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

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.

Figure 1

(a) Lattice vectors of Bi${}_2$Se${}_3$ primitive cell, 2$\times$2$\times$1 supercell, and 2$\times$2$\times$2 supercell. (b) Corresponding bulk Brillouin zone and its surface projection.

Figure 2

Ground-state energies vs impurity concentration $x$ for (Bi${}_{1-x}$In${}_x{)}_2$Se${}_3$ supercells. Here, $\Delta =E_g\left(x\right)-(1-x)E_1+x{E}_2$, where $E_g\left(x\right)$, $E_1$, and $E_2$ are the ground-state energies per five-atom cell for the alloy supercell, host material, and dopant material, respectively. Filled and open circles denote ${\mathbb{Z}}_2$-odd and -even states, respectively. Solid (red) and dashed (dark blue) lines follow the most and least In-clustered configurations, respectively.

Figure 3

(a) Ground-state energies vs impurity concentration $x$ for (Bi${}_{1-x}$Sb${}_x{)}_2$Se${}_3$ supercells, following the same conventions as in Fig. 2. (b) Band gap at the center of the Brillouin zone vs impurity concentration $x$ computed within the virtual crystal approximation. Positive and negative values of band gap denote the topological and normal phases, respectively.

Figure 4

(a) Local DOS of (Bi${}_{1-x}$In${}_x{)}_2$Se${}_3$ at $x=12.5$% for the $s$ and $p$ orbitals on the substituted In atom and the $p$ orbitals on first-neighbor Bi and Se atoms. (b) Local DOS of (Bi${}_{1-x}$Sb${}_x{)}_2$Se${}_3$ at $x=12.5$% for the $p$ orbitals on the substituted Sb atom and the $p$ orbitals on first-neighbor Bi and Se atoms.

Figure 5

(a) Wannier-interpolated band structure of In${}_2$Se${}_3$, with color code indicating In $5s$ character. (b) Same but with In $5s$ levels shifted upward by 0.79 eV.

Figure 6

Surface band structures for (Bi${}_{1-x}$In${}_x{)}_2$Se${}_3$ slabs, plotted in the 2D surface Brillouin zone. (Surface states are shown in red.) (a) 8QL slab for $x=0.125$. (b) 4QL slab for $x=0.125$ but with In $5s$ orbitals removed. (c) 12QL slab for energetically favored configuration C${}_{0.25}^{\prime }$ at $x=0.25$. (d) 8QL slab for C${}_{0.25}^{\prime }$ with In $5s$ orbitals removed. Note split Dirac cones arising at $\overline{\Gamma }$ in (b) and (d).

Figure 7

Disordered spectral functions of (Bi${}_{1-x}$Sb${}_x{)}_2$Se${}_3$ unfolded into the primitive-cell BZ. (a) $x=68$%. (b) $x=78$%. (c) $x=83$%. (d) $x=88.5$%. (e) $x=99$%.

Figure 8

Disordered spectral functions of (Bi${}_{1-x}$Sb${}_x{)}_2$Se${}_3$ at the $\Gamma$ point in the primitive-cell BZ for (a) $x=68$%, (b) $x=78$%, (c) $x=83$%, and (d) $x=88.5$%. Vertical (red) line indicates Fermi energy. Distinct VBM and CBM peaks are still visible in the topological phase in (a), merge in (b) and (c), and reappear in (d) as the gap reopens in the normal phase.

Figure 9

Disordered spectral functions of (Bi${}_{1-x}$In${}_x{)}_2$Se${}_3$ unfolded into the primitive-cell BZ. (a) $x=8.3$%. (b) $x=12.5$%. (c) $x=14.6$%. (d) $x=16.7$%. (e) $x=18.8$%.