Deterministic Switching of Perpendicular Ferromagnets by Higher-order Spin-orbit Torque in Noncentrosymmetric Weyl Semimetals

TL;DR

The paper tackles the problem of achieving field-free deterministic switching of perpendicular ferromagnets without breaking in-plane symmetry. It introduces a symmetry-based framework using vector spherical harmonics to classify higher-order spin-orbit torque contributions, and demonstrates that these torques can create off-equator fixed points in magnetization dynamics. Through a toy LLG model and first-principles calculations on PrAlGe, it shows that higher-order torques can become comparable to conventional terms, enabling reliable, symmetry-preserving switching and reversible control. The findings suggest a new route for electric-field control of magnetization in topological materials and offer guidance for material discovery and experimental validation.

Abstract

Field-free deterministic switching of perpendicular ferromagnets is a central challenge for spintronics applications, typically requiring explicit symmetry breaking. Here we show that deterministic switching can instead be achieved through higher-order (in magnetization angles) spin-orbit torques, even in systems that preserve in-plane mirror symmetry. Using a vector spherical harmonics expansion, we demonstrate that higher-order torque terms naturally give rise to additional out-of-equator fixed points, enabling reliable magnetization reversal when their magnitude is comparable to conventional lowest-order torques. We illustrate this mechanism with first-principles calculations on the noncentrosymmetric Weyl ferromagnet PrAlGe, where the combination of Weyl-node band topology and strong spin-orbit coupling produces sizable higher-order torque components. Because the Fermi surface is small, the conventional lowest-order torques are relatively weak, allowing the higher-order harmonics to compete on equal footing and strongly reshape the magnetization dynamics. The resulting spin dynamics confirm deterministic switching without additional symmetry breaking. Our results establish higher-order spin-orbit torque as a key ingredient for understanding and controlling magnetization dynamics in topological and spintronic materials.

Figures (7)

• Figure 1: Angular dependence of representative spin-orbit torque components for an applied electric field along $\hat{\mathbf{x}}$. Arrows indicate the torque direction on the unit sphere of magnetization $\hat{\mathbf{m}}$, while the color scale denotes torque magnitude. Panels (a) and (b) show the conventional lowest-order dampinglike ($\mathrm{Im}\,\mathbf{Y}^{\mathrm{D}}_{1,1}\propto \hat{\mathbf{m}}\times(\hat{\mathbf{y}}\times\hat{\mathbf{m}})$) and fieldlike ($\mathrm{Im}\mathbf{Y}^{\mathrm{F}}_{1,1}\propto \hat{\mathbf{y}}\times\hat{\mathbf{m}}$) torques, which stabilize only equatorial fixed points. Panels (c)-(h) illustrate selected higher-order symmetry-allowed torques, including $\mathrm{Re}\,\mathbf{Y}^{\mathrm{F}}_{2,1}$, $\mathrm{Re}\,\mathbf{Y}^{\mathrm{D}}_{2,1}$, $\mathrm{Im}\,\mathbf{Y}^{\mathrm{D}}_{3,3}$, $\mathrm{Im}\,\mathbf{Y}^{\mathrm{F}}_{3,3}$, $\mathrm{Im}\,\mathbf{Y}^{\mathrm{D}}_{5,5}$, and $\mathrm{Im}\,\mathbf{Y}^{\mathrm{F}}_{5,5}$, Regions in blue correspond to vanishing torque, indicating fixed points. These higher-order contributions introduce off-equator fixed points that play a crucial role in enabling deterministic switching when comparable in strength to the conventional terms.
• Figure 2: LLG simulations of magnetization dynamics with conventional lowest-order torques [Fig. \ref{['fig:AllowedTorque']}(a,b)] and the selected higher-order torque component [Fig. \ref{['fig:AllowedTorque']}(f)]. Panels (a) and (b) show representative trajectories for ratios of higher- to lower-order torque amplitudes of $0.1$ and $0.4$, respectively. For a small ratio (a), the magnetization is driven to the equatorial fixed point, resulting in nondeterministic switching. For a larger ratio (b) the trajectory crosses the equator and relaxed into a reversed state, demonstrating deterministic switching. Panel (c) presents the phase diagram of switching regimes—deterministic, nondeterministic, and no switching—as functions of the applied electric field $E$ relative to the anisotropy field $H_{\rm A}$ and the ratio of higher-order contributions. Panel (d) shows that starting from the south pole, the same electric-field direction drives the magnetization to the symmetry-related north-hemisphere fixed point, illustrating symmetry-preserving deterministic switching without reversing the field polarity.
• Figure 3: (a) Crystal structure of PrAlGe with out-of-plane Pr magnetic moments. (b) Spin-resolved density of states calculated with Hubbard $U=4$ eV, highlighting the Pr $f$ orbitals. (c,d) Electronic band structures of PrAlGe from plane-wave DFT: (c) collinear calculation without spin-orbit coupling (SOC) and (d) noncollinear calculation including SOC. (e) Comparison of symmetrized Wannier-interpolated bands with the collinear plane-wave results (c). (f) Comparison of Wannier-interpolated bands with SOC added at the atomic level to the collinear model, benchmarked against the noncollinear plane-wave results (d). Panels (e,f) are shown in a reduced energy range near the Fermi level to highlight the quality of the Wannier fitting.
• Figure 4: Dependence of selected vector spherical harmonics coefficients on chemical potential $\mu$. (a) Conventional lowest-order terms $C^{\rm D}_{1,1}$ and $C^{\rm F}_{1,1}$. (b) Dominant higher-order terms $C^{\rm F}_{3,3}$, $C^{\rm D}_{3,3}$, and $C^{\rm D}_{5,5}$.
• Figure 5: Angular dependence of spin-orbit torque in PrAlGe at $\mu=0.02$ eV. (a) spin-orbit torque in units $\tfrac{\hbar}{ea_0}$, evaluated for an applied electric field along $\hat{\mathbf{x}}$. $a_0$ is the atomic Bohr radius. The color scale indicates the torque magnitude on the magnetization unit sphere, while arrows denote the torque direction. (c,d) Magnetization trajectories obtained by solving the LLG equation with the ab initio torque of panel (a) for $E \parallel \hat{\mathbf{x}}$, starting from initial states in the (c) northern and (d) southern hemispheres at the same electric field. Trajectories are drawn on the unit sphere; color encodes time. (b) For a larger electric field than in (c,d), the magnetization no longer relaxes to the off-equator fixed points but instead exhibits sustained precessional oscillations. The oscillation frequency is tunable with electric-field strength; for $E = 14$ V/nm the frequency is approximately $0.85$ THz.