Topological surface states on ${\mathrm{Bi}}_{1\text{−}x}$${\mathrm{Sb}}_x$: Dependence on surface orientation, termination, and stability

Topological insulators support metallic surface states whose existence is protected by the bulk band structure. It was predicted early that the topology of the surface-state Fermi contour should depend on several factors, such as the surface orientation and termination, and this raises the question to what degree a given surface state is protected by the bulk electronic structure upon structural changes. Using tight-binding calculations, we explore this question for the prototypical topological insulator ${\mathrm{Bi}}_{1\text{−}x}$${\mathrm{Sb}}_x$, studying different terminations of the (111) and (110) surfaces. We also consider the implications of the topological protection for (110) surfaces for the semimetals Bi and Sb.

Figure 1

Topological predictions for the surface electronic structure of ${\mathrm{Bi}}_{1\text{−}x}$${\mathrm{Sb}}_x$(111) and (110) ($x\gtrsim 0.09$). The bulk Brillouin zone and the projected surface Brillouin zones are shown for both surface orientations. Filled black circles mark the positions of time-reversal-invariant momenta (TRIMs). Calculation of surface fermion parity values $\pi$ for surface TRIMs according to (1) is illustrated. Darker-shaded (blue) areas denote an odd number of closed Fermi contours around a TRIM as derived from the surface Fermion parity in Eq. (1). Lighter-shaded (yellow) areas would be expected for the first atomic layer removed from either surface.

Figure 2

(a) Calculated surface electronic structure for Bi${}_{0.86}$Sb${}_{0.14}$(111). Red lines are the projected bulk band structure and the image is the imaginary part of the Green's function. (b) The same for Bi${}_{0.86}$Sb${}_{0.14}{\left(111\right)}^{\prime }$. (c) Surface structure for the two surfaces. For the ${\left(111\right)}^{\prime }$ surface the (red) top-layer atoms are missing. Black lines are nearest-neighbor bonds.

Figure 3

(a) Fermi contour for Bi${}_{0.86}$Sb${}_{0.14}{\left(111\right)}^{\prime }$ together with the surface Brillouin zone (or the imaginary part of the Green's function taken at the Fermi energy). (b) The same, but with the Fermi level assumed to be 27 meV higher. This shift results in a closed pocket around $\overline{M}$. Inset: Magnification of the closed pocket.

Figure 4

(a) Calculated surface electronic structure for Bi${}_{0.86}$Sb${}_{0.14}$(110). Red lines are the projected bulk band structure and the image is the imaginary part of the Green's function. (b) The same for Bi${}_{0.86}$Sb${}_{0.14}{\left(110\right)}^{\prime }$. Inset: Magnification of the electronic structure near the ${\overline{X}}_2$ point. (c) Structure of the (110) surface. For the ${\left(110\right)}^{\prime }$ surface the first-layer (red) atoms are missing. The green line (pseudo–mirror line) would be a mirror line if the crystal structure was simple cubic instead of rhombohedral A7. Black lines are the nearest-neighbor bonds.

Figure 5

(a) Fermi contour for Bi${}_{0.86}$Sb${}_{0.14}$(110) together with the surface Brillouin zone (or the imaginary part of the Green's function taken at the Fermi energy). The small (yellow) square contains a very small electron pocket. (b) The same for Bi${}_{0.86}$Sb${}_{0.14}{\left(110\right)}^{\prime }$. Inset: Magnification of the hole pocket around ${\overline{X}}_2$.

Figure 6

(a)–(c) Prediction of the surface-state topology for (110) surfaces of Bi, ${\mathrm{Bi}}_{1\text{−}x}$${\mathrm{Sb}}_x$, and Sb. Blue areas denote an odd number of closed Fermi contours around a TRIM. Red areas are the projected bulk Fermi surface. Arrows on the Fermi contours around $\overline{\Gamma }$ and $\overline{M}$ denote the calculated spin directions. (d)–(f) ARPES measurements of the electronic structure of these surfaces along the $\overline{\Gamma }$-$\overline{M}$ line [dashed line in (a)–(c)]. Dark areas correspond to a high photoemission intensity. Black lines are the projection of the bulk bands onto the (110) surface. Data for (a) and (b) taken from Ref. [11]; for (c), from Ref. [25].