Edge spin transport in the disordered two-dimensional topological insulator ${\mathrm{WTe}}_2$

The spin conductance of two-dimensional topological insulators (2D TIs) is not expected to be quantized in the presence of perturbations that break the spin-rotational symmetry. However, the deviation from the pristine-limit quantization has yet to be studied in detail. In this paper, we define the spin current operator for the helical edge modes of a 2D TI and introduce a four-terminal setup to measure spin conductances. Using the developed formalism, we consider the effects of disorder terms that break spin-rotational symmetry or give rise to edge-to-edge coupling. We identify a key role played by spin torque in an out-of-equilibrium edge. We then utilize a tight-binding model of topological monolayer ${\mathrm{WTe}}_2$ and scattering matrix formalism to numerically study spin transport in a four-terminal 2D TI device. In particular, we calculate the spin conductances and characteristic spin decay length in the presence of magnetic disorder. In addition, we study the effects of interedge scattering in a quantum point contact geometry. We find that the spin Hall conductance is surprisingly robust to spin symmetry-breaking perturbations, as long as time-reversal symmetry is preserved and interedge scattering is weak.

Figure 1

Schematic diagrams demonstrating the voltage setups considered in this paper. Each diagram corresponds to a particular voltage arrangement indicated by the corner values. The edge states along with their propagation directions and spin orientations are depicted by the solid lines; the Dirac cones indicate the filling of these states. The relation between the left and right currents in the presence of $S_z$ nonconserving disorder can be deduced using the transmission and reflection coefficients from Eqs. (7) and (8) and is shown in the center of each diagram. (a) Voltage setup of a 2D TI nanoribbon with a quantized spin Hall conductance $G_H^s=I_{z,L}^s/V=I_{z,R}^s/V$. Only perturbations, which cause bulk conduction or couple the top and bottom edges will cause a deviation from the quantized conductance value. In the absence of such perturbations, the spin current is conserved since each edge is at a local equilibrium and spin torque vanishes. (b) Voltage setup producing nonquantized spin conductances when $S_z$ nonconserving disorder is present. Due to the non-equilibrium distribution on each edge, there is a nonzero spin torque, which breaks the conservation of spin current, $I_{z,L}^s\ne I_{z,R}^s$. The lack of spin current conservation requires the definition of separate incident and transmitted spin conductances given by $G_I^s=I_{z,L}^s/V$ and $G_T^s=I_{z,R}^s/V$, respectively. (c) Standard setup used to define the two-terminal charge conductance $G_{2T}^c=I_L^c/V=I_R^c/V$. Here, voltage distribution of each edge is the same, resulting in no net horizontal spin current.

Figure 2

(a) Schematic depiction of a four-terminal TI device of length $L$, width $W$, and lead width $W_{\text{lead}}$. The dotted region denotes disorder localized between two clean transition regions of length $L_{\text{trans}}$, where $L_{\text{trans}}=0$ indicates a fully disordered sample. We evaluate the spin conductances from the spin currents entering the terminals, see Eq. (26). (b) In order to measure the spin current entering each terminal, we consider the terminals to be composed of two closely spaced ferromagnetic leads with magnetization axes parallel and anti-parallel to the quantized axis of the TI, see Eq. (24). (c) Depiction of spin currents in the spin Hall setup, Fig. 1, with spin-nonconserving disorder between leads 3 and 4. The spin Hall current $I_H^s$ passing through a cross section of the TI sample is conserved and equal to the sum of the currents along the top and bottom edges, $I_H^s=I_{\text{top}}^s+I_{\text{bottom}}^s$. However, spin-nonconserving disorder and a non-equilibrium distribution lead to a spin torque on the edge connecting terminals 3 and 4, see also Fig. 1. Due to the spin torque, an additional current $\delta I_H^s=I_1^s-I_2^s$ is generated and flows to leads 3 and 4, resulting in a total spin current $I_H^s+\delta I_H^s$ entering the terminals and a nonquantized $G_H^s$.

Figure 3

Straight-edge terminated ${\mathrm{WTe}}_2$ band structure in a pristine sample (left) and leads (right), corresponding to the TI and metallic phases, respectively. The dashed line shows the chemical potential; the lead bands are shifted by 400 meV relative to the sample.

Figure 4

Conductances versus disorder strength $w$ for spin-conserving TR-symmetric perturbations. The system dimensions are $L=20$ nm, $W=30$ nm, $L_{\text{trans}}=0$, and $W_{\text{lead}}=12$ nm (see Fig. 2). Conductances are measured in units of the charge and spin conductance quanta. Note that the lowest 3 curves, $G_T^s, G_I^s$, and $G_D^s$, are overlapping over the full range of $w$ in both figures. (a) On-site scalar disorder. (b) Spin-conserving SOC disorder.

Figure 5

Transmitted spin conductance $G_T^s$ with on-site scalar disorder of width $w=150\mathrm{meV}$ or $w=300\mathrm{meV}$ vs sample length $L$. The other sample dimensions are $W=30$ nm, $L_{\text{trans}}=3$ nm, and $W_{\text{lead}}=12$ nm (see Fig. 2).

Figure 6

Conductances versus spin-nonconserving SOC disorder width $w$. The system dimensions are $L=20$ nm, $W=30$ nm, $L_{\text{trans}}=0$ nm, and $W_{\text{lead}}=12$ nm (see Fig. 2). Each data point represents the average of 500 samples. Conductances are measured in units of the charge and spin conductance quanta. Note that the $G_T^s$ and $G_D^s$ curves are overlapping over almost the full range of $w$.

Figure 7

Conductances versus on-site magnetic disorder width $w$. For both plots $L=20$ nm, $W=30$ nm, and $W_{\text{lead}}=12$ nm (see Fig. 2). Conductances are measured in units of the charge and spin conductance quanta. (a) A $L_{\text{trans}}=2.5$ nm wide clean transition region is added to the ends of the TI to ensure no disorder at the lead-TI interfaces. In this case the spin current entering the terminals is approximately conserved and $G_H^s$ stays quantized up to large $w\lesssim 200\mathrm{meV}$. (b) No such transition region is added, $L_{\text{trans}}=0$. In this case there is a spin torque that prevents the quantization of $G_H^s$, see Fig. 2.

Figure 8

Transmitted spin conductance $G_T^s$ as a function of sample length $L$ and on-site magnetic disorder width $w$ for a 32 by 32-mesh grid. The other sample dimensions are $W=30$ nm, $L_{\text{trans}}=3$ nm, and $W_{\text{lead}}=12$ nm [see Fig. 2]. The average value of $G_T^s$ over 50 samples was used to color each grid point; the plot was then smoothed using a Gaussian. The solid lines are contours of constant $G_T^s$ and roughly follow a $L\propto w^{-2}$ dependence (black-dashed line), as is predicted by the relation $ln{G}_T^s\propto -L{w}^2$. Inset: Logarithmic plot of $G_T^s$ averaged over 300 samples vs $L$ for a fixed $w=150\mathrm{meV}$ slice (green-dashed line in the main figure). The slope of the best fit line (solid green) is $-1/\left(9.7\text{nm}\right)$.

Figure 9

Conductance deviations $\delta G=G_0-G$ of a four-terminal QPC system made of a $L=30$ nm by $W=30$ nm rectangular sample cut such that the width smoothly transitions to a narrowed region of length $L_{\text{QPC}}=200$ nm and varying width $W_{\text{QPC}}$. A scalar disorder term with standard deviation $w=300\mathrm{meV}$ is then added to extend the effective decay length and increase edge-to-edge coupling. Conductance deviations are measured in units of the charge and spin conductance quanta. The slopes of the corresponding best fit lines are $-1/\left(13.2\text{nm}\right)$ for $G_{2T}^c, -1/\left(13.6\text{nm}\right)$ for $G_H^s$, and $-1/\left(13.5\text{nm}\right)$ for $G_T^s$. Inset: Diagram of the QPC device demonstrating the definitions of the various dimensions.

Figure 10

Normalized differences between 300 sample and 1000 sample averages. The disorder term is a localized magnetic perturbation as discussed in Sec. 3c and shown in Fig. 7.

Figure 11

Disorder-averaged conductance components versus the number of samples used at fixed $w$. The disorder term is a localized magnetic perturbation as discussed in Sec. 3c and shown in Fig. 7.