Micromagnetic analysis and optimization of spin-orbit torque switching processes in synthetic antiferromagnets

Robust and efficient data encoding in various types of materials via spin currents has been the main challenge of spintronics. Previous studies of carefully engineered CoFeB/Pt,¹ Fe/Pt², and Ta/CoFe/Ox/Ta³ systems have demonstrated large anomalous Hall effect (AHE), which in conjunction with anisotropic magnetoresistance generates spin-transfer torque (STT).⁴ A large current flowing through a magnetic tunnel junction (MTJ) has been used to electrically manipulate magnetization using STT.^4–6 However, spin–orbit torque (SOT) has gained popularity as a better alternative for developing nonvolatile memory and logic technology with efficient and reliable storage capabilities.⁷ An in-plane current in heavy metals (HMs)^8–11 and antiferromagnets (AFMs)^12,13 has been shown to provide a means of manipulating the magnetization of ferromagnetic (FM) systems, driving magnetization switching^8–12 and domain wall motion.^14–17 The main outstanding challenges in SOT devices include achieving thermal stability, reducing switching current, increasing write speeds, and eliminating the need for an external symmetry-breaking magnetic field.

These issues can potentially be solved by introducing novel structures where interlayer interactions as well as interfacial effects could provide mechanisms to propel magnetization switching. A promising approach to achieving field-free SOT switching of FM layers is to introduce an intrinsic in-plane exchange bias supplied by an uncompensated antiferromagnet (AFM) such as IrMn^18–20 or PtMn.¹² On the other hand, thermal stability and decreased stray fields can be achieved in synthetic antiferromagnetic (SAF) nanowires,²¹ microwires,²² and thin films^20,23–27 where domain wall motion and SOT switching can be enhanced by antiferromagnetic Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction through a nonmagnetic spacer layer. Recently, ultrathin Pt insert layers were grown around the Ru spacer in between the two ferromagnets to enhance perpendicular magnetic anisotropy (PMA) and also provide SOT.²⁵ However, oscillatory antiferromagnetic coupling through a Pt spacer thicker than 25 Å has also been achieved.²⁸ Thus, it is possible to make Pt SAF stacks where the heavy metal spacer layer provides antiferromagnetic RKKY coupling, as well as spin torque to the adjacent ferromagnets.

Recently, several successful field-free SAF switching studies have been done.^29,30 Our approach is slightly different in that we aim to exploit the exchange interaction between an antiferromagnet and a PMA ferromagnet to design a small canting of the spins [see the schematic in Fig. 1(a)]. The canting eventually breaks the symmetry of the problem and makes it possible to achieve field-free switching. The other key feature of the structure is the use of Pt as a dual spin Hall metal and a spacer for the SAF. Due to the symmetry of the generated spin current at the top and bottom surfaces, charge current flowing in Pt can torque both FM1 and FM2 simultaneously. Nonetheless, the heterostructure exhibits several competing phenomena, namely, coupling between AFM and FM1, coupling between FM1 and FM2, and the dynamics of these two ferromagnets arising from the spin–orbit torque. This necessitates a rigorous micromagnetic study to understand if a large enough phase space is available to make this heterostructure work.

In this work, GPU-accelerated micromagnetic simulations were performed in the MuMax³¹ framework to solve the time-and-space-dependent Landau-Lifshitz-Gilbert equation. The original MuMax source code was modified to explicitly include Slonczewski and fieldlike spin–orbit torque terms, as well as interlayer RKKY interactions and exchange bias. Simulation details are found in the supplementary material. The structure shown in Fig. 1(a) is simulated, where the two PMA ferromagnets, FM1 and FM2, are separated by a Pt spacer layer. The simulation mesh consists of $256×256×1$ cells of size $4×4×0.8$ nm$3$⁠. These dimensions were taken to be similar to previous experimental studies of spin torque switching dynamics.^11,20,21,32 Since FM1 is underneath the heavy metal layer, a current in the x-direction would produce a spin polarization $σ=−y^$ at the interface and vice versa for FM2. The AFM underlayer supplies the exchange bias field $BEB$ only to FM1. To compensate for the lack of symmetry-breaking magnetic field acting on FM2, the FM2 layer has a weaker magnetic anisotropy, characterized by the field $|Banis|=2Ku1μ0Msat$⁠.³¹ For this reason, the saturation magnetization $Msat$ for FM2 was taken to be larger than that of FM1: $MsatFM1=9×105$ A m$−1$⁠, while $MsatFM2=9.3×105$  A m$−1$⁠.

Deterministic switching polarity is determined by the scalar product of the current and symmetry-breaking magnetic field, $J⋅B$⁠.³³ In the given FM-HM-FM trilayer configuration with $Jx>0$⁠, if FM1 experiences a positive field (⁠$Bx>0$⁠), $J⋅B>0$ would give up-down switching for FM1. Applying the same magnetic field to FM2 would instead cause down-up switching since it is subject to “opposite” spin polarization at the HM interface. Since the two ferromagnetic layers are vertically antialigned and experience antisymmetric SOT effects, deterministic SAF switching requires that FM1 and FM2 experience the “same” (not opposite) in-plane fields for the odd parity to hold. However, in this case, the structure is designed such that the exchange bias field is only experienced by FM1. This point will be discussed further with regard to exchange bias and RKKY interplay.

The main challenge in the switching of our SAF stack is to achieve deterministic switching of FM2, which does not experience a symmetry-breaking field. As mentioned earlier, FM2 is designed to have a weaker anisotropy, requiring a smaller switching current than FM1. Still, FM2 must be switched entirely by spin–orbit torque and antiferromagnetic coupling to the evolving magnetization of FM1, as well as FM2’s intrinsic spin exchange interactions, anisotropy, demagnitization, and thermal effects. Of those mentioned mechanisms, only RKKY torque is not “symmetry invariant” with respect to up-down states of FM2, as it tends to align FM2 opposite to FM1, thus could uniquely determine the final state of FM2. However, the time-dependent RKKY coupling field is not always conducive to deterministic switching of either FM1 or FM2, as will be discussed further.

Figure 1(b) shows a schematic diagram of the extreme cases where deterministic switching is not possible for one or both FMs. The horizontal axis represents the magnitude of the in-plane exchange bias field (⁠$Bx$⁠) acting on FM1. The regions where successful switching can actually be achieved are discussed further. Note that if $Bx$ is taken to be too large, FM1 will simply be saturated in the x-direction and no switching will be achieved, while a very weak $Bx$ is not strong enough to switch FM1. Thus, for the purpose of this argument, it is assumed that FM1 alone can be switched deterministically without RKKY coupling for all considered values of bias field $Bx$⁠.

The effects of RKKY coupling are considered for the entire SAF stack (FM1 and FM2) for this range of bias fields acting only on FM1. First, in the bottom region of the diagram, RKKY is not strong enough to achieve any “communication” between the two ferromagnetic layers and does not provide the symmetry-breaking mechanism for FM2 switching. Second, in the top region, a strong RKKY interaction completely overpowers spin–orbit torque, leading to what we call the “lock-out” state in which neither of the FM layers can be fully switched. Third, a strong exchange bias field under significant RKKY interaction (middle right region) leads to canting of FM2 in the $−x^$⁠, leading to the incorrect $J⋅B$ product for the desired switching scheme. These undesirable scenarios are analyzed further in the context of switching phase diagrams.

Next, a micromagnetic analysis of a successful switching event is performed. The initial state is FM1-up and FM2-down, where the FM1 is slightly canted along the $x^$ direction due to the in-plane bias field. An in-plane current pulse of $Jx=4.4×108$ A m$−1$ is applied for 3 ns and then the magnetization is allowed to evolve for an additional 2 ns. The Dzyaloshinskii–Moriya interaction (DMI) was modeled in MuMax with $D=1.2$ mJ/m$2$⁠, in accordance with previous simulation studies.²¹ In Figs. 2(a)–2(f), the vertical magnetization component $mz$ is plotted for both ferromagnetic layers at several time steps to show the spatial switching dynamics. Additionally, the movie files in the supplementary material demonstrate $mz$ magnetization dynamics for the entire switching event.

FM1 fully switches from up to down during the current pulse (by the end of 3 ns). Our results are in agreement with previous analysis of reverse domain-wall nucleation in SOT switching of ferromagnetic thin films, specifically Fig. 3 of Baumgartner et al.³⁵ In our case, the current polarity is reversed, since FM1 is positioned underneath the Pt layer, thus for the configuration [$mz>0$⁠, $Jx>0$⁠, $BEB>0$], we expect the reverse domain to be nucleated at the top left corner of the film. This is evident at 0.2 ns, Fig. 2(b). Investigating the in-plane magnetization components shows that this is a Néel domain wall, consistent with previous SOT reversal studies.^21,35 However, instead of simple domain motion across the film, the switching process is better described as reverse domain penetration diagonally from left to bottom right, aided by demagnetization and nucleation of reversed “bubbles.” Analogously, the domain wall formed at the bottom right corner of FM1 at 1 ns, Fig. 2(c), is also consistent with the domain-wall analysis.³⁵ The bottom right edge spins are slightly canted inward (⁠$mx<0$ and $my>0$⁠), which reduces the effect of exchange bias field and slows down the switching in that region. However, the strong antiferromagnetic coupling field causes full magnetization reversal of FM1 by 3.6 ns, Fig. 2(e).

FM2 does not experience the same domain-wall motion due to the absence of an in-plane field, instead, domain nucleation governs its reversal process. After FM1 has mostly switched, the FM2 film experiences a magnetic field of $BRKKY=17.4$ mT along the z-axis. This scenario is similar to the selective switching observed previously³⁶ in antiferromagnetically coupled PMA bilayers under an external vertical magnetic field. It was claimed that the domain nucleation process is dependent on the sweep rate of the magnetic field: nucleation-driven reversal dominates when $dH/dt$ is high and overcomes domain-wall speed, while propagation dominates at lower rates.³⁶ Similarly, we propose that as soon as FM1 has nearly switched, the rate of change $BRKKY$ acting on FM2 is reduced, thus domain growth dominates the reversal of FM2. Note, however, that since the pulse is off and no symmetry-breaking field acts on FM2, the domains are random islands, not structured domain walls.

Finally, we present an analysis of the switching “phase space,” investigating the interplay of RKKY coupling strength $JRKKY$⁠, exchange bias field $BEB$⁠, and current density on magnetization dynamics. Due to its lack of a symmetry-breaking field, FM2 can only be reversed as a result of FM1 switching; thus, the final z-magnetization of FM2 is used as the ultimate outcome of an SAF switching event. Evidently, deterministic SAF switching is only achieved for certain parameter combinations, and nonswitching regions are discussed below.

First, RKKY coupling (⁠$JRKKY<0$⁠) and current density (⁠$Jx>0$⁠) are varied, while keeping the exchange bias coupling constant at $JEB=0.04$ mJ/m$2$ (⁠$BEB≈56$ mT). As shown in Fig. 3, three separate switching regimes can be identified:

• A: Current is too weak to compensate for RKKY strength, this is what we call the “lock-out” state in which neither of the FM layers can be fully switched.

• B: Weak RKKY does not provide a symmetry-breaking mechanism for FM2.

• C: Switching of the entire SAF (both layers).

Second, we present simulation results that provide a more defined picture of the complex RKKY-bias relationship briefly discussed earlier [Fig. 1(b)]. Here, current density was held fixed at $Jx=4.4×108$ A/cm$2$ while varying RKKY and exchange bias coupling parameters, shown in Fig. 4. In addition to regions A–C, area D was identified, corresponding to the right nonswitching region of Fig. 1(b). As discussed, high bias fields cause more FM1 canting, which, in turn, drives FM2 in the $−x^$ direction, impeding switching at the low RKKY regime. However, instead of complete lack of switching, this region displays “conditional switching,” depending on the current pulse duration. The large $BEB$ slows down domain propagation in FM2, thus a longer current pulse is required to fully switch FM2. Additionally, a shorter current pulse would increase region D, even causing it to be merged with region A, the “locked-out” state. A further discussion of region D dynamics for various current pulses is present in Sec. S3 of the supplementary material.

It should be noted that increasing the RKKY strength to 0.017–0.036 mJ/m$2$ actually causes deterministic switching to occur (entering region C). This can be explained from the domain motion observed in both FM1 and FM2: as RKKY is increased, FM2 begins to “respond” via the antiferromagnetic interaction to FM1 domain formation and movement earlier in the pulse. This interaction causes FM2 to break into domains before FM1 is completely switched down, while the subsequent strong RKKY field $BRKKY,z>0$ causes FM2 to fully switch to the up state.

We attribute the patterns that emerge in the second simulation scenario to the unequal access of FM1 and FM2 to the symmetry-breaking bias field. In Fig. 3, both $Jx$ and $JRKKY$ have direct effect on FM1 and FM2, and the two axes are effectively decoupled from each other. However, in the second case, FM2 experiences $BEB$ only indirectly via the antiferromagnetic RKKY interaction. As a result, region D emerges—a strong exchange bias impedes the switching of FM2, even though FM1 actually switches much more easily.

It has been shown that in nanowires and racetrack devices, conventional spin-transfer torque (both adiabatic and nonadiabatic) can induce domain wall motion when the wall is oriented perpendicular to the in-plane current.^21,37–39 However, in our case, no perpendicular domain walls are being injected into FM1 or FM2, while the square geometry does not allow for the simple domain-wall motion guided by STT. Thus, we mainly focus on SOT. As discussed previously, the switching dynamics are in fact much less ordered and characterized by domain nucleation and spreading.

Previous studies have shown that antiferromagnetic materials, including FeMn, PdMn, and IrMn have nontrivial positive spin Hall angles (SHA),¹³ while a negative spin Hall angle was measured in epitaxial IrMn.⁴⁰ Thus, depending on the type of AFM used as the underlayer, the effects of SOT supplied by the Pt will be slightly increased or reduced for FM1 specifically due to the AFM. In fact, for the case of the negative SHA material, SOT supplied by Pt to FM1 would be slightly enhanced by the presence of the AFM. Note that FM1 already experiences an exchange bias field, which makes the conditions inequivalent for FM1 and FM2. We would see a small reduction in FM1 switching time; also the yellow regions of Figs. 3 and 4 would move leftward, to smaller bias fields, since the “lock-out” state would be slightly reduced.

In summary, we showed that, in an appropriately designed synthetic antiferromagnet exchange-coupled to another antiferromagnet, it is possible to achieve a field-free switching by spin–orbit torque. The spatially nonuniform magnetization evolution governed by Heisenberg and Dzyaloshinskii–Moriya spin interactions, as well as interlayer antiferromagnetic coupling, are crucial for reversal processes. A reasonable phase space remains in terms of exchange bias and RKKY interaction strength, which should allow for sufficient flexibility in designing experimental structures. By combining FM2 as part of an MTJ, it should be possible to directly integrate a READ mechanism to the device. The proposed structure should also be achievable by combining conventional materials. Challenges remain with respect to integrating a high quality MTJ with the synthetic antiferromagnet discussed here. The use of Pt as the spacer layer could complicate stabilizing the appropriate phase of CoFeB for the highest possible TMR ratio. We note, however, that this challenge is not unique to the specific stack under study, rather it is present for SOT switching in general, as the spin Hall metal may not be compatible with the optimization needed for the MTJ.

See the supplementary material for further micromagnetic simulation details, analysis of effective magnetic field components, and successful switching events.

This work was supported by the National Science Foundation (NSF) E3S Center at UC Berkeley.