Analytical Solutions for the Surface States of Bi${}_{1-x}$Sb${}_x$ ($0\le x\lesssim 0.1$)

Abstract

Analytical solutions for the surface state (SS) of an extended Wolff Hamiltonian, which is a common Hamiltonian for strongly spin-orbit coupled systems, are obtained both for semi-infinite and finite-thickness boundary conditions. For the semi-infinite system, there are three types of SS solutions: (I-a) linearly crossing SSs in the direct bulk band gap, (I-b) SSs with linear dispersions entering the bulk conduction or valence bands away from the band edge, and (II) SSs with nearly flat dispersions entering the bulk state at the band edge. For the finite-thickness system, a gap opens in the SS of solution I-a. Numerical solutions for the SS are also obtained based on the tight-binding model of Liu and Allen [Phys. Rev. B, 52, 1566 (1995)] for Bi${}_{1-x}$Sb${}_x$ ($0\le x\le 0.1$). A perfect correspondence between the analytic and numerical solutions is obtained around the $\bar{M}$ point including their thickness dependence. This is the first time that the character of the SS numerically obtained is identified with the help of analytical solutions. The size of the gap for I-a SS can be larger than that of bulk band gap even for a "thick" films ($\lesssim 200$ bilayers $\simeq 80$ nm) of pure bismuth. Consequently, in such a film of Bi${}_{1-x}$Sb${}_x$, there is no apparent change in the SSs through the band inversion at $x\simeq 0.04$, even though the nature of the SS is changed from solution I-a to I-b. Based on our theoretical results, the experimental results on the SS of Bi${}_{1-x}$Sb${}_x$ ($0\le x\lesssim 0.1$) are discussed.

11 pages, 12 figures

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Cited by in corpus (5)

• Boundary conductance in macroscopic bismuth crystals
• A field-induced reentrant insulator state of a gap-closed topological insulator (Bi_{1-x}Sb_x) in quantum-limit states
• Topological blocking at the Bi(111) surface due to surface relaxation
• High-field immiscibility of electrons belonging to adjacent twinned bismuth crystals
• Long-range permeation of wave function and superficial surface state due to strong quantum size effects in topological Bi/BiSb heterojunctions