Tight-binding surface states in finite crystals

The electronic states of a finite crystal are studied using Goodwin's model of a tight-binding linear chain of N one-level atoms with nearest-neighbou r overlap. Using a transfer matrix approach we obtain the explicit form of the secular equation which correctly yields N eigenvalues in the interval ( 0, pi) of wavenumber q, unlike Goodwin's equation which involves spurious s olutions at q = 0 and q = pi. We present a new general analysis of bulk- an d surface-state eigenvalues as a function of the parameter epsilon(0/)gamma describing the difference (epsilon(0)) of Coulomb integrals for surface an d bulk atoms relative to the overlap integral gamma. We identify four disti nct domains of values of \epsilon(0)/gamma \ in three of which one or two s urface states of different origins exist, which we determine explicitly. Ou r discussion is valid for both signs of epsilon(0)/gamma and differs consid erably in detail from Goodwin's analysis. In particular, it does not requir e distinct analyses for chains with even and odd numbers of sites.