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Analytical solutions for the surface state (SS) of an extended WolffHamiltonian, which is a common Hamiltonian for strongly spin-orbit coupledsystems, are obtained both for semi-infinite and finite-thickness boundaryconditions. For the semi-infinite system, there are three types of SSsolutions: (I-a) linearly crossing SSs in the direct bulk band gap, (I-b) SSswith linear dispersions entering the bulk conduction or valence bands away fromthe band edge, and (II) SSs with nearly flat dispersions entering the bulkstate at the band edge. For the finite-thickness system, a gap opens in the SSof solution I-a. Numerical solutions for the SS are also obtained based on thetight-binding model of Liu and Allen [Phys. Rev. B, 52, 1566 (1995)] forBi$_{1-x}$Sb$_x$ ($0\le x \le 0.1$). A perfect correspondence between theanalytic and numerical solutions is obtained around the $\bar{M}$ pointincluding their thickness dependence. This is the first time that the characterof the SS numerically obtained is identified with the help of analyticalsolutions. The size of the gap for I-a SS can be larger than that of bulk bandgap even for a "thick" films ($\lesssim 200$ bilayers $\simeq 80$ nm) of purebismuth. Consequently, in such a film of Bi$_{1-x}$Sb$_x$, there is no apparentchange in the SSs through the band inversion at $x\simeq 0.04$, even though thenature of the SS is changed from solution I-a to I-b. Based on our theoreticalresults, the experimental results on the SS of Bi$_{1-x}$Sb$_x$ ($0\le x\lesssim 0.1$) are discussed.

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