Deterministic spin-orbit torque switching including the interplay between spin polarization and kagome plane in ${\mathrm{Mn}}_3\mathrm{Sn}$

Previous research has demonstrated the spin-orbit torque (SOT) switching of $\mathrm{M}{\mathrm{n}}_3\mathrm{Sn}$ in configuration I, where the spin polarization σ resides within the kagome plane. However, this configuration has yielded several unexpected outcomes, giving rise to debates concerning the fundamental physics governing the switching process. Alternatively, in configuration II, σ is perpendicular to the kagome plane, which bears greater resemblance to the ferromagnetic system. In this study, we show successful SOT switching of $\mathrm{M}{\mathrm{n}}_3\mathrm{Sn}$ in configuration II, demonstrating behaviors more akin to ferromagnets, e.g., the critical switching current density ($J_{\mathrm{crit}}$) and external field ($H_{\mathrm{ext}}$) are in the order of ${10}^{10}\mathrm{A}/{\mathrm{m}}^2$ and tens of Oersted, respectively. The switching result is also independent of the initial state. We further show that the distinctive spin structure of $\mathrm{M}{\mathrm{n}}_3\mathrm{Sn}$ leads to unique switching characteristics, including $J_{\mathrm{crit}}$ increasing linearly with $H_{\mathrm{ext}}$ and the opposite switching polarity to ferromagnetism. A switching phase diagram is further provided as a guideline for experimental demonstrations, offering a clear physical picture for the observed phenomena.

Figure 1

(a) Atomic structure of $\mathrm{M}{\mathrm{n}}_3\mathrm{Sn}$. The red and yellow circles denote Mn atoms on the different layers. The black and silver circles denote Sn atoms on the same layer of yellow and red Mn, respectively. (b) The six equivalent states of $\mathrm{M}{\mathrm{n}}_3\mathrm{Sn}$, which are defined as states $a$, $b$, $c$, $d$, $e$, and $f$, respectively. The corresponding angles are φ $={30}^{\circ }$, ${90}^{\circ }$, ${150}^{\circ }$, ${210}^{\circ }$, ${270}^{\circ }$, ${330}^{\circ }$. (c) The three device configurations which are defined as configurations I, II, and III, respectively. In configuration I, the line connecting $m_1$ and $m_3$ points to $+\mathbf{y}$ direction.

Figure 2

The dynamics of octupole moment in configuration II. Before turning on ${\mathbf{J}}_{\mathrm{c}}$ and ${\mathbf{H}}_{\mathrm{ext}}$, the system is relaxed for 1 ns to reach the equilibrium state. At 4 ns, both ${\mathbf{J}}_{\mathrm{c}}$ and ${\mathbf{H}}_{\mathrm{ext}}$ are removed, and the system is relaxed to the new stable state. The insets show the spin structure for the initial and final states.

Figure 3

(a)–(d) Illustration of effective fields from spin-orbit torque and external field. (e) The switching phase diagram when $J_{\mathrm{c}}$ and $H_{\mathrm{ext}}$ are varied. Region 1 denotes successful switching from state $d$ to state $a$. The octupole moment aligns to ${\mathbf{H}}_{\mathrm{ext}}$ in region 2. In region 3, the magnetic moments remain in the initial state. In region 4, the magnetic moments develop into oscillation.

Figure 4

Dynamics of the (a) octupole moment in $\mathrm{M}{\mathrm{n}}_3\mathrm{Sn}$ and (b) magnetization in ferromagnets. Both devices have ${\theta }_{\mathrm{SH}}>0$. ${\mathbf{J}}_{\mathrm{c}}$ and ${\mathbf{H}}_{\mathrm{ext}}$ in both cases are along $+y$ direction.

Figure 5

(a) When ${\mathbf{J}}_{\mathrm{c}}=+5\times {10}^{10}\mathrm{A}/{\mathrm{m}}^2$, the dynamics of octupole moment under different $H_{\mathrm{ext}}$. Starting from different initial states, the dynamics of octupole moment when (b) ${\mathbf{J}}_{\mathrm{c}}=+5\times {10}^{10}\mathrm{A}/{\mathrm{m}}^2$ and (c) ${\mathbf{J}}_{\mathrm{c}}=-5\times {10}^{10}\mathrm{A}/{\mathrm{m}}^2$ with ${\mathbf{H}}_{\mathrm{ext}}=+1000$ Oe.