Stacking-dependent energetics and electronic structure of ultrathin polymorphic ${\mathrm{V}}_2{\mathrm{VI}}_3$ topological insulator nanofilms

Topological insulators represent a paradigm shift in surface physics. The most extensively studied ${\mathrm{Bi}}_2{\mathrm{Se}}_3$-type topological insulators exhibit layered structures, wherein neighboring layers are weakly bonded by van der Waals interactions. Using first-principles density-functional theory calculations, we investigate the impact of the stacking sequence on the energetics and band structure properties of three polymorphs of ${\mathrm{Bi}}_2{\mathrm{Se}}_3,{\mathrm{Bi}}_2{\mathrm{Te}}_3$, and ${\mathrm{Sb}}_2{\mathrm{Te}}_3$. Considering their ultrathin films up to 6 nm as a function of its layer thickness, the overall dispersion of the band structure is found to be insensitive to the stacking sequence, while the band gap is highly sensitive, which may also affect the critical thickness for the onset of the topologically nontrivial phase. Our calculations are consistent with both experimental and theoretical results, where available. We further investigate tribological layer slippage, where we find a relatively low energy barrier between two of the considered structures. Both the stacking-dependent band gap and low slippage energy barriers suggest that polymorphic stacking modification may offer an alternative route for controlling the properties of this new state of matter.

Figure 1

The atomic structures of ${\mathrm{V}}_2{\mathrm{VI}}_3$ polymorphs: the top (ABC-CAB-BCA), fcc (ABC-ABC-ABC), and normal (ABC-BCA-CAB) stacking sequences. Large spheres (blue) are used to represent the group-V atoms, while the small ones (green, yellow, and white) for group-VI atoms. The stacking sequence is labeled by the group-VI atomic layer according to the numbers (given in red).

Figure 2

QL-QL interaction energy $E^{\mathrm{QL}}$ of normal, fcc, and top polymorphic nanofilms as a function of increasing number of QLs $N^{\mathrm{QL}}$ for (a) ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, (b) ${\mathrm{Bi}}_2{\mathrm{Te}}_3$, and (c) ${\mathrm{Sb}}_2{\mathrm{Te}}_3$. As an example, the bulk limit of this interaction energy $E_{\mathrm{bulk}}^{\mathrm{QL}}$ is shown as the horizontal dotted line for each ${\mathrm{V}}_2{\mathrm{VI}}_3$ compound.

Figure 3

Energy profile for slippage: Relative energy differences between the normal and fcc structures for 2 QL nanofilms for two different paths, namely, along the $\langle \overline{1}100\rangle$ (energy $E_2)$ or $\langle 1\overline{1}00\rangle$ (energy $E_3)$ directions. Schematic energy profile along the $\langle \overline{1}100\rangle$ direction for (b) ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, (c) ${\mathrm{Bi}}_2{\mathrm{Te}}_3$, and (d) ${\mathrm{Sb}}_2{\mathrm{Te}}_3$. All energies are given with respect to the reference total energy of normal films.

Figure 4

$\text{PBE}+\text{SOC}$ DFT-derived QL-resolved electronic band structure of ${\mathrm{V}}_2{\mathrm{VI}}_3$ nanofilms (from 1 to 6 QL): (a) normal structure of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, (b) fcc structure of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, (c) normal structure of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$, (d) fcc structure of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$, (e) normal structure of ${\mathrm{Sb}}_2{\mathrm{Te}}_3$, and (f) fcc structure of ${\mathrm{Sb}}_2{\mathrm{Te}}_3$. Only the band edges due to the highest occupied crystal orbital (HOCO) and the lowest unoccupied crystal orbital (LUCO) are shown for clarity.

Figure 5

$\Gamma$-point band-gap energies $E_{\mathrm{g}}$ as a function of the number of QL, $N^{\mathrm{QL}}$: (a) ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, (b) ${\mathrm{Bi}}_2{\mathrm{Te}}_3$, and (c) ${\mathrm{Sb}}_2{\mathrm{Te}}_3$. normal (PBE) and fcc (PBE) denote the $\text{PBE}+\text{SOC}$ value, while normal (HSE) and fcc (HSE) denote that calculated by the $\text{HSE}06+\text{SOC}$ hybrid functional, respectively. The absolute values for these $E_{\mathrm{g}}$ are listed in Table 1 of the Supplemental Material [51].

Figure 6

$\Gamma$-point electron density distributions (namely, the HOCO and LUCO states): (a) 5 and 6 QL normal structured films of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, (b) 5 and 6 QL fcc structured films of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, (c) 3 and 4 QL normal structured films of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$, (d) 2 and 3 QL fcc structured films of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$, (e) 3 and 4 QL normal structured films of ${\mathrm{Sb}}_2{\mathrm{Te}}_3$, and (f) 3 and 4 QL fcc structured films of ${\mathrm{Sb}}_2{\mathrm{Te}}_3$.