Analytical Solutions for the Surface States of Bi1−xSbx (0 ≤ x \\( \\lesssim \\) 0.1)

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 two types of SS solutions: (I-a) linearly crossing SSs in the direct bulk band gap, and (I-b) SSs with linear dispersions entering the bulk conduction or valence bands away from 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 Bi1−xSbx (0 ≤ x ≤ 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 t...