Solid State Communications 151 (2011) 952–955

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Solid State Communications

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

Thickness dependence of positive exchange bias inferromagnetic/antiferromagnetic bilayersXiao-Yong Xu a,b,∗, Yu-Jie Gao a, Yei-Li Wang a, Jing-Guo Hu a,∗

a School of Physics Science and Technology, Yangzhou University, Yangzhou 225002, Chinab School of Electronic Science and Engineering, Southeast University, Nanjing 210096, China

a r t i c l e i n f o

Article history:Received 10 December 2010Received in revised form29 March 2011Accepted 31 March 2011by E.L. IvchenkoAvailable online 16 April 2011

Keywords:A. FM/AFM bilayersC. Spin configurationD. Exchange bias

a b s t r a c t

For the ferromagnetic (FM)/antiferromagnetic (AFM) bilayers, both negative and positive exchange biasHE have been observed for low and high cooling field HCF , respectively. The thickness dependence of HEand coercivity HC have been investigated for the cases of negative and positive HE . It is found that thenegativeHE and the positive one have similar FM thickness dependence that is attributed to the interfacialnature of exchange bias. However, the AFM thickness dependence of positive HE is completely contraryto that of the negative one, which clearly demonstrates that the AFM spins play different roles for thecases of positive and negative HE . In particular, the AFM thickness of positive HE was first highlighted byan AFM spin canting model. These results should be attributed to the interfacial spin configuration afterfield cooling procedure.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

In ferromagnetic (FM)/antiferromagnetic (AFM) bilayers, hys-teresis loops will be shifted along the magnetic field axis and thecoercivity (HC ) is usually enhanced after the exchange bias (HE) isestablished [1,2]. This phenomenon originates from exchange in-teractions between FMandAFM layers. During the past decade, theHE in FM/AFM bilayers has been extensively investigated due to itsfundamental role in the development of spintronic devices, such asspin-valve sensors and tunnel junctions [3–5].

The main feature of normal HE is the displacement of thehysteresis loop to field that points opposite to the applied coolingfield (HCF) and is known as negative HE . Surprisingly, Nogués et al.in 1996 observed not only the usual negative HE but also anunexpected positive HE under large HCF [6]. Such an interestingphenomenon constitutes an important clue to understand themechanism that governs HE in general. In the previous studies,it has been found that the magnitude and sign of HE dependstrongly on the interfacial coupling [7–9], the strength [10–12]and direction [9] of HCF. Zhou et al. in 2005 concluded twoadditional distinguished features. First, HE has a crossover from

∗ Corresponding address: Yangzhou University, No. 180, Siwangting Road,Yangzhou City, Jiangsu Prov., China. Tel.: +86 0514 87975466; fax: +86 051487975467.

E-mail addresses: xxy@yzu.edu.cn, xuxiaoyong001@163.com (X.-Y. Xu),jghu@yzu.edu.cn (J.-G. Hu).

0038-1098/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.ssc.2011.03.035

negative values to positive ones at a critical value of the coolingfield (H0

CF). Second, HC has a maximum at H0CF, in addition to the

normal enhancement due to the HE [10]. However, Kirk et al.found three distinct coercivity behaviors during the transitionfrom positive to negative bias when varying the cooling field [13].The crossover from negative to positive HE was reproducedsuccessfully by different theories, which predict the formation ofAFM domains [14], canting of the AFM spins [9], uncompensatedspins in the AFM [15], or the symmetry breaking of AFM spins [11].However, HC as a function of HCF was investigated by fewtheories. For example, Du et al. concluded that HC depends weaklyon HCF, which is apparently different from the experimentalobservation [8]. Up until now, the microscopic origin and morefeatures related to positive HE are still discussed controversially.

In addition, it is well known that HE is an interfacial effect,which strongly depends on the constituent layer thickness. It hasbeen concluded from the effect of FM layer thickness (NF ) on HEthat due to the interfacial nature, HE is inversely proportional toFM layer thickness (NF ) and HC decreases with increasing NF [1].For example, Zhou et al. found that positive HE induced by thelarge HCF and HC are proportional to the inverse NF [10]. Recently,Tripathy et al. have also found that both HE and HC decreasewith the increase of FM layer thickness by using the anisotropicmagnetoresistance technique [16]. In contrast, for AFM layer, theevolution of AFM spins is closely related to the mechanism of HE ,but since very few direct techniques can be employed to probe theAFM spin motion due to the zero net magnetization in the AFMlayer, it is difficult to reveal the effect of AFM spins. Therefore, theeffect of the AFM layer thickness (NAF) on HE was investigated

X.-Y. Xu et al. / Solid State Communications 151 (2011) 952–955 953

extensively in recent years because it can provide rich informationto understand the role of the AFM spins [17–22]. For example,Zhou et al. found that with the increase of NAF, HE increasesmonotonously and HC goes through a maximum in NiFe/FeMnbilayers [21]. More recently, Bisio et al. have taken the thickness-dependent variation of AFM spin structure into account and arguedthat theHE evolves fromanuniaxially anisotropic state, inwhich itsmagnitude is a function of the field cooling direction, to anisotropicbehavior with the increasing Cr thickness in Fe/Cr bilayers [22].

In this paper, we have studied the HCF dependence of HEand HC for FM/AFM bilayers with various NF and NAF, thenhave presented the thickness dependence of positive HE and HC .Particularly, the key point different from previous micromagneticcalculations, is addressing the spin motion in AFM layer ratherthan only in exchange coupling interface. Our results on thicknessdependence of HE and HC are in excellent agreement with theexperimental phenomenons [17–22], which should be attributedto the interfacial spin configuration after field cooling procedure.

2. Model

The total magnetic energy of the system in ourmodel is writtenas

E = −

N−i=j

JijSi · Sj −N−i=1

[Ki(Si · ei)2 + µBgiSi · H], . (1)

The first term above represents the nearest neighbor exchangeinteraction, with constants Jij = JAF, JF and JF/AF for the AFM, FMfilms, and for the interface, respectively. The second term is theuniaxial anisotropy, with the easy axis in the ei direction, whereKi denotes the uniaxial anisotropy constant of FM or AFM. Thelast term is the Zeeman energy with an external magnetic fieldH , where gi (i = gF or gAF) denotes their respective gyromagneticratio, and µB is Bohr magneton.

We extend the previous model by considering spin canting ineach monolayer of AFM layer, rather than only in the interfacemonolayer [9]. The two magnetic spin vectors of the ith AFMmonolayer are labeled as Sα

i and Sβ

i , NAF denotes the number ofmonolayer in the AFM slab, and i = 1 labels the AFM interfacemonolayer. The vectors Sk are the classical spin vectors of the kthFM monolayer, NF denotes the number of monolayer in the FMslab, and k = 1 labels the FM interface monolayer. Thus, E =

EAF + EF/AF + EF may be written as

EAF = −JAF

NAF−1−i=1

Sαi · Sβ

i+1 +

NAF−1−i=1

Sβ

i · Sαi+1 + 2

NAF−i=1

Sαi · Sβ

i

−12KAF

NAF−i=1

(Sα

i · eAF)2 + (Sβ

i · eAF)2

−12µBgAF

NAF−i=1

Sαi + Sβ

i

· H, (2)

EF/AF = −JF/AF

Sαi=1 + Sβ

i=1

· Sk=1, (3)

EF = −2JFNF−1−k=1

Sk · Sk+1 −

NF−k=1

µBgF Sk · H. (4)

In Eq. (4) the FM anisotropy is ignored because it is very small.We allow FM spins Sk to rotate in each monolayer, parallel toeach other and to the interface. During the cooling process, allFM spins parallel to HCF. For the FM/compensated AFM bilayersin numerous experimental works, there is not one easy axis inthe AFM but rather a distribution of easy axes, and the bulkmagnetic order of the FM and AFM spins are perpendicular toeach other [14], which implies that the AFM easy axis direction

Fig. 1. Schematic diagram of the angles related to the FM magnetization, AFMsublattice magnetization and applied field in a ferromagnetic/antiferromagneticbilayer.

eAF is perpendicular to the direction of the applied magnetic field.Therefore, during the cooling process, theAFMspins are frozen intoa canted spin configuration, and the canting angle θα

i = θβ

i of eachmonolayer of AFM layer due to symmetry consideration. Fig. 1 isthe schematic diagram of the spin configuration of the FM/AFMbilayer after field cooling under the Néel temperature (TN ). Asshown in Fig. 1, under the cooling field HCF, the magnetizationof ith AFM monolayer will depart from the AFM anisotropy axisto approach HCF called canting. Here, the canting angle θα

i (θβ

i ) isthe angle between the magnetization of ith AFM monolayer andthe cooling field. At equilibrium, the canting angle θα

i (θβ

i ) can bedetermined by minimizing ε in Eq. (5).

When the determination of the hysteresis loop is carried out, forT < TN , the non-constant terms are related only to the FM, sincethe AFM spins are assumed to be frozen, this allows to write Eq. (1)as

ε = −

NF−k=1

cos(θk+1 − θk) − hNF−k=1

cos θk − κ1 cos θk=1

− κ2 sin θk=1, (5)

where θk is the angle between the spins in the kth FM monolayerand the HCF direction. We introduce the following dimensionlessquantities: ε = E/2JF , the applied field h =

12µBgFH/JF and the

effective interface coupling can be written as

κ1 cos θk=1 + κ2 sin θk=1 = −JF/AF

2JF

(cos θα

i=1 + cos θβ

i=1)

× cos θk=1 + (sin θβ

i=1 − sin θαi=1) sin θk=1

, (6)

where JF/AF ≈ JAF for AFM-type interface coupling. The magnetiza-tion curve, and subsequently the values of HE and HC , are obtainedby minimizing ε in Eq. (5). Using micromagnetic calculation basedon Eq. (5), the roles of AFM and FM thickness are addressed by con-sidering their effect on spin canting in the each monolayer of AFMlayer during field cooling process and on spin reversal of the FMlayer during field-measuring process, respectively [23].

3. Results and discussion

Fig. 2(a) shows the variation of HE as a function of HCFfor FM/AFM bilayers with different NF . With increasing HCF, HEchanges always from negative values to positive values at a criticalvalue of the cooling field H0

CF. Moreover, this H0CF for the crossover

954 X.-Y. Xu et al. / Solid State Communications 151 (2011) 952–955

a

b

Fig. 2. HE (a) and HC (b) versus HCF for FM/AFM bilayers with several different NF .

a

b

Fig. 3. HE (a) and HC (b) versus 1/NF for FM/AFM (NAF = 10) bilayers whenHCF = 0.3(2JF/µBgF ).

is almost independent of NF . Fig. 2(b) shows that for all NF , HCincreases with initially increasing HCF and reaches a maximum,then it decreases slowly to approach a constant as HCF is furtherincreased. Fortunately, the maximum HC is located near H0

CF. Fig. 3shows the FM thickness dependence of positive HE and HC whenHCF = 0.3 (2JF/µBgF ) and NAF = 10. HE is proportional to 1/NF ,demonstrating an interfacial nature of the exchange bias, as shownin Fig. 3(a). From Fig. 3(b), one can know that HC increasesnonlinearly with increasing 1/NF because the combined action ofboth the interfacial pinning and intrinsic coercivity of FM.

To clarify the influences of NAF on HC , HE and its crossover, wedescribe the HCF dependence of HE in Fig. 4(a) and HC in Fig. 4(b)for FM/AFM bilayers with several NAF. As shown in Fig. 4(a), HEshows also a crossover from negative values to positive values

a

b

Fig. 4. HE (a) and HC (b) versus HCF for FM/AFM bilayers with several different NAF .

a

b

Fig. 5. HE (a) and HC (b) versus NAF for FM (NF = 10)/AFM bilayers whenHCF = 0.3 (2JF/µBgF ).

withHCF increasing.Moreover, the critical value of the cooling fieldH0

CF for the crossover of HE decreases obviously with increasingNAF. Fig. 4(b) shows that for a certain NAF, HC undergoes thenonmonotonic variation with increasing HCF, which is similar tothat in Fig. 2(b). Moreover, the maximum HC is also located nearH0

CF, though H0CF decreases with increasing NAF. Fig. 5(a) shows the

AFM thickness dependence of HE when HCF = 0.3 (2JF/µBgF ) andNF = 10. With increasing NAF, HE , as a negative value, decreasesand becomes a positive value, then increases sharply and finallyapproaches to saturation. Moreover, the HE decreases linearlywith increasing 1/NAF, as shown in the inset. Fig. 5(b) shows the

X.-Y. Xu et al. / Solid State Communications 151 (2011) 952–955 955

effect of NAF on HC at HCF = 0.3 (2JF/µBgF ) and NF = 10. Withincreasing NAF, the HC reaches a maximum near the crossover ofHE , then decreases sharply. And the HC increases nonlinearly withincreasing 1/NAF, as shown in the inset. From Figs. 4 and 5, onecan find that in the case of negative HE , the HE decreases but HCincreases with increasing NAF, however for the positive HE , the HEincreases while HC decreases with increasing NAF. It indicates thatthe pinning effect of AFM layer plays different roles for the cases ofpositive exchange bias and negative one.

Below we will analyze the above features and reveal themechanism behind them. THE FIRST FEATURE: The H0

CF for thecrossover of HE is independent of NF , while decreases markedlywith increasing NAF. For the FM/compensated AFM bilayers, theAFM easy direction eAF is perpendicular to the direction of theapplied magnetic field in our model, and the sign change of HEwith HCF can be explained as a result of the competition of theZeeman energy of the AFM spins under HCF and the AFM couplingat the interface [8–10]. Therefore, we can explain why the H0

CFfor the crossover of HE is independent of NF , whereas dependsstrongly on NAF. For the larger NAF, the Zeeman energy of AFMspins under increasing HCF overcomes more easily the interfaceAFM coupling, thus align faster the AFM interface magnetizationalong HCF. The AFM magnetization along the FM magnetization isof ameta-stable state, i.e., in high energy state at low temperatures,then the positive HE can be induced. Hence, the increasing NAFmay reduce markedly H0

CF. THE SECOND FEATURE: Though H0CF

decreases with increasing NAF, the maximum HC occurs alwaysvery close to the point at which HE(H0

CF) = 0. Moreover, themaximum HC increases markedly with increasing NAF, whiledecreases with increasing NF . The coercivity maxima can generallybe observed at magnetic phase transitions [24]. It is in theintermediate region where HE is close to zero that we observe theenhancement of the coercivity. Here the interfacial AFM couplingis balanced just by the Zeeman energy of AFM spins and somefraction of the spins are aligned with the cooling field, whileothers remain in the low potential energy, AFM-coupled state.It is at this point of maximum frustration that we observe thelargest HC , whereas such frustration is extremely detrimental toHE . In other words, the AFM surface net magnetizations (MAF) splitinto an intermediate region, which are aligned either with FMmagnetizations (MF ) or in the original AFM-coupled configuration.Therefore, the enhancement of HC at H0

CF should be attributedto increased pinning of the propagating domain wall in the FMlayer resulting from the interfacial magnetic frustration. Thisunderstanding demonstrates further the previous result that thespin-flop coupling at interface can weaken or eliminate HE , butalways enhances HC [25]. Furthermore, it is inferred that theincrease of NAF can make interfacial magnetization faster accessto larger frustration. The maximum HC decreases with increasingNF , which is due to that the additional coercivity is an interfacialeffect. THE THIRD FEATURE: The dependences of the positive andnegative HE on NF are similar to each other, but dependences ofthe positive and negative HE on NAF are contrary to each other.The negative (or positive) HE and HC increase as NF decreases,which is due to the interfacial nature of the exchange biasing.But, they are influenced differently by NAF. In the case of negativeHE , the AFM coupling dominates, thus the AFM magnetization isaligned antiparallel to the FM magnetization and a low energystate is ‘‘frozen in’’ at TN . The increase of the Zeeman energy ofthe AFM spins resulting from increasing NAF can reduce the AFMnet magnetization antiparallel to HCF (negative M//

AF), then induceboth the decrease of negative HE and the enhancement of HC . Onthe other hand, for the positive HE , the Zeeman energy of the AFMspins dominates and AFM net magnetization along HCF (positiveM//

AF) increaseswith the enhancement of Zeemanenergy of theAFMspins resulting from increasing NAF, thus the positive HE increasesand HC decreases. These results confirm further our previousconclusion [9] that the positive/negative HE increases with the

positive/negative M//

AF increasing. Moreover, It is implied that theadditional coercivity increases with the positive (or negative) M//

AFdecreasing, except for some structural factors such as intrinsicatomic structure, microtopography and anisotropy style.

4. Conclusions

In summary, the cooling field dependence of HE , HC and theeffects of both ferromagnetic and antiferromagnetic layers thick-ness on them have been investigated, respectively. For FM/AFM bi-layers with AFM interface coupling, HE can change from negativevalues to positive ones with increasing HCF and HC acquires a max-imum near the crossover of HE , which is in good agreement withexperimental observation [10].Moreover, the critical value of cool-ing field H0

CF, in which HE changes from negative values to positiveones, is independent ofNF , however it decreasesmarkedlywith in-creasing NAF. The results also show that the positive HE increaseslinearly with increasing 1/NF but decreases linearly with increas-ing 1/NAF. However,HC always nonlinearly increases with increas-ing of 1/NF or 1/NAF. The detailed results are implied that boththe increased coercivity and the exchange bias depend stronglyon the interfacial spin configuration after field cooling procedure.Namely, with the AFM positive/negative net magnetization M//

AF

(positiveM//

AF denotes AFM layer net magnetization along FM layermagnetization MF , and negative one is of opposite) increasing,the positive/negative HE increases, but HC decreases. On the otherhand, the increase of perpendicular AFM layer net magnetizationM⊥

AF (M⊥

AF denotes AFM layer net magnetization perpendicular toFM layer magnetization MF ) can weaken HE but enhance HC . As aresult, the pinning effect of AFM layer plays different roles for thecases of positive exchange bias and negative one.

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China Grant No. 10974170, the Scientific ResearchLaunching Foundation of Ministry of Education for returnedoverseas scholar Grant No. 2007/1108 and the Innovation NurtureFoundation of Yangzhou University Grant No. 2010CXJ005.

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