Change of the density of electron states caused by the surface of a layered crystal structure

Starting from the notion of the ‘‘${\mathrm{k}}_{?}$-resolved local density of states,’’ the properties of the ‘‘${\mathrm{k}}_{?}$-resolved integrated density of states’’ are analyzed. While this quantity remains finite when integrated over films or crystal slabs of bounded thickness (leading to the well-known representation of the density of states via Green’s functions), it will diverge when the range of integration is extended across a semi-infinite layered structure. It is shown that this divergency can be separated in a suggestive way by introducing the concept of the ‘‘change in the density of states,’’ which moreover turns out to be of essential importance in the formulation of a criterion ensuring overall charge neutrality. Explicit expressions for each of the above quantities are derived, which are easy to be evaluated in terms of the dispersion of surface states, the bulk band structure (and the associated Bloch waves) of the substrate material, and the reflection coefficients of the bulk states at the surface.