Topological Insulators in Ternary Compounds with a Honeycomb Lattice

We investigate a new class of ternary materials such as LiAuSe and KHgSb with a honeycomb structure in Au-Se and Hg-Sb layers. We demonstrate the band inversion in these materials similar to HgTe, which is a strong precondition for existence of the topological surface states. In contrast with graphene, these materials exhibit strong spin-orbit coupling and a small direct band gap at the $\Gamma$ point. Since these materials are centrosymmetric, it is straightforward to determine the parity of their wave functions, and hence their topological character. Surprisingly, the compound with strong spin-orbit coupling (KHgSb) is trivial, whereas LiAuSe is found to be a topological insulator.

Figure 1

The lattice of honeycomb compound $XYZ$ (the ZrBeSi structure type) shown in (a) the $ab$ plane and (b) the $ac$ plane. Thus, the ternary semiconductor (with 18 valence electrons) can be viewed as $X$ stuffed ($X=\mathrm{Li},\mathrm{Na},\mathrm{K}$) graphene $YZ$ lattice consisting of alternating late transition metals $Y$ (Ag, Au, Zn, Hg) and main group elements $Z$ (Se, Te, P, As, Sb).

Figure 2

Band structure of LiAuSe (a),(b) and KHgSb (c),(d) calculated without (a),(c) and with (b),(d) SOC.

Figure 3

Evolution of the band structure at the $\Gamma$ point starting from atomic orbitals for KZnP (a). The chemical bonding is introduced in step I. In step II the crystal field is switched on. These $p_{x,y,z}$ orbitals split into $p_{x,y}$ and $p_z$ due to the hexagonal symmetry. Then SOC is turned on in step III. This splits the energy levels further. “$\uparrow \downarrow$” represents spin labels. The parity labels “$\pm$”are also shown. The blue dashed line marks the $E_F$. One notices that the Zn $s$ bands are above the P $p$ bands for KZnP. Similarly, the schemes of LiAuSe and KHgSb are shown in (b) and (c), respectively. The main difference to KZnP is that the Au (or Hg) $s$ orbital and the Se (or Sb) $p$ orbital are inverted. KHgSb has a second BI in step III, while LiAuSe does not. Panels (d)–(f) show the schematic band structure for KZnP, LiAuSe, and KHgSb near $\Gamma$, respectively. The black curves mark the $s$ bands. Gray dashed (red) and gray solid (green) lines stand for the $p$ bands with “$-$” and “$+$” parities, respectively.

Figure 4

Surface states of semi-infinite (0001) surfaces corresponding to topologically trivial (a) KZnP, (b) KHgSb, and nontrivial (c) LiAuSe insulators (strained case) calculated along $\overline{K}\mathrm{\text{−}}\overline{\Gamma }\mathrm{\text{−}}\overline{M}$ in the Brillouin zone. In contrast to KZnP and KHgSb, LiAuSe exhibits the surface states seen in the bulk energy gap, zoomed in the inset (d).