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AFM bilayers with interface coupling frustrations
Solid State Communications 309 (2020) 113842
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Exchange bias enhancement in FM/AFM bilayers with interface coupling frustrations Xiaozhi Zhan a, b, c, *, Zhongquan Mao a, Xi Chen a, ** a
Department of Physics, South China University of Technology, Guangzhou, 510640, China Spallation Neutron Source Science Center, Dongguan, 523803, China c Institute of High Energy Physics, Chinese Academy of Sciences (CAS), Beijing, 100049, China b
A B S T R A C T
Extensive Monte Carlo simulations have been carried out to investigate the exchange bias effect in ferromagnet/antiferromagnet system where bond dilution is introduced into the interface coupling. The bias field and its blocking temperature are enhanced by the interface bond dilution, suggesting a novel way to tune the exchange bias effect for spintronics applications. By investigating the interfacial spin configurations at the antiferromagnet layer, this enhancement is ascribed to an increase in the effective pinning field due to interface modifications.
1. Introduction The exchange bias (EB) effect manifests itself as a shift of the hys teresis loop along the magnetic field axis after field-cooling a ferro magnet (FM)/antiferromagnet (AFM) system to below the N� eel temperature of the AFM. Since EB effect is widely employed in infor mation storage technology [1,2], it has attracted an enormous amount of research after its discovery in CoO/Co particles [3,4]. Meiklejohn and Bean proposed a phenomenological model [3,4] with a totally uncom pensated AFM interface and gave a fascinating insight into EB effect. However, only a small amount of interface pinned spins have been observed experimentally [5,6]. The domain state model involving for mations of domains in AFM layer due to roughness [7,8] or defects [9–14] suggests a smaller net magnetization at the interface pinning the FM layer and a better agreement between experiments and theoretical calculations was reached. On the other hand, a number of reports sug gest that spin disorder exists at the interface [15–19]. It has even been shown that a FM layer coupled to a spin glass layer can show EB [20,21]. In a previous article [22] it has already been shown that diluting FM bonds randomly into an AFM system would strongly enhance the EB effect, indicating that the degree of spin disorder in AFM layer is important for the occurrence of EB effect. To provide a further insight into EB, the interface effect has been widely studied. One way is to introduce roughness at the interface in a controlled manner. The presence of interface roughness can affect the exchange interactions between FM and AFM spins [23,24]. For instance,
Leighton et al. reported that in MnF2/Fe bilayers the interfacial ex change coupling evolved from antiferromagnetic to ferromagnetic with increasing interface roughness [24]. The roughness-dependence of ex change coupling between FM and AFM layer provides a novel way to study the interface effect of EB. Another series of experiments focus on modifying the interface exchange interactions by inserting a spacer layer [25–36] into the FM/AFM interface. However, controversial results have emerged in different samples, with some reporting that the ex change bias field was found to decay exponentially with the spacer layer thickness in NiFe/CoO [25], while other showing that the biasing could be enhanced with a dusting of certain magnetic impurities [33]. It seems that no agreement has yet been reached due to the fact that the intro duction of magnetic or nonmagnetic atoms brings with itself a variety of factors which affect the EB effect [35]. Further investigations on the interface effect are required for a better insight into the EB effect. As pointed out by Ali et al., the interface coupling can be modified by introducing magnetic elements at the interface and tune the EB effect [33]. Meanwhile, Gamino et al. [37]. reported a great EB enhancement was observed in Co/IrMn bilayers by inserting Py space layer with appropriate thickness. They proposed that the stronger exchange bond across the interface induced by the inserted spacer layer should be an important factor that influences EB field. Therefore, to capture the physics nature of introducing magnetic dusting layer or spacer layer at the FM/AFM interface, Usadel et al. proposed an interface bond dilution model, in which the strengths of exchange couplings across the interface are varied accordingly [35]. Moreover
* Corresponding author. Department of Physics, South China University of Technology, Guangzhou, 510640, China. ** Corresponding author. E-mail addresses: [email protected] (X. Zhan), [email protected] (X. Chen). https://doi.org/10.1016/j.ssc.2020.113842 Received 26 July 2019; Received in revised form 29 December 2019; Accepted 1 February 2020 Available online 5 February 2020 0038-1098/© 2020 Elsevier Ltd. All rights reserved.
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Solid State Communications 309 (2020) 113842
Fig. 1. Sketch of interface bond dilutions with Pint ¼ 0.0 (a), Pint ¼ 0.5 (b) and Pint ¼ 1.0 (c). The plus signs and minus signs represent the FM bonds and AFM bonds respectively. (d) The Pint dependence of EB field in FM/AFM system at t ¼ 0.1 (open squares) and t ¼ 0.3 (filled circles). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
Moran et al. [23] and Leighton et al. [24] proposed that the interface exchange coupling could be modified not only in magnitudes but also in signs by structural disorder at the FM/AFM interface. Thus, we expanded the interface bond dilution model by diluting FM bonds into the AFM interface coupling to simulate the evolvement of interface ex change coupling resulting from the introduction of roughness or mag netic defects. Therefore the effect of interface coupling frustrations on EB can be studied systematically. It is found in our simulations that the bond dilution in interface coupling will enhance the EB field and in crease the blocking temperature. In addition, the AFM interface frozen spin distributions were further investigated and revealed that AFM domain boundaries become thickened and divide AFM layer into small-size domains resulting in large interface pinning field which strengthens the EB effect. All these findings suggest an efficient way to tailor the EB effect by interface modifications for spintronics devices.
“�J model” introduces randomness of exchange bonds with a fixed nearest-neighbor exchange coupling strength yet randomized signs [39]. This simple model has been widely used in studying the glassy behavior in magnetic materials [40–42]. To include disordered spins and break the compensated AFM interface following the same strategy, FM bonds are introduced into an AFM with a probability distribution as � � � (2) P Jpin ði; jÞ ¼ P0 δ Jpin ði; jÞ J0 þ ð1 P0 Þδ Jpin ði; jÞ þ J0 where the dilution concentration of positive bonds randomly distributed in the pinning layers is denoted by the parameter P0 (�0.5). As pointed out in a previous paper [22], AFM domain state emerges with an enhancement in EB effect for increasing bond dilution from P0 ¼ 0.0 to P0 ¼ 0.2. After that, a spin glass state can be established for P0 � 0.3. In this letter we focus on the 20% bond-diluted AFM layers which refers to a domain state antiferromagnet. In addition, we vary the interface coupling distribution to investigate its effect on EB in our FM/AFM system. The value of interface exchange interaction Jint(k) is determined by the following probability distribution
1.1. Model The Monte Carlo model we considered comprises of a FM monolayer and four-layer pinning layers on a simple cubic lattice (only in the sec tion of thickness dependence the thickness of pinning layers N will be varied). Lateral extension in the xy plane is chosen to be a fairly large scale of L � L (L ¼ 100) with periodic boundary conditions. The FM spin ! S i and the pinning layer spin σ k are modeled as the classical Heisenberg spin and the Ising spin respectively [12,13,21], where i and k indicate site index. Only the nearest-neighbor interaction is considered with exchange constant JFM (JFM > 0), Jpin and Jint for spin interactions in the FM layer, in the pinning layers and at the interface, respectively. In addition, to obtain a well defined hysteresis loop and imitate the shape anisotropy, an easy axis along the x axis (i.e., dx ¼ 0.1 JFM) and an in-plane anisotropy (i.e., dz ¼ 0.1JFM) are introduced in the FM layer [21]. With an external magnetic field B applying along the x axis, the Hamiltonian of the system can be written as [21]. X ! ! X � H¼ dx S2ix þ dz S2iz þ BSix JFM S i ⋅ S j
X
i
Jpin ði; jÞσi σ j
X Jint ðkÞSkz σ k :
X Bσ i i
PðJint ðkÞÞ ¼ Pint δðJint ðkÞ
J0 Þ þ ð1
Pint ÞδðJint ðkÞ þ J0 Þ
(3)
where the parameter Pint denotes the concentration of positive bonds which are randomly distributed in the interface coupling. Fig. 1 illus trates the introduction of bond dilutions at the FM/AFM interface, where Pint ¼ 0 and 1.0 refer to antiferromagnetic interface coupling (AFMIC) and ferromagnetic interface coupling (FMIC), respectively. In the ! following we set J0 ¼ JFM/2. In order to simplify the calculations, S i is set to be a unit vector and σi ¼ �1. Meanwhile, reduced field b ¼ B/JFM and temperature values t ¼ kBT/JFM are employed. A heat-bath algorithm and single-spin flip dynamics are employed in our Monte Carlo simulations. We slowly cooled the system from a temperature above the critical temperature of the domain state pinning layers to a low temperature, where the hysteresis loop was measured to obtain the coercive fields (i.e., the left coercive field b and the right coercive field bþ) of FM layer. Thus the exchange bias field bEB can be ascertained by bEB ¼ (bþ þ b )/2. 2. Results and discussions
(1)
Fig. 1(d) shows the EB field as a function of the bond-diluted con centration in the interface couplings. At a low temperature of t ¼ 0.1, a small EB field with magnitude of about 0.01 is obtained for Pint ¼ 0.0, i.e., AFMIC. With increasing ferromagnetic bond dilution in the interface coupling, a significant enhancement in EB is observed. The maximum EB fields appear in the dilution range between Pint ¼ 0.4 and 0.7 with
k
It is well acknowledged that the Edwards-Anderson (EA) model is widely applied to mimic the spin disorder in magnets (e.g., CuMn) with randomized FM and AFM bonds [21,38]. As one type of the EA model, 2
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Solid State Communications 309 (2020) 113842
Fig. 2. The temperature dependence of EB field in FM/AFM system with Pint ¼ 0.5. The inset shows the hysteresis loops measured at different temperatures. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
Fig. 4. The thickness (N) dependence of EB field in FM/AFM system with Pint ¼ 0.5. The inset shows the hysteresis loops of FM/AFM bilayers with different AFM thickness. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
Fig. 3. The Pint dependence of EB field in FM/AFM(N) systems with AFM thickness N ¼ 2 (open squares) and N ¼ 6 (filled circles). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
temperature dependence of EB field in FM/AFM bilayers systems with random interface coupling Pint ¼ 0.5 are shown. The EB decreases monotonously with increasing temperature and vanishes at around t ¼ 0.45, which is higher than the blocking temperature of tB ¼ 0.35 for AFMIC case [45]. Our simulation result is in agreement with a previous experimental report where significant increases in the EB field and the blocking temperature are observed with interfacial Cu dusting at the IrMn/Co interface [46]. Since the thickness of AFM exercises great influence on AFM domain structures which would further affect EB fields [12], it is of great importance to investigate the AFM thickness dependence of the enhancement effect. Fig. 3 displays the Pint dependence of EB field in FM/AFM(N) systems with AFM thickness of N ¼ 2 and N ¼ 6. With increasing bond dilution, the EB fields for both cases are enhanced initially and then decrease after going over maximum values. The en hancements in both cases imply that random interface coupling tends to obtain larger EB than AFMIC and FMIC despite of the different thickness of pinning layers. In addition, the enhancements are of different amounts for different AFM thicknesses. The maximum EB fields for systems with N ¼ 2 and N ¼ 6 are about 0.04 and 0.02 respectively, while according to Fig. 1 the maximum EB for N ¼ 4 lies between these two values, suggesting that large AFM thickness suppresses the EB enhancement effect induced by interface bond dilution. This is mainly due to the fact that large AFM domains tend to form in the bulk of thick AFM layer [12] and the coupling between large domains and the interface further confirmed in Fig. 4, where the thickness dependence of EB field in FM/AFM bilayers systems with random interface coupling Pint ¼ 0.5 is shown. As the thickness N increases, the EB field experiences a rapid increase for N < 2, then decreases after reaching a maximum value at N ¼ 2, and tends to level out for N � 4. The nontrivial depen dence can be understood as a result of the competition between the stability of the AFM interface spins and the magnitude of AFM interface irreversible magnetization [47]. As mentioned before, the large thick ness of AFM leads to suppression of EB enhancement induced by bond dilution, however, the system with N ¼ 1 interestingly does not show the maximum EB value of all but the minimum. It is demonstrated that though the bond dilution in the interface coupling introduces largest amount of disordered spins in the AFM monolayer, the stability of the interface spins are the lowest [47], thus most of the spins follow the rotation of FM spins resulting in a small EB effect. For a larger AFM thickness of N ¼ 2, despite of the reduction of uncompensated magne tization due to the formation of larger domains, the stability of AFM
magnitude of about 0.025, doubling the EB field for Pint ¼ 0.0. With a further increase in bond dilution concentration, the EB effect decreases slowly to around 0.005 for Pint ¼ 1.0, i.e., FMIC. The bond dilution dependence of EB suggests that random interface coupling between AFM layer and FM layer would give rise to much stronger EB fields than that for AFMIC and FMIC. The observation of enhanced EB with random interface coupling is consistent with experimental findings in FM/AFM systems where spacer layer [33,37,43,44] or interface roughness [23] are introduced. T. J. Moran et al. [23] proposed that the significant EB field enhancement in CoO/Py system after ion bombardments or me chanical treatments is related to the nonuniform interface couplings, agreeing well with our simulations. As also shown in Fig. 1, the enhancement in EB field by the bond dilution can still be observed at a higher temperature of t ¼ 0.3, where the EB fields of AFMIC and FMIC cases almost vanish. Though the enhancement is less pronounced at t ¼ 0.3, its existence suggests that random interface coupling may increase the blocking temperature, which is defined as the temperature where the EB field tends to zero [10]. This is confirmed in Fig. 2, where the 3
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Solid State Communications 309 (2020) 113842
Fig. 5. The configurations of reversible spins (white points) and frozen spins (black points) at the AFM interface layer for Pint ¼ 0.0 (a), 0.5 (b) and 1.0 (c) with exactly the same AFM pinning layers. The numbers of frozen spins are 7714, 6521 and 6994 for Pint ¼ 0.0, 0.5 and 1.0, respectively, while the numbers of reversible spins are 2286, 3479 and 3006 respectively. The red curves indicate some of the domain boundaries. (For interpretation of the ref erences to colour in this figure legend, the reader is referred to the Web version of this article.)
interface spins are strengthened [47], thus a large EB field is observed. To investigate the mechanism of how the interface bond dilution gives rise to an enhancement of EB effect, we further carried out studies on the interface frozen spin distributions. It is well established that interface frozen spins, which stay unchanged during a hysteresis cycle, play an important role in the occurrence of the EB effect [48]. Theo retically, the interface frozen spin distribution should be determined by comparing the interface spin configurations under all hysteresis scan ning fields. However, this would require too high capacity of memory storage. Thus we choose the following six cases which should include most of the reversible spins during a hysteresis measurement: after annealing, under maximum external field, at minimum external field, back to maximum external field, under certain external fields with the maximum and minimum interface magnetizations. Spins with un changed direction under all the above situations are considered to be frozen spins. According to Ref. 48, the bias field should be proportional to the interface coupling strength and the frozen spins at the AFM interface. Therefore, the effective pinning field (named bfrozen after wards) acting on each FM spin from the interface frozen spins can be obtained by P Jint ðkÞσ k k2frozen bfrozen ¼ : (4) L2 where the frozen spins and the interface bond dilution can both be taken into consideration. As an example, the calculated frozen spin distribu tions for Pint ¼ 0.0, 0.5, 1.0 at t ¼ 0.1 with exactly the same AFM pinning layers have been shown in Fig. 5. In agreement with an earlier report [45], the reversible spins can be found mostly in domain boundaries and small AFM domains, while the frozen spins distribute mainly in large AFM domains. Comparing to systems with AFMIC (Fig. 5(a)) and FMIC (Fig. 5(c)), in systems with random interface coupling Pint ¼ 0.5 (Fig. 5 (b)), the thickening of AFM domain boundaries and an increase in the number of reversible spins are observed, which results in the shrinkage of AFM domain size and a reduction in the number of frozen spins for Pint ¼ 0.5. However, a stronger pinning field bfrozen ¼ 0.0355 is obtained for Pint ¼ 0.5 comparing to bfrozen ¼ 0.0098 for AFMIC and bfrozen ¼ 0.0085 for FMIC. Since a strong pinning field exerted by AFM would result in a strong EB effect in FM layer, the enhancement in EB field for Pint ¼ 0.5 is expected. Note that the pinning field for AFMIC is slightly stronger than that for FMIC, which is the reason why a stronger EB effect can be obtained for the former. Following the same strategy, the interface frozen spin distributions of all the systems we discussed above were obtained and the corresponding frozen fields have been calculated. All the obtained bfrozen are found to be positive indicating their directions parallel to the cooling field after the field-cooling process. As shown in Fig. 6, EB fields are almost linearly proportional to the pinning field bfrozen, suggesting that even though the introduction of interface bond dilution reduces the number of frozen spins at the interface, it leads to an enhancement in the effective pinning field exerted by AFM layer and thus gives rise to an increase of EB effect. This result agrees well with the earlier report on domain state model proposing that small-size AFM domains may inspire surplus magneti zation and enhance EB [12]. Our simulations suggest a possible way to enhance the EB field and the blocking temperature by tailoring the interface couplings, e.g., by introducing dusting layer or roughness at the FM/AFM interface. For example, J. Wang et al. reported that the
(caption on next column)
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Solid State Communications 309 (2020) 113842 [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
Fig. 6. The EB field as a function of the interface frozen field bfrozen for different systems. The red line is an linear best-fit function |bEB| ¼ a*bfrozen, where a ¼ 0.71947 � 0.01751. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
[18] [19]
interfacial coupling can be modified by tuning the interface roughness in Co/CoO systems and give rise to a stronger bias field [49]. J. B. Salazar et al. proposed that inserting Fe or Py spacer layers in IrMn/Co bilayers could enhance the bias field due to modifications of the interface cou plings [50]. These experimental reports agree with our simulations and suggest that modifications of the interface coupling are important from an application perspective for the EB effect in spintronics technology.
[23] [24] [25] [26] [27]
[20] [21] [22]
[28] [29]
3. Conclusion
[30] [31]
In conclusion, the interface bond dilution would lead to an enhancement in both of the EB field and the blocking temperature. The EB enhancement is more significant in systems with thinner antiferro magnet layer, indicating that the domain structure of the bulk of the pinning layers can mediate the EB effect by influencing the interface spins. The mechanism of enhanced EB by interface bond dilution has been proposed to root in the fact that AFM domain boundaries are thickened due to increasing number of reversible spins and divide AFM layer into small-size domains, which results in a large interface pinning field and thus strengthens the EB effect. All results indicate that a proper interface modification could be an efficient way to tailor the EB effect for spintronics applications.
[32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42]
Acknowledgments
[43] [44]
The work is supported by the National Natural Science Foundation of China (No. 11174083 and No. 11304098), the Fundamental Research Funds for the Central Universities, SCUT (No. 2012ZM0064) and the PhD Start-up Fund of Natural Science Foundation of Guangdong Prov ince, China (No. 2016A030310004).
[45] [46] [47] [48] [49] [50]
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Exchange bias enhancement in FM/AFM bilayers with interface coupling frustrations Xiaozhi Zhan a, b, c, *, Zhongquan Mao a, Xi Chen a, ** a
Department of Physics, South China University of Technology, Guangzhou, 510640, China Spallation Neutron Source Science Center, Dongguan, 523803, China c Institute of High Energy Physics, Chinese Academy of Sciences (CAS), Beijing, 100049, China b
A B S T R A C T
Extensive Monte Carlo simulations have been carried out to investigate the exchange bias effect in ferromagnet/antiferromagnet system where bond dilution is introduced into the interface coupling. The bias field and its blocking temperature are enhanced by the interface bond dilution, suggesting a novel way to tune the exchange bias effect for spintronics applications. By investigating the interfacial spin configurations at the antiferromagnet layer, this enhancement is ascribed to an increase in the effective pinning field due to interface modifications.
1. Introduction The exchange bias (EB) effect manifests itself as a shift of the hys teresis loop along the magnetic field axis after field-cooling a ferro magnet (FM)/antiferromagnet (AFM) system to below the N� eel temperature of the AFM. Since EB effect is widely employed in infor mation storage technology [1,2], it has attracted an enormous amount of research after its discovery in CoO/Co particles [3,4]. Meiklejohn and Bean proposed a phenomenological model [3,4] with a totally uncom pensated AFM interface and gave a fascinating insight into EB effect. However, only a small amount of interface pinned spins have been observed experimentally [5,6]. The domain state model involving for mations of domains in AFM layer due to roughness [7,8] or defects [9–14] suggests a smaller net magnetization at the interface pinning the FM layer and a better agreement between experiments and theoretical calculations was reached. On the other hand, a number of reports sug gest that spin disorder exists at the interface [15–19]. It has even been shown that a FM layer coupled to a spin glass layer can show EB [20,21]. In a previous article [22] it has already been shown that diluting FM bonds randomly into an AFM system would strongly enhance the EB effect, indicating that the degree of spin disorder in AFM layer is important for the occurrence of EB effect. To provide a further insight into EB, the interface effect has been widely studied. One way is to introduce roughness at the interface in a controlled manner. The presence of interface roughness can affect the exchange interactions between FM and AFM spins [23,24]. For instance,
Leighton et al. reported that in MnF2/Fe bilayers the interfacial ex change coupling evolved from antiferromagnetic to ferromagnetic with increasing interface roughness [24]. The roughness-dependence of ex change coupling between FM and AFM layer provides a novel way to study the interface effect of EB. Another series of experiments focus on modifying the interface exchange interactions by inserting a spacer layer [25–36] into the FM/AFM interface. However, controversial results have emerged in different samples, with some reporting that the ex change bias field was found to decay exponentially with the spacer layer thickness in NiFe/CoO [25], while other showing that the biasing could be enhanced with a dusting of certain magnetic impurities [33]. It seems that no agreement has yet been reached due to the fact that the intro duction of magnetic or nonmagnetic atoms brings with itself a variety of factors which affect the EB effect [35]. Further investigations on the interface effect are required for a better insight into the EB effect. As pointed out by Ali et al., the interface coupling can be modified by introducing magnetic elements at the interface and tune the EB effect [33]. Meanwhile, Gamino et al. [37]. reported a great EB enhancement was observed in Co/IrMn bilayers by inserting Py space layer with appropriate thickness. They proposed that the stronger exchange bond across the interface induced by the inserted spacer layer should be an important factor that influences EB field. Therefore, to capture the physics nature of introducing magnetic dusting layer or spacer layer at the FM/AFM interface, Usadel et al. proposed an interface bond dilution model, in which the strengths of exchange couplings across the interface are varied accordingly [35]. Moreover
* Corresponding author. Department of Physics, South China University of Technology, Guangzhou, 510640, China. ** Corresponding author. E-mail addresses: [email protected] (X. Zhan), [email protected] (X. Chen). https://doi.org/10.1016/j.ssc.2020.113842 Received 26 July 2019; Received in revised form 29 December 2019; Accepted 1 February 2020 Available online 5 February 2020 0038-1098/© 2020 Elsevier Ltd. All rights reserved.
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Fig. 1. Sketch of interface bond dilutions with Pint ¼ 0.0 (a), Pint ¼ 0.5 (b) and Pint ¼ 1.0 (c). The plus signs and minus signs represent the FM bonds and AFM bonds respectively. (d) The Pint dependence of EB field in FM/AFM system at t ¼ 0.1 (open squares) and t ¼ 0.3 (filled circles). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
Moran et al. [23] and Leighton et al. [24] proposed that the interface exchange coupling could be modified not only in magnitudes but also in signs by structural disorder at the FM/AFM interface. Thus, we expanded the interface bond dilution model by diluting FM bonds into the AFM interface coupling to simulate the evolvement of interface ex change coupling resulting from the introduction of roughness or mag netic defects. Therefore the effect of interface coupling frustrations on EB can be studied systematically. It is found in our simulations that the bond dilution in interface coupling will enhance the EB field and in crease the blocking temperature. In addition, the AFM interface frozen spin distributions were further investigated and revealed that AFM domain boundaries become thickened and divide AFM layer into small-size domains resulting in large interface pinning field which strengthens the EB effect. All these findings suggest an efficient way to tailor the EB effect by interface modifications for spintronics devices.
“�J model” introduces randomness of exchange bonds with a fixed nearest-neighbor exchange coupling strength yet randomized signs [39]. This simple model has been widely used in studying the glassy behavior in magnetic materials [40–42]. To include disordered spins and break the compensated AFM interface following the same strategy, FM bonds are introduced into an AFM with a probability distribution as � � � (2) P Jpin ði; jÞ ¼ P0 δ Jpin ði; jÞ J0 þ ð1 P0 Þδ Jpin ði; jÞ þ J0 where the dilution concentration of positive bonds randomly distributed in the pinning layers is denoted by the parameter P0 (�0.5). As pointed out in a previous paper [22], AFM domain state emerges with an enhancement in EB effect for increasing bond dilution from P0 ¼ 0.0 to P0 ¼ 0.2. After that, a spin glass state can be established for P0 � 0.3. In this letter we focus on the 20% bond-diluted AFM layers which refers to a domain state antiferromagnet. In addition, we vary the interface coupling distribution to investigate its effect on EB in our FM/AFM system. The value of interface exchange interaction Jint(k) is determined by the following probability distribution
1.1. Model The Monte Carlo model we considered comprises of a FM monolayer and four-layer pinning layers on a simple cubic lattice (only in the sec tion of thickness dependence the thickness of pinning layers N will be varied). Lateral extension in the xy plane is chosen to be a fairly large scale of L � L (L ¼ 100) with periodic boundary conditions. The FM spin ! S i and the pinning layer spin σ k are modeled as the classical Heisenberg spin and the Ising spin respectively [12,13,21], where i and k indicate site index. Only the nearest-neighbor interaction is considered with exchange constant JFM (JFM > 0), Jpin and Jint for spin interactions in the FM layer, in the pinning layers and at the interface, respectively. In addition, to obtain a well defined hysteresis loop and imitate the shape anisotropy, an easy axis along the x axis (i.e., dx ¼ 0.1 JFM) and an in-plane anisotropy (i.e., dz ¼ 0.1JFM) are introduced in the FM layer [21]. With an external magnetic field B applying along the x axis, the Hamiltonian of the system can be written as [21]. X ! ! X � H¼ dx S2ix þ dz S2iz þ BSix JFM S i ⋅ S j
X
i
Jpin ði; jÞσi σ j
X Jint ðkÞSkz σ k :
X Bσ i i
PðJint ðkÞÞ ¼ Pint δðJint ðkÞ
J0 Þ þ ð1
Pint ÞδðJint ðkÞ þ J0 Þ
(3)
where the parameter Pint denotes the concentration of positive bonds which are randomly distributed in the interface coupling. Fig. 1 illus trates the introduction of bond dilutions at the FM/AFM interface, where Pint ¼ 0 and 1.0 refer to antiferromagnetic interface coupling (AFMIC) and ferromagnetic interface coupling (FMIC), respectively. In the ! following we set J0 ¼ JFM/2. In order to simplify the calculations, S i is set to be a unit vector and σi ¼ �1. Meanwhile, reduced field b ¼ B/JFM and temperature values t ¼ kBT/JFM are employed. A heat-bath algorithm and single-spin flip dynamics are employed in our Monte Carlo simulations. We slowly cooled the system from a temperature above the critical temperature of the domain state pinning layers to a low temperature, where the hysteresis loop was measured to obtain the coercive fields (i.e., the left coercive field b and the right coercive field bþ) of FM layer. Thus the exchange bias field bEB can be ascertained by bEB ¼ (bþ þ b )/2. 2. Results and discussions
(1)
Fig. 1(d) shows the EB field as a function of the bond-diluted con centration in the interface couplings. At a low temperature of t ¼ 0.1, a small EB field with magnitude of about 0.01 is obtained for Pint ¼ 0.0, i.e., AFMIC. With increasing ferromagnetic bond dilution in the interface coupling, a significant enhancement in EB is observed. The maximum EB fields appear in the dilution range between Pint ¼ 0.4 and 0.7 with
k
It is well acknowledged that the Edwards-Anderson (EA) model is widely applied to mimic the spin disorder in magnets (e.g., CuMn) with randomized FM and AFM bonds [21,38]. As one type of the EA model, 2
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Fig. 2. The temperature dependence of EB field in FM/AFM system with Pint ¼ 0.5. The inset shows the hysteresis loops measured at different temperatures. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
Fig. 4. The thickness (N) dependence of EB field in FM/AFM system with Pint ¼ 0.5. The inset shows the hysteresis loops of FM/AFM bilayers with different AFM thickness. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
Fig. 3. The Pint dependence of EB field in FM/AFM(N) systems with AFM thickness N ¼ 2 (open squares) and N ¼ 6 (filled circles). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
temperature dependence of EB field in FM/AFM bilayers systems with random interface coupling Pint ¼ 0.5 are shown. The EB decreases monotonously with increasing temperature and vanishes at around t ¼ 0.45, which is higher than the blocking temperature of tB ¼ 0.35 for AFMIC case [45]. Our simulation result is in agreement with a previous experimental report where significant increases in the EB field and the blocking temperature are observed with interfacial Cu dusting at the IrMn/Co interface [46]. Since the thickness of AFM exercises great influence on AFM domain structures which would further affect EB fields [12], it is of great importance to investigate the AFM thickness dependence of the enhancement effect. Fig. 3 displays the Pint dependence of EB field in FM/AFM(N) systems with AFM thickness of N ¼ 2 and N ¼ 6. With increasing bond dilution, the EB fields for both cases are enhanced initially and then decrease after going over maximum values. The en hancements in both cases imply that random interface coupling tends to obtain larger EB than AFMIC and FMIC despite of the different thickness of pinning layers. In addition, the enhancements are of different amounts for different AFM thicknesses. The maximum EB fields for systems with N ¼ 2 and N ¼ 6 are about 0.04 and 0.02 respectively, while according to Fig. 1 the maximum EB for N ¼ 4 lies between these two values, suggesting that large AFM thickness suppresses the EB enhancement effect induced by interface bond dilution. This is mainly due to the fact that large AFM domains tend to form in the bulk of thick AFM layer [12] and the coupling between large domains and the interface further confirmed in Fig. 4, where the thickness dependence of EB field in FM/AFM bilayers systems with random interface coupling Pint ¼ 0.5 is shown. As the thickness N increases, the EB field experiences a rapid increase for N < 2, then decreases after reaching a maximum value at N ¼ 2, and tends to level out for N � 4. The nontrivial depen dence can be understood as a result of the competition between the stability of the AFM interface spins and the magnitude of AFM interface irreversible magnetization [47]. As mentioned before, the large thick ness of AFM leads to suppression of EB enhancement induced by bond dilution, however, the system with N ¼ 1 interestingly does not show the maximum EB value of all but the minimum. It is demonstrated that though the bond dilution in the interface coupling introduces largest amount of disordered spins in the AFM monolayer, the stability of the interface spins are the lowest [47], thus most of the spins follow the rotation of FM spins resulting in a small EB effect. For a larger AFM thickness of N ¼ 2, despite of the reduction of uncompensated magne tization due to the formation of larger domains, the stability of AFM
magnitude of about 0.025, doubling the EB field for Pint ¼ 0.0. With a further increase in bond dilution concentration, the EB effect decreases slowly to around 0.005 for Pint ¼ 1.0, i.e., FMIC. The bond dilution dependence of EB suggests that random interface coupling between AFM layer and FM layer would give rise to much stronger EB fields than that for AFMIC and FMIC. The observation of enhanced EB with random interface coupling is consistent with experimental findings in FM/AFM systems where spacer layer [33,37,43,44] or interface roughness [23] are introduced. T. J. Moran et al. [23] proposed that the significant EB field enhancement in CoO/Py system after ion bombardments or me chanical treatments is related to the nonuniform interface couplings, agreeing well with our simulations. As also shown in Fig. 1, the enhancement in EB field by the bond dilution can still be observed at a higher temperature of t ¼ 0.3, where the EB fields of AFMIC and FMIC cases almost vanish. Though the enhancement is less pronounced at t ¼ 0.3, its existence suggests that random interface coupling may increase the blocking temperature, which is defined as the temperature where the EB field tends to zero [10]. This is confirmed in Fig. 2, where the 3
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Fig. 5. The configurations of reversible spins (white points) and frozen spins (black points) at the AFM interface layer for Pint ¼ 0.0 (a), 0.5 (b) and 1.0 (c) with exactly the same AFM pinning layers. The numbers of frozen spins are 7714, 6521 and 6994 for Pint ¼ 0.0, 0.5 and 1.0, respectively, while the numbers of reversible spins are 2286, 3479 and 3006 respectively. The red curves indicate some of the domain boundaries. (For interpretation of the ref erences to colour in this figure legend, the reader is referred to the Web version of this article.)
interface spins are strengthened [47], thus a large EB field is observed. To investigate the mechanism of how the interface bond dilution gives rise to an enhancement of EB effect, we further carried out studies on the interface frozen spin distributions. It is well established that interface frozen spins, which stay unchanged during a hysteresis cycle, play an important role in the occurrence of the EB effect [48]. Theo retically, the interface frozen spin distribution should be determined by comparing the interface spin configurations under all hysteresis scan ning fields. However, this would require too high capacity of memory storage. Thus we choose the following six cases which should include most of the reversible spins during a hysteresis measurement: after annealing, under maximum external field, at minimum external field, back to maximum external field, under certain external fields with the maximum and minimum interface magnetizations. Spins with un changed direction under all the above situations are considered to be frozen spins. According to Ref. 48, the bias field should be proportional to the interface coupling strength and the frozen spins at the AFM interface. Therefore, the effective pinning field (named bfrozen after wards) acting on each FM spin from the interface frozen spins can be obtained by P Jint ðkÞσ k k2frozen bfrozen ¼ : (4) L2 where the frozen spins and the interface bond dilution can both be taken into consideration. As an example, the calculated frozen spin distribu tions for Pint ¼ 0.0, 0.5, 1.0 at t ¼ 0.1 with exactly the same AFM pinning layers have been shown in Fig. 5. In agreement with an earlier report [45], the reversible spins can be found mostly in domain boundaries and small AFM domains, while the frozen spins distribute mainly in large AFM domains. Comparing to systems with AFMIC (Fig. 5(a)) and FMIC (Fig. 5(c)), in systems with random interface coupling Pint ¼ 0.5 (Fig. 5 (b)), the thickening of AFM domain boundaries and an increase in the number of reversible spins are observed, which results in the shrinkage of AFM domain size and a reduction in the number of frozen spins for Pint ¼ 0.5. However, a stronger pinning field bfrozen ¼ 0.0355 is obtained for Pint ¼ 0.5 comparing to bfrozen ¼ 0.0098 for AFMIC and bfrozen ¼ 0.0085 for FMIC. Since a strong pinning field exerted by AFM would result in a strong EB effect in FM layer, the enhancement in EB field for Pint ¼ 0.5 is expected. Note that the pinning field for AFMIC is slightly stronger than that for FMIC, which is the reason why a stronger EB effect can be obtained for the former. Following the same strategy, the interface frozen spin distributions of all the systems we discussed above were obtained and the corresponding frozen fields have been calculated. All the obtained bfrozen are found to be positive indicating their directions parallel to the cooling field after the field-cooling process. As shown in Fig. 6, EB fields are almost linearly proportional to the pinning field bfrozen, suggesting that even though the introduction of interface bond dilution reduces the number of frozen spins at the interface, it leads to an enhancement in the effective pinning field exerted by AFM layer and thus gives rise to an increase of EB effect. This result agrees well with the earlier report on domain state model proposing that small-size AFM domains may inspire surplus magneti zation and enhance EB [12]. Our simulations suggest a possible way to enhance the EB field and the blocking temperature by tailoring the interface couplings, e.g., by introducing dusting layer or roughness at the FM/AFM interface. For example, J. Wang et al. reported that the
(caption on next column)
4
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Solid State Communications 309 (2020) 113842 [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
Fig. 6. The EB field as a function of the interface frozen field bfrozen for different systems. The red line is an linear best-fit function |bEB| ¼ a*bfrozen, where a ¼ 0.71947 � 0.01751. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
[18] [19]
interfacial coupling can be modified by tuning the interface roughness in Co/CoO systems and give rise to a stronger bias field [49]. J. B. Salazar et al. proposed that inserting Fe or Py spacer layers in IrMn/Co bilayers could enhance the bias field due to modifications of the interface cou plings [50]. These experimental reports agree with our simulations and suggest that modifications of the interface coupling are important from an application perspective for the EB effect in spintronics technology.
[23] [24] [25] [26] [27]
[20] [21] [22]
[28] [29]
3. Conclusion
[30] [31]
In conclusion, the interface bond dilution would lead to an enhancement in both of the EB field and the blocking temperature. The EB enhancement is more significant in systems with thinner antiferro magnet layer, indicating that the domain structure of the bulk of the pinning layers can mediate the EB effect by influencing the interface spins. The mechanism of enhanced EB by interface bond dilution has been proposed to root in the fact that AFM domain boundaries are thickened due to increasing number of reversible spins and divide AFM layer into small-size domains, which results in a large interface pinning field and thus strengthens the EB effect. All results indicate that a proper interface modification could be an efficient way to tailor the EB effect for spintronics applications.
[32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42]
Acknowledgments
[43] [44]
The work is supported by the National Natural Science Foundation of China (No. 11174083 and No. 11304098), the Fundamental Research Funds for the Central Universities, SCUT (No. 2012ZM0064) and the PhD Start-up Fund of Natural Science Foundation of Guangdong Prov ince, China (No. 2016A030310004).
[45] [46] [47] [48] [49] [50]
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