Geometric treatment of conduction electron scattering by crystal lattice strains and dislocations

TL;DR

This paper develops a geometric theory for electron scattering by lattice strains and dislocations, using a covariant Schrödinger equation to distinguish elastic and plastic effects, and estimates resistivity in copper aligning with experiments.

Contribution

It introduces a covariant geometric formalism for electron scattering in deformed lattices, applicable to elastic strains and dislocations, improving understanding of resistivity contributions.

Findings

Resistivity due to edge dislocations in Cu matches experimental data.

The formalism distinguishes elastic and plastic strain effects.

Electrons are modeled as moving in a non-Euclidean space.

Abstract

A theory for conduction electron scattering by inhomogeneous crystal lattice strains is developed, based on the differential geometric treatment of deformations in solids. The resulting fully covariant Schr\"odinger equation shows that the electrons can be described as moving in a non-Euclidean background space in the continuum limit of the deformed lattice. Unlike previous work, the formalism is applicable to cases involving purely elastic strains as well as discrete and continuous distributions of dislocations --- in the latter two cases it clearly demarcates the effects of the dislocation strain field and core and differentiates between elastic and plastic strain contributions respectively. The electrical resistivity due to the strain field of edge dislocations is then evaluated using perturbation theory and the Boltzmann transport equation. The resulting numerical estimate for Cu shows good agreement with experimental values, indicating that the electrical resistivity of edge dislocations is not entirely due to the core, contrary to current models. Possible application to the study of strain effects in constrained quantum systems is also discussed.