Ab initio studies of adatom- and vacancy-induced band bending in ${\mathrm{Bi}}_2{\mathrm{Se}}_3$

We investigate the influence of potassium adsorption and selenium vacancies in the surface layer on the electronic properties of the prototypical topological insulator ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. These modifications of the surface give rise to oscillations in the charge density that extend deep into the crystal. They result in a long-ranged potential perpendicular to the surface (also referred to as band bending) and new states in the band structure that are reminiscent of the states of a two-dimensional electron gas. Very similar effects have been observed in several experiments. The reorganization of the charge deep inside the crystal as a reaction to the surface modification constitutes a remarkable property of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ and is closely related to its layered structure. The emergence of the long-ranged potential as a direct consequence of the charge reorganization turns out to be a generic property of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. However, calculations without spin-orbit coupling show that the band bending is not related to the nontrivial topological character of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$.

Figure 1

Top view of the ${\mathrm{Bi}}_2{\mathrm{Se}}_3$(111) surface. The three topmost layers are shown and possible adsorption sites for adatoms are indicated.

Figure 2

The band structures of a pristine slab (left) and of K-covered ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ slabs with 1 ML (middle) and 1/3 ML coverage (right) are shown. The gray areas show the projected bulk band structure. Red (blue) dots indicate a negative (positive) Rashba component of the spin expectation value. For the definition of the Rashba-component $P_{\perp }$ at a given wave vector ${\mathbf{k}}_0$, please see the inset in the left panel. The size of the dots is proportional to the absolute value of the Rashba component. The bands from the upper and lower surfaces of the slab have opposite Rashba components. Here, only the values of the bands from the upper surface are shown. This applies to all figures. In the middle and right panels, green lines indicate states that are predominantly located on the potassium atoms, as resulting from a Mulliken population analysis. The arrows mark the upper part of the modified TSS (see text for explanation). The thickness of the lines is proportional to the Mulliken population. The valence-band maximum (VBM) of the projected bulk band structure is taken as the zero of the energy scale.

Figure 3

The upper part shows the averaged effective potentials $V\left(z\right)$ of a 13 QL slab with and without adsorbed K. $V\left(z\right)=0$ represents the Fermi energy. The lower part shows the difference $V_{\mathrm{diff}}\left(z\right)$ of these two potentials. For comparison, $V_{\mathrm{diff}}\left(z\right)$ for K adsorbed on Si(001) is also shown. Vertical dotted lines indicate the midpoints of the vdW gaps between two neighboring quintuple layers. $z=0$ is the position of the outermost selenium layer of the slab. The label “K” indicates the position of the K layer. The inset shows the same data on an enlarged energy scale.

Figure 4

The model potential $V_M\left(z\right)$ in comparison with $V_{\mathrm{diff}}\left(z\right)$ from Fig. 3 and the resulting band structure (see text for details of the calculation).

Figure 5

The upper part shows $\Delta \rho \left(z\right)$ for K-covered ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ as defined in Eq. (5). The colored vertical lines mark different points $z_0$ for which we compute ${\rho }_c\left(z\right)$ (see text for further explanation). The lower part shows the Coulomb potential of ${\rho }_c\left(z\right)$ for different $z_0$ in the corresponding colors. The black line shows the potential of the original $\Delta \rho \left(z\right)$ of K-covered ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. The dotted lines mark the QLs, the surface, and the K position as in Fig. 3.

Figure 6

The “decay-corrected” charge density difference $f\left(z\right)$ (see text for explanation) for K-covered ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ with and without SOC. Dotted vertical lines mark the midpoints of the vdW gaps. The underlying periodic structure in $\Delta \rho \left(z\right)$ for $z<-8$ Å is nicely revealed.

Figure 7

The same picture as Fig. 5, but for K-covered artificial BiSe instead of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$.

Figure 8

$\Delta \rho \left(z\right)$ and $V_{\mathrm{coul}}\left(z\right)$ for K-covered van der Waals silicon (see text for explanation).

Figure 9

$\Delta \rho \left(z\right)$ and $V_{\mathrm{diff}}\left(z\right)$ for K-covered ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ with reduced (left) and increased (right) vdW gap $d_{\mathrm{vdW}}$. The real crystal has $d_{\mathrm{vdW}}=2.43$ Å. The gray areas indicate the vdW gaps.

Figure 10

The band structure of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ with selenium vacancies in the surface layer. The blue and red dots mark the spin expectation value as in Fig. 2.