Metastable Random Field Ising model with exchange enhancement: a simple model for Exchange Bias

We present a simple model that allows hysteresis loops with exchange bias to be reproduced. The model is a modification of the T=0 random field Ising model driven by an external field and with synchronous local relaxation dynamics. The main novelty of the model is that a certain fraction f of the exchange constants between neighbouring spins is enhanced to a very large value J_E. The model allows the dependence of the exchange bias and other properties of the hysteresis loops to be analyzed as a function of the parameters of the model: the fraction f of enhanced bonds, the amount of the enhancement J_E and the amount of disorder which is controlled by the width sigma of the Gaussian distribution of the random fields.


I. INTRODUCTION
Hysteresis and metastability are intriguing phenomena with implications in both fundamental and applied physics. Magnetic systems are the prototypical example of thermodynamic systems exhibiting hysteresis cycles for which different theoretical approaches have been proposed [1]. Besides the classical micro-magnetic analysis, based on a continuous description of the magnetic properties of the system, more recently a lot of efforts have been devoted to the study of lattice models. For example, the zero-temperature Random Field Ising model (RFIM) driven by an external field with convenient metastable dynamics has been very successful in qualitatively explaining some basic properties of rate independent hysteresis loops [2,3,4,5]. The most important achievement of this model has been to give a simultaneous explanation of the effect of disorder on the hysteresis loops and the existence of Barkhaussen noise with critical properties. Less attention has been paid to the use of such models for understanding other interesting features of the hysteresis loops such as remanence, coercivity, minor loop properties or exchange bias (EB) [6,7], which is the property we will focus our attention on here.
We present a modification of the zero-temperature RFIM that allows magnetic hysteresis loops with EB to be reproduced. The main characteristic of EB is that the hysteresis loops, represented as magnetization m versus external applied field H, are not centred on H = 0 but exhibit a displacement in the field axis by an amount H EB (exchange bias field) [6,7]. This property has received a lot of attention recently, since the possibility of finding systems with large EB has enormous technological interest [8]. Experimentally EB has been found in different magnetic systems [6,7]. The basic ingredient for EB is the existence of interfaces between ferromagnetic (FM) and antiferromagnetic (AFM) systems, where coupling can be induced after field cooling from above the Neel temperature of the AFM. This heat treatment freezes some of the magnetic moments at the interface which are supposed to be responsible for the occurrence of EB. The prototype is a FM/AFM bilayer, for instance Co/CoO [9], NiFe/NiMn [10], Fe/FeF 2 [11] and Fe/MnF 2 [12]. This effect is also observed in granular systems formed by small particles with a ferromagnetic core covered by their native antiferromagnetic oxide [13].
Different models have been proposed to understand EB. Although a basic qualitative explanation was given 40 years ago [14] a deep understanding of the phenomenon has not yet been achieved [15,16]. Different features remain unclear: the role played by the AFM thickness [9,17], the formation of domain walls [18,19], whether the frozen spins belong to the FM or to the AFM layer [20], etc... Especially intriguing is the fact that EB not only occurs, in uncompensated AFM layers which exhibit a net magnetization after being cooled, but also in compensated AFM layers with zero net interface magnetization [21].
The aim of the present paper is to introduce a very simple model with a new mechanism for the explanation of EB in totally compensated layers. The model is based on a lattice spin system with metastable dynamics for which some of the exchange interactions show a marked enhancement. In section II the Hamiltonian and the detailed mestastable dynamics are presented. In section III we show the results of the numerical simulations. In section IV we discuss the possible physical origin of the exchange enhancement. In section V we compare with available experimental data and, finally, in section VI we summarize and conclude.

II. MODEL
The model is intended to reproduce the properties of the ferromagnetic layer only, and regards the AFM part to be totally quenched and thus does not contribute to the net magnetization (compensated AFM layer). Consequently, we consider the 2d RFIM on the square lattice, although a generalization to bulk ferromagnets or thin layers could easily be implemented. Note, however, that the AFM plays an indirect role, as will be discussed later.
The mathematical formulation of the model is very similar to the RFIM on a square lattice with size N = L × L. On each lattice site we define a spin variable S i which takes values ±1. The Hamiltonian, in reduced units, reads: The first sum is the ferromagnetic exchange contribution that extends over nearest neighbour pairs (J ij > 0). The second sum accounts for the interaction with quenched random fields h i , which stands for the disorder present in any ferromagnetic system. The last term is the interaction with the external driving field H. The random fields h i are independent and distributed according to a Gaussian probability density: where σ is the standard deviation of the random fields and controls the amount of disorder in the system. The novelty of the model is in the values of the exchange constants J ij , which are not equal for all spin pairs: we consider that J ij = J except for a fraction f of the bonds (selected at random) for which J ij = J E >> J. This fraction of bonds is supposed to contain the effect of the quenched antiferromagnetic layer. A physical reason for this local exchange enhancement will be discussed in section IV. Such a bond distribution can be mathematically expressed as: We have focused our study in the region of small values of f (f < 0.06). The magnetization of the system is defined For the analysis of the hysteresis loops we use the so called synchronous local relaxation dynamics. This is the standard dynamics used in previous studies of the zero-temperature RFIM [2]. Each spin S i flips according to the sign of its local field H i given by: where the first sum extends over the four neighbours of S i . We start with a value of H large enough so that the stable situation is given by all the spins S i = 1. We decrease the external field H until H i vanishes on a certain spin. The spin is then reversed keeping H constant. This reversal may destabilize some of the neighbouring spins which are then reversed simultaneously. This is the beginning of an avalanche. The avalanches proceed until a new stable situation with all the spins S i aligned with their respective local fields H i is reached. We can then continue decreasing the external field H.
Most of the calculated properties are averaged over a large number (∼ 10 3 ) of different realizations of disorder. Averages are indicated by the symbol · . We will consider, without loss of generality, that J = 1. Therefore, from now on, magnetic fields and energies are given in units of J.

III. RESULTS
In Fig. 1 we present an example of a hysteresis loop obtained with the numerical simulation of a system with L = 50, J E = 20, f = 0.03 and σ = 1.65. The external field is cycled between H = ±2.7. As can be seen, the loop exhibits remarkable EB. At first glance it may look surprising that the model defined in the previous section displays such asymme-try, since the Hamiltonian is totally symmetric under the changes S i → −S i and H → −H. The reason is that the hysteresis loop shown in figure 1 corresponds, strictly speaking, to a minor loop. This is revealed in Fig. 2 where the system is cycled between H = ±19.4 (which is a field that is one order of magnitude larger than the coercive field H c1 ). Due to the existence of a tiny fraction of very large exchange interactions, the total loop exhibits long, flat plateaux in which the system behaves reversibly exactly as if it was saturated. Only when cycling between extremely large values of the external field, does one obtain the symmetric hysteresis loop. Therefore, the loops with EB are incomplete loops and are accompanied by a magnetization shift. In order to perform a quantitative analysis of this behaviour, averaging over different realizations of disorder is carried out. The hysteresis loops are systematically obtained according to the following protocol: decreasing the field from H = +∞ to 2H c1 < 0 and increasing the field again to H = +∞, where H c1 is the coercive field in the decreasing branch. We also compute the pseudo coercive field H c2 in the increasing branch (see figure 1).
The criteria for choosing the value 2H c1 as a returning point is similar to the criteria used in many experimental cases. One could easily change this limit to 3H c1 or 4H c1 without changing the results provided that J E is large enough. This can be easily understood from the flat tails in the full hysteresis loop shown in figure 2.
The EB field H EB , the coercivity ∆H and the magnetization shift m B are defined as: According to this definition, the loops shifted to the left on the H axis (as occurs with the loops in the present paper) will have negative exchange bias field. In the case of coercivity (Fig. 4) two important results should be emphasized: the increase (almost 40 % in certain cases) in coercivity for intermediate values of J E and the saturation at a constant value (which depends on f ) for large J E . Such limiting values at large J E however, are smaller than the coercivity of the system without exchange enhancement.
In order to analyze the dependence of the system properties on the amount of disorder σ, we choose a value of J E that is large enough so that |H EB | has reached the limiting maximum value (see Fig. 3). behaviour with σ, first decreasing until a minimum is reached, but which increases slowly for large amounts of disorder.
The marked variation of |H EB | and ∆H for values of f between 0.015 and 0.025 when σ → 0 is associated with the fact that the ascending part of the hysteresis loop is very sensitive to the existence of nuclei of unreversed spins in the negative magnetized state. For very low values of f we expect that all the nuclei of unreversed spins will be formed by two positive spins joined by an enhanced bond. The negative spins surrounding such a nucleus will flip (in the σ → 0 limit) around H = 2 (see Eq. 4). However, for larger values of f , larger nuclei will exist. For instance, a nucleus formed by three spins joined by two perpendicular bonds, both with exchange enhancement, acts as a nucleating seed which triggers the avalanches towards the positive magnetization phase when H = 0. When f is large enough, such that the probability for such nuclei is significantly different from zero, the coercive field for the ascending branch decreases from 2 to 0, thus increasing H EB and decreasing ∆H.
The non-monotonic behaviour of the EB when disorder is increased better seen by plotting |H B | / ∆H , which is a dimensionless quantity and is more relevant from the experimental point of view. Note that for large f and σ, we can find strongly biased hysteresis loops for which H c1 and H c2 are negative. An example, obtained by sweeping the field between H = ±4.5 is shown in figure 8.

IV. DISCUSSION
One of the reasons for the present work was to obtain hysteresis loops with EB by modifying the zerotemperature RFIM as little as possible. The straightforward, naive idea would be to consider non-symmetric distributions of random fields, for which the average h i = 0 will create the displacement of the loop. From our point of view, this will correspond to the effect of an uncompensated AFM layer. Constraining ourselves to the inclusion of compensated disorder, we have found that any symmetric distribution of random fields cannot give EB. To understand this, suppose that a certain fraction of spins is pinned by very positive and (symmetrically) very negative random fields. Fig. 9(a) shows the schematic hysteresis loop corresponding to such a system. The spins with more positive random fields, which are the last to reverse in the decreasing branch, will be the first to flip in the increasing branch of the full hysteresis loop. Therefore, the full hysteresis loop (symmetric, without EB) will overlap with the minor hysteresis loop, as shown in Fig. 9a. In contrast, when the bond distribution is distorted, as done in the model presented in this work (and which preserves the symmetry properties of the Hamiltonian in Eq. 1), the spins with larger ferromagnetic coupling will be the last to reverse in the decreasing branch and also the last to be reversed in the increasing branch. Thus, minor loops will not coincide with the full loops and will easily exhibit large EB as indicated in Fig. 9b.
The justification of the proposed model would require a physical explanation for the exchange enhancement phenomena. Recently it has been suggested that spin-waves in the FM/AFM interface could be responsible for an enhancement of the exchange coupling [22]. Here we propose a different mechanism based on the existence of quenched disorder in the AFM layer. The most common case is the existence of antiphase domain boundaries as considered in Refs. 9, 23, 24 and 25. One can assume that the exchange interaction between the magnetic moments in the FM layer has two contributions: the first comming from the direct overlap of the electronic wave functions of the atoms in the FM layer and a second arising from a superexchange interaction through the overlap  with the electrons in the AFM layer. The existence of quenched disorder in the AFM layer can modify this second contribution of the exchange interaction giving rise to the exchange enhancement. Since the energies associated with the broken AFM bonds can be higher than the FM exchange energies (for instance due to the existence of strong anisotropy), it is plausible to imagine that the defects in the AFM can influence the FM exchange interactions. We would like to give a possible mathematical formulation for such a physical mechanism within the framework of lattice models. Let us consider that the interaction between two neighbouring spins S i and S j of the FM layer is given by: where σ i and σ j are the spin variables describing the magnetic moments in the AFM layer that sit exactly below the S i and S j spins of the FM layer. The constant J 0 > 0 accounts for the exchange interactions in the free FM layer and K > 0 accounts for the coupling between the two layers. This coupling term can include, in an effective form, the interactions with many atomic layers in the AFM. Figure 10 shows a schematic representation of the spins of the FM and the AFM layer in two situations: (a) corresponds to the normal situation for an ordered AFM layer, whereas (b) corresponds to the case in which the AFM exhibits an antiphase domain boundary.
(a) · · · Si Sj · · · ← → ← → (b) · · · Si Sj · · · ← → → ← In case (a), the energy of the i, j pair is E ij = −(J 0 − K)S i S j , whereas in case (b) (or on any antiphase boundary) the energy is given by The ratio between the two exchange constants is J E = (J 0 + K)/(J 0 − K), which can be much larger than 1 when K is close but smaller than J 0 .

V. COMPARISON WITH EXPERIMENT
The idea that exchange bias could be a minor loop effect was already suggested to explain the influence of the AFM spin-flop in EB [26]. Moreover, it has been shown experimentally that the uncompensated spins which have been suggested to result in exchange bias [27] and which could result in the proposed exchange enhancement, can be reversed at high enough fields [17] (much higher than those usually used in exchange bias studies). Hence, in this case EB could also be considered as a minor loop effect to a certain extent.
Moreover, the peak observed in the behaviour of the coercivity with J E (see fig. 4) could be correlated with the experimental observation of peaks in the coercive field H c (t AF M ) (where t AF M is the thickness of the AFM layer) close to the critical AFM thickness for the onset of EB or in H c (T ) close to the Néel temperature (see figures 10 and 13 in Ref. 6). One could consider that as the AFM thickness or the temperature increases, the FM-AFM coupling, which causes the enhancement of J E , decreases and is consequently the origin of the observed H c increase, in agreement with theory. Similarly, the changes in H EB and ∆H with J E could also be correlated with the behaviour experimentally observed in Fe/MnF 2 under a cooling field [12].
Another interesting result of this model is the behaviour of EB with the amount of disorder σ, as seen in Figs. 5 and 7. Experimentally the role played by disorder in EB is not clear. In some cases there is evidence that increasing disorder increases H EB , whereas in other cases the opposite effect has been found [28]. Our model is able to explain both possibilities. Moreover, it is noteworthy that experimental non-monotonic dependence of H EB on disorder (e.g. roughness, irradiation damage or structural disorder) have also been reported [9,29,30].
Finally, another remarkable result from our model is that the loops exhibiting EB also exhibit a vertical shift in the magnetization axis. This effect has been observed experimentally [24,28,31,32,33]. However, since the fraction f is very small, the shift can also be very small and in some cases, would be difficult to observe experi-mentally. Fig. 11 shows the dependence of the magnetization shift m B on H EB for different values of f . The points correspond to different values of σ ranging from 0 to 2. In view of these results, it would be interesting to measure such displacements in different experimental systems.

VI. SUMMARY AND CONCLUSIONS
We have presented a simple model for the study of Exchange Bias. The model is based on the RFIM driven by an external field with metastable dynamics. The key ingredient is the existence of a tiny fraction of ferromagnetic bonds, which are strongly enhanced. This creates a long reversible plateaux in the decreasing branch. When this pseudo-saturated state is reached, the reversal of the field gives rise to an EB loop. Different properties of these loops have been computed as a function of the fraction of enhanced bonds and the amount of disorder in the system. We have suggested a possible physical mechanism to justify the existence of such an exchange enhancement in a FM layer on a AFM layer with antiphase domain boundaries, based on the existence of superexchange coupling between the two layers.
The main conclusions of the paper are: (i) EB is due to a minor loop effect; (ii) compensated AFM layers can exhibit exchange biased loops with a concomitant magnetization shift; (iii) many experimental phenomena related to exchange bias such as: peaks in the coercive field, magnetization shifts, marked coercivity increase or non-monotonic dependence of H EB on disorder can successfully be reproduced with this model.