Abstract: Surface properties of a two−band semiconductor

Abstract

It has been shown 1−3 that very simple model calculations can give useful insight into the properties of solid surfaces. This is due largely to the fact that the calculations can be carried quite far analytically so that only simple numerical computations are needed. Previous work 1−3 has been confined to the study of one−band crystals of various structure and has been used to simulate the surface properties of metals. In this work, we present results of a two−band calculation which are appropriate for the study of semiconductor surfaces. Using the LCAO or the tight−binding formalism, we have calculated the band energies for a two−band semiconductor with a CsCl structure. One orbital is assumed for each atom in the crystal with the orbitals being different for the two basis atoms in the unit cell. With these bands, the free surface properties of the (001) surface are calculated using the Green’s function and the phase shift techniques, 1,2 which are similar to that used to study point defects in a crystal. The phase shift technique allows us to determine the change in the density of states due to the perturbation caused by creating the surface. A major advantage in the present calculation is that the Green’s function is obtained analytically. We have calculated the free surface properties for three different models. In the first, we consider only the nearest neighbor−hopping integral (resonance integral), γ1. In the second calculation, we extend the first to include one of the second neighbor interactions, γ2, the hopping integral between atoms of the same type in adjacent unit cells. In these two calculations, we do not consider the variation of the local potential in the vicinity of the surface. In the third calculation, we have added to the first model a change in the self energies of the surface atoms, U0. With only γ1 included in the calculation, no surface states are found. With γ2 added, Shockley surface states appear inside the band gap. With U0 added, Tamm surface states appear inside the band gap. In addition, surface states occur above the conduction band. The surface states appear for arbitrarily small γ2 and U0, and their separation distance from the top of the valence band (and the top of the conduction band, in the case with U0) is proportional to the magnitude of γ2 and U0. Since the energy states below the band gap are completely filled for a semiconductor, Friedel’s sum rule is satisfied, and all three of our calculations are self consistent. Within the nearest−neighbor−interaction framework, we have calculated the variation in the surface tension as a function of the filling of the band. We have also found the change, due to the surface, in the electronic specific heat and electronic surface entropy as a function of temperature. These properties are significantly different from those calculated with a filled one−band model, 3 and are consistent with the known properties of a semiconductor. Thus, we show that the presence of the second unoccupied band is crucial in studying semiconductor surfaces. The present calculations can be easily extended to the study of the free surface of the alkali metals with the b.c.c. crystal structure by setting the energy gap to zero. When this is done, our calculations agree with previous results. 2