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: John.McCloy@pnnl.gov.

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.

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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-

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and grain demagnetization factors were fixed to the beta A distribution.

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