Intrinsic spin-orbit torque mechanism for deterministic all-electric switching of noncollinear antiferromagnets

Using a pure electric current to control kagome noncollinear antiferromagnets is promising in information storage and processing, but a full description is still lacking, in particular, on intrinsic (i.e., no external magnetic fields or external spin currents) spin-orbit torques. In this work, we self-consistently describe the relations among the electronic structure, magnetic structure, spin accumulations, and intrinsic spin-orbit torques in the magnetic dynamics of a noncollinear antiferromagnet driven by a pure electric current. Our calculation can yield a critical current density comparable with those in the experiments, when considering the boost from the out-of-plane magnetic dynamics induced by the current-driven spin accumulation on individual magnetic moments. We stress the parity symmetry breaking in deterministic switching among magnetic structures. This work will be helpful for future applications of noncollinear antiferromagnets.

Figure 1

(a) A kagome noncollinear antiferromagnet, e.g., ${\mathrm{Mn}}_3\mathrm{Sn}$. The red, green, and blue arrows stand for the three Mn moments $\mathbf{m}$ (1–3 on layer A, $1^{\prime }–3^{\prime }$ on layer B) in the unit cell of a single kagome layer. (b) and (c) The spin-orbit coupling converts a pure electric current into spin accumulations (dashed arrows for in-plane ${\mathbf{s}}_{\left|\right|}$ and $\otimes$ for ut-of-plane ${\mathbf{s}}_{\perp }$). The spin accumulations induce an out-of-plane component $m_{\perp }$ of $\mathbf{m}$, via the field-like torque $\mathbf{m}\times {\mathbf{s}}_{\left|\right|}$ for ${\mathbf{s}}_{\left|\right|}$ in (b) or via the damplike torque $\alpha \mathbf{m}\times (\mathbf{m}\times {\mathbf{s}}_{\perp })$ for ${\mathbf{s}}_{\perp }$ in (c). (d) For a moment $\mathbf{m}$, its antiferromagnetic interaction with other two moments acts like an effective field ${\mathbf{H}}_{\mathrm{AFM}}$ and $m_{\perp }\times {\mathbf{H}}_{\mathrm{AFM}}$ gives the torques (orange arrows and $\otimes$) that rotate the magnetic structure in the kagome ($x\text{−}z$) plane. The fieldlike torque in (b) is dominant, because of the small damping constant $\alpha$ in front of the damplike torque in (c).

Figure 2

The flowchart of the numerical simulation. We start by assuming an initial magnetic structure described by Eq. (1), which serves as local magnetic fields in the $s-d$ model Eq. (7). Then, we use Eqs. (4) and (5) to calculate the spin accumulations on the Mn atoms at a given injecting current Eq. (6) and use Eq. (3) to calculate the magnetic torques of the spin accumulations in the Landau-Lifshitz-Gilbert equation, Eq. (2), which generates a new magnetic structure. The steps are iterated until the magnetic structure converges, then the sign of the Hall signal can be determined by ${\phi }_2$.

Figure 3

Parity symmetry has to be broken for an electric current to deterministically manipulate the Mn moments, because the same magnetic structure is allowed for opposite electric currents (black arrow) under parity symmetry. We list all possible linear-momentum Hamiltonian terms that break parity symmetry and their resulting magnetic structures under the switching current.

Figure 4

(a) Simulated magnetic dynamics of ${\mathrm{Mn}}_3\mathrm{Sn}$ in terms of the moment angle ${\phi }_2$ (inset) of the Mn atom 2, driven by a pure electric current below ($7.5\times {10}^6$), well above ($54\times {10}^6$), and at the critical current density ($j\sim 9.0\times {10}^{6}A/{\mathrm{cm}}^2$, comparable with those in the experiment [12]), for the Weyl model (${\mathrm{\Delta }}_p^{\mathrm{I}}=0.075\mathrm{eV}, {\mathrm{\Delta }}_p^{\mathrm{II}}=0$) in Eq. (8); From 2 through 30 ns (from 50 through 75 ns), $j$ is along the $+z$ ($-z$) direction. In absence of $j, {\phi }_2$ tends to relax at one of the six stable positions marked on the right axis. (b) The total out-of-plane magnetic moment of all three Mn atoms ${({\mathbf{m}}_1+{\mathbf{m}}_2+{\mathbf{m}}_3)}_{\perp }$ for the three current densities in (a). The parameters are $t=0.25\mathrm{eV}, J_{sd}=0.125\mathrm{eV}, {\alpha }_1={\alpha }_2=\pi /5, \theta =\pi /4$ [16], $\tau =\hslash /2\mathrm{\Gamma }$ with $\mathrm{\Gamma }=1.25\mathrm{meV}, m^{*}=9.1\times {10}^{-31}\mathrm{kg}, n_{3D}=6\times {10}^{23}/{\mathrm{cm}}^3$; $J_m=23\mathrm{meV}, D=1.6\mathrm{meV}, K=0.17\mathrm{meV}, \alpha =0.003$ [19].