Theory of glide symmetry protected helical edge states in a ${\mathrm{WTe}}_2$ monolayer

Helical edge states in quantum spin Hall (QSH) materials are central building blocks of topological matter design and engineering. Despite their principal topological protection against elastic backscattering, the level of operational stability depends on manifold parameters such as the band gap of the given semiconductor system in the “inverted” regime, temperature, disorder, and crystal orientation. We theoretically investigate electronic and transport properties of QSH edge states in large gap 1-$T^{\prime } {\mathrm{WTe}}_2$ monolayers. We explore the impact of edge termination, disorder, temperature, and interactions on experimentally addressable edge state observables, such as local density of states and conductance. We show that conductance quantization can remain surprisingly robust even for heavily disordered samples because of an anomalously small edge state decay length and additional protection related to the large direct gap allowed by glide symmetry. From the simulation of temperature-dependent resistance, we find that moderate disorder enhances the stability of conductance by localizing bulk states. We evaluate the edge state velocity and Luttinger liquid parameter as functions of the chemical potential, finding prospects for physics beyond linear helical Luttinger liquids in samples with ultraclean and well-defined edges.

Figure 1

Structural and electronic properties of ${\mathrm{WTe}}_2$ for two types of edge termination. (a) Schematic arrangement of atoms inside the unit cell in the $xy$ plane. Top view of ${\mathrm{WTe}}_2$ ribbon with (b) zigzag and (c) armchair edge types. Rectangles and circles denote W and Te atoms, respectively. Contacts $G_{1-4}$ define the two-terminal setup. Edge states are schematically shown as blue and red arrows. The dashed line and gray arrows help to visualize glide reflection. (d) Band structure of infinite ${\mathrm{WTe}}_2$ along the $X\text{−}\mathrm{\Gamma }\text{−}M\text{−}Y$ line in the rectangular Brilloiun zone. Note three gaps ${\mathrm{\Delta }}_{X\text{−}\mathrm{\Gamma }}$ (fundamental), ${\mathrm{\Delta }}_{\mathrm{\Gamma }}$, and ${\mathrm{\Delta }}_{M\text{−}Y}$. (e), (f) Band structures of (e) the zigzag and (f) the armchair of 20-nm-wide ribbons, with edge and bulk states colored as in (a). Exclamation mark (!) denotes the position of 1D Dirac points.

Figure 2

The energy-position resolved local density of states near one edge of the ribbon. Top panels are for zigzag and bottom panels are for armchair-type ribbon, respectively. The left panel shows a zoom into the dispersion of edge states in a clean system. The vertical axis of the LDOS maps (middle and right panels) begins at the edge of the system: $y=0$ for zigzag, $x=0$ for armchair, with coordinates consistent with those in Figs. 1 and 1. The middle panel presents LDOS for a clean system ($W=0.0$ eV). The solid black line represents edge state penetration depth calculated using wave functions for the clean systems. The right panel shows the corresponding maps for the disordered case ($W=1.5\mathrm{eV}$). Note that the color scale encoding LDOS is logarithmic and changes between clean and disordered maps.

Figure 3

Transport in disordered ribbons. (a) Two-terminal conductance $G$ versus disorder strength $W$ for zigzag (black squares) and armchair (red squares) ribbons. The system size here is $20\times 20 {\mathrm{nm}}^2$ and $E_F=0.0825\mathrm{eV}$. Contact configurations are shown in Fig. 1. The blue curve shows the corresponding $G\left(W\right)$ characteristic for HgTe QW, as described in the text. (b) Scaling of longitudinal resistance $R=1/G$ with increasing sample length up to $1\mu \text{m}$ for two values of disorder strength for both types of ribbon edges. Note the logarithmic scale on the $y$ axis. (c), (d) Evolution of resistance with respect to increasing disorder strength $W$ as a function of the Fermi energy tuned between the valence and the conduction bands for (c) zigzag and (d) armchair terminations.

Figure 4

Temperature dependence of conductance for clean and disordered samples. Different colors of $G(T,W)$ for zigzag (a) and armchair (b) denote different values of disorder strength.

Figure 5

Luttinger liquid parameter $K$ estimation in ribbons. Top panels correspond to zigzag termination and bottom ones to armchair. From left to right: $E_F$ vs $k_F$ dispersion, corresponding Fermi velocity $v_F$, penetration depth ${\lambda }_{\text{pen}}$, and $K$ parameter are presented for different Fermi energies $E_F$ on the $y$ axis. Two different lengths of channels used for $K$ estimation are given by red and blue curves on the rightmost panels. Blue crosses represent $K$ estimation using microscopic ribbon wave functions, as described in the text.

Figure 6

Atom arrangement in $1T^{\prime }$ unit cell of the ${\mathrm{WTe}}_2$ monolayer for (a) $xy$, (b) $xz$, and (c) $yz$ projections. (d) Lattice of atoms in a W-terminated “zigzag” ribbon; the gray region shows one “stripe” of the ribbon. (e) Corresponding arrangement of atoms and stripes in an “armchair” -type ribbon. The red and blue arrows on (d) and (e) denote counterpropagating quantum spin Hall states along the edges.

Figure 7

Band structure of bulk and nanoribbons. (a) Band structure near the Fermi level of ${\mathrm{WTe}}_2$ obtained from a tight-binding Hamiltonian. (b) Joint energy-resolved density of states in a periodic system (without edge states). Band structures of (c) zigzag and (d) armchair ribbons. The color bar on (c) and (d) describes where along dimension perpendicular to periodicity (from one edge to the other) the wave-function density is localized (green = bulk states, red and blue = edge states on opposite edges).

Figure 8

Evolution of the band structure with spin-orbit coupling along $X\text{−}\mathrm{\Gamma }\text{−}M$ line for different values of the glide symmetry breaking perturbation $V_{\text{GB}}$. Closure of the gap for $V_{\text{GB}}=0.1\mathrm{eV}$ signals the topological phase transition from the $Z_2=1$ to the $Z_2=0$ insulator.

Figure 9

Zigzag- (top panels) and armchair- (bottom panels) type nanoribbons for two values of glide symmetry breaking perturbation strength: $V_{\text{GB}}=0.05\mathrm{eV}$ (left) and $V_{\text{GB}}=0.2\mathrm{eV}$ (right). The color bar denotes the localization of ribbon eigenstates, with the red one representing edge states and the blue one representing bulk nanoribbon states.

Figure 10

Localization properties of edge states. Red circles in top panels of (a) and (b) represent choice of purely in-gap, nonoverlapping edge states. The lower panels show the corresponding numerically obtained penetration depths compared with different models ${\lambda }_{\text{pen}}$ for (a) zigzag and (b) armchair ribbons.

Figure 11

Energy-position LDOS maps for two different values of disorder, (a) $W=0.5\mathrm{eV}$ and (b) $W=1.0\mathrm{eV}$. From left to right the panels show the corresponding band structure of the zigzag ribbon, the energy-position resolved LDOS map for the zigzag, then the armchair band structure and the LDOS map for the armchair. The LDOS scale is given by a pair of $(C_{\text{min}},C_{\text{max}}), (C_{\text{min}}=0,C_{\text{max}}=4)$ for (a) and $(C_{\text{min}}=0.5,C_{\text{max}}=3.0)$ for (b).

Figure 12

Comparison of conductance $G$ in zigzag (black) and armchair (red) ribbons for the clean (solid line) and disordered ($W=1.5\mathrm{eV}$, rectangles) case. The left panels show the corresponding band structure of clean ribbons.

Figure 13

Role of glide symmetry breaking perturbation on conductance. (a) Fermi energy dependence of $G$ in $20\times 20$ nm system with zigzag edge for three values of $V_{\text{GB}}$: 0, 0.05 eV (weak glide symmetry breaking), and 0.2 eV (strong glide symmetry breaking). (b) Corresponding analysis for the armchair type of edge. For all curves the disorder strength is set to $W=1.5\mathrm{eV}$.

Figure 14

Scaling of $G\left(W\right)$ curves as a function of the length of the samples (20–1000 nm, constant width = 20 nm) for (a) zigzag and (b) armchair types of the edge. The vertical cut for some disorder strength $W$ gives values of $G$ for different lengths used to calculate longitudinal resistance in Fig. 3 in the main text.

Figure 15

Transmission $\mathcal{T}$ (black solid lines) of a clean sample at $T=0$ K and $\mathit{f}\left(E\right)$ function describing thermal broadening in (a) clean zigzag, (b) clean armchair, (c) disordered zigzag, and (d) disordered armchair cases. In all cases we compare the broadening for two temperatures: $T=150\mathrm{K}$ (rectangles) and $T=300\mathrm{K}$ (circles).

Figure 16

Conductance dependence on temperature in the range 10–300 K for (a) zigzag and (b) armchair ribbons with different widths (20–160 nm). The dotted line shows our linear interpolation to the system width equal to 1000 nm. The open circles on both graphs are experimental values extracted from Ref. [54].

Figure 17

Dependence of the $K$ parameter on the channel radius $R$ and the length of the channel $L$ for the zigzag edge.

Figure 18

Coupling constant ${\lambda }_2$ dependence on the Fermi wave vector for two types of edge termination.