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Analytical ferromagnetic hysterons with various anisotropiesIulian Petrila andAlexandru StancuCitation: J. Appl. Phys. 109, 083937 (2011); doi: 10.1063/1.3579448View online: http://dx.doi.org/10.1063/1.3579448View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v109/i8Published by theAIP Publishing LLC.Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/Journal Information: http://jap.aip.org/about/about_the_journalTop downloads: http://jap.aip.org/features/most_downloadedInformation for Authors: http://jap.aip.org/authors

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Analytical ferromagnetic hysterons with various anisotropies

Iulian Petrilaa) and Alexandru StancuAlexandru Ioan Cuza University of Iasi, Department of Physics & CARPATH, Bd. Carol I, nr. 11, Iasi,700506, Romania

(Received 2 March 2011; accepted 16 March 2011; published online 28 April 2011)

A new critical reflection on the anisotropic constraints of the ferromagnetic particles allow us to

analytically describe the behavior of complex ferromagnetic systems. The anisotropic constraints

of each individual ferromagnetic particle such as magneto-crystalline, shape, interface, defects,

domain wall, or other induced influences are described in a simplified manner. The first

approximation of anisotropy free energy density provides an analytical description of various

magnetization processes even in the case of very complex anisotropic influences. The hysteretic

behavior described by this model, including both reversible and irreversible processes, is presented

and discussed for the typical anisotropy cases observed in ferromagnetic materials: uniaxial,

biaxial, cubic, and orthorhombic. This practical method to model hysteresis for various types of

anisotropy could be fundamentally important for many studies that demand very efficient

algorithms at the level of single-domain magnetic elements. VC 2011 American Institute of Physics.

[doi:10.1063/1.3579447]

I. INTRODUCTION

The concept of a single-domain ferromagnetic particle

has been known in magnetism for many years.13 The critical

volume under which the magnetization processes of a ferro-

magnet are essentially linked to the rotation of the total mag-

netic moment vector of the particle can be calculated with

Browns micromagnetic theory.4 This theory, published

more than 50 years ago, gives an estimation of the nucleation

field when the coherent rotation is the first magnetization

mode which is activated at the highest value of the applied

field starting from positive saturation. It is remarkable to

note that the coherent rotation magnetization model for the

single-domain particle was given before Browns result was

published in the famous paper of Stoner and Wohlfarth.5

They have used a stronger condition for the single-domain

particle behavior that constrains the moments dynamics

only by coherent rotations in any applied field. The Stoner

Wohlfarth model (SW) was intensively used in many theo-

retical approaches as the most simple and efficient hysteresis

model. For uniaxial ferromagnetic single-domain particles

the critical curve approach introduced by Slonczewski6 is

widely used today, even if it gives the values of the magnet-

ization in certain applied fields only as a solution of a mathe-

matical equation that can only be numerically solved.

The development of nanotechnologies in recent years

has improved the experimental capacity to measure magnet-

ization processes even at the level of one single-domain fer-

romagnetic particle and a number of discrepancies with the

SW approach have been observed.79

Even if these discrepancies were expected, if one takes

into account the strong simplifications made in the SW

model, new fundamental discussions concerning the physical

basis of the model are rarely published and no fundamental

evolution can be noticed in this area.

Ideally, what we need in many areas of ferromagnetism

and in the modeling of devices using single-domain particles

is a more accurate model that can be solved mathematically

in a simpler way, if possible, with an analytical solution.10,11

These conditions are contradictory and thus, it is very diffi-

cult to simultaneously fulfill them.

In this paper we offer a possible solution to the

previously mentioned problem which is atthe same time

numerically efficient and relevant from the physical point of

view as an improvement of the uniaxial case12,13 and that

offers a straightforward generalization for other anisotropy-types.1416

To present the basis of this approach, we had to revisit

the fundamental discussion on the symmetry in the expres-

sion of the free energy density for the single-domain ferro-

magnetic particle. The anisotropic terms in this expression

can be developed in a series expansion in at least two ways;

one that gives the SW solution and one that can provide the

simpler solution we present in this paper.

In the following sections, we present the founding prin-

ciples of the method. Then we exemplify the model used for

different anisotropic influences: uniaxial, biaxial, cubic, and

orthorhombic. Finally we present the conclusions of our

study.

II. ANISOTROPY

The anisotropic influences on the equilibrium states of

the magnetic moment of a single-domain ferromagnetic par-

ticle are usually included, in a general way, by the phenome-

nological expressions of the anisotropy free energy.5,17,18

The orientation versor m M=M of the ferromagnetic par-ticles magnetization vector M, relative to the coordinate

axes, is given by the direction cosines ai as

a)Author to whom correspondences should be addressed. Electronic mail:

[email protected].

0021-8979/2011/109(8)/083937/6/$30.00 VC 2011 American Institute of Physics109, 083937-1

JOURNAL OF APPLIED PHYSICS 109, 083937 (2011)

http://dx.doi.org/10.1063/1.3579447http://dx.doi.org/10.1063/1.3579447http://dx.doi.org/10.1063/1.3579447http://dx.doi.org/10.1063/1.3579447

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m a1; a2; a3 sin h cos/; sin h sin/; cos h (1)

with h; / the spherical angles.Usually,5,19,20 the magneto-crystalline free energy den-

sity Wa is described by a power series expansion of the com-

ponents of magnetization

Wa Xi;j13A

0

ij aiaj ; (2)

where A0 represents the anisotropy coefficients for differentorders of approximations. As the anisotropy free energy den-

sity is invariant to the reversal ofM

WaM WaM; (3)then each term in the series expansion as in Eq. (2), must

include only even powers of any direction cosine. The free

term is independent of particle orientation and is usually

ignored, because normally we are only interested in the

change in the free energy when the M vector changes its ori-

entation. The classical series expansion of the anisotropy

energy (2), even in the simplest case of the uniaxial systems,

does not provide an analytical description of the magnetiza-

tion processes. We analyze the anisotropy energy series

expansion with the aim offinding a simpler description for

the characteristic unit of hysteresis called an hysteron.12,13

A ferromagnetic particle as a macro-spin can have mul-

tiple anisotropic constraints. The anisotropic constraints act

symmetrically on different directions which are named ani-

sotropic directions.

First we observe that, in the series expansion we also

have to consider the terms in the modulus of the direction

cosines of the magnetic moment, gij j

, relative to each anisot-

ropy direction. Consequently, the series expansion for the

anisotropy free energy density is given by

Wa X

i1NaAi gij j ; (4)

with Ai the anisotropy coefficients and Na the number of ani-

sotropy axes. The anisotropy directions can be related to any

kind of anisotropic influences such asmagneto-crystalline,

shape, interface, irregularities (defects), etc. With these con-

siderations, even a very complex case of a particle under the

influence of different types of anisotropic factors can be ana-

lytically described and the hysteretic behavior of the system

can be, in this way, handled properly. This method offers a

simple but still sufficiently realistic way to describe the mag-

netization processes of ferromagnetic particles with various

magneto-crystalline anisotropy types with a wide range of

external physical anisotropic constrictions. In the next sec-

tion, the general framework of the model is presented in

detail.

III. MAGNETIZATIONS PROCESSES

Any magnetization process of a single-domain ferro-

magnetic particle is the result of the interaction between the

applied magnetic field and the magnetic moment of that par-

ticle. In the quasistatic approximation, one looks for the

equilibrium states at a given applied field. Besides the anisot-

ropy free energy density term presented in the previous sec-

tion one also has to consider the interaction between the total

dipolar magnetic moment of the particle and the external

field which is given by the Zeeman energy density

WZ l0M H l0MSHa1b1 a2b2 a3b3 (5)with bi coswi and i 1 3 the direction cosines of theapplied field, H, and MS is the saturation magnetization.

The versor of the applied field can be written as

h H=H b1; b2; b3: (6)The magnetization processes are provided by the equilib-

rium,21 dWh; / 0, or@Wh; /

@h 0; @Wh; /

@/ 0; (7)

and the stability (the states where all the near variations oftotal energy density are positives)

dWh; / Wh dh; / d/ Wh; / > 0; (8)are conditions of the total energy density, W Wa WZ.

The normalized projection of magnetization on the

applied field direction is defined by m M=MS. With the an-isotropy term given by (4) the orientation of the particles

total magnetic moment, m, can be described analytically for

most anisotropy types.

Once the magnetic moment orientation at equilibrium it

is known, the hysteresis loops which represent the projection

of magnetization along the applied magnetic field have ananalytical description and are given in normalized form by

m m h a1b1 a2b2 a3b3: (9)

Generally, a system with multiple equilibrium states may

suffer transitions from one state to another with less energy.

These critical (transition) states correspond to the states

where the sign of near variation of total energy density

dWh; / is undefined. By considering the first term in (4),the switching conditions can easily be identified as the condi-

tions when the modulus terms in anisotropy free energy den-

sity (4) are zero (the nonderivability points)

gi 0; (10)

with i 1 Na.Because the particles moment on each hysteresis

branch is switched, the number of irreversible transitions on

each hysteresis branch can be up to the number of anisotropy

axes, Na (or the number of the switching conditions).

These general results allow us to calculate various mag-

netization processes like the major hysteresis loop (the most

typical measurement used in ferromagnetism) in a number of

particular cases. Because of the existence of the multiple

equilibrium states of the system, when a magnetization pro-

cess is analyzed, it is important to start from a well-defined

083937-2 I. Petrila and A. Stancu J. Appl. Phys. 109, 083937 (2011)


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state such as the saturation state in the case of the hysteresis

loop.

IV. UNIAXIAL HYSTERON

The uniaxial anisotropy represents the simplest case of

the ferromagnetic systems.

5,12,13,22

In the case of the present approximation (4), if one con-

siders Oz the symmetry axis one has Na 1 andbz is the ani-sotropy axis (g1 a3). The first approximation of theanisotropy energy (4) becomes

Wua Ku a3j j; (11)

with Ku A3 the anisotropy constant. The main charac-teristics of the uniaxial hysteron based on the anisotropy

energy (11) are listed in Table I.

Some hysteresis characteristics of the uniaxial hysteron

are described in Ref. 12 and a comparative analysis between

the classical SW hysteron and the uniaxial vector hysteroncan be found in Ref. 13.

In the case of uniaxial systems,13 the magnetization

processes can be reduced to a 2D form in the plane of zOH

and the reduced anisotropy energy becomes wua cos hj j.Considering w w3 the angle between the external field, H,and the symmetry axis (Oz), the hysteresis loops can be

described by m h6 cosw=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

h26 2h cosw 1p

,12 and

the hysteresis curves can be shown in Fig. 1.

By adding the second uniaxial anisotropy term in (4),

the anisotropy energy

Wu2a Ku cos hj j Ku2 cos2 h; (12)

can be used to describe a variety of the uniaxial struc-

tures8,10,23 and represents, in fact, a mixture of the present

analytical description (first approximation of the anisotropy

free energy) and the Stoner-Wohlfarth model.5

V. BIAXIAL HYSTERON

In the case of a biaxial hysteron we have Na 2, fbx; bzgthe anisotropy axes (g1 a1, g2 a3), and the first approxi-mation of the anisotropy energy becomes

Wba Kb ja1j ja3j (13)

TABLE I. Uniaxial hysteron proprieties: a) free anisotropy energy density,

b) free energy density, c) anisotropy field, d) magnetic moment orientation,

e) switching conditions, f) hysteresis loop, and g) switching fields.

a) wua Wua =Ku a3j jb) wu a3j j ha1b1 a2b2 a3b3c) Hua Ku=l0MS

d)^

m hb1; hb2; hb361

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffih262hb3 1pe) hb361 0f) m h6b3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffi

h262hb3 1p

g) hs 61=b3

FIG. 1. (Color online) Hysteresis loops for different external field

orientations.

TABLE II. Biaxial hysteron proprieties: a) free anisotropy energy density,


e) switching conditions, f) hysteresis loop, g) switching fields, and h)

notations.

a) wba Wba =Kb a1j j a3j jb) wb a1j j a3j j ha1b1 a2b2 a3b3c) Hba

Kb=

l0MS

d) m hb1 s1; hb2; hb3 s3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffih2 2hs1b1 s3b3 2

pe) fhb1 s1 0; hb3 s3 0gf) m h s1b1 s3b3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

h2 2hs1b1 s3b3 2p

g) hs f61=b1;61=b3gh) si sgnai 61

FIG. 2. (Color online) Hysteresis loops of biaxial systems for relevant

crystallographic directions are described by: m100 h61=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffi

h262h 2p ,m010 h=

ffiffiffiffiffiffiffiffiffiffiffiffiffih2 2p , m110 h61=

ffiffiffi2

p =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

h26ffiffiffi

2p

h 2p

, m101 h6ffiffiffi

2p =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffi

h262ffiffiffi

2p

h 2p

, a nd m111 h62=ffiffiffi

3p =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffih262=

ffiffiffi3

ph 2

q. For e ach

direction, the switching fields are: hs100 61, hs010 0, hs110 6ffiffiffi

2p

,

hs101 6 ffiffiffi2p , and hs111 6 ffiffiffi3p .



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with Kb the anisotropy constant. The main characteristics

of the biaxial hysteron are presented in Table II.

For different crystallographic directions of the applied

field the hysteresis loops are shown in Fig. 2.

In the biaxial plane xoz b2 0, b1 sinw, andb3 cosw, the hysteresis loops are described by

m h s1 sinw s3 coswffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffih2 2hs1 sinw s3 cosw 2

p ; (14)with hs f61=cosw;61=sinwg, the switching fields.

As shown in Fig. 3, for an applied field in the biaxial

plane, there are up to two switches on each hysteresis branch

and the system has no pure reversible behavior.

VI. CUBIC HYSTERON

The first approximation of the anisotropy free energy

density (2) for the cubic systems is given by5,17,22

Wc0a Kc0 a21a22 a21a23 a22a23

: (15)

Usually, the expression (15), developed for the simple cubic

systems or simple cubic with Kc0 > 0,20 is used for body-centered cubic systems with Kc0 KBCC0 > 0 and forface-centered cubic systems with Kc0 KFCC0 < 0, andalso with easy axis and hard axis interchanged. In the presentapproximation, for the cubic symmetry one has Na 3 withfbx; by; bzg the anisotropy axes (g1 a1, g2 a2, g3 a3)and the first approximation of the anisotropy energy (4)

becomes

Wca Kc ja1j ja2j ja3j ; (16)

with Kc A1 A2 A3 the anisotropy constant.The differences between the two approaches of the nor-

malized anisotropy energy can be visualized in Fig. 4. The

main characteristics of the cubic hysteron are presented in

Table III.

From the switching conditions one can always extract aswitching field, because cubic systems have no pure reversi-

ble directions.

FIG. 3. (Color online) Hysteresis loops of the biaxial hysteron for different

directions of applied field in the xOz plane.

FIG. 4. (Color online) Comparative normalized cubic anisotropy energy.

TABLE III. Cubic hysteron proprieties: a) free anisotropy energy density,


e) switching conditions, f) hysteresis loop, g) switching fields, and h)

notations.

a) wca Wca =Kc a1j j a2j j a3j jb) wc a1j j a2j j a3j j ha1b1 a2b2 a3b3c) Hca

Kc=

l0MS

d) m hb1 s1; hb2 s2; hb3 s3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih2 2hs1b1 s2b2 s3b3 3

pe) fhb1 s1 0; hb2 s2 0; hb3 s3 0gf) m h s1b1 s2b2 s3b3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

h2 2hs1b1 s2b2 s3b3 3p

g) hs f61=b1;61=b2;61=b3gh) si sgnai 61

FIG. 5. (Color online) Hysteresis loops of cubic systems for relevant crystallo-

graphic directions are described by: m100 h61=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffi

h262h 3p ,m110 h6

ffiffiffi2

p=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffi

h262ffiffiffi

2p

h 3p

, and m111 h6ffiffiffi

3p =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffih262

ffiffiffi3

ph 3

p.

For each crystallographic direction, the switching fields are: hs001 61,hs110 6 ffiffiffi2p , and hs111 6 ffiffiffi3p .



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In the case of the crystallographic directions the hys-teresis loops can be described analytically by adapting

Table III(f) to determine the direction of applied field.

For the crystallographic directions represented in Fig. 5

on each hysteresis branch, the hysteresis loops of the cubic

ferromagnetic particle presents one irreversible transition.

In the xOz plane (b2 0, b1 sinw, b3 cosw) thehysteresis loops are given by

m h s1 sinw s3coswffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffih2 2hs1 sinw s3cosw 3

p ; (17)with hs fs1= sinw;s3=coswg, the switching fields.

As shown in Fig. 6, the hysteresis loops are always irre-versible and on each hysteresis branch, for a given direction

of applied field, the particle can have one of two irreversible

transitions. This 2D representation of the 3D cubic system

can be used in some situations (as in a traditional way) as a

2D representation of biaxial anisotropy (see also, Fig. 3).

For an arbitrary orientation of the applied field, h

1=72; 3; 6, the switching fields are given by hs

f67=2;67=3;67=6

gand the hysteresis loop is described

by Table III(f).As shown in Fig. 7 the cubic ferromagnetic particle can

have up to 3 irreversible transitions on each hysteresis

branch.

From practical considerations, Hc Hcaffiffiffi

3p Kc ffiffiffi3p =Ps

can be used as a more realistic definition for the anisotropy

field of the cubic ferromagnetic particle.

VII. ORTHORHOMBIC HYSTERON

For the orthorhombic (triaxial) anisotropy,24 the usual

anisotropy free energy is given by W0a K01 a21 K

0

2 a2

2 K0

3 a2

3. From the present symmetry consideration(4) the orthorhombic system has: Na 3, and fx; y; zg theanisotropy axes (g1 a1, g2 a2, g3 a3). The expressionof the anisotropy energy of the orthorhombic system

becomes Woa A1ja1j A2ja2j A3ja3j. One may definethe effective anisotropy constant by Ko

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiA21 A22 A23

pand the reduced components of the anisotropy constants by

ai Ai=Ko; i 1 3. The main characteristics of theorthorhombic hysteron are presented in Table IV.

As in the cubic systems, the orthorhombic systems can

have up to 3 irreversible transitions on each hysteresis

branch.

These results can be extrapolated to other sources of ani-

sotropy, such as exchange bias systems which show asym-metries2527 or the magneto-elastic anisotropy5,2830 which

represent an example of a system with mixed anisotropy and

also with two nonorthogonal anisotropy axes.

VIII. CONCLUSIONS

From the symmetry considerations, the magnetization

processes of very complex ferromagnetic particles can be

analytically described in a first approximation.

For a given direction of the applied field, the numbers of

irreversible transitions on the hysteresis branch can be up to

the numbers of the anisotropy axis.

FIG. 6. (Color online) Hysteresis loops of a cubic system for different direc-

tions of applied field in the xOz plane.

FIG. 7. (Color online) Hysteresis loop of the cubic ferromagnetic particle

for an arbitrary orientation of applied field.

TABLE IV. Orthorhombic hysteron proprieties: a) free anisotropy energy

density, b) free energy density, c) anisotropy field, d) magnetic moment ori-

entation, e) switching conditions, f) hysteresis loop, g) switching fields, and

h) notations.

a) woa Woa =Ko a1ja1j a2ja2j a3ja3jb) wo 3i1 aijaij haibi c) Hoa

Ko=

l0MS

d) m hb1 s1a1; hb2 s2a2; hb3 s3a3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffih2 2hs1a1b1 s2a2b2 s3a3b3 1

pe) fhb1 s1a1 0; hb2 s2a2 0; hb3 s3a3 0gf) m h s1a1b1 s2a2b2 s3a3b3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffi

h2 hs1a1b1 s2a2b2 s3a3b3 1p

g) hs fs1a1=b1;s2a2=b2 s3a3=b3gh) si sgnai 61



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For a more accurate description one must use higher

terms for the approximation of anisotropy energy, but with

the price of a nonanalytical description.

ACKNOWLEDGMENTS

The work was supported by HiFi 12-093 Grants of

Romanian ANCS and was facilitated by the rAMONa com-

puter cluster of the AMON Interdisciplinary Platform ofAlexandru Ioan Cuza University of Iasi.

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