Symmetry enhanced spin-Nernst effect in honeycomb antiferromagnetic transition metal trichalcogenide monolayers

We investigate systematically the spin-Nernst effect in Néel and zigzag ordered honeycomb antiferromagnets. Monolayers of transition-metal trichalcogenides, $\mathrm{Mn}\mathrm{P}{\mathrm{Se}}_3$, $\mathrm{Mn}{\mathrm{PS}}_3$, and ${\text{VPS}}_3$ show an antiferromagnetic Néel order while $\mathrm{Cr}\mathrm{Si}{\mathrm{Te}}_3$, $\mathrm{Ni}{\mathrm{PS}}_3$, and $\mathrm{Ni}\mathrm{P}{\mathrm{Se}}_3$ show an antiferromagnetic zigzag order. We extract the exchange and Dzyaloshinskii-Moriya interaction parameters from ab initio calculations. Using these parameters, we predict that the spin-Nernst coefficient is at least two orders of magnitude larger in zigzag compared to the Néel ordered antiferromagnets. We find that this enhancement relies on the large band splitting due to the symmetry of magnetic configuration in the zigzag order. Our calculations indicate that the Dzyaloshinskii-Moriya interaction is the underlying factor for the spin-Nernst effect in both cases, although with different microscopic mechanisms. In the case of Néel antiferromagnets, magnon bands already possess a Berry curvature and introducing the Dzyaloshinskii-Moriya interaction splits the magnon bands with the opposite helicity throughout the Brillouin zone which results in an unbalanced population of magnons carrying opposite spins. In the case of zigzag antiferromagnets, magnon bands do not possess the Berry curvature but they are split for opposite helicity magnons due to symmetry of the system. In this case, introducing the Dzyaloshinskii-Moriya interaction induces the Berry curvature and results in the spin-Nernst effect. Due to large magnon band splitting, the spin-Nernst effect in zigzag antiferromagnets is stronger than Néel antiferromagnets.

Figure 1

(a) Top and (b) side views of transition-metal trichalcogenide ${ABX}_3$ monolayer. Blue, red, and green filled circles represent $A$, $B$, and $X$ atoms, respectively. In our DFT calculations we used a $3\times 3$ supercell as shown in (a) where the unit cell is indicated by black dashed lines.

Figure 2

(a) Full view and (b) cross section of the geometry considered to evaluate SNE signal. A uniform temperature gradient along the $\widehat{\mathbf{y}}$ direction generates a total transverse spin current, ${\mathbf{J}}_s$, along the $\widehat{\mathbf{x}}$ direction.

Figure 3

Magnon band structures of (a) $\mathrm{Mn}\mathrm{P}{\mathrm{Se}}_3$, (b) $\mathrm{Mn}{\mathrm{PS}}_3$, (c) ${\text{VPS}}_3$, (d) $\mathrm{Cr}\mathrm{Si}{\mathrm{Te}}_3$, (e) $\mathrm{Ni}{\mathrm{PS}}_3$, and (f) $\mathrm{Ni}\mathrm{P}{\mathrm{Se}}_3$. Blue and red solid lines (dashed lines) represent the magnon bands in the absence (presence) of the DMI interaction. Magnon energies are in units of meV. In the Néel phase materials, when the DMI parameter is zero, the modes overlap with each other. However, in the zigzag phase materials, two split bands can be seen. Magnification of some parts of the bands, magnetic unit cells and symmetry points for magnon band calculations are shown in the inset. Convention of the magnetic sublattice index used for this work is also illustrated inside the unit cell.

Figure 4

The band splitting for (a) $\mathrm{Mn}\mathrm{P}{\mathrm{Se}}_3$ and (b) $\mathrm{Cr}\mathrm{Si}{\mathrm{Te}}_3$ vs percent of variation in $|J_3/J_1|$ and $|D_{2z}/J_1|$, respectively. The results are given at a point where the maximum band splitting is observed ($K$ and $\mathrm{\Gamma }$ points for the Néel and zigzag materials, respectively).

Figure 5

The Berry curvature of the Néel order materials. (Top) $\mathrm{\Omega }\left(\mathbf{k}\right)$ and (bottom) ${log}_{10}\left(\right|\mathrm{\Omega }\left(\mathbf{k}\right)\left|\right)$. $k_x$ and $k_y$ are in units of the lattice constant inverse.

Figure 6

Same as Fig. 5 but for the zigzag ordered materials.

Figure 7

The SNE coefficient (blue dashed lines) and SNE signal (red dashed lines) vs temperature for (a) $\mathrm{Mn}\mathrm{P}{\mathrm{Se}}_3$, (b) $\mathrm{Mn}{\mathrm{PS}}_3$, (c) ${\text{VPS}}_3$, (d) $\mathrm{Cr}\mathrm{Si}{\mathrm{Te}}_3$, (e) $\mathrm{Ni}{\mathrm{PS}}_3$, and (f) $\mathrm{Ni}\mathrm{P}{\mathrm{Se}}_3$. The SNE coefficient of $\mathrm{Cr}\mathrm{Si}{\mathrm{Te}}_3$ is at least two orders of magnitude larger than that in Néel order materials. The experimental and Monte Carlo Néel temperatures are also marked with vertical dashed lines. The results are valid only below $T_N^{\mathrm{MC}}$.

Figure 8

Schematic of four spin states to calculate $J_1$ in the four-state method: (a) state $zz$, (b) state $z\overline{z}$, (c) state $\overline{z}z$, and (d) state $\overline{z}\overline{z}$, where $+z$ and $-z$ directions are denoted by $⨀$ and $⨂$, respectively.

Figure 9

Schematic of four spin states to calcaulate $D_{2z}$ in the four-state method: (a) state $xy$, (b) state $x\overline{y}$, (c) state $\overline{x}y$, and (d) state $\overline{x}\overline{y}$, where $+x$, $-x$, $+y$, $-y$ and $+z$ directions are denoted by $\to$, $\leftarrow$, $\uparrow$, $\downarrow$ and $⨀$, respectively.

Figure 10

Temperature dependent evolution of normalized sublattice magnetization $\langle \mathbf{m}\rangle$ (blue circles) and the normalized susceptibility $\tilde{\chi }=(\chi /{\chi }_m)$ (red squares) for (a) $\mathrm{Mn}\mathrm{P}{\mathrm{Se}}_3$, (b) $\mathrm{Mn}{\mathrm{PS}}_3$, (c) ${\text{VPS}}_3$, and (d) $\mathrm{Cr}\mathrm{Si}{\mathrm{Te}}_3$. (Inset) Normalized sublattice magnetization at low temperatures starting from a random configuration of magnetic moments for two different easy-axis anisotropy values of 0.1 meV (black circles) and 0.01 meV (gray circles). The linear fit to the results are plotted with dashed lines. (e) $\mathrm{Ni}{\mathrm{PS}}_3$ and (f) $\mathrm{Ni}\mathrm{P}{\mathrm{Se}}_3$. The Néel temperature is also marked with a verical dashed line for each material.