Calculation of the Energy Band Structures of the Diamond and Germanium Crystals by the Method of Orthogonalized Plane Waves

Abstract

The mathematical procedure employed in recent exploratory calculations of the energy band structures of the diamond and germanium crystals is described. Some of the symmetry properties of the eigensolutions of diamond-type lattices and the construction of an approximate potential for the diamond crystal are reviewed.The relative order of the crystal eigenvalues ${E}_{\ensuremath{\gamma}}(\mathrm{k})$ for a particular k was found to depend more upon the symmetry of the crystal potential than upon the detailed nature of this potential. On the other hand, the curvatures of the energy surfaces calculated by perturbation theory were found to depend in a rather critical manner upon the exact form of the crystal potential.It would appear that more reliable estimates of the energy band structures of actual crystals can be obtained with the aid of Herring's method of orthogonalized plane waves than by means of other approximational methods requiring comparable effort provided (1) reliable crystal potentials are employed, (2) calculations are carried to the point where the eigensolutions are satisfactorily "convergent," and (3) eigensolutions for more than just the points of high symmetry in the reduced zone are investigated.A more elaborate calculation of the energy band structure of the germanium crystal has been undertaken; the work is now in progress.