Physica B 406 (2011) 906–910

Contents lists available at ScienceDirect

Physica B

0921-45

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/physb

Hysteresis characteristics of an analytical vector hysteron

Iulian Petrila, Alexandru Stancu n

Alexandru Ioan Cuza University of Iasi, Department of Physics and CARPATH, Blvd. Carol I, Number 11, Iasi 700506, Romania

a r t i c l e i n f o

Article history:

Received 27 August 2010

Received in revised form

30 November 2010

Accepted 10 December 2010Available online 15 December 2010

Keywords:

Vector hysteresis

Magnetic anisotropy

Micromagnetics

Stoner–Wohlfarth model

26/$ - see front matter & 2010 Elsevier B.V. A

016/j.physb.2010.12.025

esponding author.

ail address: alstancu@uaic.ro (A. Stancu).

a b s t r a c t

The analytical way to describe uniaxial single-domain particles, using new symmetry considerations on

the magnetic anisotropy, as a vector hysteresis unit (vector hysteron), is presented. The main

characteristics of the vector hysteron such as the longitudinal hysteresis loops, transversal hysteresis

loops and rotational hysteresis loops are presented. An extension of the vector hysteron and a vector

hysteron that can be applied to the ferroelectric hysteresis are also presented.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

A vector hysteresis model for the ferromagnetic materials,should describe accurately both the magnitude of the magnetiza-tion and its direction when a magnetic field is applied to such amagnetic medium. Vector hysteresis models have been developedusing different methods [1–4] but still with significant limits inaccuracy, physical relevance or computational efficiency. Essen-tially, the main element in the development of a high-quality vectorhysteresis model is to have an appropriate vector hysteresis unit(vector hysteron) associated with each physical particle or to agroup of particles (in the second case the hysteron is also namedpseudo-particle [2,5–7]). The vector hysteron must describe effi-ciently the behavior of the ferromagnetic domain (or single-domain particle) as an isolated particle but also as the representa-tive part (subsystem) of the multi-particulate ferromagnetic sys-tem as an assembly of hysterons. A numerically inefficient vectorhysteron will certainly produce a very inefficient model for anensemble of a large number of hysterons [8,9]. Consequently, it is ofparamount importance that the fundamental brick of a model forlarge systems is a vector hysteron model with a considerablenumerical efficiency and as much accurate as possible, from thephysical point of view.

The most relevant vector hysteresis unit used to describe thebehavior of a uniaxial ferromagnetic particle is provided by theStoner–Wohlfarth model (SW) [10]. However, the SW hysteron hassome inconveniences for description of multi-particulate systems.Even for an isolated ferromagnetic particle the SW hysteron is not

ll rights reserved.

providing an accurate switching field [11–14]. Another observedlimitation is the cross-over problem indicating that hysteresisbranches may intersect if the angle of applied field relative to theanisotropy axis is close to 901 [15]. Also, for an applied fieldperpendicular to the easy axis, the SW particle reaches thesaturation (becomes perfectly aligned on the applied field direc-tion) for a finite applied field and not asymptotically for intensefields as observed in many experiments. As an element in a multi-particulate ferromagnetic system the SW hysteron is computa-tionally slow because the particle equilibrium orientation must beobtained numerically [16,17].

By reviewing symmetry considerations concerning the mag-netic anisotropy, as it is presented in this paper, one obtains ananalytical way to describe uniaxial single-domain particles andimplicitly one provides a relevant vector hysteron from the point ofview of the criteria discussed previously. Because in the case of thevector hysteron obtained in this way the magnetization orientationand the switching fields or switching orientations are analyticallycalculated, all the hysteresis characteristics of the uniaxial ferro-magnetic particle associated with the vector hysteron can be easilyobtained. One can even develop from this vector hysteron modelcharacterized by a constant macrospin (coherent rotations) anapproximation that may also account for non-coherent rotations byincluding in a simple way the deviation of some individualmoments from the particle moment orientation. This vectorhysteron model can be applied for example to describe thehysteresis of ferroelectric particles that are not obeying to the rulethat the total moment of one particle is constant during anypolarization process, as in the case of single-domain ferromagneticparticles [1].

In the next sections we present the implementation of thevector hysteron model, the main characteristics of the vector

I. Petrila, A. Stancu / Physica B 406 (2011) 906–910 907

hysteron, the hysteresis curves (longitudinal, transverse and rota-tional) and the non-coherent rotations extension for ferromagneticor ferroelectric particles.

Fig. 1. Particle’s magnetic moment equilibrium angle for different external field

orientations.

2. Vector hysteron model

The ferromagnetic particles’ hysteresis in the macrospinapproximation is essentially anisotropic. The anisotropic influ-ences are usually included with phenomenological expressionsthat take into account the magnetic symmetry, the geometricsymmetry (in the case of shape, surface and interface anisotropy)and the crystal symmetry (in the case of magneto-crystallineanisotropy) [10,18,19]. The anisotropic influences on the ferro-magnetic systems are accounted by the anisotropy free energydensity Wa, as a function of the magnetization M, which must obeythe symmetry condition

WaðMÞ ¼Wað�MÞ ð1Þ

which means that, there is no energy difference for oppositemagnetized systems. Because, in the simple case of the uniaxialparticle, there is a single symmetry direction, the anisotropy freeenergy density must depend on the direction cosine of themagnetization orientation relative to the anisotropy direction by

WaðcosyÞ ¼Wað�cosyÞ ð2Þ

where y is the angle between magnetization orientation andanisotropy direction. This is solved usually as a series expansionin the even powers of cosine (or sine) functions. However, themagnetic symmetry conditions (2) allow to expand the anisotropyenergy also as a power series expansion of the modulus of thedirection cosine 9cos y9 by

Wa ¼�X1n ¼ 1

Kn cosy�� ��n ð3Þ

where Kn is the anisotropy constant and the sign is present fromphysical considerations. By using Eq. (3) the first approximationterm in the free energy density series expansion is

Wa ¼�K cosy�� �� ð4Þ

which is the main assumption in the development of our model forthe analytical uniaxial vector hysteron.

The interaction between particle and applied field is given bythe Zeeman energy

WZ ¼�PUH¼�PSHcosðc�yÞ ð5Þ

wherec is the angle between the external field H and the easy axis.For the sake of simplicity one normalizes all the energetic

expressions at the anisotropy constant K. Also one defines theanisotropy field as HK ¼ K=PS and the projection of the normalizedmagnetization or polarization on applied field direction bym¼M=MS ¼ p¼ P=PS ¼ cosðc�yÞ. The anisotropy field HK isdefined as the value of the switching field for the rectangularhysteresis (c¼ 0). The normalized magnetic field is defined byh¼H=HK and the expression of total normalized free energybecomes

w¼� cosy�� ���hcosðc�yÞ ð6Þ

From the equilibrium condition

wu¼ @w=@y¼ 0 ð7Þ

one obtains the orientation of the magnetization, which is given by

y¼ arctansinc

coscþs=hþarccoss ð8Þ

with s¼ sgnðcosyÞ ¼ 71. From the non-derivability points of theanisotropy energy y¼ 7p=2, using Eq. (8), one obtains the

switching condition

hcoscþs¼ 0 ð9Þ

Once the orientation of the particle at equilibrium and theswitching condition are known [20], all the magnetization processesof the present uniaxial ferromagnetic particle can be obtained.

3. Longitudinal hysteresis curves

The longitudinal hysteresis loops are obtained as the projectionof the magnetic moment on the applied field constant directionwhen the field value is modified. In fact, this describes the classicalmeasurement of the hysteresis loop of a ferromagnet using atypical magnetometer.

In this case, the orientation of the particle moment relative tothe easy axis, given by Eq. (8), has a hysteretic behavior and tends tobe aligned asymptotically for intense fields (see Fig. 1).

The equilibrium orientation hysteresis is rectangular for c¼ 01and perfectly reversible for c¼ 901. The longitudinal hysteresisloops can be described analytically using Eq. (8) and the normalizedmagnetization expression

m¼h7coscffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

h272hcoscþ1p ð10Þ

with the plus sign for the upper branch and the minus sign for thelower branch. From the critical conditions (9) one obtains thecritical fields

hs ¼ 71=cosc ð11Þ

The switching field expression (11) is similar to the expressionobtained for the bulk material with a hysteretic behavior essentiallydue to the domain-wall motion [21] but in our case we haveconsidered a single-domain particle. The switching field dependenceon the angle between the easy axis and the applied field directionshows a minimum when the field is applied along the easy axis and amonotonous increase as the angle increases towards 901. As it is wellknown, a classical Stoner–Wohlfarth approach gives a dependenceof the critical field on the orientation angle with a minimum at 451and with equal maximum values at 01 and 901 (see Fig. 2).

However, even if the SW model was really successful indescribing in a simple form the hysteresis of a single ferromagneticparticle, it was observed repeatedly [12,13,22–24] that the model isnot able to describe correctly the angular dependence of the critical

Fig. 2. Switching fields for the present vector hysteron (VH) and for Stoner–

Wohlfarth hysteron (SW).

Fig. 3. Longitudinal hysteresis loops for different external field orientations.

Fig. 4. Transverse hysteresis loops for different external field orientations.

I. Petrila, A. Stancu / Physica B 406 (2011) 906–910908

field, which shows in many experiments a monotonic increase withthe angle. This behavior is more in agreement with our result andcan be an indication that there is a more significant physical basisfor this simple hypothesis, idea which should be analyzed moreprofoundly. We consider however that this aspect is beyond themain purpose of this paper and shall be analyzed in thefuture paper.

The longitudinal hysteresis loops for different external fieldorientation are transformed from the rectangular form (c¼ 01) toreversible form (c¼ 901), as can be seen in Fig. 3. For c¼ 901, thehysteresis curve is given by mðc¼ 901Þ ¼ h=

ffiffiffiffiffiffiffiffiffiffiffiffiffih2þ1p

(purereversible case).

When the applied field reaches the critical field given by Eq. (11)the particle magnetization orientation is changed (see Fig. 1) by

Dy¼ yþ ðhsÞ�y�ðhsÞ ¼ arccot1

2

sinccosc

� �ð12Þ

which is reflected in a jump on hysteresis loop given by

Dm¼mþ ðhsÞ�m�ðhsÞ ¼1þcos2cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ3cos2c

p �sinc ð13Þ

The jump from lower to upper branch of the hysteresis loop ispositive and unlike Stoner–Wohlfarth case [8] the cross-over effectis not present.

4. Transverse hysteresis curves

The transverse hysteresis loops measures the projection ofmagnetization perpendicular to the applied field directionm? ¼M?=MS ¼ sinðc�yÞ. Using the equilibrium orientation ofthe magnetization given by Eq. (8), the normalized transversecomponent of the magnetization becomes

m? ¼ 7sincffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

h272hcoscþ1p ð14Þ

with the plus sign for the upper branch and the minus signfor the lower branch (see Fig. 4). For c¼ 901 the transversemagnetization is decreasing asymptotically for intense fieldsm?ðc¼ 901Þ ¼ 71=

ffiffiffiffiffiffiffiffiffiffiffiffiffih2þ1p

.

The switching field is the same as in the longitudinal case (11)because the process is essentially the same (just the projection ofthe magnetization differs).

5. Rotational hysteresis curves

Another important magnetization process, which evidencesessentially the vectorial magnetic properties, is the rotationalhysteresis loop obtained by measuring the magnetic momentumon the field direction while a constant magnetic field is rotated.The particle equilibrium orientation is given by Eq. (8) butin this case the orientation of the applied field is continuouslychanging.

When a rotational constant magnetic field is applied, theparticle has a small reversible deviation around the easy axis forweak fields and follows the applied field orientation for intensefields as it can be seen in Fig. 5.

Fig. 5. Particle angle orientation function by the field orientation for different

applied fields in the case of rotational hysteresis loop.

Fig. 6. The rotational hysteresis loops for different values of the applied fields.

Fig. 7. Hysteresis loops for different external field orientations in the case of

ferroelectric particle.

I. Petrila, A. Stancu / Physica B 406 (2011) 906–910 909

The rotational hysteresis curves can also be described by Eq. (10)but with c variable and hconstant (see Fig. 6).

For hr1 the switch is not present (the pure reversible case) andfor h41 the switch is always present

cs ¼ fcs0,p7cs0,2p�cs0g ð15Þ

with cs0 ¼ arccosð1=hÞand the moment jump is given by

Dm¼mþ ðcsÞ�m�ðcsÞ ¼1

h

h2þ1ffiffiffiffiffiffiffiffiffiffiffiffiffih2þ3p �

ffiffiffiffiffiffiffiffiffiffiffiffih2�1

p� �ð16Þ

For h¼ 0 one has the pure cosine curves (m¼ cosc) and forh¼1 (very intense fields) the particle is perfectly aligned onapplied field direction (m¼ 1).

6. Non-coherent rotations extension

For the real particles, the concept of coherent rotations ormacrospin consideration is an idealization because not all theelementary moments within a particle are aligned at the same time.For instance, the moments from the particle’s surface are notaligned in the same way as the moments from the core of the

particle. Because for an intense applied field, all the individualsmoments are aligned on fields directions one may considerfor the particle magnetization a small dependence on the appliedfield

MsðhÞ ¼Ms0RðhÞ ð17Þ

with RðhÞ representing saturation function with a reversiblebehavior and with a small variation near to unity (RðhÞ ¼ 1 forcoherent rotation case).

A simple extension accounting for non-coherent rotations of thevector hysteron can be obtained using for the normalized projec-tion of the magnetization the expression

m¼ RðhÞh7coscffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

h272hcoscþ1p ð18Þ

If one makes a statistics of the non-aligned particles, one mayput the saturation function RðhÞ, for instance, in a Gaussian form. Inthis way one obtains a more accurate description of the single-domain particle’s hysteresis.

This extension of our model is very appropriate for example tothe ferroelectric hysteresis. If one refers at a ferroelectric particle(or domain) [25], for each electric dipole, the displacement of theelectrical charge centers depends on the applied field and conse-quently can be modelled in the same way. If one considers a lineardependence of the polarization versus the effective field:P:¼ P0ð1þx etotÞ with etot the normalized effective field, onehas for the projection of the normalized polarization on appliedfield direction the expression

p¼ ðe7coscÞ1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

e272ecoscþ1p þx

!ð19Þ

where e¼ E=Ea is the normalized applied field, Ea is the anisotropyfield, x a corrective parameter (x¼ 0 for pure constant macrospincase) and the double-signs pack the upper branch (plus sign) andlower branch (minus sign) into the same expression.

The hysteresis loops in the case of vector hysteron used as vectorhysteresis unit for a ferroelectric system are shown in Fig. 7, andcan be used as a simple way to model ferroelectric structures [26].The present approach describes also the effects of the boundarieson the ferroelectric domains.

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7. Conclusions

From very simple symmetry considerations we have obtained ageneralization of the free anisotropy energy for the uniaxialferromagnetic particle. In the first approximation an analyticalformula was obtained for the hysteresis of a single ferromagneticparticle. This vector hysteron allows also a simple extension toaccount for non-coherent rotations and that can be applied todescribe for example the hysteretic behavior of ferroelectricparticles. This analytic vector hysteron can be used in a veryefficient manner to develop a fully vectorial model of a multi-particulate system.

Acknowledgments

This research was supported by NANOMAT 72-186/2008 CNMP-ANCS Romania Grant and was facilitated by the rAMONa computercluster of the AMON Interdisciplinary Platform of Alexandru IoanCuza University of Iasi.

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