The theory of microwave propagation is here described in a transversely magnetized composite medium. On the assumption of symmetry, the tensor components of the magnetic susceptibility of a composite medium are defined. The effective magnetic parameters of the composite medium are further introduced, in which magnetic particles with form of ellipsoid are placed into a nonmagnetic dielectric material. The resonance phenomena are studied in the case of the electromagnetic wave propagation in a magnetized composite medium. The numerical calculations of the transmission and reflection coefficients, as well as their magnetic field dependences, are carried out. The magnetic field dependences of the wavenumber are drawn as well.
3.1.1. Propagation of Electromagnetic Waves in Transversely Magnetized Medium
Let the electromagnetic wave propagate in an infinite macroscopically uniform medium along the 0y axis. The magnetic field
H is directed along the 0z axis. Following to [
9], let us consider the solution of Maxwell’s equations:
jointly with the constitutive relations:
where
and
are the electric and magnetic constants, respectively. Note that the second of Equation (1) takes into account both the conduction and displacement currents when we are using the complex dielectric permittivity
.
In Equations (1) and (2),
b and
d are the magnetic and electric induction and
e and
h are the electric and magnetic field, respectively. It is assumed that the dielectric permittivity of the medium
ε is a scalar value and it does not depend on magnetic field. The magnetic permeability is defined by the tensor
. At chosen directions of the fields, this tensor has a view [
9]:
We will assume here that all the fields do not change in the direction of the
x axis and the electromagnetic wave propagates along the
y axis. Thus, the vectors
h, b, e and
d can be expressed in the form
,
,
and
, where
,
,
and
are the vector complex amplitude factors for the corresponding alternating fields,
is the cyclic frequency and
is the propagation constant. The solution of the system of Equations (1) and (2) in a coordinate view is reduced to the following equations:
where
k is the wavenumber in the magnetized medium and
,
is the speed of electromagnetic wave in vacuum. The condition of consistency of the algebraic equations systems (4) and (5) relatively to
and
is the equality to zero of the determinant of the following form:
From here, two solutions for the wavenumbers are the following: The relation
corresponds to the first solution. This solution is related to the wave in which the polarization of the alternating magnetic field
h coincides with the direction of the DC magnetic field
H. The wavenumber of this wave does not depend on the DC magnetic field. The second solution corresponds to the wave in which the alternating magnetic field is perpendicular to the magnetizing field. From (6), it follows that
where the effective magnetic permeability is introduced as:
Equation (8) can be used for both the uniform magnetic medium and the composite medium. The following relations are valid for the classical Polder tensor:
,
and
. Equation (8) takes the known view in this case [
9]:
3.1.2. Propagation Tensor of Magnetic Permeability of the Media with a Single Particle
Before turning to the composite medium, let us consider an alone magnetic particle in the non-magnetic medium. Assume that the particle has the form of an ellipsoid with arbitrarily directed axes in relation to the DC and microwave fields. Denote
the DC magnetic field inside the ellipsoid and
differs from the external magnetic field
H. The connection between these fields is given by:
where
M is the magnetization and
is the tensor of the demagnetizing factors. This tensor, written for any particle, defines the difference of the magnetic field
inside it from the outer magnetic field
H in the surrounding non-magnetic matrix. Its off-diagonal elements obey to the following conditions:
at
,
; the diagonal ones to the other condition:
, i.e.,
.
Let us write the Landau–Lifshitz equation for the lossy magnetic medium [
9]:
where
is the length of the DC magnetization vector
,
and
m are the alternating magnetic field and magnetization,
is the dissipation parameter and γ is the gyromagnetic ratio. Equation (11) can be rewritten as:
with the following elements of the matrix
:
The solution of Equation (12) lets one disclose the connection between the elements of the vectors
and
as
, therefore, define the elements of the magnetic susceptibility tensor:
its elements have the view:
Here, the definitions are used. They are the minors of the matrix , which are determined by the relation . It is assumed, here, that and .
If
and
, where
is the field of magnetic saturation, the inequalities
and
are valid for the projections of the vector
. Therefore, in this case, one can suppose that
, that is,
, at least at first approximation. Then the tensor (13) becomes:
where
,
.
Using the magnetic susceptibility tensor
, it is possible to find the magnetic permeability tensor:
Inserting (14) into (15), one gets, according to (3):
The designation is used here, at the definition .
3.1.3. Tensor of Magnetic Permeability of Composite Medium
Let us now discuss the problem of the effective magnetic parameters of the composite medium, which consists of identical magnetic particles in the form of ellipsoid with the same spatial orientation placed into the non-magnetic matrix. This composite medium is regarded as a macroscopically uniform medium, since the sizes of any elementary volume are significantly less than the characteristic scales, such as the wavelength of the propagating wave and the dimensions of the sample, but larger than the sizes of any magnetic particle. Magnetization of the composite medium is characterized by a volume fraction of the ferromagnetic phase, which is assumed constant for any elementary volume of the medium.
The tensor
, in accordance with (16), takes into account the demagnetizing fields both for the DC and alternating fields. Following to [
3], the expression for permeability tensor of composite
can be represented as the Silberstein formula:
As is shown in [
23,
24], variations of the field in the composite medium at
can be taken into account by introducing the effective demagnetizing tensor
, which has the following limiting cases: at
, the approximate equality
is valid, where the tensor
is defined for the single particle of a given form. At
, the effective demagnetizing tensor is close to zero,
, that corresponds to the unbounded magnetic medium. The paper [
25] describes a mathematical model enabling the calculation of the effective permeability tensor of heterogeneous magnetic materials. The model uses a definition similar to Equation (17) and gives all the complex components of the permeability tensor. According to [
23,
24], the effective demagnetizing tensor can be determined as:
Taking (17) into account, one gets the equation for
:
Equation (19) shows that
is, generally speaking, dependent on
. In the first approximation for
and
, it is reasonable to consider this dependence as negligible and Equation (19) takes the simpler view:
Therefore, the magnetic permeability tensor
of the composite medium with identically directed particles is determined by Expression (20), where the components of the tensor
are the same as in (19), but with replacement of the components of
by the components of
. Finally, it can be written:
The foregoing formulas for the components of the magnetic permeability tensor are a generalization of formulas for a composite medium specified in [
9]. They describe the resonance variations of permeability near FMR. Of course, these resonance features of frequency or magnetic field dependences of the component of the tensor (21) have to manifest themselves in the dependences of transmission and reflection coefficients and absorption of the wave. Besides, variations of non-resonant type are possible, which are caused by magnetization of the composite. These variations allow, for example, the so-called low field absorption [
26]. In order to take into account non-resonant variations, the factor
is introduced, which corrects the values of the elements of the magnetic permeability tensor. One of the simplest ways to introduce
is the expression:
where
is the diagonal component of the susceptibility tensor at
. The contribution to dynamic magnetic permeability coming from the domain walls motion is calculated [
27]. The magnetic field dependence of this contribution is very similar to our Equation (22). The components of the tensor
can be rewritten as:
Notice that, according to (22), under condition =1, the equality is valid. The value of is frequency dependent. Under magnetic saturation, at , almost equals to 1.
3.1.4. Magnetic Permeability of Ensemble of Arbitrarily Directed Particles
Let us set the vector of rotation angles of a ferromagnetic particle relative to the axes
as
. It conditions, first, variation of the components of demagnetizing tensor
of the particle if its orientation varies and, second, variation of
, therefore,
. In order to get the effective magnetic permeability tensor
of a composite medium in the case of arbitrarily oriented particles, it is necessary to perform the statistical averaging of its components:
The effective magnetic permeability of the transversely magnetized medium
, corresponding to a given value of the vector
, is expressed by a relation similar to (8), which includes the components of the tensor
. The averaged value of the effective magnetic permeability for the transversely magnetized medium is determined by the formula:
If particles have the form of flakes, their space orientation can be characterized by the direction of normal to its plane. If the normal is directed along to the y axis, then it can be defined by the vector . Let us believe that this the initial or base orientation of a ferromagnetic particle, for which the vector of rotation angles can be defined as . All other possible space orientations of particles have to be determined by the vectors . In the case of arbitrary orientation of a great number of particles, every element from the multitude of vectors belongs to the independent multitudes of uniformly distributed random numbers which get into the intervals: , and . The discrete number of the vector of rotation angles corresponds to the discrete multitudes of numbers , and .
The calculation of the effective magnetic permeability of a magnetized composite has been carried for 10,000 particles made of material with the saturation magnetization
Ms = 900 kA/m and magnetic damping constant
α = 0.05. Every particle has the form of ellipsoid with the axes
a =
b = 25 µm and
c = 1 µm. The calculation is carried out for frequency
f = 32 GHz. The results for several values of
are shown in
Figure 4. The variations of the effective permeability caused by FMR are present in the graphs. It is evidently seen, from
Figure 4, that the resonance occupied a wide region of magnetic fields despite the small value of
α. Increase in the volume fraction of ferromagnetic particles leads to more pronounced resonance variations.
Let us consider the case when a group of particles with definite orientation is present in the composite besides the group of arbitrarily oriented particles. Denote the whole number of ferromagnetic particles as
; the number of arbitrarily oriented particles as
; the number of particles with the definite orientation (in the plane of a sample, for example) as
,
. In the limiting case when
, all particles are oriented arbitrarily, when
, all particles are oriented in the plane of a sample. The formula for calculation of the components of the tensor
can be written for the presence of several groups of differently oriented particles. Their mean values are determined by the equation:
The results of the calculation of the effective magnetic permeability following to Equation (27) are shown in
Figure 5 for the composite with 5000 particles of ellipsoidal form with the same sizes as in
Figure 4. The saturation magnetization of particles is
= 900 kA/m, the magnetic damping constant
α = 0.19 and the volume fraction of ferromagnetic particles
. In
Figure 5, curve 1 corresponds to the case
= 1, that is, the non-resonant absorption is not taken into account. It is assumed, in this case, that
L1 = 4000,
L2 = 1000 and, namely, 20% particles are oriented by the manner when the DC magnetic field is in parallel to their plane. The remainder 80% particles are oriented arbitrarily. Curve 2 corresponds to the case when all particles are oriented to the plane of the sample and the DC magnetic field lies in this plane. The resonant-type variations are expressed much more strongly in this case. The values
= 1.4 and
κ = 1.184 × 10
−9 (m/A)
2 are chosen for curve 3, so the non-resonant contribution is taken into account. The calculation is carried out for frequency
f = 32 GHz.
The results in
Figure 4 and
Figure 5 show that well-known procedures of homogenization based on the constitutive parameters and concentrations of its components are insufficient in order to describe the resonance phenomena in the heterogeneous magnetic medium; see [
3] for example. The conception of the mean demagnetizing factor is sometimes used instead of the procedure of averaging of contributions in the components of the magnetic permeability tensor from particles with different demagnetizing factor [
16]. This conception is capable, for instance, to describe the FMR spectrum of the composite material containing spherical particles [
24]. For media in which the value of the demagnetizing factor of particles varies essentially, it is prescribed to apply the afore-mentioned calculation procedure. Notice that division of particles into separate groups with identical value of the demagnetizing factor used in our calculation in
Figure 5 is similar to the method applied in [
16].