Many-electron model of equilibrium metal-semiconductor contacts and semiconductor heterojunctions

Abstract

A many-electron model is proposed to describe the electronic structure of metal-semiconductor interfaces and semiconductor heterojunctions. The model is utilized to calculate the self-consistent one-electron potential in the vicinity of the interface. This potential arises from two sources: a short-range dipole contribution due to valence-electron rearrangements at the interface relative to the corresponding vacuum surfaces, and a long-range space-charge contribution caused by band-bending effects extending thousands of angstroms from the interface.The short-range, valence-electron dipole contribution is evaluated by minimizing the interface energy calculated within a local-density description of the valence-electron charge density. The long-range space-charge contribution is determined both by the valence-electron rearrangements, being included in the interface energy through an electrostatic energy term, and by the thermodynamic equilibrium boundary conditions. The boundary conditions ensuring thermal, mechanical, and electron-transfer equilibrium are imposed explicitly. The chemical potentials required to satisfy the electron-transfer boundary condition are obtained from the local-density model at the constant temperature and pressure characteristic of the equilibrium interface.For numerical calculations the ion-core charges are approximated by a uniform positive charge density. The interfaces between the metal and semiconductor as well as the metal-vacuum and semiconductor-vacuum surfaces are defined by abrupt discontinuities in this uniform positive charge density. A version of local-density theory suitable for both metals and semiconductors is utilized to describe the valence-electron charge densities and is extended to incor- porate the dopants in the semiconductor as well. Consequently, a unified model of both the microscopic interface dipole and macroscopic space-charge region is achieved. Numerical calculations reveal that the microscopic interface dipole is independent of the doping in the space-charge region for dopant densities N\ensuremath{\lesssim}${10}^{17}$ ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}3}$.These calculations further predict the validity of Schottky's phenomenological boundary condition for metal-semiconductor contacts, the applicability of the electron-affinity rule for semiconductor heterojunctions, and systematic correlations between Schottky-barrier heights and heterojunction band offsets. The latter three results emerge as consequences of the rigorous and complete imposition of thermodynamic equilibrium boundary conditions and of the use of a model of the ion-core charge density in which atomic reconstructions at the interface relative to the vacuum surfaces are not considered explicitly.