A relativistic green matrix method to calculate the surface states in a three-dimensional crystal slab

In this article we generalize the relativistic Green matrix method, which we developed in an earlier paper for calculating the relativistic surface states in a semi-infinite three-dimensional crystal, to the case of a crystal slab with one atom in a unit cell, the extension to more complicated crystals being straightforward. We use the muffin-tin model for the crystal potential and constant potential energy in the vacuum regions. We demonstrate the existence of four different wave vectors. Our theory avoids the difficulties arising from the matching problem across the boundaries, as they occur in the augmented plane wave method. A two-dimensional relativistic surface bandstructure is found.