Simple Mechanism for a Positive Exchange Bias

We argue that the interface coupling, responsible for the positive exchange bias (HE) observed in ferromagnetic/compensated antiferromagnetic (FM/AF) bilayers, favors an antiferromagnetic alignment. At low cooling field this coupling polarizes the AF spins close to the interface, which spin configuration persists after the sample is cooled below the Néel temperature. This pins the FM spins as in Bean’s model and gives rise to a negative HE. When the cooling field increases, it eventually dominates and polarizes the AF spins in an opposite direction to the low field one. This results in a positive HE . The size of HE and the crossover cooling field are estimated. We explain why HE is mostly positive for an AF single crystal, and discuss the role of interface roughness on the magnitude of HE, and the quantum aspect of the interface coupling. PACS numbers: 75.70.Cn, 75.30.Gw Typeset using REVTEX 1 In recent experiments by Schuller et al. a ferromagnetic (FM) film is grown on a compensated antiferromagnetic (AF) surface under large cooling field, and the hysteresis loop is observed to shift along the positive side of the field axis. This phenomena belongs to the general category of exchange anisotropy, first discovered more than 40 years ago by Meiklejohn and Bean. However, different from the original observation and later theories2–4, a compensated (i.e., no net magnetization) surface was used and the sign of the bias was unexpected. The compensated part is resolved by a recent micromagnetic calculation by Koon, but the sign remains only speculations. Without knowledge of the detail structure at the interface or confirmation of the existence of AF domains, we try to build a simple intuitive theory. It not only can explain the main features of the phenomena, but also gives reasonable quantitative estimations. Quantum aspect of the interface coupling is analyzed in the second half of the paper. Experimentally the exchange bias, HE , decreases with increasing temperature and vanishes at the Néel temperature. This shows that the coupling of the FM spins to the ordered AF spins is crucial for the exchange bias. Furthermore, the plot of lnHE v.s. ln tF is found to fit nicely by a straight line with slope= −1 where tF denotes the thickness of the FM film. This can be viewed as another support to concentrate on the interface coupling for the source of exchange bias. The interpretation is based on, if the F spins at the interface (of number N) are stabilized each by an energy, E, due to their coupling to the AF spins, we need to divide the total change, NE, by the total number of FM spins in the film, ≈ NtF , when converting to the shift in the hysterisis loop. This gives HE ≈ E/tF which explains the linearity and its slope in the ln-ln plot. Interface roughness will increase the interface area (while the total spin number remains unchanged) and introduce an extra factor α > 1 into HE ≈ αE/tF . However, when the easy axes of FM and AF are parallel, surface roughness may also introduce frustrations (see Fig. 1) which will diminish the coupling. This does not happen when the easy axes are perpendicular since there is no preferred direction for any FM spin from its neighboring AF spins. We shall distinguish these two orientations, parallel/perpendicular easy axes, and assign them separately to the negative/positive HE 2 cases. Such a 90 degree rotation of the FM easy axis for Fe/(110)FeF2 single crystal due to the AF ordering was indeed observed by examining the hysteresis loops. That is, the easy axis of FM spins, originally in the (001) direction at T = 300 K, rotates to (11̄0) at T = 10 K for which a positive HE was measured. For our theory, the interface coupling, ∑ Jc~ SF · ~ SAF , is assumed to favor an antiferromagnetic alignment with Jc ≈ JAF (the coupling constant between AF spins). This will be justified if, take Fe/FeF2 for instance, the fluoric ions happen to lie at the interface and mediate the coupling between neighboring irons from either sides (the superexchange mechanism). However, if the irons across the interface build a direct chemical bond, presumably their coupling will of the same order and sign as JF in the bulk Fe. But since JF is twenty times stronger than JAF , this ~ SAF will be locked rigidly parallel to ~ SF and can be treated as an extension of the ferromagnetic film. The relevant interface will now be between this first layer and the next layer of the antiferromagnetic, which of course favors an antiferromagnetic alignment and agrees with our assumption. At low cooling field for which HE is negative, the easy axes of FM and AF spins are assumed to be parallel. Using the mean field analysis, we estimate the deviation from the positive z-axis (which results in a nonzero magnetization for AF) of each spin-up ~ SAF at the interface due to its antiferromagnetic coupling with a spin-up ~ SF neighbor is of the order of 1 − tanh [(JAF · q − Jc)/kBT ] where q is the number of nearest neighbors for each ~ SAF at the surface. At the usual operating temperature, say T = 10 K, the magnetization is approximately −2 · exp [ − (JAF · q − Jc)/kBT ]. To obtain the total energy change for the system, we need to multiply it by Jc and N/2 (number of up-spin ~ SAF at the interface). Note that this magnetization points antiparallel to the FM spins. When the cooling field is large enough to cause a positive HE, we assume that the easy axis of FM spins rotates and becomes perpendicular to the AF easy axis. Different from the previous case, polarization of the AF spins is now mainly due to the cooling field and, not just those ~ SAF at the interface but, all spins are involved. A physical justification for making such a rotation may lie in the fact that the perpendicular magnetic susceptibility 3 of AF spins (≈ 1/JAF ) is much larger than the parallel one at low temperatures. By canting the AF spins more effectively towards the field direction the system can gain more energy from the Zeeman effect. Note that the polarization here points parallel to the external field, i.e., the easy axis of FM spins, and is opposite to that caused by the interface coupling. We can estimate the minimum strength of cooling field, Hcool, required to obtain a positive magnetization by comparing these two energy changes: H cool JAF · tAFN = Jc e −(JAF ·q−Jc)/kBT ·N (1) This gives Hcool ≈ 0.2 T for tAF = 90 nm, the right magnitude to cause the sign change of HE experimentally . Had the easy axes been perpendicular at low cooling field, RHS of the above equation would become J cN/JAF and give too high a threshold field Hcool ≈ 5 T. The thickness tAF becomes very large for a single crystal, which implies an easier entrance into the positive-HE scenario. This is again consistent with observations 8 that HE is mostly positive when an AF single crytal is used (the fact that its surface is much rougher than in films also contributes). The appearance of an exchange bias due to the locking of FM spins by the polarization is the same as in Bean’s original model, except that an uncompensated AF surface is not required here and HE can become positive when the cooling field is strong. We do not know how the polarization survives below the Néel temperature. This could be the place where possible AF domains or impurities need to be introduced. Experimental evidence for this ”memory” is found when putting samples, field cooled in 2 kOe, under 70 kOe magnetic field at low temperatures (10K). HE is found to remain unchanged to within 5% of the Hcool = 2 kOe value. Aside from possible instability due to finite temperature fluctuations, the main conclusion of Koon that FM orders perpendicular to the AF easy magnetization axis was checked to be correct by Kiwi using a Monte-Carlo calculation. We shall examine the validity of this conclusion against a full quantum mechanical treatment, i.e., we analyze the change of vacuum energy, Evac, due to the virtual process of FM spins emitting and reabsorbing AF spin waves via the interface coupling. Suhl and Schuller have considered the special case 4 when the FM and AF easy axes are parallel, and found a negative Evac. We extend their calculations to a general angle, φ, between these two easy axes (see Fig.2) in order to find the most stable spin orientation. Quantum mechanically the interface coupling, ∑ Jc~ SF · ~ SAF where the summation runs over all sites at the interface, can be decomposed into raising and lowering operators as [S F S − AF + S − F S + AF ]/2 + S z FS z AF . Since the easy axis of ~ SF is now in the (0, sinφ, cosφ) direction, we need to reexpress S F and S ± F in terms of the new projection and raising/lowering operators: P z ≡ S F sinφ+ S z F cos φ P ≡ S F ± i(S y F cosφ− S z F sin φ). (2) In the mean time follow the standard spin wave derivation in rewriting the AF spin operators in terms of boson operators a and a which create and destroy spin deviations, S l ≈ √ s 2 (al + a + l ), S l ≈ −iσl √ s 2 (al − a + l ), (3) S l = σl(s− a + l al), where l is the site label and σl = 1/− 1 at the spin-up/down ~ SAF site, and s/S is the size of the AF/FM spin. Since there is no confusion now between the different spin notations, we shall drop the subscripts F and AF from now on. The interface coupling becomes Jc times