Custodial glide symmetry of quantum spin Hall edge modes in monolayer ${\mathrm{WTe}}_2$

A monolayer of ${\mathrm{WTe}}_2$ has been shown to display quantum spin Hall (QSH) edge modes persisting up to 100 K in transport experiments. Based on density-functional theory calculations and symmetry-based model building including the role of correlations and substrate support, we develop an effective electronic model for ${\mathrm{WTe}}_2$ that fundamentally differs from other prototypical QSH settings: we find a remarkably strong transverse localization of QSH edge modes in ${\mathrm{WTe}}_2$ related to the glide symmetry due to which the topological gap opens away from high-symmetry points in momentum space. While the indirect bulk gap is much smaller, a large direct gap of up to 1 eV in the Brillouin zone region of the dispersing edge modes determines their properties.

Figure 1

(a) Lattice structure and unit cell of monolayer ${\mathrm{WTe}}_2$. Dashed (solid) circles indicate Wannier functions with $d_{x^2-y^2}$ ($p_x$) symmetry contributing to the low-energy physics. The other two Te atoms in the unit cell do not play a role in the effective low-energy electronic structure. A termination parallel to the $\widehat{x}$ ($\widehat{y}$) direction corresponds to the zigzag (armchair) edge. (b) Binding energy of monolayer ${\mathrm{WTe}}_2$ (height $h$) deposited on $d$-distant bilayer graphene (BLG) computed via a van der Waals corrected functional.

Figure 2

(a) Comparison of DFT HSE06 band structure (red) and tight-binding band structure (blue) including SOC. The tight-binding model (1) with SOC term of strength $V=0.115\mathrm{eV}$ was used, while ${\mathcal{H}}_{\mathrm{R}}^{\mathrm{SOC}}$ was neglected. The inset shows the evolution of the indirect band gap induced by SOC as a function of the Hartree-Fock exchange. The red curve corresponds to the HSE06 value ${\alpha }_{\mathrm{exx}}=0.25$. (b) Left: Band structure of monolayer ${\mathrm{WTe}}_2$ ($2\times 2$ unit cells) on BLG ($3\times 6$ unit cells) along the high-symmetry lines of the unfolded primitive Brillouin zone. Middle: Same as left, where the red circles highlight the unfolding coefficients originating from the unfolding procedure. These weights are defined from the scalar product $\langle KN|kn\rangle$, where upper (lower) -case symbols $K$ and $N$ ($k$ and $n$) identify supercell (primitive cell) momenta and band indices, and they are interpreted as proper spectral weights that allow us to efficiently map the band structure of the primitive cell out of the band structure of the supercell. A derivation of the unfolding strategy can be found in Refs. [27, 28, 29]. Right: Zoom around the Dirac cone dispersion. The small splitting of the bands is due to Rashba SOC.

Figure 3

(a) Spectral function defined in Eq. (4) showing the topological edge states of tight-binding model (1) on a ribbon of 100 sites width with open boundary conditions in the $y$ direction. (b) Same as (a), but for open boundary conditions in the $x$ direction. (c) Localization length (solid lines) and inverse gap (dashed lines) as a function of $k_{\parallel }=k_x$ for the case of (a), and $k_{\parallel }=k_y$ for the case of (b). The relevant bands are highlighted by white arrows in (a) and (b). Around $k_x=0$, the edge states are localized within a fraction of the unit cell due to the large direct bulk gap.