Local fields in electron transport: Application to electromigration

Abstract

A quantum-mechanical expression is given for the local electric field in dc electron transport. The local field is shown to provide the driving for the migration of atoms during dc electron transport (electro-migration). We express the local field in terms of the electron charge density, which we obtain by solving the Liouville equation for the single-particle density matrix. The solutions are found within a self-consistent field weak-scattering approximation scheme for an impurity in a jellium background. Electron-phonon interaction is included via a phenomenological relaxation time. It is shown that the local field arises from both static and dynamic screening. The static screening is associated with the screening of the external field near the impurity, while the dynamic screening is associated with the electron-wind force, or the momentum transferred by the electrons in collisions with the impurity. It is found that the electron-wind force dominates when ${k}_{F}l\ensuremath{\gg}1$, where ${k}_{F}$ is the Fermi wave vector and $l$ is the electron mean free path. Only when ${k}_{F}l$ is of order unity are the static and dynamic screening contributions comparable. Landauer's residual-resistivity dipoles and carrier-density-modulation effect are investigated, and are found to contribute to the local field. The carrier-density modulation effect is shown to lead to deviations from Matthiessen's rule by shifting the Fermi energy relative to the band bottom. Within the local-field framework, we discuss the distinction between the external field and long-range macroscopic fields in quantum-mechanical formulations of dc electron transport.