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0304-8853/

doi:10.1016

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Journal of Magnetism and Magnetic Materials 301 (2006) 238–244

www.elsevier.com/locate/jmmm

Measurement of exchange anisotropy in exchange-bias bilayers

Jing-guo Hua,b,�, Guojun Jina, R.L. Stampsc, Yu-qiang Maa

aNational Laboratory of Solid State Microstructures, Nanjing University, Nanjing, 210093, ChinabCollege of Physics Science and Technology, Yangzhou University, Yangzhou, 225002, China

cSchool of Physics, University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia

Received 5 March 2005; received in revised form 15 June 2005

Available online 8 August 2005

Abstract

Recently, experimental results for ferromagnetic (FM) and antiferromagnetic (AFM) bilayers show that the

measured exchange anisotropy depends on what measurement is done. In this article, we theoretically investigate how

the interface exchange coupling including bilinear and biquadratic terms manifests itself in different measurements such

as hysteresis loop, ac susceptibility, and ferromagnetic resonance in FM/AFM bilayers. In the semi-infinite limit for the

AFM layer, the analytical expressions for measured exchange anisotropy from the hysteresis loop, ac susceptibility and

ferromagnetic resonance are derived. It is found that the intrinsic magnetic parameters, such as FM anisotropy

constant, AFM domain wall energy, and the interface exchange coupling including the bilinear and biquadratic

contributions, can play different roles in different measurement techniques. However, by comparing different

measurements the intrinsic magnetic parameters, the domain energy sAFM and the bilinear and biquadratic exchange

coupling constants JE1 and JE2, can be determined, respectively.

r 2005 Elsevier B.V. All rights reserved.

PACS: 75.70.�i; 75.30.Gw; 76.50.+g

Keywords: FM/AFM bilayers; Exchange anisotropy; Bilinear and biquadratic couplings

1. Introduction

In more than one decade, magnetic multilayersystems have attracted considerable interest due totheir potential applications such as recording mediaand switching devices. It is of vital importance to

$ - see front matter r 2005 Elsevier B.V. All rights reserve

/j.jmmm.2005.06.028

onding author.

address: hujgyz@263.net (J.-g. Hu).

understand the fundamental physical phenomenain multilayer systems for improving their perfor-mance. Recently, considerable interest has beenshown in ferromagnetic (FM) and antiferromag-netic (AFM) exchange coupling systems, because oftheir application to giant magnetoresistive spin-valve heads for high-density recording systems[1–5]. The exchange bias effect [6], which arisesfrom the interfacial exchange coupling between

d.

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J.-g. Hu et al. / Journal of Magnetism and Magnetic Materials 301 (2006) 238–244 239

a ferromagnet and an antiferromagnet, can bemodeled as an exchange anisotropy field addedvectorially to external fields. However, a satisfac-tory understanding of the phenomenon has notyet been fully developed. Initially, the exchange-bias effect was assumed to arise from the exchangecoupling at an uncompensated interface betweenthe FM and AFM layers [6]. This argument leads toan exchange field two orders of magnitude largerthan experimental measurements. Two models, therandom-field model by Malozemoff [7] and theplanar-domain-wall model by Mauri et al. [8], wereproposed to explain this difference. But these twomodels and their extensions [9–13] have onlyattained different degrees of agreement with exist-ing experimental results. In order to understandthe origin of exchange bias, several measurementtechniques, such as ac susceptibility and ferromag-netic resonance (FMR), other than the conven-tional hysteresis loop measurement where the shiftof the loop corresponds to the exchange bias, havebeen used to characterize its effect. The experi-mental results of the exchange anisotropy werefound to depend strongly upon the measurementtechniques.

In this paper, we will present a model for FM/AFM bilayers with interfacial exchange couplingincluding bilinear and biquadratic terms [10].From this model, we can derive analytical expres-sions for exchange anisotropy defined by hysteresisloop, ac susceptibility, and FMR. The effect ofbiquadratic coupling on the exchange anisotropyis studied in particular.

The paper is organized as follows: Section 2 isdevoted to the basic model and the total equili-brium energy for a FM/AFM bilayer system.Expressions for the exchange anisotropy constantare derived in Section 3, and in Section 4 thenumerical results are presented. A brief summaryis given in Section 5.

2. Model and formulation

We consider a perfect uncompensated interfacebetween the FM layer and the AFM layer, andassume that the FM layer with thickness tF isuniaxially anisotropic in the layer plane and thin

enough so that the FM moments can rotateuniformly in the presence of an applied field. TheAFM layer is also uniaxial anisotropic and itsthickness tAF can be regarded to be semi-infiniteallowing a domain wall to form at the AFM sideof the interface. The energy of the system per unitarea can be written as

E ¼ ½2pðMFM � nÞ2�H �MFM

� KFMðMFM � u=MFMÞ2�tFM

� sAFMMAFM � u=MAFM

� JE1MFM �MAFM=ðMFMMAFMÞ

þ JE2½MFM �MAFM=ðMFMMAFMÞ�2, ð1Þ

where the interaction of the external field with theAFM layer is neglected. The first three termsrepresent the demagnetization, the zeeman, andthe FM anisotropy energies, respectively, withMFM as the FM magnetization, KFM the uniaxialFM anisotropy constant and H the appliedmagnetic field; The fourth term refers to thedomain wall energy in the AFM layer withsAFM ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAAFMKAFM

p[10], where AAFM and

KAFM are the exchange stiffness constant anduniaxial anisotropy constant, respectively, but theMAFM is the net sublattice AFM magnetization atthe interface between FM and AFM layers; Thelast two terms are the bilinear and biquadraticexchange coupling energies with JE1; JE2 being thebilinear and biquadratic interfacial coupling con-stants, respectively. The magnetization are takento lie in the xy plane with the z axis normal tothe layer plane, and the FM easy magnetizationaxis is chosen to coincide with the AFM one. Twounit vectors n and u are used to represent thedirections normal to the layer surface (i.e., the z

axis) and of easy magnetization (along the x axis),respectively.In spherical coordinates, the total energy per

unit area of the FM/AFM bilayer can be written as

E ¼ 2pM2FMt2FM cos2 y�MFMHtFM sin y cosðf� bÞ

þ KFMtFM sin2 y sin2 f� sAFM cos a

� JE1 sin y cosðf� aÞ

þ JE2 sin2 y cos2ðf� aÞ. ð2Þ

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Here, the magnetic field is assumed to apply inthe xy plane with an angle b from the x axis. Theangle a denotes the AFM moment direction at theinterface between the FM and AFM layers relativeto the x axis in the xy plane. The angles y and fare the polar angle from the z axis and azimuthalangle from the x axis for the FM moment,respectively.

At equilibrium, the first derivatives of E withrespect to y, f and a must be equal to zero. Atrivial solution of the equilibrium equations isy ¼ p=2. It means that in general, MFM must lie inthe layer plane due to the strong demagnetizingfield of the thin FM layer. The angles f and a willbe determined from the following equations

MFMHtFM sinðf� bÞ þ KFMtFM sin 2f

þ JE1 sinðf� aÞ � JE2 sin 2ðf� aÞ ¼ 0, ð3Þ

and

sAFM sin a� JE1 sinðf� aÞ

þ JE2 sin 2ðf� aÞ ¼ 0. ð4Þ

At equilibrium, the energy of the system is thengiven by

E ¼ �MFMHtFM cosðf� bÞ þ KFMtFM sin2 f

� sAFM cos a� JE1 cosðf� aÞ

þ JE2 cos2ðf� aÞ, ð5Þ

where the angles f and a are determined byEqs. (3) and (4). Based on Eqs. (3)–(5), we canobtain the exchange anisotropy constant from acsusceptibility measurements, hysteresis loop mea-surements, and FMR measurements.

3. Exchange anisotropy constant and exchange bias

Experimentally, different measurement techni-ques are based on different physical entities.Hysteresis loop measurements may introducecomplications of the magnetic structure of theFM layer on exchange-bias energy measurement inthe interpretation of the exchange anisotropymeasurement since they may involve the irrever-sible switching of the magnetization of the FM

layer. Here the exchange coupling can be calcu-lated from Eq. (5) based on the Stoner–Wohlfarthapproach described in the paper of Mauri et al. [8].The ac susceptibility measurement is a reversiblemeasurement for the exchange anisotropy, inwhich only small rotations of the magnetizationare involved in the presence of a small in-planeapplied field. The FMR measurement is also areversible measurement for the exchange aniso-tropy, but assuming the demagnetization fieldð4pMFMÞ is much larger than the resonance field.Here, we will derive the exchange anisotropyconstant from hysteresis loop, ac susceptibilityand FMR measurements. Customarily, threeexchange anisotropy constants Jhl, Jac, and JFMR

are defined from hysteresis loop, ac susceptibility,and FMR measurements, respectively [14,15]. Inthe following, we show how the measured ex-change anisotropy constant depends on JE1, JE2

and sAFM in different measurement techniques.The ac susceptibility is defined in terms of the

oscillatory component of magnetization alongthe applied field. The ac susceptibility can bewritten as

w �qqH½MFM cosðf� bÞ�, (6)

where the value of f can be found from Eqs. (3)and (4). According to the definition of Strom et al.[16], the exchange anisotropy for this measurementJac is written as

Jac ¼ ðM2FMtFM sin2 bÞ=w, (7)

hence, we can obtain

Jac ¼sAFMðJE1 � 2JE2Þ

sAFM þ JE1 � 2JE2þ 2KFMtFM. (8)

The hysteresis loop measurement of the ex-change field HEB is usually carried out with theapplied field along the easy axis of FM layer. Inthe present model, we can define an effectiveexchange bias for the magnetization MFM perpen-dicular to the applied fieldH corresponding to f ¼p=2 and b ¼ 0 [17]. Therefore, HEB ¼ HjM¼0. Theexchange anisotropy Jhl is defined as

Jhl � ðMFMtFMÞHEB. (9)

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From the above definitions, and Eqs. (3) and (4),the exchange anisotropy of the hysteresis loopmethod is obtained directly

Jhl ¼

sAFMJE1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðsAFM þ 2JE2Þ

2þ J2

E1

q if JE1osAFM þ 2JE2;

sAFMðJE1 � 2JE2Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2AFM þ ðJE1 � 2JE2Þ

2q if JE14s

AFMþ 2JE2:

8>>>>><>>>>>:

(10)

Based on the energy described by Eq. (5), we canuse the method of Smit and Beljers [18] to calculatethe FMR frequency o as

og

� �2

¼ HR cosðf� bÞ þHE1 cos a cosðf� aÞ �HE2 cos a cos 2ðf� aÞcos aþ J 0E1 cosðf� aÞ � J 0E2 cos 2ðf� aÞ

þHFM cos 2f� �

� HR cosðf� bÞ þHFM cos2fþ 4pMFM

�

þHE1 cos a cosðf� aÞ �HE1sin

2ðf� aÞ½J 0E1 � 2J 0E2 cosðf� aÞ�

cos aþ J 0E1 cosðf� aÞ � J 0E2 cos2ðf� aÞ

�HE2 cos

2ðf� aÞ½J 0E2 sin2ðf� aÞ þ cos a�

cos aþ J 0E1 cosðf� aÞ � J 0E2 cos2ðf� aÞ

�, ð11Þ

where g is the gyromagnetic ratio, HE1 ¼ JE1=ðMFMtFMÞ, HE2 ¼ JE2=ðMFMtFMÞ are the ex-change coupling fields, HFM ¼ KFM=MFM theFM anisotropy field, J 0E1 ¼ JE1=sAFM, J 0E2 ¼

JE2=sAFM the reduced exchange coupling, andHR the FMR field [19–21]. The angles f and a arestill determined by Eqs. (3) and (4). For H alongthe exchange bias direction, the FMR measure-ment of exchange anisotropy can be defined as

JFMR ¼ ðMFMtFMÞHFMR

� ðMFMtFMÞ12 ½HRð0Þ �HRðpÞ�. ð12Þ

Combining with Eq. (11), one readily obtains

JFMR ¼

s2AFMJE1

ðsAFM þ 2JE2Þ2� J2

E1

if JE1osAFM þ 2JE2;

sAFMðJE1 � 2JE2Þ2

ðJE1 � 2JE2Þ2� s2AFM

if JE14sAFM þ 2JE2:

8>>><>>>:

(13)

One finds that the exchange anisotropy constantdetermined from the ac susceptibility measurementdepends on the FM layer anisotropy parameters,

but the results of the hysteresis loop and FMRmeasurements are independent of the FM layeranisotropy parameters while they all depend oninterface exchange coupling and energy of AFMdomain wall. This means that the KFM could bedetermined by the ac susceptibility measurementbased on comparing the results of two differentthickness samples. Namely, KFM ¼ ðJac1 � Jac2Þ=2ðtAFM1 � tAFM2Þ, where the Jac1ð2Þ stands forexchange anisotropies values by ac susceptibilitymeasurement with respect to FM layer beingtAFM1ð2Þ under the same preparation, respectively.

When the FM layer is isotropic and the biqua-dratic coupling at FM/AFM interface is absent,our formulae agree with the numerical results ofXi et al. [14].

4. Numerical results and discussion

Using Eqs. (7), (9) and (12), the dependences ofexchange anisotropies obtained by hysteresis loop,ac susceptibility and FMR measurements on JE1

and JE2 can be shown by numerical calculations.We take 2KFMtFM ¼ 1 and sAFM ¼ 1 for all thenumerical calculations in the following. Numericalresults in Fig. 1 show that, both Jac and Jhl

increase with increasing JE1 but decrease with JE2.The behavior of Jac and Jhl dependence uponJE1ðJE2Þ is similarity, but Jac is larger than Jhl

due to that the ac susceptibility measurementalways gives a value combining the exchangeanisotropy and the FM uniaxial anisotropy, whichcan explain why in Co/CoO bilayers the experi-mental exchange anisotropy obtained from ac

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0.0 0.1 0.2 0.3 0.4 0.50.0

0.5

1.0

1.5

2.0

2.5

3.0

Exc

hang

e an

sotr

opy

Biquadratic exchange coupling

Jac :

JFMR :

JE1=0.6;

JE1=0.6;

JE1=1.2;JE1=1.8

JE1=0.6;

JE1=1.2;

JE1=1.8

JE1=1.8

Jhl :

Fig. 2. Dependence of exchange anisotropy Jac, Jhl, and JFMR

on JE2 obtained from hysteresis loop, ac susceptibility, and

ferromagnetic resonance measurements, respectively.

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.5

1.0

1.5

2.0

Exc

hang

e an

sotr

opy

Bilinear exchange coupling

JE2=0; JE2=0.1; JE2=0.2JE2=0.2JE2=0.2

JE2=0.1;JE2=0.1;

JE2=0;JE2=0;JFMR :

Jac :Jhl :

Fig. 1. Exchange anisotropies Jac, Jhl and JFMR versus JE1

obtained from hysteresis loop, ac susceptibility, and ferromag-

netic resonance measurements, respectively.

J.-g. Hu et al. / Journal of Magnetism and Magnetic Materials 301 (2006) 238–244242

susceptibility measurement is so large [17]. How-ever for JFMR, there are two branches of exchangeanisotropy depending on relative magnitude ofJE1, JE2 and sE2.

In the weak interfacial exchange couplingregion, JE1osAFM þ 2JE2, JFMR increases rapidlywith increasing JE1 but decreases with JE2. On theother hand, JFMR decreases obviously with in-creasing JE1 but increases with JE2 in stronginterfacial exchange coupling region, JE14sAFM

þ2JE2. When JE1 ¼ sAFM þ 2JE2 called the criti-cal point, it distinguishes the weak and stronginterfacial exchange coupling regions, an irrever-sible transition occurs where the helical structureof AFM layer is no longer stable and resultsin the divergence of JFMR. At this point, theassumption that the behavior of AFM layer isreversible cannot be applied. When the biquadraticcoupling JE2 is considered, the critical pointwill be relaxed due to biquadratic coupling JE2

can damp effect of bilinear coupling JE1. More-over, we find that JFMR and Jhl always approachzero when JE1! 0 independent of JE2. It meansthat only biquadratic coupling JE2 cannot result inthe exchange bias. However, Jac depends on

biquadratic coupling JE2, because it is alwaysrelated to the FM uniaxial anisotropy. In theextreme cases, i.e., JE15sAFM and JE1bsAFM, themeasured exchange anisotropy Jhl and JFMR

approach JE1 and sAFM, respectively, but Jac ¼

JE1 þ 2KFMtFM or sAFM þ 2KFMtFM.In order to investigate the effect of biquadratic

coupling JE2, we plot the dependence of exchangeanisotropy on biquadratic coupling JE2 as shownin Fig. 2. We find tendencies that are similar forJac and Jhl. Both Jac and Jhl always decrease withincreasing JE2, but increase with JE1. Also withincreasing JE2, the decrease of Jac and Jhl becomesslow. However, for JFMR, it behaves differentlyand depends upon JE1. It can be classified abouttwo cases, one of them is that the exchangeanisotropy decreases with increasing JE2 for thesmaller JE1; the behavior of JFMR for the largerJE1 is contrary to that for the smaller JE1, in whichthe exchange anisotropy rapidly increases withincreasing JE2. The critical point, which distin-guishes their different behavior, depends on theJE2, as well as JE1. Further, when JE2 is largerthan JE1, the relative exchange bias can becomepositive from negative, because the JE2 contributesto positive exchange bias when JE1a0 [10].

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Table 1

The comparison between our results and the related experimental results

Samples Experimental results Our results

sAFM tFMHex Interfacial exchange

coupling

Interfacial exchange

coupling

NiFe=FeMn 0:4 erg=cm2 103Oenm 0:07 erg=cm2 0:078 erg=cm2

NiFe=NiO 0:42 erg=cm2 390Oenm 0:036 erg=cm2 0:031 erg=cm2

Fe=FeF2 4:1 erg=cm2 4:7� 103Oenm 1:1 erg=cm2 0:92 erg=cm2

Fe=FeF2 JFMR=Jhl ¼ 1:25 [20] JFMR=Jhl ¼ 1:12

J.-g. Hu et al. / Journal of Magnetism and Magnetic Materials 301 (2006) 238–244 243

5. Summary

In the limit of a semi-infinite AFM layer, wehave studied the exchange anisotropy of FM/AFM bilayer for hysteresis loop, ac susceptibility,and ferromagnetic resonance measurements. Ana-lytical expressions for measured exchange aniso-tropy were derived for the different measurements.We find that if a domain wall forms in the AFMlayer, the exchange anisotropy values from theabove three kinds of measurements are generallydifferent. However, when FM/AFM interfacialexchange coupling is very strong or very weakcompared with sAFM, the difference between themwill go to zero. In addition, the different behaviorof JFMR dependence upon the JE1 or JE2 can beused to identify the strong exchange interfacialcoupling from weak exchange interfacial coupling.Importantly, according to Eqs. (8), (10) and (13)the intrinsic magnetic parameters sAFM, JE1 andJE2 can all be determined by comparing differentmeasurements, respectively.

As examples, we would like to make a compar-ison between our analytic results and experimentsby taking NiFe=FeMn given by Refs. [8,22]NiFe=NiO by Refs. [23,24] bilayer, and Fe=FeF2

given by Refs. [25,26]. As shown in Table 1, ourresults are well in agreement with the experiments.As results, our suggestion about measurement ofthe related magnetic parameters are reasonable.

Acknowledgements

One of authors, Jingguo Hu, really appreciatesChina Scholarship Council for supporting me

working as academic visitor in University ofWestern Australia. This work was also supportedby the Provincial Natural Science Foundation ofJiangsu BK2002086, 03KJB140153, the State KeyProgram of China 2001CB610602, 10347118, andthe Natural Science Foundation 60371013.

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