Topological interface states in the natural heterostructure (PbSe)${}_5$(${\mathrm{Bi}}_2{\mathrm{Se}}_3$)${}_6$ with ${\mathrm{Bi}}_{\mathrm{Pb}}$ defects

We study theoretically the electronic band structure of (${\mathrm{PbSe})}_5$(${\mathrm{Bi}}_2{\mathrm{Se}}_3$)${}_6$, which consists of an ordinary insulator PbSe and a topological insulator ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. The first-principles calculations show that this material has a gapped Dirac-cone energy dispersion inside the bulk, which originates from the topological states of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ layers encapsulated by PbSe layers. Furthermore, we calculate the band structures of (${\mathrm{Bi}}_x{\mathrm{Pb}}_{1-x}\mathrm{Se}$)${}_5$(${\mathrm{Bi}}_2{\mathrm{Se}}_3$)${}_6$ with ${\mathrm{Bi}}_{\mathrm{Pb}}$ antisite defects included in the PbSe layers. The result shows that a high density of ${\mathrm{Bi}}_{\mathrm{Pb}}$ defects can exist in real materials, consistent with the experimentally estimated $x$ of more than 30%. The ${\mathrm{Bi}}_{\mathrm{Pb}}$ defects strongly modify the band alignment between ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ and PbSe layers, while the topological interface states of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ are kept as a gapped Dirac-cone-like dispersion.

Figure 1

(a) Crystal structure of (PbSe)${}_5$(${\mathrm{Bi}}_2{\mathrm{Se}}_3$)${}_6$ and (b) its Brillouin zone with the $\mathbf{k}$ labels defined in this study. In the crystal structure, small (green), medium (gray), and large (purple) spheres show Se, Pb, and Bi atoms, respectively, and the symmetrically equivalent Pb-site labels are shown.

Figure 2

Layer-resolved band structure of (PbSe)${}_5$(${\mathrm{Bi}}_2{\mathrm{Se}}_3$)${}_6$. The energy origin is taken at the Fermi level. The wave-function characters spatially dominated by ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ (black), PbSe (red), and both layers (magenta) are shown in the respective color.

Figure 3

Band structures of the two 2QL-(${\mathrm{Bi}}_2{\mathrm{Se}}_3$)${}_6$ slab (black solid circles) and the two bilayer-(PbSe)${}_5$ slab (red solid circles). The Fermi energy for the ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ slab at $-0.42$ eV and the valence-band top energy for the PbSe slab at +0.20 eV are shown by dashed lines, and the relative energy position is adjusted so as to match with Fig. 2.

Figure 4

Charge-density difference between (PbSe)${}_5$(${\mathrm{Bi}}_2{\mathrm{Se}}_3$)${}_6$ and the superposition of the building-block slabs used for Fig. 3. Isosurface densities of $\pm 0.65\times {10}^{-3}$ electrons/Bohr ${}^3$ are plotted, and the positive (red) and negative (blue) values show that the electron density increases and decreases in (PbSe)${}_5$(${\mathrm{Bi}}_2{\mathrm{Se}}_3$)${}_6$, respectively.

Figure 5

Defect formation energy of ${\mathrm{Bi}}_{\mathrm{Pb}}$ antisite defects in the (${\mathrm{Bi}}_x{\mathrm{Pb}}_{1-x}\mathrm{Se}$)${}_5$(${\mathrm{Bi}}_2{\mathrm{Se}}_3$)${}_6$ models.

Figure 6

Layer-resolved band structures of the (${\mathrm{Bi}}_x{\mathrm{Pb}}_{1-x}\mathrm{Se}$)${}_5$(${\mathrm{Bi}}_2{\mathrm{Se}}_3$)${}_6$ models for (a) $x=0$, (b) $x=0.2$, and (c) $x=0.4$. The wave-function characters spatially dominated by ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ (black), ${\mathrm{Bi}}_x{\mathrm{Pb}}_{1-x}\mathrm{Se}$ (red), and both layers (magenta) are shown in their respective color. The Fermi energy is set to be the energy zero. (d) The energy levels at $\mathrm{\Gamma }$ of the ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ Dirac-like points $E_{{\mathrm{Bi}}_2{\mathrm{Se}}_3}^{\mathrm{DP}\left(\mathrm{\Gamma }\right)}$ (black triangles), the PbSe valence-band top $E_{\mathrm{PbSe}}^{\mathrm{VB}\left(\mathrm{\Gamma }\right)}$ (red solid squares), and the PbSe conduction-band bottom $E_{\mathrm{PbSe}}^{\mathrm{CB}\left(\mathrm{\Gamma }\right)}$ (red open squares). The energy levels at $C$ of the ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ highest occupied state $E_{{\mathrm{Bi}}_2{\mathrm{Se}}_3}^{\text{VB}\left(C\right)}$ (black solid circles) and the lowest unoccupied state $E_{{\mathrm{Bi}}_2{\mathrm{Se}}_3}^{\text{CB}\left(C\right)}$ (black open circles) are also shown.

Figure 7

Layer-resolved band structures magnified near the Fermi energy of the (${\mathrm{Bi}}_x{\mathrm{Pb}}_{1-x}\mathrm{Se}$)${}_5$(${\mathrm{Bi}}_2{\mathrm{Se}}_3$)${}_6$ models for (a) $x=0$, (b) $x=0.2$, and (c) $x=0.4$. The color coding of the bands corresponds to Fig. 6. Note that band symmetries are not considered for the line connections.