Topological phase driven by confinement effects in Bi bilayers

A topological insulator is characterized by protected Dirac states on the edges due to the spin-orbit coupling. In this work we show that the topological phase can be driven by nanosize effects in Bi bilayers stacking. The nontrivial phase is a result of the interaction of Rashba-type Fermi gases coming from the Bi bilayer surfaces with opposite spin-orbit coupling. The surface states form a two-dimensional topological protected conductor, in contrast to the one-dimensional conduction observed in a single Bi bilayer. The three-dimensional (3D) topological phase takes place for a few Bi bilayers, where confinement effects open a band gap. The topological protected Dirac states, the helical spin texture, and Fermi velocities reveal a 3D topological insulator character, which is in agreement with first-principles results. Our findings help us to understand topological properties of nanostructured stacking of Rashba-type spin-orbit Fermi gases, as well provide a tool to find or construct new 3D topological insulators.

Figure 1

Schematic representation of the interaction between Bi bilayers. A single Bi bilayer is described by two atomic planes, each one with opposite Rashba spin-orbit coupling $\pm {\alpha }_{\mathrm{R}}$, and two identical massive bands $M_{\mathrm{N}}$. Intrabilayer $D\left(\mathbf{k}\right)$ quantum tunneling and interbilayer hopping $T$ between Rashba and non-Rashba bands are considered.

Figure 2

Band structure for 2, 3, 6, 9, 10, and 11 Bi bilayers (BL) from first principles. Blue and red symbols represent opposite spin alignments. The size of the symbols is proportional to the spin polarization.

Figure 3

Band structure around the Fermi level for a single Bi bilayer obtained from (a) first principles and (b) the Rashba Hamiltonian model. (c) First-principles band structure for two Bi bilayers, (d) results from the Rashba model including interactions between Rashba and non-Rashba bands, and (e) results when the interactions between Rashba and non-Rashba bands are turned off. Blue and red symbols represent opposite spin alignments. The size of the symbols is proportional to the spin polarized. The Hamiltonian parameters are indicated.

Figure 4

(left) Bulk Brillouin zone of Bi and its projection on the (111) surface. (right) A constant-energy surface in $k$ space for occupied Dirac states taken at $E_{\mathrm{F}}-0.5$ eV (purple line) and empty states at $E_{\mathrm{F}}+0.5$ eV (green line) for two Bi bilayers. The arrows indicate the spin helicity.

Figure 5

Schematic band diagram as a function of the Bi bilayer (BL) stacking. The green $\overline{M}$ point (projection of the bulk $L$ point) and the purple $T$ point are the ones that suffer stronger confinement effects. The red curves are the bands that come from the $\Gamma$ point. The filled areas are the bands that form the bulk valence- and conduction-band edges.