Integration of Poisson Equation for the Diamond-Structure Semiconductors by the Green Function Cellular Method (GFCM)

Abstract

Solving the Poisson equation for the electrostatic potential in a solid is an integral part of a modern electronic structure calculation. In this work, the Poisson equation for the diamond-structure semiconductors is solved using the Green Function Cellular Method. The charge density was obtained from a first principle consideration of the atomic wave functions for the electrons. The real solid harmonics JL, and HL are used in expanding the Poisson equation Green function, therefore, most of the calculations were done in the spherical coordinates system. The maximum value of the index L considered is six. The Poisson equation is solved only for the state k = 0 in this work. The results show that the total electron-electron Coulomb potential, VCE(rn) is given mainly by the central cell contribution and thus varies in a similar fashion. From a certain value at the origin, the total potential initially increases with the distance up to a certain point and then decreases as the distance is further advanced. The peak potential is roughly at about 0.05d in carbon, 0.03d in silicon and germanium, and 0.02d in gray tin. The rate of convergence of the external cells potential contribution over the l summation is faster than the central cell contribution, while the total potential rate is the slowest. In general the rate of convergence at a given point depends on the distance and direction of the point. The mathematical formulation and the results of the calculations are presented and discussed