Effects of domain, grain, and magnetic anisotropy distributionson magnetic permeability: Monte-Carlo approach
Jaehun Chun, A. Mark Jones, and John S. McCloya)
Pacific Northwest National Laboratory, Richland, Washington 99352, USA
(Received 1 May 2012; accepted 14 June 2012; published online 23 July 2012)
We have investigated the effects of domain and grain anisotropy on spin-resonance in magnetic
permeability, implementing a Monte-Carlo approach and a coupled Landau-Lifshitz-Gilbert
equation. The Monte-Carlo approach provides great flexibility by employing different probability
density functions, allowing modeling of material texture differences that may occur due to
different preparation methods. Changes in the permeability tensor result from variations in grain
demagnetization and domain demagnetization as well as the anisotropy field relative to saturation
magnetization. Experimental permeability measurements on demagnetized polycrystalline yttrium
iron garnet show for the first time that the best fit to measured data requires a complex distribution
of both grain and domain demagnetization factors. Assuming that grain and domain
demagnetizations are decoupled, it was found that the grain structure (i.e., grain demagnetization
distribution) has a smaller effect on the frequency-dependent permeability than does the same
distribution of domains (i.e., domain demagnetization distribution). Implications for modeling
experimental data assuming particular phenomenological loss coefficients or linewidths are also
offered. VC 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4737417]
I. INTRODUCTION
An accurate prediction of the permeability tensor for
polycrystalline ferrites is of great interest for developing
optimal designs of nonreciprocal microwave devices (e.g.,
circulators and isolators) and reciprocal power phase shifters
that often operate in a partially magnetized state. In a com-
pletely saturated state, all magnetic moments are aligned and
the magnetic permeability is described by the Polder formu-
lation.1 However, determining the permeability of a poly-
crystalline ferrite in a partially magnetized state is more
challenging due to the complexity of the orientations and
shapes of both domains and grains. Moreover, it is difficult
to estimate the internal field of each domain and grain while
including the effects of their respective boundaries. Existing
models, such as those from Schloemann,2 Gelin and Ber-
thou-Pichavant,3 and Gelin et al.,4 are inappropriate to pro-
vide the permeability of polycrystalline ferrites with
distributions of shapes of both domains and grains, espe-
cially in demagnetized and partially magnetized cases below
the natural spin resonance frequency. This is the region of
the so-called Polder-Smit resonance, or low-field loss, which
occurs due to dynamic interaction of oppositely magnetized
domains present in the non-saturated case.5,6 The coupling
between domains with opposite direct current (DC) moment
occurs through the off-diagonal component of the permeabil-
ity tensor, resulting in an out-of-phase magnetization in the
direction perpendicular to the exciting magnetic field.7 The
out-of-phase component creates uncompensated magnetic
charges on the domain boundaries, behaving as capacitances
and inductances in the system, and significantly spreading
the resonance frequency.7,8
To predict the permeability tensor of polycrystalline fer-
rites under such conditions, Gelin and Queffelec9 used the
generalized permeability tensor model. This model incorpo-
rated the interactions between adjoining domains/grains by
taking into account demagnetizing dynamic fields associated
with those interactions. Although the model is based on a
self-consistent theoretical approach and captures the essen-
tial physics necessary to calculate the permeability of fer-
rites, it still uses conventional distribution laws to describe
magnetic anisotropy, magnetic domains, and grain shapes.
As a result, this approach may not be readily applicable to
study various non-uniform distributions (and especially,
coupled distributions) of magnetic anisotropy, magnetic
domains, and grain shapes on the permeability tensor. In
fact, variations in microstructure are expected for polycrys-
talline ferrites, depending strongly on preparation methods
of the material, leading to complex correlations among
domains and grains.
We take essentially the theoretical formulation of Gelin
and Queffelec,9 but implement a classical Monte-Carlo
approach to ensure the greatest flexibility in obtaining the (en-
semble) averages for the permeability tensor of polycrystalline
ferrites. The Monte-Carlo computational algorithm uses
repeated random sampling to calculate multi-dimensional
integrals and is ideal for computation of permeability where
there can be a large number of coupled degrees of freedom
with complicated boundary conditions. Utilizing the Monte-
Carlo approach, various non-uniform (and/or coupled) distri-
butions of magnetic anisotropy, magnetic domains, and grain
shapes within the material can be realized by different (joint)
probability density functions (PDFs). Therefore, the Monte-
Carlo approach can provide a simple but versatile tool to
model variations in polycrystalline ferrites and calculate the
average tensor permeability components.a)Email: [email protected].
0021-8979/2012/112(2)/023904/8/$30.00 VC 2012 American Institute of Physics112, 023904-1
JOURNAL OF APPLIED PHYSICS 112, 023904 (2012)
We investigate the permeability of polycrystalline fer-
rites over different conditions (i.e., different distributions of
magnetic domains, grain shapes, and magnetization states)
by utilizing the Monte-Carlo approach. Key features of the
formulation and Monte-Carlo approach are explained in
Secs. II–III. Calculated results based on different magnetic
characteristics of polycrystalline ferrites are shown as exam-
ples. Additionally, comparison to measured experimental
data for demagnetized polycrystalline yttrium iron garnet
(YIG) is provided to illustrate the utility of this approach.
II. BACKGROUND THEORY
The theoretical formulation shown in this section is
well-established and summarized here for completeness indi-
cating sufficient detail to understand where the Monte-Carlo
approach is inserted in the computation. It should be pointed
out that we are only interested in the effects of permeability
due to spin resonance, and no physics of domain wall reso-
nance will be considered.6
Assume a polycrystalline ferrites consisting of many
grains with the same crystallographic structure. Each grain
may contain multiple magnetic domains. When the spin
wavelength is large compared to the lattice constant, one can
use a macroscopic description for the dynamics of the mag-
netic moment. Assuming that the exchange interaction of the
atoms is only important, the equilibrium magnetization is
thereby fixed by the exchange interaction; the direction of
the magnetic moment varies slowly in space and its magni-
tude remains constant.10 A resultant conservation equation
for the magnetic moment in a magnetic domain can be
described by a first order differential equation known as the
Landau-Lifshitz equation. This equation can be extended to
the Landau-Lifshitz-Gilbert (LLG) equation when a phenom-
enological higher order loss term is included.11,12 The inter-
actions between adjacent domains/grains within ferrites in
the presence of an external magnetic field, the phenomena of
interest in this paper, can be further described by a coupled
Gilbert equation as described by Gelin et al.,4
dM1
dt¼ cM1 � H1 þ h� ndðm1 �m2Þ � ng
m1 þm2
2� M
MShmig
� �� �þ a
MSM1 �
dM1
dt;
dM2
dt¼ cM2 � H2 þ h� ndðm2 �m1Þ � ng
m1 þm2
2� M
MShmig
� �� �þ a
MSM2 �
dM2
dt:
(1)
Here a and c are, respectively, the phenomenological loss
coefficient and gyromagnetic ratio, and M and MS are the
macroscopic magnetization and saturation magnetization.
Demagnetizing coefficients nd and ng are due to domain and
grain shape effects and have values ranging from 0 to 1, these
limits corresponding to a needle-like ellipsoid, parallel (0), or
perpendicular (1) to the magnetization field.13,14 It is impor-
tant to note that the demagnetizing fields associated with nd
include the Polder-Smit effect, a coupling of magnetic poles
in adjacent domains originating from different orientations of
the magnetization in the domain structure.8 The magnetic
moments M1 and M2 have dynamic components m1 and m2,
noting that Mi ¼ MSui þmi where ui is the unit vector for
the equilibrium direction of the local magnetic vector Mi.
Hið¼ HiuiÞ is the effective DC magnetic field in domain i,and h is the local variable radio-frequency (RF) magnetic
field. The governing equation incorporates the demagnetizing
dynamic fields produced by the magnetic domain shapes (the
term associated with nd in Eq. (1)) and grain shapes (the term
associated with ngin Eq. (1)), as well as the magnetic interac-
tions between adjoining domains (i.e., domains 1 and 2) and
between adjoining grains. Note that the use of the coupled
Gilbert equation in this way implicitly assumes that each
domain or grain in the distribution can be described by a
scalar demagnetization factor, but the completely arbitrary
three-dimensional case can be solved by making suitable
replacements in Eq. (1) allowing nd and ng to be tensors.
Following the effective medium approximation (see
Bouchaud and Zerah7), hmig, the average of the dynamic
magnetic moments over all possible grains in the material
is
hmig ¼1
21þ M
MS
� �m1 þ
1
21� M
MS
� �m2: (2)
Here, hmig is identical to a vectorial average of dynamic
magnetic moments in a completely demagnetized state (i.e.,
M ¼ 0), but is not the simple average in a partially demagne-
tized state (i.e., M 6¼ 0). According to Stoner and Wohl-
farth15 and Gelin and Queffelec,9 the equilibrium direction
of the domain i (hi) for a uniformly magnetized ellipsoid can
be obtained by minimizing the energy with respect to hi,
@g@hi¼ Ha
2sin 2ðhi � hÞ þ H0 sin hi ¼ 0;
@2g
@h2i
¼ Ha cos 2ðhi � hÞ þ H0 cos hi > 0:
(3)
The first and second terms in the right-hand side correspond
to the energies associated with the demagnetizing field in the
domain and the applied field, respectively. Here, g is the total
energy of a uniformly magnetized ellipsoid, Ha is the
magneto-crystalline anisotropic field, and H0ð¼ Hext � NMÞis the static net DC magnetic field (i.e., equal to the external
field Hext minus a macroscopic demagnetizing effect deter-
mined by the demagnetization factor N and the magnetiza-
tion M along the applied field direction). Note that h1 ¼ hand h2 ¼ p� h for H0 ¼ 0, implying that ferrites under zero
023904-2 Chun, Jones, and McCloy J. Appl. Phys. 112, 023904 (2012)
net DC magnetic field present an internal structure divided
into domains of parallel and anti-parallel magnetization. As
pointed out by Stoner and Wohlfarth,15 the energetic argu-
ment introduces important critical values of H0=Ha and hi
for a given magnetic anisotropy angle, h. The scalar effective
DC magnetic field in domain i, Hi, can be obtained from the
second derivative of the total energy,
Hi ¼ Ha cos 2ðhi � hÞ þ H0 cos hi: (4)
One can show that H1 ¼ Ha but H2 ¼ Ha cos 2ðp� 2hÞ with
H0 ¼ 0, which suggests that H1 6¼ H2 unless h ¼ 0, p=2,
or p.
To obtain the permeability tensor of polycrystalline fer-
rites, one needs to solve the coupled Gilbert equation (i.e.,
Eq. (1)), assuming the small signal approximation (i.e., keep-
ing terms only up to the first order of mi). As explained in a
previous study,3 this involves various steps: a Laplace trans-
form, several coordinate transformations, and multiple inte-
grations over distributions of magnetic domains, grains, and
magnetic anisotropy angles. The permeability tensor of poly-
crystalline ferrites, assuming the magnetic field is applied
along the z-direction, can be obtained via the tensorial rela-
tion between hmi and hhi,
hmi ¼hli jhki 0
�jhki hli 0
0 0 hlZi
0@
1Ahhi; (5)
where hi denotes an ensemble average over distributions of
magnetic domains, grains, and magnetic anisotropy angles.
hli and hlzi are diagonal components, hki is an off-diagonal
component, and j ¼ffiffiffiffiffiffiffi�1p
. Note that hli, hki, and hlzi are
complex quantities. The proportional tensor in Eq. (5) is the
magnetic susceptibility tensor, rather than the magnetic
permeability tensor. Therefore, in SI or MKS units, two diag-
onal components of the non-dimensional relative magnetic
permeability are represented by 1þhli and 1þhlzi. hkidenotes an off-diagonal component of the relative magnetic
permeability, resulting in the nonreciprocal behavior of
microwaves into magnetized ferrite used for the design of
circulators and isolators.
III. EXPERIMENTAL AND NUMERICAL APPROACHES
A. Ferrite measurements
In order to test the utility of the Monte-Carlo approach
for modeling experimental data, a set of samples of YIG (Pa-
cific Ceramics, Sunnyvale, CA, USA) were machined for a
coaxial line experiment for determining the permittivity and
permeability. Samples in the as-received demagnetized state
were measured with a vector network analyzer (PNA
E8361A, Agilent, Santa Clara, CA, USA) with a 7 mm outer
diameter coaxial line fixture (M07T, Damaskos Inc., Con-
cordville, PA, USA). The sample permittivity and permeabil-
ity were calculated by custom software using well-known
methods.16,17 This device can in principle provide precision
electromagnetic measurements up to 18 GHz, but due to geo-
metrical resonances (2 mm and 5 mm thick samples), usable
data was generated from 10 MHz–14 GHz. Due to the strong
absorption in YIG caused by gyromagnetic (spin) resonance
near 5 GHz in demagnetized samples, permeability data
obtained at frequencies higher than this in coaxial specimens
are questionable. For this reason, additional measurements
were performed on 2 mm diameter YIG samples using
dielectric ring resonators (QWED, Warsaw, Poland) at
6.102, 7.071, and 7.851 GHz.
B. Monte-Carlo approach
To enable exploration of arbitrary distributions of grains
and domains (i.e., to obtain an ensemble average over arbi-
trary distributions of grains and domains), we use a distribu-
tion of their respective demagnetization factors. Following
the Monte-Carlo approach, these probability distributions are
then sampled, enabling calculation of the permeability tensor
over domain and grain shape diversity. The Monte-Carlo
approach uses a uniform random number for sampling, satis-
fying a probability distribution of variables throughout the
ferrite.18 For example, ferrites with uniform spherical mag-
netic domains can be represented by a PDF of domain
demagnetizing coefficient, PðndÞ ¼ dðnd � 1=3Þ, where ddenotes a delta function. In other words, nd will always be 1/
3, regardless of the random number generated by the Monte-
Carlo. More importantly, it is possible to easily adapt differ-
ent distributions with the same average values. For example,
ferrites with non-uniform magnetic domains but spherical
domains on average can be described by various beta proba-
bility density functions including PðndÞ ¼ 2ð1� ndÞ and
PðndÞ ¼ 1=ð2 ffiffiffiffiffindp Þ. Here, a beta function is represented by
Cð~a þ ~bÞCð~aÞCð~bÞ
x~a�1ð1� xÞ~b�1at 0 < x < 1; 0 elsewhere; (6)
where C denotes the Gamma function, ~a and ~b are positive
constants, and x is a variable.19 Note that a beta function is
one of the most reasonable choices to describe the distribu-
tions of domain and grain demagnetizing coefficients
because the variable only changes between 0 and 1 by defini-
tion.13 Similarly, a uniform distribution of magnetic anisot-
ropy angle or easy axis angle h can be used to obtain the
permeability tensor of "random" polycrystalline ferrites via
the probability density function PðhÞ ¼ sin h=2. Following
the Monte-Carlo approach, one can easily adapt a delta func-
tion to represent strong alignment of the magnetic anisot-
ropy. Note that we here assume there is no coupling between
domain and grain demagnetizations for simplicity. A benefit
to the Monte-Carlo approach is that one can use a joint prob-
ability density function of nd and ng (e.g., joint normal distri-
bution) to couple domain and grain demagnetizations.
The frequency-dependent permeability was calculated
using the Monte-Carlo approach for several cases assuming
different distributions for grain and domain demagnetization
factors. Figure 1 shows an example of four of the distribu-
tions considered. Note in particular that the distributions A,
B, and C all have a mean value of 1/3 for the demagnetiza-
tion factor. Distribution C is for uniform spherical domains,
while distribution A is more heavily weighted towards
023904-3 Chun, Jones, and McCloy J. Appl. Phys. 112, 023904 (2012)
ellipsoids with their long axis parallel to the field direction
than distribution B. Distribution D is for uniform cylindrical
domains, and similar beta distributions can be created having
mean values of 1/2 as for the cylindrical domains, for exam-
ple, PðndÞ ¼ 1, which represents an average cylindrical do-
main with a random distribution of nd.
For the simulations, the following parameters were used
unless otherwise noted. The gyromagnetic ratio and phenom-
enological loss coefficient are set to c ¼ 2:174� 105 rad�m/
A � s (1.734� 107 rad�s�1�Oe�1 in cgs Gaussian units) and
a ¼ 0:006, respectively, which are assumed to be reasonable
values based on published data for YIG (see below). The
gyromagnetic ratio (in SI units), c ¼ ge=2me, is determined
from the spectroscopic splitting factor g and the charge e,
and mass me of the electron. The phenomenological loss
coefficient is determined from the resonance line width DHby a ¼ cDH=2x0, where x0 ¼ cH0.20 Both g factor and res-
onance linewidth are typically listed in manufacturer’s speci-
fications. For YIG (TT-G113, Trans-tech, Adamstown,
Maryland), g¼ 1.97 and DH¼ 20 Oe, which give rise to the
above gyromagnetic ratio and phenomenological loss coeffi-
cient. The Ms value of 141.6 kA/m (1780 G) was taken from
the manufacturer’s data sheet (Pacific Ceramics, Sunnyvale,
CA, USA), and the Ha value of 3.98 kA/m (50 Oe) was esti-
mated from Hisatake et al.21 The frequency region f0 to
(f0þ fm) defines the low-field loss region, 0.14–4.91 GHz in
this case, where fm¼ cMs/2p, f0¼ cH0/2p, and H0 is the in-
ternal field that is sometimes assumed to be equal to Ha in
the absence of an external applied field.
For the Monte-Carlo calculations, the number of ensem-
bles (Nensem) for the approach was set to 2� 106 to ensure a
small enough uncertainty, as statistical uncertainty in Monte-
Carlo calculation is proportional to 1=ffiffiffiffiffiffiffiffiffiffiffiffiNensem
p. The increment
of frequency in the Monte-Carlo calculations was 50 MHz for
the demagnetized YIG study, from 1 GHz to 10 GHz. The pro-
cedure to obtain the best fit to experimental data was as fol-
lows. An initial distribution was fixed for grains (beta B), and
distributions were varied for domains. The qualitatively best
fit for both the real and imaginary parts of the demagnetized
YIG permeability was found to be beta A. Next, this distribu-
tion was fixed for domains, and the distributions were varied
for grain demagnetization. Table I shows a complete list of
the probability density functions used for nd and ng in each
case for modeling demagnetized YIG.
IV. RESULTS AND DISCUSSION
Figure 2 shows the comparison between the measured
and calculated frequency-dependent real and imaginary per-
meability for demagnetized YIG to illustrate the study vary-
ing the domain demagnetization factor. The measured
frequency-dependent permeability for YIG compared well
with that previously determined by similar methods.22,23 It
can be seen that the imaginary permeability for the beta A
distribution agrees quite well with the experiment. It was for
this reason that this domain distribution was fixed and the
grain distribution was subsequently varied. This latter set of
simulations is shown in Figure 3 for the study of varying the
grain demagnetization factor. Again the beta A distribution
appeared to best represent the data, though not perfectly. Fig-
ure 4 shows the best fit to the experimental data, a logarith-
mic scale of the permeability to better illustrate the
goodness-of-fit. As shown in Figures 2 and 3, the real part of
the permeability does go negative in a certain frequency
region, so the corresponding logarithmic plot in Figure 4(a)
only represents the positive values and accentuates the
region where the permeability is negative. In Figures 2–4,
the circles represent the measured single frequency resonator
data, the thick line represents the measured coaxial line
data, and the thinner lines represent the calculated data as
labeled.
FIG. 1. The probability density function (counts by bin) for nd sampling for
four distributions considered: (A) beta distribution with mean 1/3, (B) a sec-
ond, different beta distribution with mean 1/3, (C) distribution of uniformly
spherical domains, value always 1/3, and (D) uniformly cylindrical domains.
TABLE I. Conditions for simulations of demagnetized YIG described in the text and plotted in Figures 2–4.
Run ID P(nd) P(ng) Notes
1 nd 1 dðnd � 1=2Þ 2ð1� ngÞ Domains uniform cylindrical; grains average spherical (beta B)
2 nd 2 dðnd � 1=3Þ 2ð1� ngÞ Domains uniform spherical; grains average spherical (beta B)
3 nd 3 2ð1� ndÞ 2ð1� ngÞ Domains average spherical (beta B); grains average spherical (beta B)
4 nd 4 1=ð2 ffiffiffiffiffindp Þ 2ð1� ngÞ Domains average spherical (beta A); grains average spherical (beta B)
5 ng 1 1=ð2 ffiffiffiffiffindp Þ dðng � 1=2Þ Domains average spherical (beta A); grains uniform cylindrical
6 ng 2 1=ð2 ffiffiffiffiffindp Þ dðng � 1=3Þ Domains average spherical (beta A); grains uniform spherical
7 ng 3 1=ð2 ffiffiffiffiffindp Þ 2ð1� ngÞ Domains average spherical (beta A); grains average spherical (beta B)
8 ng 4 1=ð2 ffiffiffiffiffindp Þ 1=ð2 ffiffiffiffiffi
ngp Þ Domains average spherical (beta A); grains average spherical (beta A)
9 ng 5 1=ð2 ffiffiffiffiffindp Þ 1 Domains average spherical (beta A); grains average cylindrical
023904-4 Chun, Jones, and McCloy J. Appl. Phys. 112, 023904 (2012)
Though certainly our choice of distributions to model is
in some senses arbitrary, this exercise in parameterization
clearly showed the following: (1) the choice of distribution
is critical and not simply the average value of the demagnet-
ization factor (e.g., uniform spheres, beta A, and beta B, all
have the same average value but result in very different
frequency-dependent permeability) and (2) choosing a distri-
bution of demagnetization factors rather than uniform cylin-
ders or spheres, for example, results in a better fit to
experimental data. We could not find experimental data on
distributions of grains or domains in real samples, other than
the literature on texturing in barium hexaferrite, for example,
which results in very different low frequency permeability
due to domain walls.24 However, high frequency data con-
taining the spin resonance, which for barium hexaferrite is in
the millimeter wave range,25 was not collected.
One might also object to the theoretical approach which
does not take domain wall resonance into account. It is well
known that the domain wall resonance or relaxation accounts
for the large low-frequency permeability in multi-domain
samples.6,26 Recent measurements of demagnetized YIG
ceramics and fits of the imaginary permeability have shown,
however, that the component of permeability due to domain
walls is not important above 1 GHz, where we begin our sim-
ulations.23 Additionally, it was shown long ago that single
domain particles show no magnetic loss due to domain walls,
and hence, the permeability is controlled only by the spin
resonance,6,27,28 which we model in this paper. Finally, most
microwave ferrites are only used at frequencies above their
spin resonance frequency unless they are at least partially
magnetized.
It may be argued that the domain and grain distributions
are, in a sense, just fitting parameters and do not represent
any physical differences in microstructure. Typically, when
modeling permeability spectra, grain, and domain demagnet-
ization factors, if used at all, are assumed to be simple geo-
metries such as cylinders or spheres, and the damping
parameter a is used as the fitting parameter (e.g., a¼ 0.5 for
Terfenol-D (Ref. 29)). The fitting of experimental data rely-
ing on simple geometries for demagnetization results in
unrealistically large values of a (e.g., a¼ 0.725 for Y35 fer-
rite4) which do not correspond to those measured using ferro-
magnetic resonance techniques. Mallegol et al.30 point out
that the a values used to fit data in various models are at least
one order of magnitude larger than that predicted from the
Gilbert formulation based on linewidth, and the authors call
for the use of a distribution of domain and grain shapes in
order to accommodate realistic values of a. They show one
example of a computation with an unstated distribution of
grain and domain shapes and show that it fits experimental
data much better than static values (cylinders or spheres)
while maintaining an experimentally realistic damping pa-
rameter. Our work, therefore, is an extension of the need
expressed by Mallegol et al.30 The difference we emphasize
is that distributions with the same average value (e.g., beta A
and beta B) do not produce the same permeability results,
thus the choice of distribution is important.
Note, however, that there are experimental and theoreti-
cal reasons to believe that a is not constant with frequency,
although it is usually treated as such. For example, non-
oriented Co2Y hexaferrite linewidths were seen to vary from
0.7 kOe at 9.56 GHz to 2.5 kOe at 40 GHz.31 Assuming
g¼ 2, this equates to a¼ 0.103 at 9.56 GHz and a¼ 0.088 at
40 GHz. Additionally, it is well known that there are many
mechanisms for inhomogeneous broadening of linewidth of
polycrystalline ferrites, such as porosity, anisotropy, and
grain boundary and grain-to-grain magnon scattering
processes.32–34 Experimentally, high quality polycrystalline
5
4
3
2
1
0
8642Frequency (GHz)
Fixed ndChanging ng
56
7
8
Measured
(a)
9
5
4
3
2
1
0
8642Frequency (GHz)
Fixed ndChanging ng
5
Measured
6
7
8 (b)
9
µ’ µ’’
FIG. 3. Comparison of (a) real and (b)
imaginary parts of the diagonal perme-
ability for measured and calculated
demagnetized YIG where domain demag-
netization factor was fixed and grain
demagnetization factor was varied. Num-
bers correspond to the simulations listed
in Table I. Circles are the single fre-
quency resonator data.
5
4
3
2
1
0
8642Frequency (GHz)
Fixed ngChanging nd
1
234
Measured
(a) 5
4
3
2
1
0
8642Frequency (GHz)
Fixed ngChanging nd
1
Measured234
(b)µ’
µ’’
FIG. 2. Comparison of (a) real and (b)
imaginary parts of the diagonal perme-
ability for measured and calculated
demagnetized YIG where grain demag-
netization factor was fixed and domain
demagnetization factor was varied.
Numbers correspond to the simulations
listed in Table I. Circles are the single
frequency resonator data.
023904-5 Chun, Jones, and McCloy J. Appl. Phys. 112, 023904 (2012)
YIG linewidths vary from �5 to 45 Oe depending on fre-
quency,34 whereas single crystal YIG linewidth at 10 GHz is
�0.5 Oe.33 If the damping parameter a is calculated from ex-
perimental linewidths of polycrystalline ferrites, very large
damping parameters are obtained, �0.05 to 0.45 (for
Ha¼ 50 Oe, g¼ 1.97, and DH¼ 5 or 45 Oe). In order to
obtain the normally quoted value of a¼ 0.006 for YIG, one
must assume rather DH¼ 0.6 Oe, as for a single crystal.
Thus, it is important to realize that a is a phenomenological
parameter that depends on microstructure and which is
related to measured linewidth. From a physical standpoint,
the frequency dependence of the linewidth is due to various
processes such as grain-to-grain and grain-boundary magnon
scattering.34
To further compare the effect of distribution with the
change in the damping parameter a, we explored the best fit
of demagnetized YIG (beta A for both domains and grains)
by setting the damping parameter a¼ 0.0006, 0.006, 0.06,
and 0.6. These choices contain the range of values seen in
the literature from the experimental value to approximately
that used for fitting other ferrite data. The results of this
parameterization are shown in Figure 5(a). It can be seen
that the choice of a has a qualitatively different effect on
the permeability than that of domain and grain distribu-
tions, in which it leaves the low frequency (1 GHz) value of
the real permeability unchanged, and changes the minimum
point.
The choice of specific algorithm in the Monte-Carlo has
an effect on the results for any finite number of samples.
Even with Nensem¼ 2� 106, we found, particularly with the
beta A distribution, that the actual numerical method
affected the results. This is most easily seen in the real
permeability of Figure 5(b), which is the same calculation as
Figure 5(a) but with a different algorithm for computing the
distribution. The main difference between the two is in how
to select the demagnetization factor of domain (and grain)
from a uniform random number (i.e., between 0 and 1 with
an equal probability) for the Monte-Carlo. To realize the
beta A distribution, one needs to use ~N ¼ n1=2d ðor n
1=2g Þ to
determine ng (and nd) for each ensemble (note that ~N is a
uniform random number between 0 and 1). The algorithm
which generates only one random number then squares it
(algorithm 2) results in an inferior fit to the experimental
data with higher values of real permeability, compared to the
algorithm which essentially picks two random numbers and
multiplies them together (algorithm 1).
Additionally, to attempt a better fit to the experimental
data for the imaginary component in Figure 4(b), we used
the damping factor of 0.0006. With the smaller a value, it
required an order of magnitude larger number of ensembles
to get smoothly varying data, Nensem¼ 2� 107. Note that the
a¼ 0.006 case with this larger number of ensembles did not
result in better data. The real part with a¼ 0.0006 was not
significantly different than that obtained for a¼ 0.006 and
shown in Figure 4(a). However, as Figure 6 shows, the run
with a¼ 0.0006 better matches the experimental data for
imaginary part of permeability as determined by the resonant
cavity. Assuming a value of a¼ 0.0006 results in a calcu-
lated linewidth of 0.06 Oe, much smaller than the intrinsic
linewidth of single crystal YIG �0.5 Oe.35 The only other
parameters in question are the spectroscopic splitting factor
g (g¼ 1.975 per Ref. 34 which is quite similar to the typical
manufacturer specification of g¼ 1.97) and the anisotropy
field (Ha¼ 40 Oe per Ref. 34 and 50 Oe per Ref. 21). We
μ’
2.0
1.5
1.0
0.5
0.0
-0.5
8642Frequency (GHz)
P(n)=1/(2n1/2
)algorithm 1Changing α
0.006
0.6
0.06
Measured
(a)2.0
1.5
1.0
0.5
0.0
-0.5
8642Frequency (GHz)
P(n)=1/(2n1/2
)algorithm 2Changing α
0.006
0.6
0.06
Measured
(b)
μ’
FIG. 5. Real part of the diagonal com-
ponent of permeability for demagnetized
YIG where domain and grain demagnet-
ization factors were fixed to the beta A
distributions with three different damp-
ing parameters a (0.006, 0.06, and 0.6):
(a) algorithm 1, (b) algorithm 2. See the
text for details on algorithms 1 and 2.
0.01
0.1
1
10
86420Frequency (GHz)
(a)
0.001
0.01
0.1
1
10
86420Frequency (GHz)
(b)
µ’ µ’’
FIG. 4. Comparison of (a) real and (b)
imaginary parts of the diagonal perme-
ability for measured and calculated
demagnetized YIG for the best fit simu-
lated (#8). Permeability is shown on a
logarithmic scale to show the goodness-
of-fit. Real permeability, both measured
and calculated, shows a negative region
which is roughly indicated on the plot
as between �1–4 GHz (measured) and
�2–3 GHz (simulated). Measured data is
the thick solid line (coaxial line meas-
urements) and the solid circles (resonator
measurements), while the simulated data
is shown as small crosses.
023904-6 Chun, Jones, and McCloy J. Appl. Phys. 112, 023904 (2012)
conclude, therefore, that the non-physical small alpha
required to fit the data indicates that the experimental data
from the resonant cavities may be unreliable, even though it
perfectly matches previously measured YIG data using a
similar system.22 On the other hand, some expressions36
have been developed for effective linewidth which, if used
with the resonant cavity permeability data obtained here and
assuming zero off-diagonal components to the demagnetized
permeability, lead to effective linewidth estimates of �31 to
36 Oe which is very reasonable for polycrystalline YIG as
previously stated.
Likewise, the simulations indicate that the drop in imagi-
nary permeability occurs at higher frequencies than deter-
mined by coaxial line measurements. Imaginary permeability
data between �3.75 and 5.2 GHz corresponds to the ascending
part of the real permeability resonance after it has increased
above zero with increasing frequency. These discrepancies,
then, could be due to either calibration issues with the experi-
ments or numerical issues resulting from the algorithms used
to compute the permeability from the S parameter data near
the gyromagnetic resonance frequency (�5 GHz). Other sets
of demagnetization factors (e.g., simulation #2 in Table I and
Figure 2) resulted in a decrease in imaginary permeability
which occurred at lower frequencies than the experiment, but
the simulation for the real permeability was an inferior fit to
the data than was the beta A distribution. This highlights the
fact that the simulations do not present a unique solution.
There may be another combination of demagnetization factors
which result in slightly better agreement with the real and
imaginary measured permeability. Nonetheless, it is clear that
the effects of damping and of demagnetization in grains and
domains are qualitatively different, and the effects and sensi-
tivities can be explored using a Monte-Carlo model.
V. SUMMARY AND CONCLUSIONS
We present a method for determining the tensor compo-
nents of an arbitrary polycrystalline ferrite that can be
applied to study various magnetization states and hysteresis.
A Monte-Carlo approach is implemented to calculate the en-
semble averaged permeability tensor over various distribu-
tions of magnetic anisotropy, magnetic domains, and grain
shapes which may be dependent upon material preparation
methods. The versatility of the Monte-Carlo approach results
from the ease of sampling tailored probability density func-
tions. Using this method, we have found that, of the distribu-
tions investigated, the beta A probability density function for
both grain and domain demagnetization factors best
describes the measured data from the demagnetized YIG
sample. The beta A function represents shapes which are av-
erage spherical but which have a larger number of ellipsoids
with their long axis in the direction of magnetization. Addi-
tionally, we have shown that the effect of changing the
damping parameter, often used for data fitting, is different
than changing the distributions for domains and grains. We
show that investigation of the low-loss region about the
gyromagnetic resonance frequency is very sensitive to
choice of the damping parameter a in the LLG simulation,
and that the values for a used in these and other simulations
should be carefully considered as to their physical signifi-
cance and relationship to the experimentally measured
linewidth.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the partial financial
support of the Defense Threat Reduction Agency, IACRO
10-4951I. The authors thank Jonathan Tedeschi and Justin
Fernandes for help with the network analyzer measurements
and Tim Droubay and Amandeep Chhabra for helpful discus-
sions. Pacific Northwest National Laboratory is operated by
Battelle Memorial Institute for the U.S. Department of
Energy under Contract No. DE-AC05-76RL01830.
1D. Polder, Philos. Mag. 40, 99 (1949).2E. F. Schloemann, J. Appl. Phys. 41, 204 (1970).3P. Gelin and K. Berthou-Pichavant, IEEE Trans. Microwave Theory Tech.
45, 1185 (1997).4P. Gelin, P. Queffelec, and F. Le Pennec, J. Appl. Phys. 98, 053906
(2005).5J. P. Bouchaud and P. G. Zerah, J. Appl. Phys. 67, 5512 (1990).6G. F. Dionne, IEEE Trans. Magn. 39, 3121 (2003).7J. P. Bouchaud and P. G. Zerah, Phys. Rev. Lett. 63, 1000 (1989).8D. Polder and J. Smit, Rev. Mod. Phys. 25, 89 (1953).9P. Gelin and P. Queffelec, IEEE Trans. Magn. 44, 24 (2008).
10E. M. Lifshiftz and L. P. Pitaveskii, Laudau and Lifshitz Course of Theo-retical Physics: Statistical Physics Part 2 (Pergamon, 1980).
11T. L. Gilbert, Phys. Rev. 100, 1243 (1955).12T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).13J. A. Osborn, Phys. Rev. 67, 351 (1945).14R. Skomski, IEEE Trans. Magn. 43, 2956 (2007).15E. C. Stoner and E. P. Wohlfarth, Philos. Trans. R. Soc. London, Ser. A
240, 599 (1948).16W. B. Weir, Proc. IEEE 62, 33 (1974).17N. J. Damaskos, R. B. Mack, A. L. Maffett, W. Parmon, and P. L. E.
Uslenghi, IEEE Trans. Microwave Theory Tech. 32, 400 (1984).18R. E. Caflisch, Monte Carlo and Quasi-Monte Carlo Methods (Cambridge
University Press, 1998), Vol. 7.19R. L. Scheaffer and J. T. McLave, Probability and Statistics for Engineers
(PWS-KENT, Boston, 1990).20A. J. Baden Fuller, Ferrites at Microwave Frequencies (Peter Peregrinus,
London, 1987), Vol. 23.21K. Hisatake, J. Appl. Phys. 48, 2971 (1977).22J. Krupka and R. G. Geyer, IEEE Trans. Magn. 32, 1924 (1996).23T. Tsutaoka, T. Kasagi, and K. Hatakeyama, J. Appl. Phys. 110, 053909
(2011).24J. Smit and H. P. J. Wijn, Ferrites: Physical Properties of Ferrimagnetic
Oxides in Relation to Their Technical Applications (John Wiley & Sons,
New York, 1959).
0.001
0.01
0.1
1
10μ'
'
86420Frequency (GHz)
Experimentα = 0.006
α = 0.0006
α = 0.06
FIG. 6. Imaginary of the diagonal component of permeability for demagne-
tized YIG for different values of the damping parameter a, where domain
and grain demagnetization factors were fixed to the beta A distribution.
023904-7 Chun, Jones, and McCloy J. Appl. Phys. 112, 023904 (2012)
25K. A. Korolev, J. S. McCloy, and M. N. Afsar, J. Appl. Phys. 111, 07E113
(2012).26V. Zhuravlev and V. Suslyaev, Russ. Phys. J. 49, 1032 (2006).27G. T. Rado, R. W. Wright, and W. H. Emerson, Phys. Rev. 80, 273 (1950).28R. Valenzeula, Magnetic Ceramics (Cambridge University Press, Cam-
bridge, UK, 1994).29J. X. Zhang and L. Q. Chen, Acta Mater. 53, 2845 (2005).30S. Mallegol, P. Queffelec, M. Le Floc’h, and P. Gelin, IEEE Trans. Magn.
39, 2003 (2003).
31M. Obol and C. Vittoria, IEEE Trans. Magn. 39, 3103 (2003).32M. Multani, P. Ayyub, V. Palkar, M. Srinivasan, V. Saraswati, R. Vijayar-
aghavan, and D. O. Shah, Bull. Mater. Sci. 6, 327 (1984).33N. Mo, Y.-Y. Song, and C. E. Patton, J. Appl. Phys. 97, 093901 (2005).34S. S. Kalarickal, N. Mo, P. Krivosik, and C. E. Patton, Phys. Rev. B 79,
094427 (2009).35R. C. LeCraw, E. G. Spencer, and C. S. Porter, Phys. Rev. 110, 1311
(1958).36J. Krupka, IEEE Trans. Microwave Theory Tech. 39, 1148 (1991).
023904-8 Chun, Jones, and McCloy J. Appl. Phys. 112, 023904 (2012)
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