Electronic structures and surface states of the topological insulator ${\text{Bi}}_{1-x}{\text{Sb}}_x$

We investigate the electronic structures of the alloyed ${\text{Bi}}_{1-x}{\text{Sb}}_x$ compounds based on first-principles calculations including spin-orbit coupling (SOC), and calculate the surface states of semi-infinite systems using maximally localized Wannier function. From the calculated results, we analyze the topological nature of ${\text{Bi}}_{1-x}{\text{Sb}}_x$, and found the followings: (1) pure Bi crystal is topologically trivial. (2) Topologically nontrivial phase can be realized by reducing the strength of SOC via Sb doping. (3) The indirect bulk band gap, which is crucial to realize the true bulk insulating phase, can be enhanced by uniaxial pressure along $c$ axis. (4) The calculated surface states can be compared with experimental results, which confirms the topological nature. (5) We predict the spin-resolved Fermi surfaces and showed the vortex structures, which should be examined by future experiments.

Figure 1

(Color online) (a) Along the (111) direction of rhombohedral A7 structure, there are three possible atomic positions (A, B, and C) for the triangle plane. Black filled dots denote $A$ site, red filled up triangles denote $B$ site, and blue filled down triangles denote $C$ site. (b) The schematic plot of Bi-layers projected onto the plane paralleling to the (111) axis. After the dimerization of two Bi layers, the Bi-2 moves to the dashed site. $\Delta d$ denotes the magnitude of the dimerization. first and second NN hopping are the interlayer hopping between the different sublattices, while third NN hopping is the intralayer hopping within the same sublattice.

Figure 2

(Color online) (a) BZ of fcc structure. (b) 3D BZ of rhombohedral A7 structure and its projection onto the [111] surface. The A7 BZ can be obtained from the fcc BZ by two steps: (1) rotating the (111) direction of cubic to be along the $c$ axis; (2) slightly distortion of the BZ along the $c$ axis. The $L_1$ point in fcc BZ is changed to be $T$ point in the A7 BZ, which is now inequivalent to $L_{2,3,4}$ due to the distortion. $T$ $\left(\Gamma \right)$, $X\left(\text{l}\right)$, and $K$ in 3D BZ are projected to $\overline{\Gamma }$, $\overline{M}$, and $\overline{K}$ in the 2D BZ of [111] surface.

Figure 3

(Color online) Energy bands are plotted as a function of the dimerization parameter $\left(\Delta d\right)$ with SOC parameter $\lambda$ taken as 1.5 and 0.5 eV. (a) and (b) are for $T$ point while (c) and (d) for $L$ point. The solid blue lines denote the states with parity $+1$ while the dashed red ones are the states with parity $-1$. The dashed black line represents experimental dimerization $\left(\Delta d=\Delta d_0\right)$.

Figure 4

The phase diagram of the system with two variables: the dimerization parameter $\Delta d$ and SOC parameter $\lambda$. In the region where Sb locates, the parity of $L$ and $T$ points are $+1$ and $-1$, respectively, in other words, the system is in the topological nontrivial phase with $Z_2$ invariants (1;111). In the other region where Bi locates, however, the parity of $L$ and $T$ points are all $-1$, and the $Z_2$ invariants are (0;000).

Figure 5

Left panel: the surface local DOS for $\lambda =1.1\text{eV}$. The dark gray regions denote the continuous bulk bands with a small gap of about 10 meV at $\overline{M}$ point, and the white gray regions denote energy bulk gap. Two surface bands (${\Sigma }_1$ and ${\Sigma }_2$) disperse within the bulk gap. The black dashed line indicates the Fermi energy, which intersects five times with the two surface bands from $\overline{\Gamma }$ to $\overline{M}$. Right panel: the region framed by the black rectangle in the left panel is zoomed in. One surface state $\left({\Sigma }_1\right)$ goes up to merge the conduction band while the other one $\left({\Sigma }_2\right)$ goes back to the valence band.

Figure 6

Left panel: the surface local DOS for $\lambda =1.28\text{eV}$. The two surface states (${\Sigma }_1$ and ${\Sigma }_2$) disperse in a different way in the present case. At the $\overline{M}$ point, both ${\Sigma }_1$ and ${\Sigma }_2$ are connected to the valence band. Therefore the Fermi energy intersects four times with the surface bands between $\overline{\Gamma }$ and $\overline{M}$. Right panel: the region framed by the black rectangle in the left panel is zoomed in.

Figure 7

(Color online) Left panel: the Fermi surface plot for $\lambda =1.1\text{eV}$. The white gray hexagonal region is the 2D BZ of [111] surface for A7 structure. $\overline{\Gamma }$ is enclosed by a hexagonal electron pocket. There are other six hole pockets and six electron pockets surrounding. Right panel: the region framed by red rectangle in the left panel is zoomed in. We can clearly see that the outermost six small electron pockets do not enclose $\overline{M}$.

Figure 8

(Color online) Left panel: the Fermi surface plot for $\lambda =1.28\text{eV}$, which is similar to Fig. 7, except that the outermost six electron pockets enclose the $\overline{M}$ point. Right panel: the region framed by red rectangle in the upper panel is zoomed in.

Figure 9

(Color online) Schematic picture for the comparison of the surface bands obtained from (a) our ab initio calculation, (b) TB model [from the work of Teo et al. (Ref. 11)], and (d) ARPES experiment results. In ARPES experiment, an additional ${\Sigma }_3$ surface band [dotted line in (d)] becomes degenerate with ${\Sigma }_2$ band at $\overline{M}$ point. This additional band may come from the hybridization between the topological surface states and the other trivial surface states, as suggested by the red dotted line in (c).

Figure 10

(Color online) The spin-resolved Fermi surface for the semi-infinite ${\text{Bi}}_{1-x}{\text{Sb}}_x$’s top surface. The arrow in (a) indicates $\left(S_x,S_y\right)$; different colors in (a) and (b) represent $S_z$ along different directions. The green color in (a) and dark gray in (b) mean that the $S_z$ is along the $+z$ direction; the red color in (a) and white gray in (b) mean that the $S_z$ is along the $-z$ direction. The three pieces of Fermi surface $F_1$, $F_2$, and $F_3$ are marked in Fig. 7. $\lambda$ is taken as 1.1 eV here.

Figure 11

(Color online) Indirect energy band gap. We define the indirect energy gap as the difference between the CBB and the VBT. The blue line and red line present the indirect energy gap’s functions of $c/a$ for $\lambda =1.28$ and 1.1 eV. $H_1$ and $H_2$ points are not high symmetry points and locate in the mirror plane in BZ. $H_1$ is near to $T$ point and $H_2$ is near to $L$ point. In the inset, we show the schematic figures indicating the different energy band structures.