Self-consistent solution of Kohn-Sham equations for infinitely extended systems with inhomogeneous electron gas

TL;DR

This paper presents a new iterative method for solving Kohn-Sham equations in infinitely extended inhomogeneous electron systems, addressing boundary condition issues and enabling accurate calculations of electronic properties.

Contribution

The authors develop an iterative scheme that eliminates Coulomb interaction's long-range effects, improving stability and accuracy in self-consistent solutions for extended systems.

Findings

Successfully calculated energy spectrum and potential of semi-infinite electron gas.

Analyzed differences between Hartree and exchange-correlation solutions.

Applied method to metal-semiconductor tunnel contact with steady-state current.

Abstract

The density functional approach in the Kohn-Sham approximation is widely used to study properties of many-electron systems. Due to the nonlinearity of the Kohn-Sham equations, the general self-consistence searching method involves iterations with alternate solving of the Poisson and Schr\"{o}dinger equations. One of problems of such an approach is that the charge distribution renewed by means of the Schr\"{o}dinger equation solution does not conform to boundary conditions of Poisson equation for Coulomb potential. The resulting instability or even divergence of iterations manifests itself most appreciably in the case of infinitely extended systems. The published attempts to deal with this problem are reduced in fact to abandoning the original iterative method and replacing it with some approximate calculation scheme, which is usually semi-empirical and does not permit to evaluate the extent of deviation from the exact solution. In this work, we realize the iterative scheme of solving the Kohn-Sham equations for extended systems with inhomogeneous electron gas, which is based on eliminating the long-range character of Coulomb interaction as the cause of tight coupling between charge distribution and boundary conditions. The suggested algorithm is employed to calculate energy spectrum, self-consistent potential, and electrostatic capacitance of the semi-infinite degenerate electron gas bounded by infinitely high barrier, as well as the work function and surface energy of simple metals in the jellium model. The difference between self-consistent Hartree solutions and those taking into account the exchange-correlation interaction is analyzed. The case study of the metal-semiconductor tunnel contact shows this method being applied to an infinitely extended system where the steady-state current can flow.