Spin glass (SG) is commonly formed in noble metals weakly doped with 3d transition metal such as AuFe, CuMn, AgMn [[1], [2], [3]], Ni-Mn-X (X = In, Sn, Sb) Heusler alloys [[4], [5], [6]] and perovskite structure compounds such as Sr₃NiSb₂O₉ [7], Ln₃Ni₂RuO₉ (Ln = La, Nd) [8], Ba_0.8Sr_0.2FeO_2.81 [9]. Signifacantly, the SG state, distinct from a series of materials with magnetic symmetries which allow to access to physical insights and mathematical simplifications [10], characterizes quenched magnetic disorder, where spin-spin interactions between magnetic ions are either positive or negative [11,12]. Thus there is no obvious long-range order, accompanied by nonobvious phase transitions and broken symmetries [[10], [11], [12]]. Moreover, frustration as another central ingredient in SG is commonly caused and enhanced by increasing competing interactions [12,13]. So far, many experimental reports as we know have revealed the SG state induced by competing interactions. In Ni-Mn-X (X=In, Sn, Sb) [[4], [5], [6]], one has suggested that exchange interactions between Mn atoms can be modulated by tuning the composition and element doping due to displacement of moments and/or change of moment separations [[14], [15], [16], [17], [18], [19], [20], [21]], resulting in metamagnetic phase transitions into so-called reentrant SG state with decreasing temperature. Ln₃Ni₂RuO₉ (Ln = La, Nd) oxides show a SG behavior down to 2 K, which reveals the importance of competing nearest-neighbor, next-nearest-neighbor and third-nearest-neighbor interactions between the magnetic Ni²⁺ and Ru⁵⁺ ions [8]. Similarly, Ba_0.8Sr_0.2FeO_2.81 oxides are featured by SG state arising from the competition between ferromagnetic (FM) and antiferromagnetic (AFM) interactions. Moreover, the competing interactions also give rise to a remarkable exchange bias (EB) effect [9]. Therefore, in order to understand the nature of SG well, the competing interaction is of vital importance. Nevertheless, to the best of our knowledge, a precise control of the magnetic properties, especially the dynamic aspect by competing interactions in SG, is still challenging experimentally and thus experimental and theoretical studies are both scarce to some extent.

A prominent phenomenon accompanied by the occurrence of SG is EB. Using EB one enables the manufacture of multifunctional devices with strongly correlated electronic materials [22], which has drawn considerable attention and extensively investigated. Intriguingly, EB can be as a characterization tool to reveal many hidden properties in SG. For example, Li et al. [12] studied the SG irreversibility temperature and found the magnetic stabilization in SG/FM bilayer from low-temperature EB plateau. Furthermore, EB field (H_E) and coercivity (H_C) are switchable, opening an opportunity to design temperature (T)-controlled write-to-read switches. Rui et al. [23] reported cooling-field (H_cool) and T dependent H_E and H_C in FeAu/FeNi bilayer, and revealed that the SG/FM interface rather than a single SG layer is responsible for the crucial EB properties. In the same FeAu/FeNi bilayer, Chi et al. [2] demonstrated the influence of SG dynamic properties on training effect and a nonzero shifting coefficient in the time index exists in SG. Besides extrinsic factors, e.g. T and H_cool, the intrinsic factors of SG should also determine SG properties by distinct manners, just as presented in our previous paper [11] where SG properties can be tunable at will by precisely controlling magnetic parameters such as SG anisotropy (K_SG) and interfacial exchange coupling (J_IF). Meanwhile, the collective effect in SG originates from exchange couplings (J_SG) between ions and contributes to dynamic relaxation properties. Therefore, it is acceptable that the magnetization reversal may depend on J_SG as well in SG/ FM bilayer. Further, the demonstration of role of J_SG on EB is required to enrich our understanding of relationship between SG and EB, where SG properties may also be well shown.

In this work, we numerically studied the influence of J_SG on dynamic relaxation and EB in SG/FM bilayer. At first, various critical temperatures from ZFC/FC curves depend on J_SG, where superparamagnetic blocking and glassy frozen temperatures are identified to confirm existence of SG. Further, it is found that the relaxation related to J_SG is nonmonotonic. For H_E and H_C, strong controllability by J_SG is activated in different ranges of J_SG, which was unambiguously interpreted by results of microscopic net magnetization, spin textures and spin rotatabilities at the SG/FM interface. Referring to real nanocomposite materials, nonmagnetic defects may also exist inevitably and thus finally their role is discussed briefly.

On an atomic scale, it is acceptable that coarse-grained bilayer is simulated by placing spins on the nodes of a simple cubic lattice. A lateral dimension of 40 × 40 spins in a monolayer is used, with periodic boundary conditions in the film plane (set by the xy plane), while open boundary conditions along the z axis. Actually, a larger size (200 × 200 spins) was also used to confirm that finite-size effects have been minimized. Using the size scaling approach reported in our previous work [11], a 36 nm-thickness SG layer represented by 6 monolayers is coupled to a 6 nm-thickness FM layer, which is modeled by a monolayer. Simulations are performed on a Heisenberg model under an external magnetic field (H), and the Hamiltonian can be written as

(1)

where S_i(j) is unit vector of moment spin i(j). The exchange, anisotropy, and Zeeman energy terms are considered in Eq. (1), where the magnetic parameters have been discussed elsewhere [11] and thus set by J_FM = 6.53 erg/cm², K_FM = 5 × 10⁵ erg/cm³, J_IF=J_FM/2 and K_SG = 3.47 × 10⁴ erg/cm³. It is worthwhile to note that the short-range ‘±J’ model is applied to represent SG [24], and remarkably, “±” signs are taken with equal FM and AFM probabilities. Meanwhile, the magnitude of J_SG is changeable from 0.325 to 5.85 erg/cm². Furthermore, 10% nonmagnetic doping may be introduced to discuss its role. To highlight the SG feature, a bilayer model with the SG layer replaced by an AFM layer is also used for comparison. Finally, H is applied along x axis, and magnetic moment value is μ_FM=μ_SG=μ_AF = 2.2μ_B, where μ_B is Bohr magneton, equal to iron’s.

The simulation process mimics the experimental protocol. Initially, the magnetic state in SG is disordered, while saturated along the easy axis in FM. Then, the SG/FM system is cooled under a field of H_cool = 200 Oe from an initial temperature (T₀ = 1000 K) to a target temperature (T = 10 K) with a step of ΔT=-10 K. Subsequently, an isothermal magnetization is recorded by cycling field between -1 and 1 kOe with a step of |ΔH| = 10 Oe to extract H_E and H_C. A modified Monte Carlo Metropolis method is used [11,12] to update spin states and all the spin energies are calculated exactly in a Monte Carlo step (mcs, i.e., simulation time). Finally, at a field or a temperature variation, 12,000 mcs are executed where the initial 10,000 steps are discarded to equilibrate system and the last 2000 steps are used to average magnetic energy and magnetization quantities. This sweep rate is slow enough to guarantee quasi-equilibrium to obtain convincing simulation results.

Magnetization curves in SG/FM bilayer with selected J_SG are measured as a function of temperature (M-T) to grasp SG magnetic properties, as shown in Fig. 1(a-c). Herein, the M-T curves are performed under field-cooling (M^FC) and zero-field-cooling (M^ZFC) at the temperature range 0.0001<T/T_C<2.0 under an applied magnetic field (200 Oe). It clearly shows that a complete transition is observed within the whole temperature range, which is characterized as the typical temperatures T_C and T_g. Previously, the similar typical temperatures have also been found elsewhere [25], where they are defined as the minimum of the T derivative of M^FC. Therefore, T_C is an obvious transition temperature from paramagnetic to ferromagnetic order in bilayer. With decreasing T, J_IF and H take effect and favor SG spins aligning with H, meanwhile, K_SG-driven energy barriers increase, leading to a sharp increase of M at a critical T, designated by T_g. Thus, T_g depends on interactions and resembles a transition temperature from a globally disordered state to an axially disordered state. Note that two states are both blocked, which has been studied in our previous literature [11]. With further decreasing T, M^FC and M^ZFC exhibit a pronounced bifurcation, followed by a peak of M^ZFC at T/T_C = 0.041, 0.03 or 0.015, defined as TMAXZFC, depending on J_SG. This phenomenon has been reported many times [18,26,27], implying the coexistence of disorder and frustration associating with typical SG character. And then, another hallmark signature of glassy behavior, M^FC shows a minimum or independent of T at low T [28] (seen in Fig. 1(a) and (b)). Nevertheless, it does not occur for large J_SG = 4.55 erg/cm² as shown in Fig. 1(c), supporting a disappearance of SG. However, in our simulation results of microscopic spin configurations presented later, it will be found that spin orientation randomness and frustration remain; on the contrary, dynamic properties are changed by J_SG to show a pseudo non-SG M^FC behavior.

Various critical temperatures are extracted and summarized from ZFC/FC curves at selected J_SG, and Fig. 1(d) depicts the J_SG dependence of T_C, T_g and TMAXZFC, where TCFM is the Curie temperature of single FM layer. T_C of the FM coupled to SG is slightly smaller than that without SG while shows an increase with elevating J_SG. Apparently, the stronger is the ferromagnetic order, is the higher the energy of J_SG. Competing J_SG deteriorates the domain structure in SG, and as a result, T_g decreases with increasing J_SG. It is noted that the TMAXZFC corresponding to SG freezing also shows a downward trend with increasing J_SG, indicating that a strong frustration may activate K_SG-driven blocking state at a given temperature intrinsically. Besides, another important temperature is the averaged blocking temperature <T_B>, which is estimated from the peak temperature in the curves of -d(MFC-MZFC)/dT vs. temperature [29]. Clearly, a transition from blocked to superparamagnetic state occurs at <T_B>, which was also studied systematically in our previous paper [12]. <T_B> is suppressed with increasing J_SG from 2 to 4.55 erg/cm²; otherwise there is roughly no change of <T_B> with J_SG to be observed.

To further demonstrate SG properties, we study the dynamic magnetic relaxation. The time dependent magnetization measurements [[30], [31], [32]] are conducted in the FC mode at 10 K with different values of J_SG. The following procedure is adopted. Firstly, the system is cooled from 1000 K to 10 K under FC conditions with a 2 kOe field. Then, the magnetic field is turned off when the temperature reaches 10 K, and subsequently the magnetization decay with the time is measured for 1000 mcs. Remnant magnetization as a function of time depending on J_SG is recorded, as seen in Fig. 2(a). Generally, the SG relaxation follows a power-law decay as(2)M(t)=M₀t^-βwhere M₀ and β are fitting parameters; M₀ is the initial magnetization with t = 0 and β is the decay parameter directly correlating with the decay rate [33]. It is clearly found from Fig. 2(a) that the SG magnetic relaxation of the system can be fitted well with the power law (red lines), and the obtained decay in magnetization is a distinct feature of the SG dynamics [[30], [31], [32]].

Furthermore, J_SG dependences of fitting parameters M₀ and β are extracted and depicted in Fig. 2(b) and (c). Notably, at J_SG<2.6 erg/cm², the dynamic relaxation behaviors relying on J_SG disappear along with M₀/M_S∼1.0 and β∼0 all the time, which are omitted in Fig. 2(b) and (c). From the fitting parameters, one can observe that M₀ decreases with increasing J_SG initially, indicating magnetization of the SG with strong frustration is far from saturation and not stabilized, which is coincident with ZFC-FC curves. As for the increasing of J_SG from 5.22 erg/cm², the system begins easily to be magnetized with the increase in M₀. In addition, β is opposite to the trend of M₀, which can be understood that due to the competing interaction along with strong frustration, M is bound to reach the minimum at the fastest speed after removing H. It is noted that both in ZFC-FC curves and magnetic relaxation results, the SG with large J_SG is different from the common SG state, which will be discussed later.

Next, the M-H curve measured at 10 K after field cooling under H_cool (200 Oe) (seen in Fig. 3(a) and (b)) shows clear hysteresis. Meanwhile, a percentage p = 10% nonmagnetic doping in SG is introduced to study its effect. For undoped SG (Fig. 3(a)), descending branch shifts to the positive H direction with increasing J_SG initially, and then return to negative upon J_SG = 4.55 erg/cm². By contrast, the behavior seems more complicated for ascending branch, i.e. with increasing J_SG, the branch moves along rightwards-leftwards-rightwards. As a whole, the hysteresis loop moves towards a negative direction along the H axis, implying appearance of a negative EB. For doped SG (in Fig. 3(b)), apparently, with increasing J_SG from 3.25 to 4.55 erg/cm², the trend of descending branch is found to be similar to that in Fig. 3(a), while it is different from Fig. 3(a) for ascending branch, accompanying with a unidirectional shift. Significantly, the width of hysteresis loop observed in doped SG is larger than that of undoped SG.

A further investigation has been made to study H_E and H_C, which are evaluated using H_E=(H_C2+H_C1)/2 and H_C=(H_C2-H_C1)/2, where H_C2 and H_C1 are coercive fields of ascending and descending branch at which M goes to zero, respectively. Fig. 3(c) and (d) shows the H_E and H_C results as a function of J_SG in undoped and doped SG/FM bilayer, along with the results in the AFM/FM bilayer (gray lines) for comparison. Interestingly, the J_SG can be divided into two intervals, one of which at low J_SG only affects H_E and the other at large J_SG just works on H_C. That is to say, while an evident decrease in H_E with elevating J_SG from 0.65 to 2.275 erg/cm² is observed, H_C remains unchanged and roughly approaches to that in AFM/FM bilayer. Compared with AFM/FM bilayer, an enhancement of H_E is observed in SG/FM bilayer; at the lowest J_SGH_E in SG/FM is nearly threefold larger than that in AFM/FM bilayer. Small oscillations appear in H_C from J_SG = 0.65 to 2.275 erg/cm², which may be caused by randomly pinning in closely degenerate energies at low temperatures, resembling small quantum fluctuations. Additionally, it is clear that the doping has no effects on either H_E or H_C in this case. Furthermore, in the J_SG range from 2.275 to 5.85 erg/cm², H_C behaves nonmonotonically with a valley value at J_SG = 4.55 erg/cm² in undoped SG and 3.9 erg/cm² in doped SG, while a quite small and constant H_E (nonzero) is also observed. In contrast, H_C is still constant as large as those at small J_SG and H_E has decreased down to zero in AFM/FM bilayer. H_E in doped SG/FM bilayer also remains nonzero at large J_SG while the valley value of H_C is somewhat recovered. Herein, from the viewpoint of technological application, J_SG = 2.275 erg/cm² in the spintronic devices of SG/FM bilayer is an analog to a switch of operation, below which only H_E is adjustable and above which H_C can be regulated, and doping merely quantitatively rather than qualitatively changes this property.

Finally, we have studied microscopically the net magnetization (M_net@IF), the spin textures and spin rotatability (SRI) at the SG/FM interface. The definition to spin rotatability has been presented elsewhere [3]. On one hand, as shown in Fig. 4(a), when increasing J_SG, the net magnetization at the interface decreases, designating that the unidirectional magnetic stabilization may become weakened. Meanwhile, the FM-like domains in SG (i.e. orange area in Fig. 4(a)) are broken into small pieces by AFM bonds, which can lead to the weaker domain-wall pinning ability at the interface. That is to say, the decreasing net magnetization and smaller-area domains may both determine the decreasing H_E with lifting J_SG. Duo to the stronger frustration with the increase in J_SG, the competition interaction between FM and AFM orders is enhanced; as a result, it is easy for spins in the vicinity to form small-area domains in SG. Nevertheless, for sufficiently large J_SG (J_SG>2.275 erg/cm²), the spins in SG are natural to form smaller-area domains (like FM clusters), which may spontaneously reverse and thus the energy barriers are probably smeared out, leading to a faint ability in domain-wall pinning along with a feeble H_E. On the other hand, the spins at the interface are either frozen or rotatable subsequent to magnetic-field application, where it is considered that the constant H_C may derive from the constant SRI (not shown here) as J_SG<2.275 erg/cm², though the areas of domains get smaller. In addition, although SRI still keeps constant upon J_SG>2.275 erg/cm², small-area domains in SG may support the decrease of H_C. It indicates that the domains in SG are too small to pin FM spins of FM layers during magnetization reversal, and some smaller-area domains (like FM clusters) may even drag spins of the FM layer to rotate together. That is to say, before J_SG = 4.55 erg/cm², a high susceptibility and a strong interfacial coupling may lead to FM magnetization reversal in the FM layer under weaker fields and thus a decrease in coercivity. Besides, combining the special change trend from J_SG = 4.55 erg/cm² in the ZFC-FC curves and dynamic relaxation, the re-increased H_C should be related to another physical mechanism. Therefore, SRI as a function of H at J_SG = 4.55 and 5.85 erg/cm² are plotted in Fig. 4(b) and the increased SRI may make a contribution to enhance H_C for too large J_SG. It indicates that magnetic frustration may extend over FM-like domains in SG meanwhile fierce energy competition occurs frequently, resembling enhanced quantum fluctuations. As a result, H_C is recovered. Nevertheless, as a result of the absence of the competition between FM and AFM order in the interfacial AFM layer, the pinning ability just stems from AFM spins rather than domains. Thus, a small H_E appears in AFM/FM bilayer at J_SG<2.275 erg/cm², together with the disappearance of H_E and constant H_C at J_SG>2.275 erg/cm². Finally, the behavior associated with dependences of H_E on J_SG is nearly unchanged even if the SG is doped, exhibiting the feeble differences between the undoped and doped SG in net magnetization. However, the doping seems to have an effect on H_C at large J_SG, accompanied by the recovery of H_C. Compared with Fig. 4(b) and (c), the recovery of H_C is likely due to the decrement of SRI in doped SG. H_C is commonly determined by the density of pinning sites, such as doping or defects, which control the nucleation and propagation of domains-walls in the FM [34,35]. Therefore, the introduction of nonmagnetic doping is bond to decrease SRI with H, equivalent to weakening the competition between AFM and FM exchange interactions at large J_SG. The decrement of SRI makes larger magnetic field be required to realize the reversal of FM spins in the FM layer, as a result a larger H_C emerges.

In summary, we have demonstrated the influence of J_SG on dynamic relaxation and EB in SG/FM bilayer. As expected, ZFC-FC curves display a complete transition, along with the irreversibility in ZFC-FC curves and the peak value in ZFC curves, probably signifying a formation of SG-like state. Additionally, typical temperatures (i.e. T_C, T_g, TMAXZFC and <T_B>) are all related to J_SG, and their behaviors demonstrate the energy competition related to competing interactions. Furthermore, the dynamic relaxation in SG follows a power-law decay of M(t)=M₀t^-β, which indicates a distinct feature of the SG dynamics. Nonmonotonic J_SG dependences of M₀ and β are also observed due to distinct mechanisms at small and large J_SG. Interestingly, the roles of J_SG on H_E and H_C take effect in distinct ranges separated by J_SG = 2.275 erg/cm², whereas the doping may play a main role on weakening frustration and recovering H_C from the induced minimum value by SG around J_SG = 4.55 erg/cm². In essence, the net magnetization, the spin textures and spin rotatability at the SG/FM interface are together responsible for EB. The physical mechanism at small J_SG is related to the unidirectional magnetic stabilization and domain-wall pinning, while at large J_SG the increase in spin rotatability plays a major role. The basic understanding of dynamic relaxation and EB phenomena by adjusting competing interactions will trigger a big step forward in SG-based magnetism.