Pseudopotential theory of covalent bonding

The electronic structure of tetrahedral semiconductors is formulated on the basis of a four-orthogonalizedplane-wave (OPW) expansion of the wave functions at the center of each Jones-zone face. The band gap is found to be approximately equal to twice the magnitude of the [111] matrix element of the pseudopotential rather than the [220] matrix element or the Heine-Jones perturbation form. This confirms the dependence of the gap upon even and odd parts of the pseudopotential assumed by Phillips and Van Vechten in formulating their ionicity theory. Fitting of this gap to the optical-absorption peak $E_2$ gives values in semiquantitative agreement with the corresponding values from the empirical-pseudopotential matrix elements of Cohen and Bergstresser. Neglecting other matrix elements allows a Penn-like calculation of the dielectric constant for polar as well as homopolar semiconductors, but one which in both cases is found to vary as the inverse cube rather than the inverse square of the gap. The form for homopolar semiconductors is only consistent with the Penn formula if, as is approximately true, the gap scales inversely with the square of the bond length; in that case it is also consistent with the bond-orbital formula with the covalent energy $V_2$ equal to the [111] pseudopotential matrix element. The form for heteropolar semiconductors is inconsistent with the direct extension of the Penn formula, but is consistent with the bond-orbital model. The change in total energy under lattice shear is computed, again retaining only the [111] matrix elements. The elastic constant is found to be proportional to the derivative of the pseudopotential matrix element with respect to wave number. Use of corresponding values from pseudopotential theory gives good estimates of the elastic constants for diamond, silicon, and germanium, and suggests the correct trend for polar semiconductors. These terms which are found adequate for understanding the dielectric constant and lattice rigidity lead to a saddle point rather than a maximum in charge density at the bond center, suggesting that the bond charge should not be thought of as the origin of structural stability of tetrahedral structures. Extending the theory beyond the four-OPW model does lead to charge accumulation in the bonds and retains the structural stability, principally through the kinetic energy of the electrons confined to the Jones zone. The fundamental difference between the pseudopotential theory of covalent solids and of metals is not the inclusion of higher-order terms in the pseudopotential, but the different treatment of zero- and first-order terms analogous to degenerate perturbation theory.