio-frequency (RF) nanowire measurements is referred to [6]. The main difficulty to perform electrical measurements on nanowires is to succeed in obtaining a good electrical contact. In this sense, several techniques have been implemented [38–40]. Here, we will restrain ourselves to the description of electrical characterization performed on single or few cylindrical nanowires. First, we describe a technique where the connected nanowires remain inside the nanowire array, thereby feeling the interaction field from their neighbors. In the second part,

Electrical characterization of nanowire arrays still embedded in porous alumina is an inter‐ esting technique because of their spatial ordering. In this case, the idea is to electrically connect nanowires both extremity, which is facilitated by the electrode already present at the nano‐ wires' bottom, used for their electrodeposition. Even if the barrier layer thinning method can be used to obtain an electrical contact [44], removing completely the barrier layer and closing the pores with a thick conductive film (usually Au) is preferred to lower the resistance. The top contact may be fabricated by filling the remaining pores length by a metallic material, like Cu [41, 43], or by mechanically polishing the top of the template before depositing an electrode

Using lithography techniques, one can limit the template region electrically connected, measuring current transport over thousands of nanowires at the same time. The main advant‐ age of this technique is to improve signal-to-noise ratio. In counterpart, the electrical charac‐ teristic in response to an electrical excitation is averaged [41], which prevents spin-transfer

**Figure 6.** (a) Schematic illustration of the single nanowire contacting process on an array of nanowires electrodeposited in supported nanoporous alumina template and (b) magnetoresistance curve for Py/Cu/Py spin-valve nanowire with 80 nm of diameter and external magnetic field in-plane with the membrane. Reprinted with permission from [40].

tomorrow's high technology.

52 Magnetic Materials

**3.1. Nanowires embedded in array**

Copyright 2007 by American Chemical Society.

by sputtering [40, 42].

experiments.

a method to measure free single nanowires is presented.

To avoid magnetic interactions between closer nanowires inside the porous membrane, which can influence the electrical and magnetic measurements of a single nanowire, one can release them from the membrane using a convenient etching solution. Two options exist to establish an electrical contact on a single nanowire, both involving the design of proper electrodes by optical or e-beam lithography. The first one consists of patterning the electrodes after dispers‐ ing the nanowires on a substrate [45]. While this method does not require nanowire manipu‐ lation, the critical step is the careful electrode alignment with the lying nanowire. Here, we will focus on the opposite technique, where the electrodes are first patterned and the free nanowires are placed afterwards between them. Depending on the nanowire manipulation technique, this method can reach large outflow.

For metallic nanowires, dielectrophoresis (DEP) technique allows to adequately position the nanowires to electrically connect them with the electrodes to further transport measurements. After chemically etching the alumina membrane, the nanowires must then be dispersed in dimethylformamide (DMF), a dielectric medium, in order to avoid nanowire cluster formation [21, 38, 41, 43]. Then, the metallic nanowires suspended in the DMF can be directly manipulated through alternating electric fields produced by a pair of electrodes separated by a gap region, as reported in [38] (**Figure 7a**). For optimized DEP parameters, one can trap one single nanowire to carry out electrical/magnetic measurements [38, 39, 41, 43] (**Figure 7b**).

As shown, DEP is an adequate tool to insert nanowire between electrodes for electrical transport measurements [38]. As already mentioned above, compared to the in-template electrical measurement, this technique removes the effects from the dipolar field from neighboring nanowires. Moreover, it permits to use a four-point resistance measurement method, since the electrode geometry is more versatile. Therefore, it makes possible to pattern a series of electrodes on a single nanowire in order to follow a domain-wall propagation. However, when the nanowire touches the electrodes, a large contact resistance is sometimes present due to native oxide or organic residues, leading to a Schottky-like behavior (nonlinear). Several options are available to reduce the contact resistance, such as depositing a thin metallic layer on the nanowire extremities by focused ion beam-induced deposition (FIBID) (**Figure 7c**) [38]. Another option consists of improving the contact resistance by passing a low current into the nanowire, allowing subsequent temperature-dependent electrical resistivity and magnetoresistance measurements (**Figure 7d**) [28].

**Figure 7.** Electrical characterization of free Ni nanowires manipulated through DEP technique, as presented in [38]. (a) Schematic illustration of the sample preparation, (b) SEM image of one single nanowire deposited on electrodes, (c) current versus voltage curves before (black, left, and down axes) and after (red, right, and up axes) 10-nm-thick Pt lay‐ er deposition by FIBID (a and c reprinted with permission from [38]. Copyright 2015 by American Vaccum Society), and (d) resistivity evolution with temperature of one single nanowire, showing metallic behavior. Inset: magnetoresist‐ ance measurement with the current flowing perpendicular to the applied field at 300 K. Reprinted with permission from [28]. Copyright 2013 by Brazilian Microelectronics Society.

Whatever the techniques used to connect a single nanowire, its electrical characterization is subjected to a special care about the current that can flow through it. Above a certain limit, it begins to damage the nanowire due to heat dissipation. Therefore, in addition to control the current during the electrical measurements, one needs to also prevent electrical discharge when preparing the sample.

## **4. Magnetic characterization**

Obviously, the investigation of magnetic nanowires cannot be complete without probing their magnetic properties. Magnetometry, which is the magnetization measurement, yields the basic magnetic behavior, typically through the acquisition of a major hysteresis curve. For an array, it can be performed on a vibrating sample magnetometer (VSM) or a superconducting quantum interference device (SQUID), while micro-SQUID and magneto-optical Kerr effect (MOKE) are suitable for single nanowire. All these magnetization measurement techniques can also be used to perform specific field routine, such as first-order reversal curves (FORCs), which give the statistical distribution of hysteresis operators, or the angular dependence, to probe the magnetization-reversal processes. On the other hand, useful magnetic characteri‐ zation is not limited to the magnetization value acquisition. Nanowire array magnetization reversal can be probed by means of magnetic force microscopy (MFM), while imaging the domain walls and/or magnetic structure along a single nanowire is enabled by MFM and electron holography, for example. For its part, ferromagnetic resonance (FMR) is an adequate tool to probe the material anisotropy.

## **4.1. Magnetometry techniques**

present due to native oxide or organic residues, leading to a Schottky-like behavior (nonlinear). Several options are available to reduce the contact resistance, such as depositing a thin metallic layer on the nanowire extremities by focused ion beam-induced deposition (FIBID) (**Figure 7c**) [38]. Another option consists of improving the contact resistance by passing a low current into the nanowire, allowing subsequent temperature-dependent electrical resistivity and

**Figure 7.** Electrical characterization of free Ni nanowires manipulated through DEP technique, as presented in [38]. (a) Schematic illustration of the sample preparation, (b) SEM image of one single nanowire deposited on electrodes, (c) current versus voltage curves before (black, left, and down axes) and after (red, right, and up axes) 10-nm-thick Pt lay‐ er deposition by FIBID (a and c reprinted with permission from [38]. Copyright 2015 by American Vaccum Society), and (d) resistivity evolution with temperature of one single nanowire, showing metallic behavior. Inset: magnetoresist‐ ance measurement with the current flowing perpendicular to the applied field at 300 K. Reprinted with permission

Whatever the techniques used to connect a single nanowire, its electrical characterization is subjected to a special care about the current that can flow through it. Above a certain limit, it begins to damage the nanowire due to heat dissipation. Therefore, in addition to control the current during the electrical measurements, one needs to also prevent electrical discharge

Obviously, the investigation of magnetic nanowires cannot be complete without probing their magnetic properties. Magnetometry, which is the magnetization measurement, yields the basic

magnetoresistance measurements (**Figure 7d**) [28].

54 Magnetic Materials

from [28]. Copyright 2013 by Brazilian Microelectronics Society.

when preparing the sample.

**4. Magnetic characterization**

While typical nanowire arrays exhibit a magnetic signal relatively easy to measure with conventional equipment, due to the high nanowire density, single nanowires require more sensible technique. Furthermore, the sample manipulation does not present specific challenge for measuring an array, apart from the magnetic field orientation with respect to the nanowire axis. However, the first problem to overcome before measuring single nanowire magnetization is their proper positioning. These differences naturally divide the magnetometry techniques between the array and single nanowire-oriented ones.

#### *4.1.1. Array magnetometry*

Most of conventional magnetometers coupled with magnets able to apply field of at least 1 T are sufficient to measure the magnetization arising from a nanowire array. Due to the high density of nanowires, its magnetic signal typically yields several memu, which is higher than the sensibility of standard magnetometer for bulk systems. The two main equipment used for nanowire magnetic characterization are the vibrating sample magnetometer (VSM) and the superconducting quantum interference device (SQUID) magnetometer. Both are based on Faraday's law principle, which states that a variation in the magnetic flux density passing through a conductive coil induces a current in this coil to compensate the variation. The sample is therefore magnetized by an external magnetic field and its position with respect to the detecting coils is changed, thus inducing a measured voltage that is subsequently converted in magnetic moment.

The VSM detection system [46] is simpler than the one in a SQUID. The sensing coils are usually copper coils located around the sample. A relatively good sensitivity is gained by mechanically vibrating the sample at a given frequency (typically less than 100 Hz) and using a lock-in amplifier to filter the signal induced. In counterpart, the SQUID benefits of a higher sensitivity. Here, the detector is a superconducting loop, which imposes a quantization of the magnetic flux. This loop contains two Josephson junctions that break the flowing current symmetry in the presence of a magnetic flux variation. For more information about SQUID operation, the reader is referred to the study of Ramasamy et al. [47].

Apart from the proper sample centering, nanowire array magnetization acquisition is rela‐ tively straightforward. The only experimental aspect to take care is the applied magnetic field direction relative to the nanowire axis, due to the anisotropy of the system (**Figure 8a**). Since the measurement is performed on a whole array, one needs to remember that it exhibits the array magnetic properties, which may differ from the individual nanowire ones (see Section 4.2.1). For example, the coercivity obtained represents when half of the array magnetic volume reversed its magnetization, while the susceptibility is related to the interaction field between the nanowires.

**Figure 8.** Typical normalized magnetization curves for Ni nanowires (130 nm of diameter, ≈ 20-μm long). (a) Nanowire array measured both parallel (black) and perpendicular (red) to the nanowire axis on a commercial VSM. It evidences the axial easy axis of the array. In blue, the anisotropy field distribution calculated according to [48] and (b) single nanowire measured axially on a MOKE setup. The squared hysteresis loop denotes a clear axial easy axis. Reprinted with permission from [49]. Copyright 2012 by IOP Publishing Ltd.

#### *4.1.2. Single nanowire magnetometry*

To overcome this discrepancy, the direct magnetic measurement of one nanowire should deliver the individual properties. However, in addition to the difficulties created by the necessity to adequately position the individual nanowire with respect to the sensing element, the low magnetic signal arising from a single nanowire represents a complex challenge. Two different solutions have essentially been implemented.

The first one consists of fabricating a micro-SQUID detector around a free nanowire lying on a substrate [50]. The nanowire proximity with the superconducting loop, coupled to its highdetection sensibility, allows performing magnetization measurement. However, this techni‐ que has several drawbacks that explain why it is not largely widespread. Mainly, the micro-SQUID detector fabrication, made by complex lithography, is restrained to a unique nanowire, thus severely limiting the number of single nanowires that can be characterized in a reasonable period of time.

On the other side, the magneto-optical Kerr effect (MOKE) is normally used as a superficial magnetic characterization technique due to its small penetration length. Nevertheless, its penetration length of the order of the tens of nanometer is enough to probe the magnetic signal from single nanowires, while actually removing unwanted background signal, as from the substrate. Compared to the micro-SQUID, the MOKE technique is advantaged by the facts that it can be performed on a conventional MOKE setup and allows to quickly measure several nanowires, making distribution properties possible to evaluate. It consists of shining an incident polarized light on the magnetic sample and measuring the rotation of the reflected light [51]. This rotation depends on the relative orientation of the incident polarization with the sample magnetization. Therefore, there exist three different configurations (longitudinal, transversal, and polar) that allow detecting magnetization in the parallel, perpendicular, and out-of-plane directions, respectively.

In order to succeed in measuring the magnetic properties of isolated nanowires by MOKE, it is important to carefully perform the measurement. Good focus of the laser spot on the sample and cautious alignment of the lens system are primordial to avoid signal loss. Additionally, the low magnetic signal requires that the hysteresis loops must be averaged several times (up to thousands) to improve the signal-to-noise ratio. As already stated, the magnetic character‐ ization of single nanowires yields fundamental properties such as the individual switching field and squareness of nanowires [49] (**Figure 8b**). This information is essential to study the magnetization-reversal mechanisms in nanowires.

#### **4.2. Magnetometry routines**

Apart from the proper sample centering, nanowire array magnetization acquisition is rela‐ tively straightforward. The only experimental aspect to take care is the applied magnetic field direction relative to the nanowire axis, due to the anisotropy of the system (**Figure 8a**). Since the measurement is performed on a whole array, one needs to remember that it exhibits the array magnetic properties, which may differ from the individual nanowire ones (see Section 4.2.1). For example, the coercivity obtained represents when half of the array magnetic volume reversed its magnetization, while the susceptibility is related to the interaction field between

**Figure 8.** Typical normalized magnetization curves for Ni nanowires (130 nm of diameter, ≈ 20-μm long). (a) Nanowire array measured both parallel (black) and perpendicular (red) to the nanowire axis on a commercial VSM. It evidences the axial easy axis of the array. In blue, the anisotropy field distribution calculated according to [48] and (b) single nanowire measured axially on a MOKE setup. The squared hysteresis loop denotes a clear axial easy axis. Reprinted

To overcome this discrepancy, the direct magnetic measurement of one nanowire should deliver the individual properties. However, in addition to the difficulties created by the necessity to adequately position the individual nanowire with respect to the sensing element, the low magnetic signal arising from a single nanowire represents a complex challenge. Two

The first one consists of fabricating a micro-SQUID detector around a free nanowire lying on a substrate [50]. The nanowire proximity with the superconducting loop, coupled to its highdetection sensibility, allows performing magnetization measurement. However, this techni‐ que has several drawbacks that explain why it is not largely widespread. Mainly, the micro-SQUID detector fabrication, made by complex lithography, is restrained to a unique nanowire, thus severely limiting the number of single nanowires that can be characterized in a reasonable

with permission from [49]. Copyright 2012 by IOP Publishing Ltd.

different solutions have essentially been implemented.

*4.1.2. Single nanowire magnetometry*

period of time.

the nanowires.

56 Magnetic Materials

Even if major magnetization hysteresis curves, from an array or a single nanowire, give a valuable characterization, the knowledge directly extracted from them remains limited: saturation magnetization, susceptibility, coercivity, squareness, and so on. However, different magnetic behaviors can be easily probed, still using the same setups as for the major curve acquisition. The idea consists of modifying the measurement routine in order to select a specific kind of behavior to follow and investigate. For example, instead of sweeping the magnetic field between saturation values, but performing reversal curves, which begin in the hysteresis area, one is able to get access to the switching of individual nanowires while measuring the whole array. This technique, called first-order reversal curve (FORC), is very powerful when applied to nanowire systems. Another possibility is to change the angle during the measure‐ ment, since nanowire magnetic behavior depends on their anisotropy. This technique repre‐ sents an effective way to achieve to experimentally study the magnetization-reversal mechanisms occurring in the nanowires. Both procedures are easily implemented in any magnetometer and can be applied to both nanowire array and single nanowire. Therefore, they should be considered as part as basic nanowire characterization, along with major hysteresis curves, due to the richness of information obtained.

#### *4.2.1. First-order reversal curve (FORC) technique*

Major hysteresis curves yield the magnetic properties of whole nanowire arrays. However, the magnetization of the array can be viewed as the average of the nanowire magnetization. The objective of the FORC technique is to extract the individual magnetic entities characteristics while performing a measurement on the whole system. A complete description of the imple‐ mentation of the FORC method for nanowire arrays is available in [11]. It is based on the classical Preisach model, where magnetic entities are modeled as squared hysteresis operators called mathematical hysterons. In order to obtain the statistical distribution of these operators, only described by their width and field shift, Mayergoyz developed a specific field-sweeping routine [52]. It consists of minor curves beginning at a reversal field (inside the hysteretic area) and returning to the saturation (**Figure 9a**). The FORC result is calculated from these data through a second-order mixed derivative and represented as a contour plot (**Figure 9b**). If the measured system meets the required conditions for the classical Preisach model, it can be considered as the hysterons statistical distribution.

**Figure 9.** Different magnetization measurement procedure results applied to an Ni nanowire array (35-nm diameter, 3.5-μm long). (a) First-order reversal curves measurement along the nanowire axis, (b) respective FORC result. Color scale ranging from blue (null value) to red (maximum value), and (c) angular evolution of the array coercivity along with the analytic fit for a transverse domain-wall nucleation. The effective anisotropy constant was evaluated at 3.612 × 105 erg/cm3 .

Several nanowires, due to their large shape anisotropy, exhibit a squared hysteresis curves when measured axially. Therefore, the hysterons could be associated with the individual nanowire magnetic behavior, leading to a clever way to get access to the single nanowire properties without requiring to remove them from the template. This explains why the FORC technique is highly suitable to investigate nanowires. However, like most of real systems, nanowire arrays do not meet the criteria for the classical Preisach model, due to their interac‐ tion field. The FORC result analysis thus necessitates to be performed carefully. The physical analysis model, based on simulated behavior of physically meaningful hysterons, has been developed in this sense [53]. It allowed obtaining a quantitative parametrization of the FORC result in order to extract the mean and distribution width coercivity and the interaction field at saturation, among other values [54]. Even if the method is based on hysteretic operators, information about the system reversibility can also be extracted [55].

Typical axial FORC result for nanowire appears as a distribution narrow along the coercivity, while elongated along the interaction field axis (**Figure 9b**). It may be interpreted as nanowires with similar coercivity, and therefore geometrical dimensions, submitted to a large and almost uniform interaction field dependent on the magnetization. In general, a ridge may appear along the coercivity axis and is attributed to the nonuniformity of this interaction field, the measurement being carried on a finite array [56]. In addition to the homogeneous nanowires, the FORC method remains suitable for more complex systems, such as multilayer nanowires, when coupled to the physical analysis model for the result analysis.

#### *4.2.2. Angular-dependent magnetization curves*

objective of the FORC technique is to extract the individual magnetic entities characteristics while performing a measurement on the whole system. A complete description of the imple‐ mentation of the FORC method for nanowire arrays is available in [11]. It is based on the classical Preisach model, where magnetic entities are modeled as squared hysteresis operators called mathematical hysterons. In order to obtain the statistical distribution of these operators, only described by their width and field shift, Mayergoyz developed a specific field-sweeping routine [52]. It consists of minor curves beginning at a reversal field (inside the hysteretic area) and returning to the saturation (**Figure 9a**). The FORC result is calculated from these data through a second-order mixed derivative and represented as a contour plot (**Figure 9b**). If the measured system meets the required conditions for the classical Preisach model, it can be

**Figure 9.** Different magnetization measurement procedure results applied to an Ni nanowire array (35-nm diameter, 3.5-μm long). (a) First-order reversal curves measurement along the nanowire axis, (b) respective FORC result. Color scale ranging from blue (null value) to red (maximum value), and (c) angular evolution of the array coercivity along with the analytic fit for a transverse domain-wall nucleation. The effective anisotropy constant was evaluated at 3.612 ×

Several nanowires, due to their large shape anisotropy, exhibit a squared hysteresis curves when measured axially. Therefore, the hysterons could be associated with the individual nanowire magnetic behavior, leading to a clever way to get access to the single nanowire properties without requiring to remove them from the template. This explains why the FORC technique is highly suitable to investigate nanowires. However, like most of real systems, nanowire arrays do not meet the criteria for the classical Preisach model, due to their interac‐ tion field. The FORC result analysis thus necessitates to be performed carefully. The physical analysis model, based on simulated behavior of physically meaningful hysterons, has been developed in this sense [53]. It allowed obtaining a quantitative parametrization of the FORC result in order to extract the mean and distribution width coercivity and the interaction field at saturation, among other values [54]. Even if the method is based on hysteretic operators,

Typical axial FORC result for nanowire appears as a distribution narrow along the coercivity, while elongated along the interaction field axis (**Figure 9b**). It may be interpreted as nanowires with similar coercivity, and therefore geometrical dimensions, submitted to a large and almost uniform interaction field dependent on the magnetization. In general, a ridge may appear

information about the system reversibility can also be extracted [55].

considered as the hysterons statistical distribution.

105 erg/cm3 .

58 Magnetic Materials

From a general point of view, all anisotropic system behavior is highly dependent on the applied magnetic field direction. From the other side, varying the field angle allows obtaining data from which additional information may be extracted. On most of magnetometers working with an electromagnet, which produces a horizontal applied field, modifying the measurement angle is easy and generally implemented in the control software. It is more challenging for superconducting coil, since the magnetic field is vertical in this case. Manual angle variation is usually required, thus increasing the total measurement time.

For nanowires, the classical case is to perform hysteresis curves along the axial and transverse directions, to determine the easy axis direction (e.g., see **Figure 8a**). In the last years, angledependent magnetization curves turned out to be richer in information, after it proved its ability to be interpreted as a signature of the magnetization-reversal processes. Nanowires with diameter up to few hundreds of nanometers usually reverse their magnetization through the domain-wall nucleation/propagation mechanism. Depending on their properties, the domain wall nucleated may be either transverse or vortex. Assuming that a transverse domainwall nucleation can be modeled as the coherent rotation of a volume equivalent to the domain wall [57], the angular nanowire coercivity can be calculated in this case [58] (**Figure 9c**). Deviations from this model can be interpreted as the occurrence of other mechanisms (curling, fanning, coherent rotation, etc.) or domain-wall types (mainly vortex), while fitting yields quantitative values of the anisotropy constants. A special care with the precision of the field direction angle should be taken during the measurement, but can be easily overcome while performing the analytical fit.

#### **4.3. Magnetization imaging**

For nanowires, the interest in obtaining a direct or an indirect visualization of their magnetic structure depends on the resolution and area observed. High-resolution imaging allows probing a particular region of the nanowire and observes details of the magnetic domains and domain walls, which is probably the best way to understand and to control them. Since the domain-wall dynamics is very sensitive to their morphology [59, 60], it is fundamental to know their geometry for all applications involving domain-wall propagation. The domain pattern of a whole nanowire, for its part, gives valuable information about its magnetization-reversal mechanism. Finally, investigations of the nanowire array complex magnetization reversal and interaction field are helped by large-scale imaging of several nanowires, usually performed on the template top surface.

The two last cases are well resolved by using the magnetic force microscopy (MFM) technique, where a magnetized atomic force microscopy tip maps the stray field. For further local magnetic information, several methods have been developed. They are usually more compli‐ cated to operate than MFM and not always suitable for cylindrical nanowires, due to their geometry. Here, we focus on a powerful characterization that can be performed in a TEM chamber and that can yield highly valuable local information: electron holography.

The main problem concerning magnetization imaging of nanowires and nanowire arrays is that both systems are tridimensional, instead of planar. Therefore, one needs to keep in mind that the observation made is not complete and that some additional features of the internal magnetization structure may be hidden from the observer. By example, assumptions about the nanowire magnetic structure are required when imaging nanowire array top extremity. Also, the nanowires' cylindrical shape makes the domain-wall internal structure very difficult to be seen, like the core of a vortex domain wall, since it lays parallel to the nanowire axis and near its center.

#### *4.3.1. Magnetic force microscopy (MFM)*

Among several techniques, the MFM is probably one of the most used magnetic imaging tools. Here, we give a brief description of the technique, but a most complete, although compact description, can be found in [61]. As mentioned above, the MFM is an atomic force microscope

**Figure 10.** (a and b) MFM images in the remanent state. The clear and dark spots denoted outgoing and ingoing stray field, respectively. (a) Top surface of Ni nanowire arrays. The clear and dark circles represent the nanowires magne‐ tized upward and downward. Reprinted with permission from [65]. Copyright 2004 by EDP Sciences, (b) individual 35-nm Co nanowire after saturation in a field parallel (top) and perpendicular (bottom) to the nanowire axis. A sketch of the possible domain pattern is underneath. Reprinted with permission from [64]. Copyright 2000 by American Phys‐ ical Society, and (c and d) electron holography results of a multilayer Cu/CoFeB nanowire (80/230 nm) for an axialapplied magnetic field. Reprinted with permission from [66]. Copyright 2014 by AIP Publishing LLC. (c) Hologram. (d) Associated map of perpendicular magnetic field, with 0.1-T contour spacing.

with a magnetic tip, usually a thin (<50 nm) coated film with high coercivity of Ni, Co, or CoCr, among others, for the tip magnetization to be fixed during the scan. The tip lies on one end of a cantilever that raster scans the sample. The scanned area can be as large as 200 μm2 and the typical resolution is 30 nm. In alternating current (AC) mode, which provides a better resolu‐ tion, the cantilever tip is put to oscillate near its resonance frequency by piezoelectric crystals, while the tip-sample interaction changes the amplitude, frequency, and phase of the oscillating cantilever. All these changes are measurable (by a laser reflected from the cantilever) and can be used to calculate the force exerted at the tip by the sample. To consider only the magneto‐ static interaction between the tip and the stray field generated by the sample, the lift-height technique is usually employed. In this mode, the tip is brought very close to the sample and a first scan is performed. The tip is then lifted and a second scan is done, thus eliminating the topographic contribution. A lift height of 50 nm is usually sufficient to image nanowires. Due to the complicated and not necessarily fixed magnetization tip and the unsolvable problem of the reconstruction of the magnetization sample by the sensed sample stray field, MFM is usually considered as a qualitative technique. Despite this, it is very useful, especially if combined with other techniques such as magnetometry, magnetoresistance measurements, and micromagnetic simulations.

Since the samples do not require any special preparations, only to lay on a plane surface, MFM is used to image both the top of nanowire arrays and free nanowires. Due to the large axial shape anisotropy, nanowires are usually taken to remain monodomain under axial-applied magnetic field. Therefore, investigations about individual nanowire-switching field and complex magnetostatic interaction between nanowires take advantages of the large template area that MFM can scan [62, 63] (**Figure 10a**). Actually, this monodomain state can be directly observed by performing MFM imaging of free nanowires. It allows visualizing the magnetic domains along the nanowire, and thus the presence of domain walls [64] (**Figure 10b**). However, in cylindrical nanowires, the domain-wall length is typically of a few nanometers, for Co, Ni, Fe, and alloys. Therefore, the MFM resolution does not normally provide much information about the domain-wall geometry, especially if it is a vortex or a helical, that have a small stray field compared with transverse domain wall.

#### *4.3.2. Electron holography*

magnetic information, several methods have been developed. They are usually more compli‐ cated to operate than MFM and not always suitable for cylindrical nanowires, due to their geometry. Here, we focus on a powerful characterization that can be performed in a TEM

The main problem concerning magnetization imaging of nanowires and nanowire arrays is that both systems are tridimensional, instead of planar. Therefore, one needs to keep in mind that the observation made is not complete and that some additional features of the internal magnetization structure may be hidden from the observer. By example, assumptions about the nanowire magnetic structure are required when imaging nanowire array top extremity. Also, the nanowires' cylindrical shape makes the domain-wall internal structure very difficult to be seen, like the core of a vortex domain wall, since it lays parallel to the nanowire axis and

Among several techniques, the MFM is probably one of the most used magnetic imaging tools. Here, we give a brief description of the technique, but a most complete, although compact description, can be found in [61]. As mentioned above, the MFM is an atomic force microscope

**Figure 10.** (a and b) MFM images in the remanent state. The clear and dark spots denoted outgoing and ingoing stray field, respectively. (a) Top surface of Ni nanowire arrays. The clear and dark circles represent the nanowires magne‐ tized upward and downward. Reprinted with permission from [65]. Copyright 2004 by EDP Sciences, (b) individual 35-nm Co nanowire after saturation in a field parallel (top) and perpendicular (bottom) to the nanowire axis. A sketch of the possible domain pattern is underneath. Reprinted with permission from [64]. Copyright 2000 by American Phys‐ ical Society, and (c and d) electron holography results of a multilayer Cu/CoFeB nanowire (80/230 nm) for an axialapplied magnetic field. Reprinted with permission from [66]. Copyright 2014 by AIP Publishing LLC. (c) Hologram.

(d) Associated map of perpendicular magnetic field, with 0.1-T contour spacing.

chamber and that can yield highly valuable local information: electron holography.

near its center.

60 Magnetic Materials

*4.3.1. Magnetic force microscopy (MFM)*

Electron holography, performed in a TEM chamber, allows obtaining higher resolution imaging of the magnetic structure. The principle is to get access to the phase shift of the electron wave that traveled through the sample, not only to its amplitude, as in conventional TEM imaging techniques. By creating two paths for the electron beam, one passing through the sample while the other remaining undisturbed, the interference pattern creates an electron hologram, which depends on the phase shift. Therefore, the sample local magnetic properties influence the electron wave when passing through it and can be directly observed afterwards. More information about this powerful technique can be found in [67, 68], the second being specifically dedicated to magnetic material investigations.

Being performed using the high-energy electron beam from a TEM, typical electron hologra‐ phy resolution is of 5 nm and results are usually associated with micromagnetic simulations.

Due to the technique complexity, the best use of electron holography for nanowires is to probe the fine magnetic structure that is out of range for MFM equipment. More specifically, it allows to obtain a clear image of nanowire domain walls [69], as well as the detailed domain pattern created by multilayers [66] (**Figure 10c** and **d**).

#### **4.4. Anisotropy probing**

Finally, as was mentioned several times, nanowire magnetic behavior is intrinsically related to their anisotropy, both global and local variations. Even if magnetometry can be used to indirectly probe the system anisotropy properties, ferromagnetic resonance (FMR) is a powerful technique to obtain more accurate data. The FMR phenomenon is based on the resonance arising when the frequency of an applied transverse AC magnetic field is equal to the material Larmor frequency, which is the frequency of magnetization precession around the effective magnetic field. Since the anisotropy energy modifies the effective magnetic field, even the anisotropy constant distribution is accessible through FMR. A complete description of the phenomenon and the various ways to measure it is reviewed in [70].

**Figure 11.** (a) Resonance field evolution with the magnetic field angle for CoFeB, Ni, and Ni/Cu nanowires. Reprinted with permission from [74]. Copyright 2007 by AIP Publishing LLC and (b) illustration of the double-resonance phe‐ nomenon occurring in a CoFeB nanowire array (40-nm diameter, 200-μm long). Left: schematic of the microstrip line and major hysteresis curve branch used to measure FMR. Right: resulting contour plot. Reprinted with permission from [72]. Copyright 2009 by AIP Publishing LLC.

In the specific case of monodomain nanowires, both the anisotropy intensity and direction distributions are highly important since electrodeposited nanowires generally present homogeneities, which could greatly affect their magnetic behavior. On the other side, electro‐ magnetic wave propagation directly depends on the medium effective anisotropy, leading to a crucial information for all nanowire applications related with high-frequency devices (**Figure 11a**) [71]. Finally, the array magnetization reversal may lead to the interesting phenomenon of double resonance (**Figure 11b**) [72]. Several setups have been developed to measure FMR in nanowires and nanowire arrays. The reader interested to push further his knowledge in the area is referenced to the following book chapter [73].

## **5. Conclusion**

Due to the technique complexity, the best use of electron holography for nanowires is to probe the fine magnetic structure that is out of range for MFM equipment. More specifically, it allows to obtain a clear image of nanowire domain walls [69], as well as the detailed domain pattern

Finally, as was mentioned several times, nanowire magnetic behavior is intrinsically related to their anisotropy, both global and local variations. Even if magnetometry can be used to indirectly probe the system anisotropy properties, ferromagnetic resonance (FMR) is a powerful technique to obtain more accurate data. The FMR phenomenon is based on the resonance arising when the frequency of an applied transverse AC magnetic field is equal to the material Larmor frequency, which is the frequency of magnetization precession around the effective magnetic field. Since the anisotropy energy modifies the effective magnetic field, even the anisotropy constant distribution is accessible through FMR. A complete description

**Figure 11.** (a) Resonance field evolution with the magnetic field angle for CoFeB, Ni, and Ni/Cu nanowires. Reprinted with permission from [74]. Copyright 2007 by AIP Publishing LLC and (b) illustration of the double-resonance phe‐ nomenon occurring in a CoFeB nanowire array (40-nm diameter, 200-μm long). Left: schematic of the microstrip line and major hysteresis curve branch used to measure FMR. Right: resulting contour plot. Reprinted with permission

In the specific case of monodomain nanowires, both the anisotropy intensity and direction distributions are highly important since electrodeposited nanowires generally present homogeneities, which could greatly affect their magnetic behavior. On the other side, electro‐ magnetic wave propagation directly depends on the medium effective anisotropy, leading to a crucial information for all nanowire applications related with high-frequency devices (**Figure 11a**) [71]. Finally, the array magnetization reversal may lead to the interesting phenomenon of

of the phenomenon and the various ways to measure it is reviewed in [70].

created by multilayers [66] (**Figure 10c** and **d**).

from [72]. Copyright 2009 by AIP Publishing LLC.

**4.4. Anisotropy probing**

62 Magnetic Materials

In summary, the adequate use of the different characterization techniques available is essential for any researcher that is investigating magnetic nanowires, since the cost in time and money is high. Moreover, improper data analysis can lead to incorrect conclusions, while the un‐ known existence of a characterization technique can severely delay the advancement of a research project. This chapter is meant to serve as a reference guide for the specific system that constitutes magnetic nanowires.

## **Author details**

Fanny Béron\* , Marcos V. Puydinger dos Santos, Peterson G. de Carvalho, Karoline O. Moura, Luis C.C. Arzuza and Kleber R. Pirota

\*Address all correspondence to: fberon@ifi.unicamp.br

Gleb Wataghin Physics Institute, State University of Campinas, Campinas, Brazil

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