Canted Persistent Spin Texture and Quantum Spin Hall Effect in ${\mathrm{WTe}}_2$

We report an unconventional quantum spin Hall phase in the monolayer ${\text{WTe}}_2$, which exhibits hitherto unknown features in other topological materials. The low symmetry of the structure induces a canted spin texture in the $yz$ plane, which dictates the spin polarization of topologically protected boundary states. Additionally, the spin Hall conductivity gets quantized ($2e^2/h$) with a spin quantization axis parallel to the canting direction. These findings are based on large-scale quantum simulations of the spin Hall conductivity tensor and nonlocal resistances in multiprobe geometries using a realistic tight-binding model elaborated from first-principle methods. The observation of this canted quantum spin Hall effect, related to the formation of topological edge states with nontrivial spin polarization, demands for specific experimental design and suggests interesting alternatives for manipulating spin information in topological materials.

Figure 1

(a) Scheme of the crystal structure of ${\mathrm{WTe}}_2$ in the $1T_d$ phase. (b) Band structure of ${\mathrm{WTe}}_2$ around the charge pockets formed by the band inversion at $\mathrm{\Gamma }$. The conduction band minimum is at the $k$ point $\mathit{Q}=(0.266,0.0)$ with energy ${ϵ}_0\approx -27\mathrm{meV}$. The inset compares the spin textures computed from DFT and the effective model; the color represents the energy with respect to the Fermi level and the arrows the spin orientation in the $yz$ plane (spin projection along $x$ is negligible). The white dots mark the position of the $Q$ point which we have centered at $\mathit{q}=0$. All $k$ points are in units of $\pi /a_x$ with $a_x$ the lattice constant along the zigzag direction.

Figure 2

Spin Hall conductivities ${\sigma }_{xy}^y$ and ${\sigma }_{xy}^z$. The solid line shows the norm of $|{\sigma }_{xy}^{\alpha }|\equiv \sqrt{({\sigma }_{xy}^y{)}^2+({\sigma }_{xy}^z{)}^2}$, the gray area highlights the band gap; the open black circles correspond to ${\sigma }_{xy}^{z^{\prime }}$. Inset: orientation of the spin of the helical edge states. The calculations were done considering a broadening of 5 meV on a system with $1000\times 1000\times 4$ orbitals.

Figure 3

Nonlocal resistances $R_{ij,kl}=(V_k-V_l)/I_{ij}$ calculated in the six-terminal Hall-bar device shown in the inset. The two plateau values $2h/3e^2$ and $h/2e^2$ seen here unequivocally attribute the nonlocal signal to QSH edge states [60]. Solid (dashed) lines correspond to simulations with (without) Anderson disorder (with strength $U=2\mathrm{eV}$). In the inset, the solid (lattice) regions delineate the device (leads). The device is defined on a rectangular lattice (parameters $a_x=3.4607\AA$ and $a_y=6.3066\AA$). The device width, interlead separations, and lead widths are all 50 nm. The small horizontal arrows along the top and bottom edges mark the direction of the local, bond-projected spin current density ${\mathit{J}}_s^{z^{\prime }}$ arising as the response to driving charge current from lead 6 to lead 2.