$GW$ study of topological insulators Bi${}_2$Se${}_3$, Bi${}_2$Te${}_3$, and Sb${}_2$Te${}_3$: Beyond the perturbative one-shot approach

We present $GW$ calculations of the topological insulators Bi${}_2$Se${}_3$, Bi${}_2$Te${}_3$, and Sb${}_2$Te${}_3$ within the all-electron full-potential linearized augmented-plane-wave formalism. Quasiparticle effects produce significant qualitative changes in the band structures of these materials when compared to density functional theory (DFT), especially at the $\Gamma$ point, where band inversion takes place. There, the widely used perturbative one-shot $GW$ approach can produce unphysical band dispersions, as the quasiparticle wave functions are forced to be identical to the noninteracting single-particle states. We show that a treatment beyond the perturbative approach, which incorporates the off-diagonal $GW$ matrix elements and thus enables many-body hybridization to be effective in the quasiparticle wave functions, is crucial in these cases to describe the characteristics of the band inversion around the $\Gamma$ point in an appropriate way. In addition, this beyond one-shot $GW$ approach allows us to calculate the values of the $Z_2$ topological invariants and compare them with those previously obtained within DFT.

Figure 1

LDA, $GW$ (diag), and $GW$ (full) band structures of (a) Bi${}_2$Se${}_3$, (b) Bi${}_2$Te${}_3$, and (c) Sb${}_2$Te${}_3$, without SOC. The region around the $\Gamma$ point in (c), where Eq. (4) [$GW$ (diag)] yields unphysical results, is enlarged in (d). The size of the circles scales with the projections of the LDA and $GW$ wave functions onto the $p$ states of Sb (dark circles) and Te (light circles). The dashed-black lines in (d) show the $GW$ (diag) eigenvalues for comparison.

Figure 2

LDA, $GW$ (diag), and $GW$ (full) band structures of (a) Bi${}_2$Se${}_3$, (b) Bi${}_2$Te${}_3$, and (c) Sb${}_2$Te${}_3$ with SOC. In panel (b), the labels 1, 2, 3, and 4 (pointing to the respective LDA bands) and the vertical dashed lines are a reference for Figs. 3 and 4.

Figure 3

Tellurium (thin lines) and bismuth (thick lines) $p$-orbital character of the bands of Bi${}_2$Te${}_3$ labeled 1 and 2 in Fig. 2b calculated using LDA and $GW$ wave functions including SOC. The represented k space corresponds to that between the vertical dashed lines in Fig. 2b.

Figure 4

Same as Fig. 3 for the bands labeled 3 and 4 in Fig. 2b. In this case, the (a) Te and (b) Bi projections are plotted separately.

Figure 5

Projected band structures including SOC, calculated with LDA wave functions [(a), (c), and (e)] and $GW$ quasiparticle wave functions [(b), (d), and (f)] for Bi${}_2$Se${}_3$, Bi${}_2$Te${}_3$, and Sb${}_2$Te${}_3$. The size of the circles is proportional to the projections of the wave functions onto the $p$ states of the different species.