Determination of the Spin Axis in Quantum Spin Hall Insulator Candidate Monolayer ${\mathrm{WTe}}_2$

Evidence for the quantum spin Hall effect (QSHE) has been reported in several experimental systems, in the form of edge conductance approaching a quantized value at zero magnetic field. However, another fundamental feature of the QSHE, that of spin-momentum locking in the edge channel, has not been demonstrated. Here, we report that in an applied magnetic field the edge conductance in monolayer ${\mathrm{WTe}}_2$, a candidate QSHE material, is suppressed by the component of the field perpendicular to a particular axis, implying that the spin in the edge states lies along this axis. Surprisingly, the axis is independent of edge orientation, chemical potential, and sample, being fixed relative to the monolayer crystal structure, lying at $(40\pm 2)°$ to the layer normal within the mirror plane. This finding is consistent with a theoretical model in which the bulk bands nearest the Fermi energy have the same parity, leading to simple spin texture with a single, momentum-independent spin axis that is inherited by the edge modes. In addition, the results strengthen the case for spin-momentum locking and, therefore, that monolayer ${\mathrm{WTe}}_2$ is a natural two-dimensional topological insulator exhibiting the QSHE.

Popular Summary

In a 2D topological insulator, also known as a quantum spin Hall insulator, electrical currents flow only around the edge, are impervious to perturbations, and are helical (electrons moving one way along the edge have the opposite spin to those moving the other way). Recent experiments have suggested that monolayer ${\mathrm{WTe}}_2$ is a quantum spin Hall insulator, based on quantized conductance appearing along its edge. However, confirmation of the helical nature was still missing. Here, we report evidence of helicity in monolayer ${\mathrm{WTe}}_2$.

The “smoking gun” prediction for helicity is that a magnetic field applied perpendicular to the spin direction will allow spin flips, induce scattering, and reduce the conductance; a field parallel to the spin direction will not. This is precisely the behavior we see. In our experiments, we find that suppression of edge conduction strongly depends on the orientation of an external magnetic field, being minimal when the field points along an axis in the mirror plane of monolayer ${\mathrm{WTe}}_2$ at about 40 degrees from the layer normal. This special direction can be associated with the spin axis because the Zeeman coupling does not mix spin parallel and antiparallel to the field. The same behavior is seen for edges at different orientations relative to the crystal axes, implying that the bulk bands also share the same spin axis.

The results all but confirm that monolayer ${\mathrm{WTe}}_2$ is a quantum spin Hall insulator, while the surprisingly simple spin texture enhances the usefulness of ${\mathrm{WTe}}_2$ as a platform for studying and exploiting topological, spintronic, and superconducting phenomena.

Figure 1

The spin axis of monolayer ${\mathrm{WTe}}_2$. (a) Cartoon illustrating our key findings: The edge modes on monolayer ${\mathrm{WTe}}_2$ are spin polarized along an axis ${\mathit{d}}_{\mathrm{so}}$ that is independent of edge orientation and shared with the bulk conduction and valence band edges, sketched here schematically near $\mathrm{\Gamma }$. (b) Side and top views of the $1T^{\prime }$ monolayer structure indicating the three spatial symmetries ($P$, $M_x$, and $C_{2x}$) and showing the direction of ${\mathit{d}}_{\mathrm{so}}$, which is in the $y\text{−}z$ mirror plane making angle ${\phi }_{\mathrm{so}}=(40\pm 2)°$ with the $z$ axis. Tungsten atoms are shown in blue, and the two inequivalent tellurium atoms, ${\mathrm{Te}}^1$ and ${\mathrm{Te}}^2$, are different shades of orange. The screw axis (red dotted line) is along the center of a zigzag tungsten chain.

Figure 2

Sensitivity of edge conduction in monolayer ${\mathrm{WTe}}_2$ to magnetic field orientation. (a) Optical microscope image of device MW5. The monolayer ${\mathrm{WTe}}_2$ flake is outlined with a white dashed line. Inset: polar plot of the polarized Raman $A_1$ mode intensity used to determine the $x$ and $y$ crystal axes. (b) Image of the device obtained using microwave impedance microscopy to map the local conductivity ($T=4\mathrm{K}$, $V_g=0$, and $B=0$), revealing edges and cracks as thin bright lines. (c) Polar plot of conductance $G$ between contacts 1 and 2 as the magnetic field $\mathit{B}$ (of strength $B=3\mathrm{T}$) is rotated in the manner indicated in the sketch above ($T=4\mathrm{K}$ and $V_g=-2.7\mathrm{V}$). The sketch also serves to define the angles $\theta$ and $\phi$ specifying the orientation of $\mathit{B}$ relative to the crystal axes. (d) Polar plot of $G$ for rotation of $\mathit{B}$ in the $y\text{−}z$ (mirror) plane, as indicated the sketch above. Here, measurements are plotted for four contact pairs. For each, $G$ is normalized by its minimum value. In all cases, the axis of anisotropy (dashed line) is close to $\phi =38°$.

Figure 3

Dependence of the anisotropy on temperature, field strength, and gate voltage. (All measurements are for contacts 1 and 2 in device MW5.) (a) Two-terminal conductance measured while rotating $\mathit{B}$ in the $y\text{−}z$ plane (see the sketch above), at a series of temperatures ($B=3\mathrm{T}$ and $V_g=-2.7\mathrm{V}$). (b) Corresponding temperature dependence of $G$ at fixed field orientations $\phi ={\phi }_{\mathrm{so}}$ and $\phi ={\phi }_{\mathrm{so}}+90°$. (c) Effect of sweeping the field at $T=4\mathrm{K}$ for the same two orientations. (d) Polar plots of $G$ versus $\phi$ for different field strengths ($V_g=-2.7\mathrm{V}$ and $T=4\mathrm{K}$). (e) Similar plots for different $V_g$ ($B=3\mathrm{T}$ and $T=4K$). (At $V_g=+8\mathrm{V}$, values are divided by 3.).

Figure 4

Anisotropy of the nonlinear conductance. Simultaneous measurements of the field-antisymmetrized nonlinear coefficient ${\gamma }_a$ (top) and linear conductance $G$ (bottom) made as $\mathit{B}$ is rotated in the $y\text{−}z$ plane at selected temperatures (contacts 1 and 2 in device MW5, $V_g=-2.7\mathrm{V}$, and ac bias 15 mV at $f=101\mathrm{Hz}$). The field strength $B=0.3\mathrm{T}$ is chosen to maximize $\gamma$. Inset: example of the variation of $\gamma$ with $B$ (see Fig. S10 [23]).