Recently, spin glasses (SG) have attracted much attention and stimulated intensive studies because the combination of quenched spin spatial randomness and frustration create a complex free energy landscape with multiple local energy minima and finding the stable spin states formidable challenging1,2,3. In spite of many theoretical attempts of developing renormalization group1,4, mean-field approximation5 and Monte Carlo simulation3,6,7,8 based analyses, the low-temperature (T) spin phases as well as the out-of-equilibrium aging dynamics of such materials remain matters of strong debates. Not surprisingly, the complex nature of SG spin structures gives rise to the unique and diverse macroscopic properties in many fields of science. Exchange bias (EB) for ultrahigh-density magnetic recording is one of such areas9.

The EB effect in coupled ferromagnet (FM)/antiferromagnet (AFM) system has been studied extensively because of its critical role in spintronic devices and intriguing spin physics. The EB phenomenon manifests itself as a shifted and broadened magnetization-applied field (M-H) hysteresis loop along the H axis when FM/AFM systems are cooled below an EB blocking temperature (TB) under an external magnetic cooling field (HFC)10,11. Therefore, EB is both HFC- and T-dependent normally. Most of the EB studies show that large enough HFC can either saturate a negative EB field (HE) for FM typed interfacial couplings (JIF > 0)12,13 or induce a positive HE for AFM typed JIF (< 0)14,15,16. On the other hand, the thermal energy at high T will weaken JIF and thus HE often decreases monotonically with increasing T and finally vanishes at TB. Under certain circumstances, a sign inversion of HE, i.e., a small positive EB, can be observed in a narrow T region just below TB for some EB systems17,18. These abnormal phenomena are argued to be a result of the unidirectional coercivity (HC) enhancement along the HFC direction. Observation of this effect requires a modest HFC and it has been only reported in quite few AFM based materials since the origin of positive EB is very different from systems with negative JIF14. Therefore, the study of HFC together with T dependences of EB is of critical importance for the understanding of EB mechanism.

As mentioned previously, when SG is involved, the EB dependences on HFC and T can be different from conventional FM/AFM systems. Nayak et al.19 found that zero-field-cooled (ZFC, HFC = 0) HE in Heusler compound Mn2PtGa is comparable to the field-cooled (FC) value, due to the coexistence of field-induced irreversible magnetic behavior and a SG-like phase. Sabyasachi et al.20 found that even a strong HFC of 80 kOe cannot saturate the HC of nanocrystalline La1/3Sr2/3FeO3-δ at 5 K and this behavior was ascribed to the randomness of glassy magnetic phase. Especially, by using a canonical SG alloy (CuMn), Ali et al.21 first reported the sign-changeable behavior of T-dependent HE in SG/FM bilayer systems. Subsequently, Yuan et al.22 and Ali et al.23 observed the similar HET trend in other SG materials (FeAu and FeCr). Without performing any HFC-dependent studies, they interpreted those abnormal T-dependent EB behaviors either in the framework of the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction theory21,22 or by considering the existence of a T-driven SG-to-AFM phase transition23. However, due to the diverse and entangled spin interactions existed in SG and the fact these interactions can be affected by the field cooling process, the T-dependent sign-changeable HE should be greatly influenced by HFC in the SG/FM bilayers. Theoretically, Usadel and Nowak24 have attempted to reproduce the EB phenomena in the SG/FM bilayer systems by applying an Ising model considering a short-range Gaussian distribution of interactions to simulate the SG material. While they were able to predict the decrease in the magnitude of HE at low T when increasing HFC, the model failed to obtain the HET behaviors as seen in Ref. 21, 22, 23. To the best of our knowledge, the experimental observations of the HFC dependence of EB in SG/FM bilayer systems has remained unreported yet and there are no unified non-phenomenological models that can interpret both the T and HFC dependences of EB in SG based systems.

In this paper, the effects of HFC and T on EB in the FeAu/NiFe SG/FM bilayers were first investigated experimentally. Phenomena of HE sign inversion at the temperatures just below TB, the decrease (increase) in HE magnitude with increasing HFC at low T (high T), the unimodal distribution of HC against T and the HFC-independent HC have all been observed. Then using Monte Carlo technique, we exploited a short-range SG vector model, with a combination of disorder and frustration as well as thermodynamic relaxation and successfully reproduced and interpreted the EB phenomena governed by HFC and T. Distinct from the AFM based systems, this study suggests that the EB effect and especially the sign-changeable (positive EB) behavior in the narrow T region just below TB in the SG/FM bilayer systems may be inherent and exists in all exchange biased SG/FM bilayers.


The quantities of HE and HC are calculated based on HE = (HC1 + HC2)/2 and HC = (−HC1 + HC2)/2, where HC1 and HC2 denote the coercive fields at the descending and the ascending branches of the M-H hysteresis loop, respectively. The temperature dependences of HE and HC in FexAu1 − x/FeNi bilayers are shown in Fig. 1, where x = 4%, 8%, 11% and 14% and HFC is 5 kOe for these measurements. In order to confirm the SG nature of the FeAu layers, ZFC-FC curves were measured with an applied field of 50 Oe. The inset in Fig. 1 shows typical features of spin glass behaviors for a FeAu single layer with x = 11%. The value of the freezing temperature (TF) for this sample is about 30 K, below which the ZFC and FC curves become bifurcated. DC memory effect25 has also been observed in this sample (not shown), providing a further proof of the SG state. Similar SG behaviors can be found in all other FuAu single layer films with x varying from 4% to 14%.

Figure 1
figure 1

Temperature dependent HE (a) and HC (b) with a cooling field of HFC = 5 kOe for FexAu1 − x(50 nm)/FeNi(5 nm) bilayers with x = 4%, 8%, 11% and 14%.

The inset shows the ZFC-FC magnetization curves for the Fe11Au89(50 nm) film under HFC = 50 Oe with T varying between 5 K to 300 K.

Figure 1(a) shows a clear sign-change in HE versus T in the FexAu1 − x/FeNi bilayers with x = 8%, 11% and 14%. With increasing T from the lowest value of 2 K, HE increases abruptly from a negative value and changes its sign at T0, which is named the compensation temperature21,22,23. When T is increased further, HE increases to a positive maximum at another temperature defined as TP. After that, HE decreases gradually and falls below to zero when T approaches to TB. As for the HC dependence on T, it increases initially with T and also shows a concurrent peak at around TP, as shown in Fig. 1(b). It is noteworthy that HE shows a maximum positive value of about 40 Oe when x = 8%, much larger in magnitude than any other previous reported results (normally less than 10 Oe), even for a thin FM layer of 2.2 nm in ref. 21. However, the sign-changeable behavior of HE with T is absent when x = 4%, possibly because the value of T0 is lower than 2 K and beyond the present temperature measuring range.

Figure 1 unambiguously demonstrates that for the SG/FM bilayers the HET curve depends strongly on the composition of the SG layer. Previously, such sign-changeable behavior of HE against T was explained by the competition between long-range oscillatory RKKY couplings from the spins deep inside the SG layer and those close to the interface21,22. In the present work, besides reconsidering the sign-changeable behavior, we focus on studying the HE and HC influenced by HFC. First we discuss the evolution of M-H hysteresis loops versus T. Figure 2 shows the M-H hysteresis loops obtained between 2 K and 20 K for the Ta(4 nm)/Fe11Au89(50 nm)/FeNi(5 nm)/Ta(2 nm) sample, after cooled from 300 K to 2 K under HFC = 50 kOe. The M-H hysteresis loop measured at 2 K clearly shows that a negative EB was established after field cooling. As shown in Fig. 2(a), when T increases from 2 K to 7 K, the ascending branch of the hysteresis loop shifts rightwards while the descending branch keeps almost unchanged. With further increasing in T from 8 K to 20 K, as displayed in Fig. 2(b), the hysteresis loop shrinks from both sides towards the center but the shift of ascending branch is more significantly than the descending branch. Therefore, the different T-dependent variation behaviors of the two loop branches lead to the concurrent peak on both the HET and HCT curves, coinciding with the appearance of TP 7 K as shown in Fig. 1. These results indicate that the T-dependent magnetization reversal mechanisms and/or the thermodynamic spin relaxation at the descending and ascending branches may be quite different.

Figure 2
figure 2

The M-H hysteresis loops measured for a Fe11Au89(50 nm)/FeNi(5 nm) sample in the temperature range of 2 K–7 K (a) and 8 K–20 K (b).

The arrows are guides to eyes, indicating the direction of loop shift.

Experimental study of the HFC dependent EB has been performed and representative results are presented in Fig. 3. As displayed in Fig. 3(a,c), when HFC increases from 0.2 kOe to 50 kOe, the HET curve shifts toward upper-left while the HCT curves keep almost overlapped. Accordingly, T0 decreases from 5.5 K to 4.2 K. The peak position of the HET curve moves towards lower T slightly and the maximum positive value of HE increases significantly from 10 Oe to 18 Oe. Figure 3(b,d) show the corresponding simulation results. The sign inversion in HE with T can be reproduced and HET curve shift can also be repeated. It is noticed that simulated T0, TP and HE(TP) results have certain deviation from the experiments, but the theoretical HET curves agree well with experimental measurements. On the other hand, the calculated trend of HC versus T is different from the experimental results obtained both by Ali et al.21 and us, but in agreement with those reported in ref. 22. Another disagreement with the present experimental results is that the theoretical HCT curves are HFC-dependent at the temperatures just below TB, i.e., HC for larger HFC is slightly larger. These discrepancies will be addressed later.

Figure 3
figure 3

Cooling field influence on the temperature dependent HE (a) and HC (c) for a Fe11Au89(50 nm)/FeNi(5 nm) sample measured between T = 2 K and 20 K and the corresponding calculated results of HE (b) and HC (d) for the SG/FM bilayers obtained between T = 2.6 K and 19.96 K.

At low T, the HFC dependence of HE is also interesting. The experimental and simulation results as shown in Fig. 4 indicate that the entire M-H hysteresis loop moves rightwards with a strong enough HFC. As a result, HE decreases with increasing HFC while HC is insensitive to HFC. Furthermore, there is a slight vertical magnetization shift observed experimentally under strong HFC, arising from a minor magnetization in the FeAu SG. Increase in HE at low HFC (≤1 kOe) is very small because saturation in the FeNi (FM) has not been achived.

Figure 4
figure 4

(a) The M-H hysteresis loops measured under HFC = 0.2 kOe and 50 kOe at T = 2 K for a Fe11Au89(50 nm)/FeNi(5 nm) sample, (b) the calculated M-H hysteresis loops under HFC = 0.2 kOe and 50 kOe at T = 2.6 K for the SG/FM bilayers and the experimental (c) and calculated (d) cooling field dependences of low-temperature HE and HC.

Based on the consistent HE results between experiment and simulation, we interpret the above experimental phenomena relying on our simulation method. At first, we have excluded that the EB phenomena in the SG/FM bilayers depend solely on interfaces [see Supplementary Material A]. In other words, the configurations/spins inside the SG influence the FM spins through the SG/FM interface and the SG/FM interface itself also plays a significant role in establishing EB. Thus in the low T region, we calculate the interfacial exchange energy density (εIF) to interpret the relative EB behavior, which can be expressed as

where is the interfacial coupling stength between the FM and SG spins, A is the interface area and μ denotes the magnetic moment of the interfacial spin belonging to the FM or SG. Since this is the dominant energy term influencing the HE in SG/FM bilayers and meanwhile a low enough T will suppress thermal fluctuation, the change of εIF mainly determines the evolution of spin configuration at the interface during isothermally magnetizing at low T. Therefore, it provides us the opportunity to image the M-H behaviors microscopically and helps us to elucidate how HE is influcenced by HFC. Figure 5(a) shows the results calculated for T = 2.6 K. In addition, other parameters including the spin energy barrier and the x component of spin under specific fields are calculated simultaneously in order to provide a clear physical picture during isothermally magnetizing at low T, with the results shown in Fig. 5 (b,c), respectively.

Figure 5
figure 5

(a) Calculated interfacial exchange energy density (εIF) during the magnetizing process at T = 2.6 K after cooling under HFC = 0.2 kOe and 50 kOe, where solid symbols-solid lines and open symbols-dot lines correspond to the descending and ascending branches of the M-H hysteresis loops and arrows indicate the magnetizing directions. (b) Calculated energy barriers (Eb) of the SG spins at the interface at T = 2.6 K after the field cooling process and before the isothermally magnetizing. (c) Calculated x component (μx) of the SG magnetic moments at the interface under H = −5 kOe during the magnetizing process at T = 2.6 K. (d) Schematic illustrations of the energy versus phase-space coordinate during the magnetization reversal at the ascending branch of the M-H hysteresis loops.

At low T after field cooling, as shown in Fig. 5(a), εIF for HFC = 50 kOe is higher than that for HFC = 0.2 kOe and the value of εIF keeps constant with decreasing H in the positive direction. Then H changes its sign and increases in magnitude and the value of εIF begins to decrease around H = −100 Oe. For HFC = 0.2 kOe, εIF can reduce to a lower value. With further increase in H along the negative direction, εIF begins to increase, corresponding to the magnetization reversal at the descending branch and then reaches a comparable stable value both for HFC = 0.2 kOe and 50 kOe. After the reversal, εIF is unchanged approximately again so long as H is applied in the negative direction. Once H turns back to the positive direction and increases exceeding 100 Oe, εIF decreases rapidly for both HFC accompanied by magnetization reversal at the ascending branch. Moreover, the rapid decrease of εIF occurs at a larger H for HFC = 50 kOe.

The above hysteretic behavior of calculated εIF and the HFC dependent EB at low T (T < T0) can be understood as follows. After field cooling, the bilayers undergone different HFC possess different εIF, as a result of the competition between HFC and JIF. Meantime, the competition also establishes a HFC dependent energy barrier (Eb) configuration, as shown in Fig. 5(b). These energy barriers confine spin rotation and Eb for HFC = 50 kOe is apparently higher than that for HFC = 0.2 kOe. The reason is that HFC is applied along with the SG anisotropy and high HFC drags more spins to align with it, so the anisotropy energy of these spins is minimized simultaneously. This will result in a higher Eb according to the Stoner-Wohlfarth theory. When H decreases along the direction of HFC, no extra energy is required to change the configurations of bilayers and the systems are called in frozen states. When H reverses its direction to become opposite to HFC (termed negative direction), there is a critical field (H ≈ −100 Oe) beyond which the spins in the FM single layer can be reversed24. However, in the SG/FM bilayers the SG spins, via JIF, can impede this process, while the interfacial spins still tend to rearrange towards low exchange energy direction. The rearrangement of SG spin is also influenced by Eb and thus εIF for HFC = 0.2 kOe decreases to a lower value. It is worth noting that the lower the εIF is, the tighter is the bond between spins. To reverse the FM spins to the negative direction, larger H is needed for HFC = 0.2 kOe to overcome the exchange energy. Hence the magnitude of HC1 for HFC = 0.2 kOe is larger than that for HFC = 50 kOe.

When the magnetization reversal at the descending branch is accomplished, εIF is increased abruptly up to a nearly identical value for both HFC and the bilayer spin configuration become frozen again. We have to resort to other approaches to interpret the subsequent magnetizing process. Accordingly, the SG spin orientations at the interface (denoted by μx) under large negative H are calculated and shown in Fig. 5(c) and μx = 2.2 μB indicates that the spins are aligning with HFC while μx = 0 means that the spins are perpendicular to HFC. Remarkably, the SG spins for HFC = 0.2 kOe show random orientations in the film plane, whereas for HFC = 50 kOe all spins keep aligning with HFC, although in both cases εIF has almost the same value. The spin configuration difference under different HFC will influence the magnetization reversal at the ascending branch around H = 100 Oe. For HFC = 0.2 kOe, randomly oriented SG spins are more easily to drag the FM spins to deviate from the negative field direction via JIF under weak H, minimizing the interfacial exchange energy. It resembles that JIF favors the reduction in the reversal field of FM spin, as illustrated in the top panel of Fig. 5(d). In comparison, for HFC = 50 kOe, collinearly aligned SG spins couple to the FM spins via positive or negative JIF. For the FM spins pointing to the negative field direction, the interfacial coupling with negative JIF restrains the FM reversal while positive JIF favors the FM reversal. However, the FM reversal for positive JIF must overcome the ansotropy energy. Therefore, in both cases JIF demotes FM reversal and thus the FM reversal is driven only by H [see the bottom panel of Fig. 5(d)]. As a result, HC2 for HFC = 0.2 kOe is smaller than that for HFC = 50 kOe and HE for HFC = 0.2 kOe is more pronounced.

On the other hand, the influence of HFC on HC at low T will be briefly discussed here. It is well-known that HC is determined by the density of pinned sites, such as defects, which controls the nucleation and propagation of domain walls in FM26,27. In the present SG/FM bilayers, HC also depends on the SG spin behaviors at the interface28. As mentioned above, HFC can reduce |HC1| while enhance HC2. We speculate that the decrement of |HC1| and the increment of HC2 are comparable due to invariable Eb during isothermally magnetizing. Therefore, HC becomes HFC-independent.

When the M-H hysteresis loops are measured at the temperatures between T0 and TB (T0 < T < TB), HE may change its sign to become positive and its magnitude is also HFC-dependent. However, in contrast to T < T0, now HE increases with increasing HFC, as the results calculated at T = 12.52 K shown in Fig. 6(a). The M-H hysteresis loop for HFC = 50 kOe has a larger width and shifts more towards right than that for HFC = 0.2 kOe. For the phenomenon of positive HE in this T region, some mechanisms have been proposed. For example, it was attributed to the AFM interfacial coupling14,15,16 or unidirectional coercive field enhancement along the HFC direction in AFM/FM bilayers. It was also argued that the positive HE in SG/FM bilayers is directly due to the RKKY type interaction influenced by T to a different extent21,22 or the T-driven SG-to-AFM phase transition23. However, all these interpretations lack microscopic evidence and do not take into account the influence of HFC on positive HE.

Figure 6
figure 6

(a) Calculated M-H hysteresis loops at T = 12.52 K after cooling under HFC = 0.2 kOe and 50 kOe and (b) the corresponding calculated interfacial exchange fields (HJ) as a function of magnetic field, where arrows indicate the magnetizing directions.

Different from the case of low T, thermal fluctuations are enhanced at these temperatures, although the SG spins can be still frozen below TF. Especially for the FM spins with weak anisotropy, their orientations will be randomized by thermal energy, so εIF cannot well describe the behaviors of SG spins and is no longer suitable to interpret the phenomena at these temperatures. Therefore, we turn to calculate the average interfacial exchange field (HJ), which arises from the SG and interacts with the FM spins through the interface as

Here is the interfacial coupling between the FM and SG spins, A is the interface area and μj represents the magnetic moment of the SG spins at the interface that are directly coupled to the FM spin. Apparently, HJ is influenced by the SG and thus is T- and H-dependent. As shown in Fig. 6(a,b), HJ remains positive until HC1 and then changes to negative before HC2. Similar to the external magnetic field effect, HJ favors the spins to align with it and thus impedes the H-driven magnetization reversals for both branches. For HFC = 0.2 kOe or 50 kOe, |HJ| in the fourth quadrant is generally larger than that in the second quadrant with same applied magnetic field, i.e. |HJ (−HC1)| > |HJ (HC1)|. Hence a larger H is needed to realize magnetization reversal at the ascending branch, i.e., HC2 is larger than |HC1|. It results in positive HE. For larger HFC of 50 kOe, HJ before HC1 is close to that for HFC = 0.2 kOe, leading to similar HC1 for both HFC. However, in the fourth quadrant, |HJ| for HFC = 50 kOe is much larger than that for HFC = 0.2 kOe [see Fig. 6(b)], leading to a larger HC2 [see Fig. 6(a)]. Consequently, the positive HE is larger for higher HFC.


Different from conventional AFM/FM bilayers, the spin configuration can be rearranged in the present FeAu/FeNi (SG/FM) bilayers even at very low T (e.g. 2 K) due to intrinsic randomness and frustration in the SG layer. Significantly, our work reveals that this reconfiguration can be controlled by HFC and behaves quite differently in different T regions. When T < T0, the spin reconfiguration in the SG can easily occur. For small HFC, the polarization of SG is weak. In the SG/FM bilayers, a saturated FM layer will couple to the SG spins unidirectionally and result in prominent asymmetry of FM reversal at both branches of an M-H loop and a negative HE. However, for large HFC, the polarization of SG is enhanced, impeding the reconfiguration process. Thus the asymmetry of FM reversal is weakened and HE decreases. The experimental and theoretical findings are both consistent with the explanations proposed by Usadel and Nowak24. On the other hand, when T0 < T < TB, the thermal fluctuation cannot be ignored and even may smear out other energy landscapes. The εIF adopted at the low T below T0 is not suitable for the explanation of positive EB effect. Therefore, more direct effect from SG to FM, i.e. HJ is considered. Positive (negative) HJ favors the FM spins to align with (against) HFC and thus forms FM (AFM) type interfacial coupling to the FM spins along the HFC direction. As the calculated results of HJ shown in Fig. 6(b), an FM-to-AFM transition type of interfacial couplings does occur and this process is H-driven, which is different from the T-driven phase transition model suggested by Ali et al.23. Hence a positive HE appears when T0 < T < TB. Moreover, although the thermal fluctuation deteriorates thermal stability, large HFC leads to strong polarization of SG and enhance the system stability, which is beneficial to HJ and favors enhancement in HE [see Fig. 6]. As a result, the positive EB for large HFC is more obvious than that for small HFC.

Now we will discuss some of discrepancies between experiment and simulation briefly. In the actual FeAu/FeNi bilayers, when T is decreased, the exchange and anisotropy energies are enhanced gradually to induce HE below TB and this process also contributes to HC enhancement. With further decreasing T, due to the polycrystalline nature of the FM layer, the easy axes of the FM grains cannot fully lie along with the magnetizing direction, leading to reduction of HC. Therefore, the variation of HC versus T is non-monotonic. In the present simulation, a single crystal model including a uniaxial anisotropy and a saturated FM was adopted to study the EB properties and the magnetizing directions were set to be along with the easy axis. Therefore, the calculated M-H hysteresis loops are more rectangle-shaped and the calculated values of HE or HC are larger than those obtained from experiments. Significantly, the increase of anisotropy energy at low T in the simulation further contributes to the increase of HC when the M-H hysteresis loops are measured along the easy-axis direction. As a result, HC has a monotonic and sharp increase with decreasing T. Also, due to the saturated FM in the initial state, the experimental result of increase in HE with initially increasing HFC is not obtained in simulation. Since the model only considers the most dominant energy terms concerning EB, a perfect agreement between theory and experiment cannot be achieved.

To summarize, the dependence of EB properties in FeAu/FeNi SG/FM bilayers on HFC and T have been studied experimentally and numerically. A sign-changeable behavior of HE versus T is observed, accompanied by a nonmonotonic behavior of HC versus T. More significantly, a phenomenon of decrease (increase) in magnitude of HE at low (high) T with increasing HFC is first observed in experiment while HC is always insensitive to HFC below TB. It is also found that the mechanisms of negative and positive HE influenced by HFC are quite different. Therefore, the highlight of this paper includes exhibiting the HFC modulation of the T-dependent EB behaviors in SG/FM bilayers experimentally and interpreting these behaviors simultaneously by using a unified vector model.


Sample fabrication and measurement

The samples were deposited on silicon wafer by DC magnetron sputtering at room temperature with a stacking sequence of Ta(4 nm)/FexAu1 − x(50 nm)/FeNi(5 nm)/Ta(2 nm). Here, x denotes the atomic fraction of Iron in FeAu alloy. The FeAu layer was co-sputtered with tilted iron and gold guns and the sample composition was characterized by Energy Dispersive X-Ray Spectroscopy (EDX). By fixing the sputtering power of iron and varying that of Au, x was varied from 4% to 14%. A 4 nm Ta buffer layer was deposited to promote the SG/FM morphology and a 2 nm Ta capping layer was deposited to prevent sample from oxidation. FeNi represents Fe19Ni81, which is used as the magnetic pinned layer. The base pressure for sputtering was better than 7.0 × 10−6 Pa and the working Ar pressure was kept at 0.3 Pa during film deposition. Another series of FexAu1 − x single layer films with the identical composition and thickness (i.e., 50 nm) were also deposited on Kapton substrates as reference samples for the low-T SG characterization by means of the ZFC and FC M-T measurements.

Magnetic hysteresis loop measurements were performed in a SQUID-VSM (Quantum Design) with the applied magnetic field (in the range from −5 kOe to 5 kOe) parallel to the film plane. Before each M-H hysteresis loop measurement at a specific T, a magnet reset procedure (oscillatory demagnetization between ±1 Tesla) was performed in order to train the sample to an approximately equilibrium state to eliminate the EB training effect and reducing the residual magnetic field to less than 3 Oe. This will ensure that the obtained HE and HC are accurate with an error less than 3 Oe.

Theoretical model and calculation details

On atomic scale, the real coarse-grained SG/FM bilayers are simulated by a finite number of spins placed on the nodes of a simple cubic lattice. By means of effective size scaling, a lateral dimension of 40 × 40 units in the film plane and a film thickness of 5 monolayers are used with periodic boundary conditions only applying in the plane. Simulations were performed with the Heisenberg model, with the Hamiltonian written as

where μi denotes the magnetic moment of the spin at site i. The first term describes the direct exchange energy in the FM. The next two terms are the magnetocrystalline and shape anisotropy energies of the FM, where αii) is the angle between μi in the FM and the x (z) axis. The fourth term is the Zeeman energy of the FM, where H is applied along the x axis. From the fifth to seventh term represent the direct exchange, magnetocrystalline anisotropy and Zeeman energies of the SG. Here γi is the angle between μi in the SG and the x axis. And the last term is the interfacial direct exchange energy between FM and SG. Detailed scaling and parameterizing processes have been described in the Supplementary Material B.

The simulation protocol mimics the experimental process. Initially, a cooling process under an HFC is performed on the SG/FM bilayers, where the FM spins are all pointing to the HFC direction and the SG spins are randomly oriented, from T = 300 K to a target temperature. Then, an isothermal magnetization is recorded by cycling H from 5 kOe to −5 kOe to extract HE and HC. As for the update of spin state, a path-related Metropolis algorithm29 is adopted during the Monte Carlo simulation, which has been also introduced in the Supplementary Material B in details. This calculation is repeated for 104 times per spin to find the equilibrium state of the system and then an additional 105 steps are taken to measure the thermodynamic average of the magnetization. Finally, 200 independent realizations of the disorder are averaged to reduce the statistical errors.

Additional Information

How to cite this article: Rui, W. B. et al. Cooling field and temperature dependent exchange bias in spin glass/ferromagnet bilayers. Sci. Rep. 5, 13640; doi: 10.1038/srep13640 (2015).