Unified nonequilibrium dynamical theory for exchange bias and training effects
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Unified nonequilibrium dynamical theory for exchange bias and training effects
Kai-Cheng Zhang and Bang-Gui Liu
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China andBeijing National Laboratory for Condensed Matter Physics, Beijing 100190, China (Dated: December 4, 2018)We investigate the exchange bias and training effects in the FM/AF heterostructures using aunified Monte Carlo dynamical approach. This real dynamical method has been proved reliable andeffective in simulating dynamical magnetization of nanoscale magnetic systems. The magnetizationof the uncompensated AF layer is still open after the first field cycling is finished. Our simulatedresults show obvious shift of hysteresis loops (exchange bias) and cycling dependence of exchangebias (training effect) when the temperature is below 45 K. The exchange bias fields decrease withdecreasing the cooling rate or increasing the temperature and the number of the field cycling. Withthe simulations, we show the exchange bias can be manipulated by controlling the cooling rate, thedistributive width of the anisotropy energy, or the magnetic coupling constants. Essentially, thesetwo effects can be explained on the basis of the microscopical coexistence of both reversible andirreversible moment reversals of the AF domains. Our simulated results are useful to really under-stand the magnetization dynamics of such magnetic heterostructures. This unified nonequilibriumdynamical method should be applicable to other exchange bias systems.
PACS numbers: 75.75.+a.75.20.-g,75.60.-d,05.70.Ln
I. INTRODUCTION
Usually, when the heterostructure consisting of cou-pled ferromagnetic (FM) and antiferromagnetic (AF) lay-ers is cooled in field below the Neel temperature of itsAF component, it shows the asymmetric magnetization[1, 2, 3, 4, 5, 6], which is referred to as the exchangebias effect. Furthermore, the exchange bias field, definedas the average of the two coercive fields, is observed todecrease with increasing the number of the consecutivefield cycling, which is referred to as the training effect[7].The exchange bias and training effects are very interest-ing and could be used in future spintronics[8, 9, 10] anddata storage. Usually, the FM layer is taken as a wholeand the AF layer consists of many grains. The AF grainis small enough to consists of a single domain, and someuncompensated domains (or grains) may be formed bydefects or impurities[11, 12, 13] and couple with eachother and with the FM domains. As the heterostruc-ture is cooled to a low temperature, the uncompensatedspins in the grains and domains become locked-in andprefer to a unidirection in the interface, thus contributeto the magnetization shift[3]. Moreover, under the rever-sal of FM domains, the uncompensated grains or domainswill be irreversibly reorganized[14, 15, 16, 17, 18] andthus cause the training effect. The idea of domain stateswas corroborated in some Monte Carlo simulations[19].On the other hand, Hoffmann[20] considered the biaxialanisotropy of the AF sublattices and solved it by varia-tional method. Actual nonequilibrium dynamical prop-erties of the magnetization are still waiting to be eluci-dated. It is highly desirable and needed to systematicallyinvestigate the two effects in a unified theory.In this article we use a unified Monte Carlo dynam-ical approach[21] to study the FM/AF heterostructurein order to investigate the exchange bias and training effect. Our simulated result shows the obvious shift ofhysteresis loops and the cycling dependence of exchangebias. The magnetization of uncompensated AF layer isstill open after the field cycling is finished. The exchangebias fields decrease with decreasing the cooling rate orincreasing the temperature and the number of the fieldcycling. With the simulations, we shows the exchangebias can be manipulated by controlling the cooling rate,the distributive width of the anisotropy energy, or themagnetic coupling constants. Essentially, these two ef-fects can be explained on the basis of the microscopicallyirreversible reversal of the AF domains. More detailedresults will be presented in the following.The remaining part of this paper is organized as fol-lows. In next section we shall define our model anddiscuss our simulation method. In section III we shallpresent our simulated results and analysis. In sectionIV we shall discuss the microscopic mechanism for thephenomena in a unified way. Finally, we shall give ourconclusion in section V.
II. MODEL AND METHOD
According to experimental observations[22], for bothcompensated and uncompensated AF layers the easy axistends to form along external cooling field direction ratherthan later rotating field direction. In our model the AFlayer consists of many AF domains, and the FM layerconsists of one single domain. Assuming the cooling fieldis applied parallel to the AF/FM interface, then all theeasy axes of AF and FM domains lie in the plane of theinterface. The coupled bilayers of AF and FM domainsare shown in the inset of Fig. 1(a). The rectangles of thewhite pattern represent the AF domains and the largerrectangle is the single FM domain. The AF domains cou-ple to each other antiferromagnetically and the single FMdomain couples to all the AF domains ferromagnetically.We define the z axis along the common easy axis whichlies in the interface plane. We apply the external field tosaturate the magnetization of the FM layer along the z axis.For simplicity, we consider all the uncompensated spinsin the AF domains are the same. We use S ′ ~s i to denotethe spin vector of the i th AF domain and S~s to denotethat of the single FM domain, where S ′ and S are the un-compensated spin values and FM spin respectively. Thenwe write the Hamiltonian of the bilayers in an externalfield as H = − K u ( s z ) − X i k ui ( s zi ) − ~B · ( γ ′ X i ~s i + γ~s )+ J X i,j ~s i · ~s j − J X i ~s i · ~s (1)where γ ′ = gµ µ B S ′ and γ = gµ µ B S . The first and sec-ond terms represents the anisotropy of the FM domainand the AF ones, and K u and k ui are the correspondinganisotropy constants. The third term represents the Zee-man energy of the moments due to the applied externalfield. The fourth term represents the antiferromagneticcoupling among the AF domains. The last term repre-sents the ferromagnetic coupling between the FM and AFdomains.Using θ i and β to describe the angles of the i -th AFmoment and the FM moment deviating from the commoneasy axis, we can express the energies of the FM domainand the i -th AF as H FM = − ( J X i s i s + K u cos β + γBs ) cos β (2)and H AF i = ( J s i X j s j − J s i s − k ui cos θ i − γ ′ Bs i ) cos θ i (3)where both s i and s are the scalars taking either 1or -1. Thus for the i -th AF domain the energy in-crement is ∆ E i = k ui sin θ i − h i (cos θ i − h i = ( − J P j s j + J s + γ ′ B ) s i , and for the FM domainthe energy increment is ∆ E = K u sin β − h F (cos β − h F = ( J P i s i + γB ) s . We can express ∆ E and∆ E i as[21]∆ E = K u [(1 + h F K u ) − (cos β + h F K u ) ] (4)and ∆ E i = k ui [(1 + h i k ui ) − (cos θ i + h i k ui ) ] (5)As a result, to reverse its moment, the the FM layermust overcomes a barrier E Fb = K u (1 + h F / K u ) if | h F | ≤ K u , or 2 h F if h F > K u ; and the i -th AF grain a barrier E ib = k ui (1 + h i / k ui ) if | h i | ≤ k ui , or 2 h i if h i > k ui . If the condition h F < − K u or h i < − k ui issatisfied, there is no barrier for the reversal.Actually, for the distribution of the AF anisotropy en-ergy we use a Gauss function, f ( k ui ) = exp[ − ( k ui − k u ) /σ ], whose σ and k u are set to 30.0 meV and 50.0meV unless stated otherwise. The anisotropy energy ofthe FM domain is set 200.0 meV without losing mainphysics. Thus the reversal rate for a spin to reverse is R = R e − E b /k B T , where E b is the energy barrier and R is the characteristic frequency. In our simulations, R is set to 1 . × /s. We adopt a square lattice forthe AF domains and use 20 ×
20 as its size. Since we areonly interested in the exchange bias and training effect atthe nanoscale, the AF lattice is enough to capture mainphysics. Furthermore, we assume the AF domains haveuniform moment 4.0 µ B and the FM domain 2000 µ B .The coupling constant J is set to 4.0 meV, and J ν = 50 K/s. The field sweepingrate is set to 0.5 T/s with the basic increment 0.1 T foreach simulation step. III. SIMULATED RESULTS AND ANALYSIS
At first, we let the AF/FM bilayers relax under a mag-netic field of 5.0 T at a high temperature 610 K. Thistemperature is enough to make both the FM layer andthe AF layer remain paramagnetic. When the tempera-ture decreases, the average magnetization values of thetwo layers increases. The external field makes the aver-age magnetization of the FM layer have a large increasebelow 600 K, and reach nearly to the saturated value at500 K. When the temperature becomes lower than 60K, the average magnetization of the AF layer looks likethat of an antiferromagnet under an applied field and isdependent on the cooling rate ν . Then, we further coolthe bilayers under the same field. After the temperaturereaches down to 10 K, we start to change the field whilekeeping the temperature unchanged. The field decreasesfrom 5.0 T to -10.0 T and then increases back to 5.0 T forthe first hysteresis. Repeating the field cycling, we willmake the second hysteresis loop. The simulated resultsare shown in Fig. 1.As shown in Fig. 1(a), the origin of the first hysteresisis clearly shifted in the negative field direction and showsthe exchange bias. The exchange bias field is defined as H E = ( H cl + H cr ) /
2, where H cl and H cr is the coercivityof the left and right branches. The left branch of the sec-ond hysteresis moves towards the positive direction, butthe right branches of the first two loops almost coincidewith each other. Actually, any further loop almost does -10 -5 0 5 10-1.0-0.50.00.51.0 -10 -5 0 5 100.060.090.120.15 m F M P2P3 P4 m A F B (T) (b)
FIG. 1: The first two hysteresis loops of the FM (a) and AF(b) layer at 10 K. The inset in (a) shows the AF/FM bilayers.The hysteresis loop is obtained by changing the field in theorder of P1-P2-P3-P4-P1. no difference in the right branch from the second hys-teresis. The shift of the second loop clearly demonstratesthat the bilayers magnetization depends on the cyclinghistory, which is known as training effect. Fig. 1(b)shows the magnetization of the AF layer, which dropslargely and opens widely due to the irreversible reversalof the AF domains after the first field cycling is finished.The subsequent magnetization is more smooth but stillnot closed, indicating the continuing cycling dependenceof exchange bias. This is consistent with other MonteCarlo simulations[19].We study the effect of the temperature T on the ex-change bias field, H E , for different loops. Our simulatedexchange bias fields as functions of T for the first twoloops are shown in Fig. 2. For both of the two curves,the data can be fitted by the simple function − µ H E = a e − ( T/T ) b (6)where a , T , and b are fitting parameters. For thefitting in Fig. 2, the parameters a , T , and b takes4.26 T, 17.95 K, and 1.58 for the first loop, and 3.94T, 16.66 K, and 1.59 for the second loop. Our resultsare consistent with experimental observation that the ex-change bias field decreases with increasing temperature[12, 23, 24].The exchange bias field is dependent on the field cy-cling number n . Our simulated result from n =1 to n =9
10 20 30 400123 - m H E ( T ) T (K) 1st loop 2nd loop
FIG. 2: Temperature dependence of the exchange bias fieldsfor the first two loops. The exchange bias field is calculatedat a given temperature after the system is cooled from 610 Kto the temperature value. The lines are the fitting curves interms of the simple function defined in Eq. (6). - m H E ( T ) n s =30.0 s =20.0 FIG. 3: The loop-number dependence of the exchange biasfields for the Gaussian width σ =30.0 and 20.0 meV. The tem-perature is 10 K. All the data except for n = 1 can be wellfitted by a simple function − µ H E ( n ) = a ρ n + b . is shown in Fig. 3. Here, the temperature is kept at 10 K,and σ is set to 20.0 and 30.0 meV. For both of the curvesin Fig. 3, the data points excepts of n = 1 are well fittedby the simple function − µ H E ( n ) = a ρ n + b , where a , b , and ρ are the fitting parameters, taking 0.23 T,2.26 T, and 0.76 for σ = 30 . σ = 20 . − µ H E (1) usu-ally is substantially above the extrapolation of the other − µ H E ( n ) ( n > ν , we study the - m H E ( T ) n / n FIG. 4: The cooling-rate dependence of the exchange biasfields for the first two loops. The lines are the fitting curvesin terms of Eq. (7). exchange bias field as the function of quenching rate ν .The result is shown in Fig. 4. For both of the loops, thedata be well fitted by − µ H E = a ln( b νν + 1) (7)where a and b are 0.292 T and 20847 for the first loop,and 0.262 T and 16248 for the second loop. The ex-change bias field at 10 K increases logarithmically withincreasing the quenching rate.It is interesting to investigate the dependence of theexchange bias field on the coupling constants J and J .The simulated results are shown in Fig. 5. As shown inFig. 5(a), the exchange bias field decreases with increas-ing J . The training effect is nearly unchanged when J changes from 0 to 2meV, but diminishes to zero quicklywith increasing J from 2meV. When J is larger than 6meV, the exchange bias field already becomes very smalland the training effect is actually zero. In contrast, the J data points in Fig. 5(b) can be well fitted by thesimple function − µ H E = a (exp( J /b ) − a and b are the fitting parameters, taking 0.44 T and 4.01meV for the first loop, and 0.51 T and 4.49 meV forthe second loop. It is clearly shown that the exchangebias field increases exponentially as J increases. Bothof the the exchange bias field and the training effect canbe enhanced by decreasing J and increasing J , or byincreasing J /J . This is consistent with experimentaltrend[25]. In Fig. 6 we shows how the distributive width σ of the AF anisotropy affects the exchange bias fields.Clearly the exchange bias field increases with σ , and sodoes the training effect. Experimentally, the width canbe increased by the additional nonmagnetic impuritiesand the enhanced roughness of the AF crystalline phases.This implies that the rougher the AF crystalline phasesare, the larger the exchange bias and training effect. Ourresult reveals that the exchange bias field is determinedby both the coupling constants and the distributive width - m H E ( T ) J (meV) 1st loop 2nd loop(b) - m H E ( T ) J (meV) 1st loop 2nd loop(a) FIG. 5: The exchange bias fields as functions of the coupling-constants J (a) and J (b) for the first two loops.
10 15 20 25 30 350123 - m H E ( T ) s (meV) FIG. 6: The exchange bias field as a function of the Gaussianwidth σ for the first two loops. of the AF domain anisotropy. These are useful to com-pletely understand the phenomenon[25, 26]. IV. TRENDS AND MICROSCOPICMECHANISM
After being cooled down to the low temperature, theAF layer has a non-zero net FM moment M A due to thedriving of both the field and the FM layer. Assumingthere are N A AF domains, on average we have the mo-ments in part of all the N A AF domains aligning parallelalthough they are coupled with AF interactions. The ex-change bias field is determined by the effective moment M A , the difference of M A between the first two loopsdetermines the training effect. Naturally, both the ex-change bias field and the training effect increase withincreasing J and with decreasing J , as shown in Fig. 5.Actually, small J does not affect the effects, but larger J than 2meV is harmful to the effects at 10 K. In addition,it is easily understood that M A decreases with increasingthe temperature T . As a result, both the exchange biasfield and the training effect decrease with increasing T ,as shown in Fig. 2. The exponential description in Eq.(6) reflects the fact that moment reversals are thermallyactivated. It is reasonable that both the exchange biasfield and the training effect increase with increasing thecooling rate ν , as shown in Fig. 4. This is mainly becausethe average magnetization of the AF layer increases with ν when the temperature is below 50 K. When the coolingrate approaches to zero, both the exchange bias field andthe training effect should be zero. In another word, ourresults should approach to those of corresponding equi-librium systems when the cooling rate ν approaches tozero.As shown in Fig. 6, both the exchange bias field andthe training effect are nearly zero when the distributivewidth σ of the AF anisotropy energy is smaller than 15meV, but they increase substantially with increasing σ for σ >
15 meV. This means that the effects are depen-dent on a wide distribution of the anisotropy energy. Thiscan be understood in terms of the changing of the energybarriers with the external field. From P1 to P2 in Fig.1(a), the effective barrier of the FM layer decreases butis still high enough to avoid the reversal, but meanwhile,more and more spins of the AF domains are reversed dueto their lower energy barriers. At the point P2, the FMmoment is reversed with the help of the field and the re-versing of the AF domains. Anyway, some AF domainswith high energy barriers have their moments unchanged,even after the FM layer has been reversed, and thus thereis a net average moment of the AF domains parallel tothe moment of the FM layer. This net average momentincreases with the distributive width σ . This explains the increasing of the exchange bias field and training effectwith increasing σ . The more the field cycling loops, thelonger the time. Actually, this is similar to reducing thecooling rate ν in effect. As a result, the exchange biasfield decreases with increasing the number of the fieldcycling. The turning point of the time scale causes thelargest drop happens between the first loops. V. CONCLUSION
In summary, we use a unified Monte Carlo dynami-cal approach[21] to study the FM/AF heterostructure inorder to investigate the exchange bias and training ef-fect. The magnetization of uncompensated AF layer isstill open after the first field cycling is finished. Our sim-ulated result shows the obvious shift of hysteresis loops(exchange bias) and the cycling dependence of exchangebias (training effect). The exchange bias fields decreasewith decreasing the cooling rate or increasing the tem-perature and the number of the field cycling. With thesimulations, we show the exchange bias can be manip-ulated by controlling the cooling rate, the distributivewidth of the anisotropy energy, or the magnetic couplingconstants. Essentially, these two effects can be explainedon the basis of the microscopical coexistence of both re-versible and irreversible moment reversals of the AF do-mains. Our simulated results are useful to really under-stand the magnetization dynamics of such magnetic het-erostructures which should be important for spintronicdevice and magnetic recording media [25, 26, 27]. Thisunified nonequilibrium dynamical method should be ap-plicable to other exchange bias systems.
Acknowledgments
This work is supported by Nature Science Founda-tion of China (Grant Nos. 10874232, 10774180, and60621091), by the Chinese Academy of Sciences (GrantNo. KJCX2.YW.W09-5), and by Chinese Department ofScience and Technology (Grant No. 2005CB623602). [1] S. Bruck, G. Schutz, E. Goering, X. S. Ji, and K. M.Krishnan, Phys. Rev. Lett. 101, 126402 (2008).[2] M. Gruyters and D. Schmitz, Phys. Rev. Lett. 100,077205 (2008).[3] Y. Ijiri, T. C. Schulthess, J. A. Borchers, P. J. van derZaag, and R. W. Erwin, Phys. Rev. Lett. 99, 147201(2007).[4] J. Eisenmenger, Z. P. Li, W. A. A. Macedo, and I. K.Schuller, Phys. Rev. Lett. 94, 057203 (2005).[5] S. Brems, D. Buntinx, K. Temst, and C. V. Haesendonck,Phys. Rev. Lett. 95, 157202 (2005). [6] J. Nogues and I. K. Schuller, J. Magn. Magn. Mater. 192,203 (1999).[7] A. Hochstrat, C. Binek, and W. Kleemann, Phys. Rev.B 66, 092409 (2002).[8] T. Zhao, A. Scholl, F. Zavaliche, K. Lee, M. Barry, A.Doran , M. P. Cruz, Y. H. Chu, C. ederer, N. A. Spaldin,R. R. Das, D. M. Kim, S. H. Baek, C. B. Eom, and R.Ramesh, Nat. Mater. 5, 823 (2006); R. Ramesh and N.A. Spaldin, Nat. Mater. 6, 21 (2007).[9] V. Laukhin, V. Skumryev, X. Mart, D. Hrabovsky, F.Sanchez, M. V. Cuenca, C. Ferrater, M. Varela, U. Lud-[1] S. Bruck, G. Schutz, E. Goering, X. S. Ji, and K. M.Krishnan, Phys. Rev. Lett. 101, 126402 (2008).[2] M. Gruyters and D. Schmitz, Phys. Rev. Lett. 100,077205 (2008).[3] Y. Ijiri, T. C. Schulthess, J. A. Borchers, P. J. van derZaag, and R. W. Erwin, Phys. Rev. Lett. 99, 147201(2007).[4] J. Eisenmenger, Z. P. Li, W. A. A. Macedo, and I. K.Schuller, Phys. Rev. Lett. 94, 057203 (2005).[5] S. Brems, D. Buntinx, K. Temst, and C. V. Haesendonck,Phys. Rev. Lett. 95, 157202 (2005). [6] J. Nogues and I. K. Schuller, J. Magn. Magn. Mater. 192,203 (1999).[7] A. Hochstrat, C. Binek, and W. Kleemann, Phys. Rev.B 66, 092409 (2002).[8] T. Zhao, A. Scholl, F. Zavaliche, K. Lee, M. Barry, A.Doran , M. P. Cruz, Y. H. Chu, C. ederer, N. A. Spaldin,R. R. Das, D. M. Kim, S. H. Baek, C. B. Eom, and R.Ramesh, Nat. Mater. 5, 823 (2006); R. Ramesh and N.A. Spaldin, Nat. Mater. 6, 21 (2007).[9] V. Laukhin, V. Skumryev, X. Mart, D. Hrabovsky, F.Sanchez, M. V. Cuenca, C. Ferrater, M. Varela, U. Lud-