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

TL;DR

This paper derives analytical solutions for the surface states of Bi$_{1-x}$Sb$_x$ using an extended Wolff Hamiltonian, compares them with numerical models, and discusses implications for experimental observations of surface state behavior.

Contribution

It provides the first identification of surface state characters in Bi$_{1-x}$Sb$_x$ through analytical and numerical solution comparison, revealing size-dependent gap effects.

Findings

Analytical solutions classify three types of surface states.

Numerical and analytical solutions show perfect correspondence near the $ar{M}$ point.

Surface state gaps can exceed bulk gaps even in thick films.

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.