Analytical Solutions For The Surface States Of Bi1-Xsbx (0 <= X Less Than Or Similar To 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 (M) over bar 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 (less than or similar to 200 bilayers similar or equal to 80 nm) of pure bismuth. Consequently, in such a film of Bi1-xSbx, there is no apparent change in the SSs through the band inversion at x similar or equal to 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 Bi1-xSbx (0 <= x less than or similar to 0.1) are discussed.