Semi-classical treatment of surface states

The electron states in a finite one-dimensional crystal are solved in the WKB approximation. No conditions are imposed on the number of atomic levels involved or on the shape of the potential wells and their matrix elements. Explicit conditions for the occurence of surface states as well as bulk states are obtained. The results are more general than, although consistent with, those from the usual tight-binding approximation. It is also shown that no Shockley surface states can arise in this semi-classical approximation. The difference with some recent work in the tight-binding model is discussed.