Quasiparticle effects in the bulk and surface-state bands of Bi${}_2$Se${}_3$ and Bi${}_2$Te${}_3$ topological insulators

We investigate the bulk band structures and the surface states of Bi${}_2$Se${}_3$ and Bi${}_2$Te${}_3$ topological insulators using first-principles many-body perturbation theory based on the $GW$ approximation. The quasiparticle self-energy corrections introduce significant changes to the bulk band structures, surprisingly leading to a decrease in the direct band gaps in the band-inversion regime as opposed to the usual situation without band inversion. Parametrized “scissors operators” derived from the bulk studies are then used to investigate the electronic structure of slab models which exhibit topologically protected surface states. The introduction of self-energy corrections results in significant shifts of the surface-state Dirac point energies relative to the bulk bands and in enlarged gap openings from the interactions between the surface states across the thin slab, both in agreement with experimental data.

Figure 1

Quasiparticle self-energy corrections as a function of LDA energies for bulk (a) Bi${}_2$Se${}_3$ and (b) Bi${}_2$Te${}_3$ calculated without taking into account SO interactions. The positions of VBM and CBM are indicated by the larger solid and open circles, respectively. The lines correspond to the fitted “scissors operators.” (c), (d) Same plots after the SO matrix elements were taken into account. Note the changes in $GW$ corrections calculated for the states which correspond to VBM and CBM in (a) and (b).

Figure 2

Band structures of bulk (a) Bi${}_2$Se${}_3$ and (b) Bi${}_2$Te${}_3$ calculated using the following theories: LDA with no SO (dotted lines), LDA with SO effects taken into account (dashed lines), and $GW$ with SO interactions (solid lines). (c), (d) Degree of band inversion for the valence band [Eq. (3)] of Bi${}_2$Se${}_3$ and Bi${}_2$Te${}_3$, respectively, calculated along the same $k$ path as in (a) and (b).

Figure 3

Evolution of the band structure (solid lines) calculated for the model Hamiltonian given by Eq. (2) upon the decrease of parameter ${\Delta }_{\mathrm{g}}$ (see text). Dotted lines represent the solution calculated in the absence of off-diagonal (SO) matrix elements.

Figure 4

Band structures calculated for 5QL slabs of (a) Bi${}_2$Se${}_3$ and (b) Bi${}_2$Te${}_3$ using LDA and $GW$ (SO interactions included in both cases). (c), (d) Band gaps at the $\Gamma$ point as a function of slab thickness for Bi${}_2$Se${}_3$ and Bi${}_2$Te${}_3$, respectively. Experimental results are reproduced from Ref. 24. The zero of energy is set at $E_{\mathrm{LDA}}\left(\mathrm{VBM}\right)$ with no SO.