VOLUME 78, NUMBER 25 P H Y S I C A L R E V I E W L E T T E R S 23 JUNE 1997 Calculations of Exchange Bias in Thin Films with Ferromagnetic Antiferromagnetic Interfaces N. C. Koon Naval Research Laboratory, Washington, D.C. 20375-5320 (Received 27 December 1996) A microscopic explanation of exchange bias in thin films with compensated ferro antiferromagnetic interfaces is presented. Full micromagnetic calculations show the interfacial exchange coupling to be relatively strong with a perpendicular orientation between the ferro antiferromagnetic axis directions, similar to the classic "spin-flop" state in bulk antiferromagnets. With reasonable parameters the calculations predict bias fields comparable to those observed and provide a possible explanation for both anomalous high field rotational hysteresis and recently discovered "positive" exchange bias. [S0031-9007(97)03407-8] PACS numbers: 75.70.Cn, 75.30.Gw The phenomena of exchange bias was discovered more thicknesses as well as orientation of the FeMn, and found than 40 years ago by Meiklejohn and Bean [1], who found critical thicknesses of the antiferromagnet for the onset that fine particles of partially oxidized Co exhibit magneti- of exchange bias as well as the fact that the F mag- zation curves with an unusual displacement along the field netization MF tended to orient perpendicular to the A axis, as though there were a "bias" field HEB in addi- easy axis. Nogues et al. [7] found for Fe on a single tion to the applied field. Bias fields are observed only if crystal FeF2 that the largest exchange bias occurred for an external magnetic field is applied while cooling the fer- the fully compensated (110) interface orientation and that romagnetic (F) Co particles below the Néel temperature increasing interface roughness decreased the bias fields. of their antiferromagnetic (A) CoO coating. Experiments The same group [8] also discovered that field cooling by Bean [2] on Co CoO thin films demonstrated that ex- Fe FeF2 in large applied fields 70 kOe can result in change bias is primarily an interface phenomena, although positive exchange bias, where the magnetization reversal it has also been observed in inhomogeneous F A mixtures occurs as the field is lowered while it still has the same [3]. Meiklejohn and Bean [1] also showed there was a sign as the cooling field rather than the opposite sign, as close connection between anomalous high field rotational normally observed. hysteresis and exchange bias, although the two do not gen- This Letter presents results of full micromagnetic (gen- erally coexist. eralized mean field) numerical calculations on a simple In recent years exchange bias in thin films has found im- model of a thin film with compensated F A interfaces. portant technological application in such devices as mag- The main difference from normal mean field calculations netoresistive sensors. However, its fundamental origin is is that each spin in the primitive magnetic cell is allowed still unclear [4,5]. The simplest model [1], which assumes to have its own direction. In spite of the simplicity of the that the F A interface occurs at an ideal uncompensated model, many of the established properties of thin film ex- (all spins aligned) plane of the antiferromagnet, predicts change bias systems appear naturally, as do some which bias fields of orders of magnitude larger than those ob- are not so well established. Fully compensated perfect served and fails to explain biasing when the interface plane interfaces are found to be favorable for exchange bias, of the antiferromagnet is fully compensated (zero net mo- and roughness does not generally play an essential role. ment). Mauri et al. [4] provided an explanation for the The mechanism for storage of magnetic exchange bias en- reduced bias fields by showing that the formation of a do- ergy is found to be parallel domain walls, as proposed by main wall parallel to the interface dramatically lowers the Mauri et al. [4]. The 90± angle between magnetization energy required to reverse the magnetization. However, directions in the ferromagnet and the antiferromagnet [6] exchange coupling across the F A interface was simply as- is predicted to be a normal occurrence and is shown to sumed, and questions regarding the origin of the F A cou- be related to the well-known "spin-flop" state in antiferro- pling or what happens at fully compensated interfaces were magnets. Irreversible magnetic transitions in the antiferro- not addressed. Malozemoff [5] interpreted exchange bias magnet are found which provide a mechanism to account in terms of random exchange fields due to interface rough- for the high field rotational hysteresis normally associated ness. However, his model has extrinsic features which de- with exchange bias [1]. Reasonable values of exchange pend on details of the microstructure [6]. and anisotropy energies lead to bias fields the same order Recently, several groups have reported experimen- of magnitude as those observed experimentally at low tem- tal results which cannot be readily explained by the peratures. Finally, an explanation for positive exchange theories discussed above. Jungblut, et al. [6] studied bias [8] is presented which may permit determination of molecular-beam epitaxy grown epitaxial wedge structures the sign of the microscopic exchange interaction JFA be- of NiFe FeMn, as a function of NiFe (F) and FeMn (A) tween F and A spins. 4865 VOLUME 78, NUMBER 25 P H Y S I C A L R E V I E W L E T T E R S 23 JUNE 1997 The model assumes a simple body centered tetragonal (A or F) film with the same magnitude of exchange and magnetic structure with exchange interactions along the same thickness. body diagonals. For the antiferromagnet this leads to the Such a strong "macroscopic" exchange coupling can simplest type of antiferromagnetic order in which spins only occur if the magnetic symmetry is broken by interac- on the two sublattices are oppositely directed and only tions at the F A interface. The nature of this broken sym- interact with spins on the other sublattice, as indicated in metry is illustrated in Fig. 3, which gives the configuration Fig. 1. The magnetic unit cell of a (110) film with this of spins near the F A interface at the minimum energy structure contains two spins per plane because of the lack angle w 90±. If JFA is antiferromagnetic the A spins of translational invariance in the [110] direction. Classical cant away from the F direction as shown. If JFA is ferro- spins with S 1 were assumed, and uniaxial anisotropy magnetic they cant toward the F direction. In both cases KU favoring the [001] was allowed only on the A sites. the canting angle decays rapidly as a function of distance Only zero temperature calculations are presented. from the interface, becoming essentially zero at 5­6 ML. Although relaxation methods can be used directly to cal- Thus it can be concluded that frustration at a fully com- culate exchange biased magnetization curves, it is instruc- pensated F A interface does not lead to zero exchange tive to first consider some simple cases. The key issue of coupling as the simplest model would suggest [1] but to a exchange coupling or "pinning" across a F A interface is macroscopic 90± coupling only somewhat weakened com- examined by considering a (110) oriented F A film with pared to homogeneous full exchange. This 90± coupling thicknesses tF and tA of 15 monolayers (ML) each. It was state is precisely analogous to the well-known spin-flop assumed that JFF 2JAA 2JFA 1 meV. In each of state of antiferromagnets in an applied field if one identi- the two outer layers (one F and one A) all the spins are fies the applied field with frustrated microscopic exchange constrained to lie along the same axis, with an angle w be- across the interface. tween the axes of the layers. The initial configuration was This result also seems to suggest that interface rough- with the axis directions parallel w 0 and with the in- ness plays a somewhat different role than usually assumed ner spins random in orientation. Relaxation methods were [5]. Because maximum frustration is already present in used to solve for the energy per unit area and spin con- a perfectly compensated F A interface plane, it appears figurations as w increased. plausible that surface roughness could only reduce frus- In Fig. 2 the energy per unit area is shown as a function tration at such an interface and thus reduce the bias field of the fixed angle w for two cases: one as described above compared to that of a perfect interface. This observation with a frustrated F A interface, and a second with either is consistent with experimental work on Fe FeF2, which all F or all A exchange interactions of the same magnitude shows that the bias fields are largest for the compensated jJj 1 meV . For pure F or A films the minimum energy (110) orientation and decrease with increasing roughness occurs at w0 0±, as expected, while the film with a of the interface [7]. frustrated F A interface has a minimum at w0 90±. In A second key issue is the nature of coupling to the lat- both cases the angular dependence of the energy is well tice, without which exchange bias could not exist. Since described by a polynomial of the form c w 2 w0 2. For anisotropy in a ferromagnet does not, by itself, lead to ex- the mixed F A film c 1.98 3 1025 meV deg22, and for change bias, it is reasonable to guess that the essential pin- pure F or A films c 2.10 3 1025 meV deg22. Since c is ning comes from uniaxial anisotropy KU favoring [001] proportional to the exchange constant J, one arrives at the striking conclusion that the effective exchange coupling between the top and bottom layers of the 30 layer film with a completely frustrated F A interface is not only shifted by 90±, it is reduced in magnitude by a relatively small amount ( 6% in this case) compared to a homogeneous FIG. 2. Energy per unit area of a 15 15 ML (110) F A film as a function of angle between the top (F) and bottom (A) FIG. 1. Magnetic structure of the model for only antiferro- magnetization axes (curve with data points). The smooth curve magnetic interactions and emphasizing the (110) planes. Ex- corresponds to a structurally identical film with only A or only change bonds are shown by the dashed lines. F spins. For both films jJj 1.0 meV. 4866 VOLUME 78, NUMBER 25 P H Y S I C A L R E V I E W L E T T E R S 23 JUNE 1997 ergy branch, blocked by an energy barrier from transition- ing to a lower energy branch until u uCR tA . The two different states of the A spins at each angle correspond to one or the other of the two degenerate ground state direc- tions of MF in zero field. Transitions between the states depend mostly on the orientation of MF (driven by H) and are weakly dependent on the interaction of H with A be- FIG. 3. Spin configuration near the interface plane for a cause the net moment of A is small for normal laboratory 15 15 ML F A film with lowest energy orientation w 90± . fields. Such discontinuous rotations, primarily in the anti- Parameters are the same as for Fig. 2. The two interface planes ferromagnet, therefore appear to be a plausible mechanism are L15 and L16. Angles are approximately to scale. for the anomalous high field rotational hysteresis observed by Meiklejohn and Bean [1]. on the A sites. To examine this issue a simplified model is More importantly, the energy curves in Fig. 4 for large considered where the F spins are locked together J t FF A uCR . 90± can lead to exchange bias. For 50 ML, for and fixed at an angle u in the (110) plane relative to example, there is a minimum energy centered at 90± 110 [001]. In accordance with the preceding calculation, one with an angular range of 344± over which MF can rotate would expect two degenerate minimum energy orientations reversibly and only a d 16± range where irreversible of M behavior occurs, as is illustrated in the inset of Fig. 4. F at 110 and 110 (u0 90± and 270±, respec- tively). To calculate the energy curves as a function of If the direction of 2H (H is along the field cooling di- t rection) falls outside the range of d, then the energy curve A in Fig. 4 it was assumed that the initial direction of MF was 110 u will be entirely reversible, regardless of the magnitude of 0 90± , while the A spins were randomized. The angle u was incremented away from 90±, with each H. The range d decreases rapidly with increasing tA w final spin configuration serving as the starting configu- greater than one, which clearly reflects the need to have a ration for the next angle. For a choice of u minimum thickness of the antiferromagnet in an exchange 0 270±, one obtains an essentially identical set of curves (not shown) bias system. shifted by 180±. For t The preceding discussion is subject to the approxima- A much less than the A domain wall width w 4J K tions that the F spins have infinite stiffness and that the u 1 2 9 ML the angular dependence of the energy is reversible and similar to that of K effect of the applied field on A can be neglected, neither U, ex- cept for the 90± interfacial shift. The curves also exhibit of which is correct in general. It is therefore essential to mirror symmetry about u 0± and 180±, reflecting the calculate an exchange bias curve with all restrictions on mirror symmetry of the 110 plane. When t the spin orientations removed and an external field applied A * w, how- ever, the energies increase smoothly as M to both the ferromagnet and antiferromagnet using realis- F rotates through u 0± and 180±, and the energy curves no longer have tic anisotropy and exchange parameters. The results are mirror symmetry. After M shown in Fig. 5 for a case where H is applied 10± from F passes through the mirror plane the spin configuration follows a metastable high en- 110 , tF tA 50 ML, and JFF 16 meV, which cor- responds approximately to Fe or Co. The angular offset from 110 was used for convenience to reduce tA and com- putation time. Other parameters are the same as for Fig. 4. The orientations of all spins were first randomized and FIG. 4. Energy per unit area as a function of angle u between MF and [001] in the (110) plane for F A films with different tA. Parameters are defined in the text. Calculated antiferromagnetic FIG. 5. Calculated magnetization curve for a 50 50 ML F A domain wall width is 9 ML. Only curves with MF initially film. Parameters are the same as for Fig. 4, except JF along 110 u 90± are shown. The inset shows the range 16 meV. For convenience, the field was applied 10± from the d where irreversible transitions occur. 110 . 4867 VOLUME 78, NUMBER 25 P H Y S I C A L R E V I E W L E T T E R S 23 JUNE 1997 then allowed to relax with a 5 kOe field applied. The field face dominated exchange bias systems. A more detailed was then decreased in small increments to 25 kOe, with discussion of these calculations as well as applications each spin configuration serving as the starting point for to real thin film exchange bias systems will be presented the subsequent one. A completely reversible (no hystere- elsewhere. sis) exchange bias curve is obtained with HEB 1.0 kOe, The author would like to thank W. Saslow, J. J. Krebs, as shown in Fig. 5. This is a larger bias field than typi- A. S. Arrott, and K. Hathaway for their advice and cally seen at room temperature, but is comparable to the encouragement in the course of this work. low temperature bias fields observed in Co CoO [1] and Fe FeF2 [8]. The physical picture emerging from this model is that exchange bias results from the formation of a mainly antiferromagnetic parallel domain wall [4] as a reverse [1] W. P. Meiklejohn and C. P. Bean, Phys. Rev. 102, 1413 field rotates M (1956); Phys. Rev. 105, 904 (1957). F away from the field cooled direction. The domain wall is made possible by strong macroscopic F A [2] C. P. Bean, in Structure and Properties of Thin Films, exchange coupling and pinning of the A by anisotropy. edited by C. Neugebauer, J. Newkirk, and D. Vermilyea If this picture is correct, then the recent observation of (Wiley, New York, 1959), p. 331. [3] See J. S. Kouvel, J. Phys. Chem. Solids 21, 57 (1961), and positive exchange bias [8] implies that in sufficiently related works. large fields the ground state of the system must contain [4] C. Mauri, H. C. Siegmann, P. S. Bagus, and E. Kay, such a domain wall, since it is the "unwinding" of the J. Appl. Phys. 62, 3047 (1987). wall which causes the magnetization to reverse before [5] A. P. Malozemoff, Phys. Rev. B 35, 3679 (1987); J. Appl. the magnitude of the field decreases to zero. Preliminary Phys. 63, 3874 (1988); Phys. Rev. B 37, 7673 (1988). calculations for the present model confirm that the ground [6] R. Jungblut, R. Coehoorn, M. Johnson, J. aan de Stegge, state can generally contain such a domain wall if J and A. Reinders, J. Appl. Phys. 75, 6659 (1994); R. Jung- FA is negative (antiferromagnetic), but not if it is ferromagnetic. blut, R. Coehoorn, M. T. Johnson, Ch. Sauer, P. J. van der Observation of positive exchange bias would then imply Zaag, A. R. Ball, Th. G. S. M. Rijks, J. aan de Stegge, and antiferromagnetic J A. Reinders, J. Magn. Magn. Mater. 148, 300­306 (1995). FA. Normal negative bias, on the other hand, is only weakly dependent on the sign of J [7] J. Nogues, D. Lederman, I. K. Schuller, and K. V. Rao, FA. Appl. Phys. Lett. 68(22), 3186 (1996). In summary, the broad qualitative agreement between [8] J. Nogues, D. Lederman, T. Moran, and I. K. Schuller, calculations utilizing this simple model and a variety Phys. Rev. Lett. 76, 3186 (1996); see also T. J. Moran, of observed exchange bias phenomena suggests that the Ph.D. thesis University of California, San Diego, 1995 model does contain much of the essential physics of inter- (unpublished). 4868