Prediction of a Large-Gap and Switchable Kane-Mele Quantum Spin Hall Insulator

Fundamental research and technological applications of topological insulators are hindered by the rarity of materials exhibiting a robust topologically nontrivial phase, especially in two dimensions. Here, by means of extensive first-principles calculations, we propose a novel quantum spin Hall insulator with a sizable band gap of $\sim 0.5\mathrm{eV}$ that is a monolayer of jacutingaite, a naturally occurring layered mineral first discovered in 2008 in Brazil and recently synthesized. This system realizes the paradigmatic Kane-Mele model for quantum spin Hall insulators in a potentially exfoliable two-dimensional monolayer, with helical edge states that are robust and that can be manipulated exploiting a unique strong interplay between spin-orbit coupling, crystal-symmetry breaking, and dielectric response.

Figure 1

Top-left panel: DFT band structure for monolayer jacutingaite with or without spin-orbit coupling (SOC); the energy zero is at the Fermi level of the DFT calculations performed without SOC. Full circles denote direct DFT calculations, while solid lines represent Wannier-interpolated states from a minimal two-band model. Green (dark grey) circles and lines represent calculations performed with SOC; orange (light grey) circles and lines represent calculations performed without SOC. Top-right panel: Top and lateral views of monolayer jacutingaite (${\mathrm{Pt}}_2{\mathrm{HgSe}}_3$); the primitive cell is marked by the grey parallelogram, while the Wigner-Seitz cell is denoted by the green hexagon. Bottom panel: Isosurface plots for the two maximally localized Wannier functions realizing the two-band model. The two MLWFs map onto each other under inversion and have the character of Hg $s$ orbitals hybridized with the three nearest-neighbor Pt $d$ orbitals; their centres compose a buckled honeycomb lattice. Red and blue isosurfaces correspond to opposite signs of the MLWFs. The low-energy physics is fully captured using this two-band model, analogous to the Kane-Mele model for graphene.

Figure 2

Left panel: DFT and $G_0{W}_0$ band structures obtained including SOC; lines are Wannier-interpolated bands, while full ($G_0{W}_0$) and empty (DFT) circles denote direct calculations. Right panel: $G_0{W}_0$ edge spectral density displaying a pair of topologically protected helical edge states crossing the bulk gap.

Figure 3

Top-left panel: DFT total energy vs displacement along a $\mathrm{\Gamma }$-unstable displacement pattern, i.e., the imaginary-frequency phonon obtained from DFPT calculations performed without SOC (the inset shows the pattern of this phonon calculated in the centrosymmetric configuration; see text). Without including SOC, the system would distort into a polar phase as shown by a double-well potential-energy curve; on the contrary, SOC stabilizes the centrosymmetric phase and restores a parabolic behavior. Bottom-left panel: Direct band gap at $K$ as a function of the displacement pattern. In the non-SOC case, inversion symmetry breaking opens a gap, while including SOC yields gap closure. Right panel: Topological phase diagram under an out-of-plane electric displacement field $D_z$. The ${\mathbb{Z}}_2$ topological invariant is marked in teal; the magnitude of the band gap is represented by the disks’ color and size. The black line stands for the projection of the field-induced ionic displacement over the 164-to-156 spacegroup reduction manifold (see text). The ionic displacement due the field acts similarly to the imaginary-frequency phonon, causing an ionic response that reduces the critical field where the topological phase transition occurs.