Topological phase transitions in (Bi${}_{1-x}$In${}_x{)}_2$Se${}_3$ and (Bi${}_{1-x}$Sb${}_x{)}_2$Se${}_3$

Abstract

We study the phase transition from a topological to a normal insulator with concentration $x$ in (Bi${}_{1-x}$In${}_x{)}_2$Se${}_3$ and (Bi${}_{1-x}$Sb${}_x{)}_2$Se${}_3$ in the Bi${}_2$Se${}_3$ crystal structure. We carry out first-principles calculations on small supercells, using this information to build Wannierized effective Hamiltonians for a more realistic treatment of disorder. Despite the fact that the spin-orbit coupling (SOC) strength is similar in In and Sb, we find that the critical concentration $x_{\mathrm{c}}$ is much smaller in (Bi${}_{1-x}$In${}_x{)}_2$Se${}_3$ than in (Bi${}_{1-x}$Sb${}_x{)}_2$Se${}_3$. For example, the direct supercell calculations suggest that $x_{\mathrm{c}}$ is below 12.5% and above 87.5 for the two alloys respectively. More accurate results are obtained from realistic disordered calculations, where the topological properties of the disordered systems are understood from a statistical point of view. Based on these calculations, $x_c$ is around 17% for (Bi${}_{1-x}$In${}_x{)}_2$Se${}_3$, but as high as 78%-83% for (Bi${}_{1-x}$Sb${}_x{)}_2$Se${}_3$. In (Bi${}_{1-x}$Sb${}_x{)}_2$Se${}_3$, we find that the phase transition is dominated by the decrease of SOC, with a crossover or "critical plateau" observed from around 78 to 83. On the other hand, for (Bi${}_{1-x}$In${}_x{)}_2$Se${}_3$, the In 5$s$ orbitals suppress the topological band inversion at low impurity concentration, therefore accelerating the phase transition. In (Bi${}_{1-x}$In${}_x{)}_2$Se${}_3$ we also find a tendency of In atoms to segregate.

12 pages, 9 figures, 3 tables

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