PHYSICAL REVIEW B, VOLUME 65, 054406

Size dependence of exchange bias in ferromagneticÕantiferromagnetic bilayers

S. Zhang and Z. LiDepartment of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211

~Received 11 October 2001; published 27 December 2001!

When the lateral size of exchange-coupled ferromagnetic-antiferromagnetic bilayers is reduced to a deepsubmicron size~below 0.2mm!, both the exchange bias and the coercive field increase compared to those ofmacroscopic samples. We attribute this size dependence to the increasing importance of the magnetostaticinteraction. By using the Landau-Lifshitz-Gilbert equation, the size dependence of the hysteresis loops ofpolycrystalline films is calculated and quantitative pictures are established to describe the crossover from smallto large samples.

DOI: 10.1103/PhysRevB.65.054406 PACS number~s!: 75.60.2d, 75.70.Cn, 75.70.Kw

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An exchange interaction at the interface of a ferromnetic ~F! and an antiferromagnetic~AF! layer results in sev-eral unique macroscopic magnetic properties. Among mdistinct experimental observations, the most interesting prerties are the shift of the hysteresis loop and the enhancoercivity of the ferromagnetic layer.1 A number of theoret-ical models have been developed to explain the originsthese phenomena.2–10 Most of these theories address the echange bias in a macroscopic sample where the magnstatic~MS! interaction in the ferromagnetic layer is expectto play insignificant roles in exchange-biased films. Indeit has been experimentally shown that the exchange biasenhanced coercivity ofF/AF layers are independent of thlateral size for lateral sizes larger than 0.5mm.11 In a largefilm, the magnetization reversal of the ferromagnetic layethe exchange-biased film is mainly controlled by interfaciexchange coupling betweenF and AF layers. The interfacialexchange interaction, which is usually random for a pocrystalline film, creates random pinning centers duringhysteresis process of magnetization reversal. This resulrandom-field-induced magnetic domains in the ferromagnlayers and the magnetic properties are strongly correlawith these domains.7 The MS energy between each domainsmall when one averages over the area of a macroscfilm. Therefore, it is justified that the theoretical modelsexchange bias do not need to include this complicatedinteraction for a large film.

As the size of bilayers is shrunk down to a few nanoeters or to a deep submicron size, which will be requirednext-generation magnetic nanostructure devices, the matostatic interaction is increasingly important in determinithe reversal behavior of the ferromagnetic layer. The Minteraction competes with other energies~interface randomenergy and ferromagnetic exchange energy!; this leads toformation of new domain structures during the magnetizatreversal, and both exchange-bias shifts and coercivitiessignificantly altered as the size of the film decreases. Inpaper, we study these new domain structures in the presof the random field and MS interaction, and determine mnetic properties of the exchange-biased film as a functionsample size. In particular, we introduce a critical sample sunder which current theoretical models that neglect magtostatic interactions fail to describe magnetic propertiesexchange-biased films.

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We consider a polycrystalline exchange-biased film csisting of antiferromagnetic grains whose lateral size isD. Tofocus our attention on the interplay between the magnestatic energy and the interfacial random field, we chooseanisotropic energy of the AF grains large enough so tirreversible transitions of the AF grains are minimized. Tenergies involved in our study are the exchange and antropy energies of theF and AF layers, the random interfaciaexchange coupling betweenF and AF layers, the MS energof the F layer, and Zeeman energy due to the external mnetic field. The magnetic hysteresis is calculated by solvthe standard Landau-Lifshitz-Gilbert~LLG! equation usingthe following procedure. The sample was laterally dividinto N3N blocks. Each block represents a grain that consof n3n atomic spins in each plane parallel to the interfacethat the grain size isD5na0 , wherea0 is the lattice con-stant. The sample size~length! is thusLN5ND. We choosethe sample to be a square shape. In the layer thickness dtion, we assume theF layer to be uniform; this assumptiomay limit our model to the cases where theF layer is thinand the anisotropy of theF layer is small. Since the AF grainsize in typical experiments is of the order of 10 nm, we neto choose grid size smaller than the grain size in orderobtain reliable results from micromagnetic computations12

In our calculation, we divide each grain into 434 grids sothat the grid size is 2.5 nm for a 10 nm grain size. Thelayer has a fixed anisotropic axis for a given grain~the di-rection of the anisotropy fordifferentgrains is random and isin the plane of the layer!. Note that we do not freeze thmagnetization of the AF layer, instead, the direction of eaatomic AF sublattice moment near the interface slightlytates during magnetization reversal of theF layer. As wehave taken the AF anisotropic constant to be large~see thecaption of Fig. 1!, these variations are, nevertheless, qusmall. The explicit forms of each energy term in the abseof magnetostatic energy have already been reported.7 Themagnetostatic energy between ferromagnetic grids is calated via the standard fast Fourier transform technique.

Before we present our numerical results on the sizependence of the hysteresis loops, it is interesting to quatively estimate the relative importance of the interfacial echange interaction and MS interaction. Let us assumethe ferromagnetic layer breaks up into domains duringmagnetization reversal. The average domain sizeL is deter-

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S. ZHANG AND Z. LI PHYSICAL REVIEW B 65 054406

mined by minimizing the ferromagnetic exchange energy,random exchange interaction at theF/AF interface, and theMS interactionEd ,

E5~1/2!tFJFLN2 /L22JsLN

2 /L1Ed , ~1!

where tF is the thickness ofF layer, JF and Js are the ex-change constants inF layer and at the interface, andLN is thesample size. Here we have discarded the anisotropic enof the F layer since it is usually very small for the somagnetic materials, e.g., Permalloy, considered here. Inabsence ofEd , the domain size is clearly independent of tsample sizeLN and one obtainsL'tFJF /Js by minimizingthe first two terms of Eq.~1!.7 In the presence ofEd , how-ever, one expects that the domain size will be reduced. W

FIG. 1. Hysteresis loops of exchange-biased films for threeferent lateral size~a! LN5400 nm, ~b! LN5250 nm and~c! LN

5140 nm. TheF layer thickness istF524 nm and the AF grain sizeD510 nm. The solid-circle and open-triangle loops are with awithout the magnetostatic interaction. The magnetization patternthe bottom panels for the coercive points atC1 andC2 as markedin the figure. Note that each arrow represents the average direof magnetization of 434 grids. The parameters are the samethose used in Ref. 5: the ferromagnetic exchange constanJF

516 mev, the average interfacial-exchange constantJs52 mev, theAF exchange constantJAF52 mev, the anisotropic constantKAF

53 mev, and the thickness of AF layer is 5 ML.

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the precise form ofEd depends on detailed domain patternwe simply estimateEd by a Neel wall whose length is thesample length and whose width is the order of the domsize L, i.e., Ed5KmtF

2L2/(tF1LN), whereKm5(1/2)m0Ms2

(Ms is the saturation magnetization!.13 By minimizing theenergy of Eq.~1!, we obtain a new domain size. If we compare the domain size due to the random interfacial eneand that due to MS energy separately, we obtain from Eq.~1!that

LN5S 2JF3Km

Js4 D 1/3

tF5/3 ~2!

is the critical sample size such that the two mechanismsequally important. Note that the critical sample size depeon the thickness of theF layer, this is because the magnetstatic energy increases when theF layer thickness increasewhile the interfacial-exchange interaction is independentthe F layer thickness. From Eq.~2! we estimate that the MSenergy becomes important when the size is of the orde200 nm fortF of the order of 10 nm.

To predict quantitatively the size dependence of the mnetic properties in exchange-biased films, we solve the Lequation with and without magnetostatic interactions. Aspected, we have found that the calculated hysteresis loopdifferent sample sizes have only small variation if we do ntake into account the magnetostatic interaction. The smvariation comes from the random fluctuations in choosthe random interfacial interaction. As long as the samcontains a sufficient number of grains~over 100!, the varia-tion is negligible, i.e., the hysteresis loops are independenthe sample size. If the MS energy is added to the calculatwe find that the calculated hysteresis loops are practicallydifference compared to those without the MS energylarge sample sizes. This justifies the current theoretical models of exchange bias by neglecting the MS energy for lafilms. In Fig. 1, we show the calculated results for a fixedFlayer thickness. ForLN5400 nm, Fig. 1~a!, the two hyster-esis loops with and without MS are almost identical. Thindicates that the random interface interaction dominatesMS is unimportant when the sample size is larger thanLN5400 nm. However, forLN5140 nm, Fig. 1~c! two hyster-esis loops are distinctly different; both the loop shift acoercivity are larger when the MS interaction is taken inaccount.

It is interesting to take a close look at the magnetizatpatterns near the coercive fields with and without MS intaction, shown at the bottom of Fig. 1. Each arrow in tdiagram represents theaveraged directionof 434 ferromag-netic grids above an AF grain. It is noted that these tpatterns are quite different. When the MS interaction domnates (C1), it becomes difficult to identify the locations olarger interfacial interaction. The strong MS interactiomakes the magnetic moments at the sample boundaries fparallel alignment to the sample surface. The boundary splies additional pinning centers to compete with randointerface-exchange interaction. Thus, we can understandenhanced coercivity as follows. In a large film, the coercivis mainly from the random interfacial interaction that ene

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SIZE DEPENDENCE OF EXCHANGE BIAS IN . . . PHYSICAL REVIEW B65 054406

getically favors the breakup of the ferromagnetic domaiThe coercivity is correlated with the size of domains: tsmaller the domain size the larger the coercivity.7 For a smallsample, the MS interaction introduces an additional pinnmechanism and results in a smaller domain size, and tthe coercivity increases.

Since the magnetostatic energy grows when the ferromnetic layer thickness increases, the influence of the magnstatic energy is more significant for the thickerF layer. InFig. 2, the hysteresis loops for a fixed sample size are shfor three different layer thicknesses. For the thinnerF layertF510 nm, Fig. 2~a!, LN5160 nm is characterized as‘‘large’’ sample since the hysteresis loop is almost unaffecby the MS interaction. For the thickerF layer tF530 nm,Fig. 2~c! the same sample sizeLN5160 nm shows significanalteration of the hysteresis and one should classifysample as a ‘‘small’’ size. This thickness dependence is csistent with our qualitative argument given by Eq.~2!. There-fore, in order to define the ‘‘small’’ or ‘‘large’’ samples onmust specify the thickness of theF layer.

To quantitatively address the critical sample size belthat one needs to include the MS interaction in modelexchange-biased films, we define a dimensionless paramj by

j5*dHuMz~H !2Mz

d~H !u*dHuMz~H !u

, ~3!

where Mzd(H) and Mz(H) are the magnetization with an

without MS interaction. Ifj is appreciable, one has to in

FIG. 2. Hysteresis loops of exchange-biased films for threeferentF layer thicknesses. The sample size isLN5160 nm and theother parameters are those in Fig. 1.

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clude MS in modeling the exchange biased films. More scifically, let us choose the critical sample size byj55%,i.e., if j is less than 5%, it is reasonable to omit the Minteraction and current theories for exchange biasing areplicable. In Fig. 3, we show the critical sample size asfunction of theF layer thickness. Clearly, the thinner is theFlayer, the smaller is the critical size. If theLN2tF ‘‘phasediagram’’ falls into the region above the ‘‘phase line’’ in Fig3, one can justify the use of models by omitting the Minteraction. Otherwise, one must include the MS in the meling. To see whether this ‘‘phase diagram,’’ Fig. 3, is sentive to the choice of the polycrystal AF grain size, we havaried the AF grain size from 10 nm to 20 nm. We found ththe phase boundary shown in Fig. 3 remain valid. Thus,critical sample size is insensitive to the grain size.

Finally, we show the change of exchange biasHe as thesample size becomes small. In Fig. 4, we compare the rtive increase ofHe for two differentF layer thicknesses.He

can be more than 30% larger than those of large films foparticular range of the sample size. We note that the effecMS disappears when the sample is too small~below 50 nm!

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FIG. 3. The critical line as defined in the text. The parametare the same as those in Fig. 1.

FIG. 4. Percentage enhancement of the exchange bias as ation of the sample size fortF510 nm ~open circles! and tF

524 nm~solid triangles!. The parameters are the same as in Fig

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S. ZHANG AND Z. LI PHYSICAL REVIEW B 65 054406

since the MS interaction is unable to break theF layer intodomains and the magnetic reversal is almost coherent rtion. Therefore, the exchange bias is simply given byaverage of the interfacial random field.

The size dependence of the hysteresis loop shift showFig. 4 leads us to conclude that the hysteresis loop shift isnotan intrinsic property to characterize the average interfainteraction betweenF and AF layers. The several competininteractions make the magnetization reversalnonrotationalso that the loop shift depends on the details of the reveprocesses. This point has also been pointed earlier whendifferent loop shifts were obtained from the experimentsusual hysteresis measurement and via rotating the direcof a large applied magnetic field to eliminate the domainsthe F layer.14 In our case, the determination ofHe in smallsize is much more difficult due to more complicated manetic domain structures. Therefore, the most commonly u

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relation between the loop shiftHe and the average interfaciaenergy3 is not valid.

In conclusion, we have shown how the boundary effethat are known to affect the magnetization reversal in a ffilm, affect the magnetization reversal and, therefore, msured exchange bias. While most of experiments wereried out for samples whose sizes are much larger thancritical size introduced here, it is certainly possible to fabcate devices smaller than the critical size with present expmental techniques.15 In fact, the present magnetoresistiv~MR! heads based on exchange-biased films have alreadthe range of submicrometers and the next generationheads will be certainly below the critical size. The theoretimodel presented here provides a prediction on the behaof exchange-biased films in small structures.

The research was partially supported by NSDMR0076171 and DARPA-MURI.

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