Quantum Spin Hall States in Stanene/Ge(111)

Abstract

For topological insulators to be implemented in practical applications, it is a prerequisite to select suitable substrates that are required to leave insulators' nontrivial properties and sizable opened band gaps (due to spin-orbital couplings) unaltered. Using ab initio calculations, we predict that Ge(111) surface qualified as a candidate to support stanene sheets, because the band structure of √3 × √3 stanene/Ge(111) (2 × 2) surface displays a typical Dirac cone at Γ point in the vicinity of the Fermi level. Aided with the result of Z2 invariant calculations, a √3 × √3 stanene/Ge(111) (2 × 2) system has been proved to sustain the nontrivial topological phase, with the prove being confirmed by the edge state calculations of stanene ribbons. This finding can serve as guidance for epitaxial growth of stanene on substrate and render stanene feasible for practical use as a topological insulator.

Figure 1

(a) Top and Side view of clean Ge(111)-(2 × 2) substrate model. T₁, T₄, H₃ and C_T mark the four adsorption sites for Sn adatoms on Ge(111) surface. (b–d) are atomic structures of possible models of Sn/Ge(111) surface; while in (d) the topmost layer Ge atom are hydrogenated. Each model is demonstrated by a top view (top half) and a stereogram view (bottom half). Sn, Ge and H atoms are represented by blue, green and pink balls, respectively.

Figure 2. Formation energies of lowest-energy structure under different Sn coverages for varying Sn chemical potentials.

Figure 3. Band structures of Sn/Ge(111) surface in configuration II without (a) and with (b) SOC, respectively.

(c,d) are corresponding ones for Sn/Ge(111) surface in configuration II-H. Fermi level is set to be zero and indicated by grey dashed line. Filled red and green circles indicate the contributions of Sn p_z orbitals and Ge p_x,y atoms respectively, while size of circles are in proportion to the contributions.

Figure 4. The n-field configuration for stanene on (a) Ge(111) and on (b) H-terminated Ge(111) substrates and the torus in the Brillouin zone spanned by the reciprocal lattice vectors G₁ and G₂.

White and black circles denote n = + 1 and −1, respectively, while the blank denotes 0. The Z₂ invariant is obtained by summing the n-field over half of the torus defined by vectors G₁ and G₂. Z₂ is 1 (nontrivial) in both (a,b).

Figure 5. Band structures along the armchair edges of stanene ribbons on (a) clean Ge(111) and (b) H-passivated Ge(111) substrates.

Atomic structures of armchair ribbons for stanene ribbons on clean Ge(111) (c,d) H-passivated Ge(111) substrates, where the supercell is outlined with black solid lines.The contribution from the edge on the right is marked with red crosses, while that from the edge on the left is marked with blue circles. Sizes of red crosses and blue circles are proportional to the contribution from the edges. The yellow filled region denotes projected 2D bulk bands on the 1D momentum space.