Two-dimensional topological phases and electronic spectrum of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ thin films from $GW$ calculations

We discuss topological and electronic properties of thin films of the topological insulator ${\mathrm{Bi}}_{\text{2}}{\mathrm{Se}}_{\text{3}}$ employing the $GW$ method. We obtain a topologically trivial phase for films of one to six quintuple layers thickness, whereas a quantum spin Hall phase is observed in neither of these films. This challenges recent contradictory reports on quantum spin Hall phases in ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ films from both theory and experiment. Moreover, we argue that one cannot conclude the topological phase from the band structures measured in those studies. The reliability of our results is supported by the excellent agreement of all calculated spectral features with photoemission experiments, i.e., the band gaps and the energy and slope of the Dirac state.

Figure 1

Band structure of a 6 QL ${\mathrm{Bi}}_{\text{2}}{\mathrm{Se}}_{\text{3}}$ slab along $\overline{\mathrm{\Gamma }}\overline{M}$ in crystal structure A, calculated within (a) DFT and (b) $GW$. The position of the Dirac point is taken as the zero of the energy scale. The cross-hatched areas show the respective projected bulk band structures. The measured dispersion of the Dirac state (thick blue line) is extracted from ARPES data for 6 QL ${\mathrm{Bi}}_{\text{2}}{\mathrm{Se}}_{\text{3}}$ shown in Fig. 1(c) of Ref. [23].

Figure 2

Calculated DFT and $GW$ band gaps $\mathrm{\Delta }$ at $\overline{\mathrm{\Gamma }}$ for crystal structure (a) A and (b) B. $N_{\text{QL}}$ is the number of QLs in a slab. For comparison, experimental data from Ref. [3] are shown as well. “DFT+$\mathrm{\Delta }{\tilde{E}}_{\text{QP}}$” refers to computed data from Ref. [24] (see text for explanation).

Figure 3

Band structures of 3 QL ${\mathrm{Bi}}_{\text{2}}{\mathrm{Se}}_{\text{3}}$ slabs from DFT [(a),(c)] and $GW$ [(b),(d)] for crystal structure A [(a),(b)] and B [(c),(d)]. (e) Zoom into (b) with colored circles encoding the contribution of the DFT wave functions to the $GW$ wave functions. Red (blue) color means that the $GW$ wave function consists purely of the wave function of the highest DFT VB (lowest DFT CB). (f) The same $GW$ data as in (e) with two different fits of Eq. (4). The parameters are given in Table 2.