Effect of Bi bilayers on the topological states of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$: A first-principles study

${\mathrm{Bi}}_2{\mathrm{Se}}_3$ is a three-dimensional topological insulator which has been extensively studied because it has a single Dirac cone on the surface, inside a relatively large bulk band gap. However, the effect of two-dimensional topological insulator Bi bilayers on the properties of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ and vice versa, has not been explored much. Bi bilayers are often present between the quintuple layers of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, since ${\left({\mathrm{Bi}}_2\right)}_n{\left({\mathrm{Bi}}_2{\mathrm{Se}}_3\right)}_m$ form stable ground-state structures. Moreover, ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ is a good substrate for growing ultrathin Bi bilayers. By first-principles techniques, we first show that there is no preferable surface termination by either Bi or Se. Next, we investigate the electronic structure of Bi bilayers on top of, or inside a ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ slab. If the Bi bilayers are on top, we observe a charge transfer to the quintuple layers that increases the binding energy of the surface Dirac cones. The extra states, originating from the Bi bilayers, were declared to form a topological Dirac cone, but here we show that these are ordinary Rashba-split states. This result, together with the appearance of a new Dirac cone that is localized slightly deeper, might necessitate the reinterpretation of several experimental results. When the Bi bilayers are located inside the ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ slab, they tend to split the slab into two topological insulators with clear surface states. Interface states can also be observed, but an energy gap persists because of strong coupling between the neighboring quintuple layers and the Bi bilayers.

Figure 1

Schematic illustration of the examined structures. The thick red blocks represent ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ QLs, while the thin blue blocks represent Bi bilayers.

Figure 2

Band structures of (a) a bare slab of five QLs ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, (b) the same five QLs but in the relaxed coordinates of the five QL system with an extra Bi bilayer on top, and (c) five QLs with a Bi bilayer on top. (d) The layer-projected electron densities of the states 1–4 labeled in (c) at $\Gamma$. (e) The charge-transfer amount of the system with five QLs and a bilayer. The right-hand-side figure in (e) is the accumulated charge-transfer amount. The red dotted vertical line denotes the middle between the QLs and the bilayer.

Figure 3

Band structures of (a) a single Bi bilayer, (b) four QLs with a bilayer on top in the relaxed atomic positions, (c) four QLs with a bilayer at 4 Å and (d) at 6 Å vertical distance.

Figure 4

Projected band structure of five QLs with a Bi bilayer on top. Here only the projection on the upper two QLs and the Bi bilayer are shown in order to compare with ARPES data. When more than $90\%$ of the electron density comes from these layers, it is indicated in dark colors; when this is between $75\%$ and $90\%$, it is indicated in light colors.

Figure 5

Band structures of (a) a bare one QL, (b) four QLs with a bilayer and a QL on top in the relaxed atomic positions, (c) four QLs with a bilayer on top and a QL at 4 Å and (d) at 6 Å vertical distance.

Figure 6

The band structure of a five QL slab with two Bi bilayers inside, which separate two QLs from three QLs. Color coding of the bands is the same as before, with colors for different QLs and the Bi bilayer as shown at the top of the figure. On the right-hand side the layer-projected electron densities of specific states at $\Gamma$ are shown.

Figure 7

The gap between the interface states as a function of the distance between the QLs next to the Bi bilayer(s).