Single and bilayer bismuthene: Stability at high temperature and mechanical and electronic properties

Based on first-principles phonon and finite temperature molecular dynamics calculations including spin-orbit coupling, we showed that free-standing single-layer phases of bismuth, namely buckled honeycomb and asymmetric washboard structures named as bismuthene, are stable at high temperature. We studied the atomic structure, mechanical, and electronic properties of these single-layer bismuthene phases and their bilayers. The spin-orbit coupling is found to be crucial in determining lattice constants, phonon frequencies, band gaps, and cohesion. In particular, phonons of 3D hexagonal crystal, as well as those of single-layer bismuthene phases, are softened with spin orbit coupling. By going from 3D hexagonal crystal to free-standing single-layer structures, 2D hexagonal lattice is compressed and semimetal is transformed to semiconductor as a result of confinement effect. On the contrary, by going from single-layer to bilayer bismuthenes, the lattice is slightly expanded and fundamental band gaps are narrowed. Our results reveals that interlayer coupling in multilayer and 3D Bi crystal is crucial for topologically trivial to nontrivial and semimetal to semiconductor transitions.

Figure 1

3D Bi crystal: (a) Hexagonal unit cell of 3D Bi crystal with lattice constants $a=b$, $c$ and relevant structure parameters $\mathrm{\Delta }$ and $\mathrm{\Delta }z$. (b) Phonon dispersion curves calculated with SOC (blue lines) and without SOC (green lines) show the effect of spin-orbit coupling on crystal vibrations. (c) Electronic energy band structure of 3D Bi crystal in equilibrium calculated by PBE with SOC (blue lines) and without SOC (green lines) are presented along major symmetry directions of the hexagonal Brillouin zone. The Brillouin zone and highlighted bands in the rectangle with direct $E_{g,d}$ and inverted indirect $E_{g,i}$ band gaps are shown by inset. (d) Electronic energy band structure calculated by HSE with SOC (blue lines) and without SOC (green lines). The atomic structure and lattice constants are optimized by PBE+SOC.

Figure 2

Single layer of buckled honeycomb structure of Bi, $b\text{−}\mathrm{Bi}$: (a) top and side views of optimized atomic structure together with 2D hexagonal lattice constants $a=b$, buckling parameter $\mathrm{\Delta }$. The 2D hexagonal unit cell is shaded. (b) Phonon dispersion curves are calculated with SOC (blue lines) and without SOC (green lines). (c) Snapshots of atomic structure calculated using ab initio MD calculations carried out at $T=300$, 500, and 700 K.

Figure 3

Single layer of symmetric washboard structure of Bi, $w\text{−}\mathrm{Bi}$: (a) Top and side views of optimized atomic structure together with lattice constants. The 2D rectangular unit cell is shaded. (b) Phonon dispersion curves are calculated with SOC (blue lines) and without SOC (green lines).

Figure 4

Single layer of asymmetric washboard structure of Bi, $aw\text{−}\mathrm{Bi}$: (a) Top and side views of optimized atomic structure together with lattice constants. The 2D rectangular unit cell is shaded. (b) Phonon dispersion curves are calculated with SOC (blue lines) and without SOC (green lines). (c) Snapshots of atomic structure calculated using ab initio MD calculations carried out at $T=300$, 500, and 700 K.

Figure 5

Electronic energy band structures of free-standing SL bismuthene phases. (a) SL $b\text{−}\mathrm{Bi}$. (b) SL $w\text{−}\mathrm{Bi}$. (c) SL $aw\text{−}\mathrm{Bi}$. Left panels are calculated by PBE (green lines) and PBE+SOC (blue lines); right panels are calculated by HSE (green lines) and HSE+SOC (blue lines). The zero of energy is set to the top of the valence bands. The gap between conduction and valence bands is shaded.

Figure 6

Atomic and electronic energy band structure of bilayers (BL) of bismuthene phases (i.e., two SL bismuthene) with minimum energy stacking sequence. (a) BL of $b\text{−}\mathrm{Bi}$ with AA stacking where similar Bi atoms in different layers are situated on top of each other. (b) BL of $aw\text{−}\mathrm{Bi}$ with AA stacking. Left: Energy band structure calculated by using PBE (green lines) and by PBE+SOC (blue lines). Right: Bands are calculated using HSE (green lines) and by HSE+SOC (blue lines).