Spin-Orbit Torque Switching of Noncollinear Antiferromagnetic Antiperovskite Manganese Nitride Mn₃GaN

Typical out-of-plane $2\theta$–$\omega$ and in-plane $2\theta_{\chi}$–$\phi$ X-ray diffraction (XRD) patterns for the 35-nm-thick MGN films are shown in Fig. 1(a). Only the MGN (002) and (200) planes exhibit Bragg peaks, in the out-of-plane and in-plane XRD patterns, respectively. In addition, epitaxial growth is confirmed by the results of $\phi$-scan measurement as shown in Fig. 1(b), showing that their epitaxial relationship is MgO(001)[100]//MGN(001)[100]. The lattice constants $c$ and $a$ are 0.38817 nm and 0.38965 nm, respectively, giving $c/a=0.9962$. The temperature dependencies of the resistivity $\rho_{\mathrm{xx}}$ and magnetization $M$ are shown in Fig. 1(c). In the $\rho_{\mathrm{xx}}$ curve, there is a clear anomaly at 200 K. Likewise, an FM-like transition is observed at 200 K in the magnetization curve. These temperature dependencies were also observed in our previous switching study of MGN/Pt films [12]. The ground state of MGN at room temperature is well known to exhibit $\Gamma_{5g}$ spin structure [27]. The coexistence of $\Gamma_{5g}$ and M-1 phase below the FM-like transition temperature has also been previously reported [28]. As the MGN/FM bilayers exhibit an exchange bias at 4 K [29], it is concluded that $\Gamma_{5g}$ order and M-1 phase coexist below 200 K.

Figure 1(d) and 1(e) show the magnetic $M$ hysteresis and anomalous Hall resistivity $\rho_{\mathrm{xy}}$ loops for the MGN films measured along the $\langle 001\rangle$ direction at various temperatures. In the $\rho_{\mathrm{xy}}$ loops, linear contribution from the ordinary Hall effect has been subtracted. The $\rho_{\mathrm{xy}}$ loops before subtract the ordinary Hall effect and a summary of the Hall coefficient values are given in Fig. 6 of Appendix A. In both loops, hysteresis is clearly exhibited not only in the coexistence phase below 200 K but also in the $\Gamma_{5g}$ single phase. Both the magnetization and the $\rho_{\mathrm{xy}}$ loops of the MGN films have similar coercive field ($H_{c}$) values, indicating that the AHE is directly related to the MGN magnetic order. The temperature dependencies of the magnetization and $\rho_{\mathrm{xy}}$ at 2 T are shown in Fig. 1(f). Above 200 K, the magnetization remains nearly constant at $\sim 0.02$ $\mu_{\rm B}/\mathrm{Mn}$, and below 200 K, it increases monotonically with decreasing temperature. $\rho_{\mathrm{xy}}$, on the other hand, increases slightly with decreasing temperature above 200 K and increases dramatically with decreasing temperature below 200 K, indicating that the increase in $\rho_{\mathrm{xy}}$ is strongly linked to the increase in magnetization. In contrast to the magnetization, however, $\rho_{\mathrm{xy}}$ begins to decrease below 75 K. As $\rho_{\mathrm{xy}}$ is found to be proportional to $\rho_{\mathrm{xx}}$ below 75 K, $\rho_{\mathrm{xy}}$ below 200 K would be dominated by a net magnetization.

According to theoretical studies on piezomagnetism of MGN with $\Gamma_{5g}$ order, a net magnetization can appear by the induction of strain [30, 31], and the magnitude of the net magnetization increases linearly with respect to the strain $\epsilon$ %, with a coefficient of 0.013 $\mu_{\rm B}/\mathrm{Mn}/\%$ [30]. Our MGN films show $\epsilon$ and remnant magnetization of approximately 0.3 % and 0.004 $\mu_{\rm B}/\mathrm{Mn}$ at 300 K, respectively. As the net magnetization observed in our MGN films is similar to the value calculated theoretically, it is considered that the canted $\Gamma_{5g}$ order is realized above 200 K. The $\Gamma_{5g}$ order does not show the AHE because it has mirror symmetry, and the symmetry operations make the Berry curvature vanish after integration over the entire Brillouin zone [15]. In antiperovskite nitride films with $\Gamma_{5g}$ order, however, the anomalous Hall conductivity (AHC) tensor is reported to be highly sensitive to strain. Although strain-free Mn₃NiN films show no AHE [18], strained Mn₃NiN films do show AHE [20]. This is explained by the reduction of the symmetry of a space group, under which nonzero Berry curvature is induced when a finite strain is applied [20]. Strained Mn₃SnN films likewise show a large AHE, but it is suggested that the biaxial strain induces $\Gamma_{4g}$ order from $\Gamma_{5g}$ order [21]. These past findings indicate that AHE cannot be accounted for by either extrinsic scattering processes or changes in magnetization; hence, nonzero Berry curvature plays an important role. In our MGN films, $\rho_{\mathrm{xy}}$ is observed to increase with decreasing $c/a$ ratio at 300 K as shown in the inset of Fig. 1(f), highlighting the fact that appearance of AHE at 300 K in MGN films is also strongly related to the film strain and/or reduced magnetic space group. Therefore, although we cannot experimentally separate the contributions from canted net magnetization and nonzero Berry curvature due to noncollinear AFM order above 200 K, we can state that part of the AHE comes from noncollinear AFM order. From the view point of SOT, we will discuss the possible origin of AHE in the later section.

For the electrical write/read operations, typical Hall bar devices of bilayers consisting of strained-MGN (20 nm) with either Pt or Ta (3 nm), with 20 $\mu$m width were prepared. From a parallel circuit model, the $\rho_{\mathrm{xx}}$ values for both the MGN/Pt and MGN/Ta bilayers can be derived using $\rho_{\mathrm{xx}}$ of each single-layer film, which enable us to estimate current density through the Pt and Ta layers ($J_{\mathrm{c},HM}$). The $\rho_{\mathrm{xx}}$ values at 300 K for MGN, Pt, and Ta single-layer films and MGN/Pt and MGN/Ta bilayers are given in Table. 1. Using these bilayers, sequential write/read operations were performed, with the results as shown in Fig. 2. Here, the top and bottom axes are the current densities derived from the Pt/Ta layer alone and the bilayer total thickness, respectively, and the left and right axes are the $\rho_{\mathrm{xy}}$ values derived from the full bilayer and the MGN layer alone, respectively. In addition, the constant offset probably due to geometrical imperfections of the Hall bar and/or thermoelectric voltage was subtracted. $\rho_{\mathrm{xy}}$ changes with respect to $J$, and a clear hysteresis loop is observed in both bilayers at 300 K with no external magnetic field, indicating success of the electrical write/read operation. The critical current densities $J_{\mathrm{c}}$ ($J_{\mathrm{c},HM}$) for MGN/Pt and MGN/Ta were obtained as $2.72\times 10^{6}$ A/cm² ($13.13\times 10^{6}$ A/cm²) and $4.81\times 10^{6}$ A/cm² ($6.13\times 10^{6}$ A/cm²), respectively. Theoretically, $J_{\mathrm{c}}$ would be proportional to $\theta_{HM}^{-1}$, where $\theta_{HM}$ is the spin Hall angle of the HM layer [32]. We find that the ratio $J_{\mathrm{c},\rm{Pt}}:J_{\mathrm{c},\rm{Ta}}=2.1:1$ is nearly equal to the ratio $|\theta_{\rm{Pt}}|^{-1}:|\theta_{\rm{Ta}}|^{-1}=1.9:1$, indicating that spin current generated in the HM layer plays an important role. We note that the different $\rho_{xy}$ switching width between MGN/Pt and MGN/Ta devices is probably due to local variations in the quality and/or strain of the thin films, which is discussed in Fig. 7 of Appendix B. Besides, the comparison of $\rho_{xy}$ width between field-sweep and SOT measurements using the same device is presented in Fig. 8 of Appendix C.

To evaluate the effect of joule heating by applying pulse current, the pulse-width dependence was investigated. Figure 3(a) shows $\rho_{\mathrm{xy}}$ as a function of $J$ with several pulse widths for MGN/Ta bilayers. Although the switching $\rho_{\mathrm{xy}}$ amplitudes remain nearly the same, the hysteresis loops become slightly narrower with increasing pulse width. $J_{\mathrm{c}}$ is plotted as a function of pulse width in Fig. 3(b). Here, $J_{\mathrm{c}}$ is fitted by the thermal activation model [33];

where $J_{\mathrm{c}0}$ is the critical current density at 0 K, $\Delta$ is the thermal stability factor, and $\tau_{0}^{-1}$ is the thermally activated switching frequency, for which we assume a frequency of $1/\tau_{0}=1$ THz. It can be seen that $J_{\mathrm{c}}$ can be fitted by the thermal activation model with $J_{\mathrm{c}0}=7.97\times 10^{6}$ A/cm² and $\Delta=58.7$. These results highlight the finding that a $J_{\mathrm{c}}$ value of the order $10^{6}$ A/cm² originates intrinsically in spin torque whereas the thermal activation, through it exists, plays a minor role.

Because heating and electromigration effects can affect $\rho_{\mathrm{xy}}$ signal amplitudes [23], the relaxation behavior after switching was investigated. Figure 4 presents the continuous write/read operation using $I_{\rm{pulse}}=\pm 13\times 10^{6}$ A/cm² with a 5 $\mu$s pulse width and subsequent relaxation measurements for MGN/Ta bilayers at 300 K. Figure 4(b) is an enlargement of the continuous $\pm I_{\rm{pulse}}$ write part shown in Fig. 4(a). As discussed for synthetic AFM materials [34] and in our previous MGN/Pt study [12], the observed asymptotic $\rho_{\mathrm{xy}}$ behavior can be fitted by an exponential decay function $y=y_{0}+A_{~{}}\mathrm{exp}\left[-(x+x_{0})/\tau\right]$ with time constants $\tau$ of 14.9 pulse number (197.0 s) for $+I_{\rm{pulse}}$ and 15.7 pulse number (207.6 s) for $-I_{\rm{pulse}}$ with continuous write/read operations in approximately 13.25 s cycles. The relaxation behavior after switching was measured after three cycles of the continuous $\pm I_{\rm{pulse}}$ write operations.

Figure 4(c) presents the $\rho_{\mathrm{xy}}$ relaxation as a function of time after continuous $\pm I_{\rm{pulse}}$ write operations. The relaxation behavior is characterized by fit to a double exponential function [23],

where $d_{0}$ is a base line and is the value reached when attenuated, $d_{1}$, $d_{2}$ are amplitude parameters, and $\tau_{1}$, $\tau_{2}$ are the relaxation times. Both relaxations after $\pm I_{\rm{pulse}}$ write operations fit well with $d_{0}=-0.0467$ $\mu\Omega$cm, $d_{1}=-0.0219$ $\mu\Omega$cm, $d_{2}=-0.0003$ $\mu\Omega$cm, $\tau_{1}=8.63$ h, $\tau_{2}=11.67$ h for after the $+I_{\rm{pulse}}$ write operation, and $d_{0}=0.0331$ $\mu\Omega$cm, $d_{1}=0.0100$ $\mu\Omega$cm, $d_{2}=0.0194$ $\mu\Omega$cm, $\tau_{1}=0.25$ h, $\tau_{2}=3.89$ h for after the $-I_{\rm{pulse}}$ write operation. With regard to the change in Hall resistance due to the effects of annealing and electromigration, the short and long decay times are reported to be approximately 4 min and 50 min, respectively [23]. Compared with these values, both the short and long decay times observed here are five to ten times longer. In addition, for the change in Hall resistance due to annealing and electromigration effects, the Hall resistance asymptotes to the initial value before pulse injection is a few minutes to a few hours [23, 35], in contrast with our MGN/Ta bilayers, for which $d_{0}$ is a rather large value. Whereas our electrical measurements do not enable us to fully distinguish SOT and other possible contributions such as thermal activation and electromigration effects, the relaxation behavior and pulse-width behavior in the MGN/Ta bilayers highlight the fact that SOT plays an important role in the present switching behavior. In the case of collinear AFM materials such as NiO, $90^{\circ}$ switching of the N$\rm{\acute{e}}$el vector is required, which can be achieved by the flow of current in two orthogonal directions. However, current flow in orthogonal directions causes inhomogeneous current density due to the current crowding effect, which induces a change in Hall resistance due to annealing and electromigration effect [23]. In the case of noncollinear AFM materials, in contrast, $180^{\circ}$ switching of each noncollinear spin is required, which can be achieved by the flow of current in a straight line; this implies that the current crowding effect in this study can be considered small.

Finally, here we discuss the memristive behavior of MGN/Ta bilayers. Figure 5 shows $\rho_{\mathrm{xy}}$–$J$ loops at 300 K, where $J_{\rm{max}}=13.0\times 10^{6}$ A/cm² was first applied and subsequently scanned from $J_{\rm{max}}$ to $J_{\rm{min}}$ and from $J_{\rm{min}}$ to $J_{\rm{max}}$. $\rho_{\mathrm{xy}}$ exhibits multiple stable signal amplitudes according to the magnitude of $J_{\rm{min}}$. The same behavior has been observed in Mn₃Sn/Pt bilayers [13] and AFM/FM bilayers [37], suggesting that the phenomenon of multiple stable magnitudes of the Hall resistance originates in the multi-AFM domain character, which allows us to tune the signal amplitude in an analog manner. In contrast to Mn₃Sn/Pt bilayers [13], MGN/Ta bilayers show a memristive behavior with no external magnetic field, highlighting the advantage of Mn${}_{3}A$N systems for use in neuromorphic computing.

According to theoretical study of SOT of $\Gamma_{4g}$ order, the noncollinear spins rotate in the (111) plane of the kagome lattice, where the injected spins are directed perpendicular to the kagome lattice. Therefore, $J_{\mathrm{c}}$ is determined by an in-plane anisotropic energy of the kagome lattice and the injected spin direction. Further improvements in SOT efficiency in noncollinear Mn${}_{3}A$N systems may be attained using (110)-oriented films of low-anisotropy materials. Theoretically, Mn₃Ga_1-xNi_xN [36] has a low anisotropy energy, implying that it would be worthwhile to investigate the dependence of SOT efficiency on $A$ atoms. On the other hand, if the AHE comes from only net magnetization (for simplicity, assume a net magnetization in the perpendicular direction like Ferrimagnet), the magnetic field parallel to the current direction is generally needed to realize the SOT switching. In contrast, SOT of noncollinear AFM theoretically satisfies even though with no external magnetic field [32, 36]. Indeed, no switching has been observed with no external magnetic field at low temperatures where the Ferrimagnetic M1 phase is dominant in the AHE [12]. Although the joule heating can affect to the Hall resistivity, the pulse-width measurement and relaxation measurement show a heating effect plays a minor role. From these results, we can conjecture that AHE is possibly related to noncollinear AFM order from the view point of SOT at 300 K.