Necessary and sufficient condition for longitudinal magnetoresistance

TL;DR

This paper establishes a precise criterion based on Fermi surface shape for the occurrence of longitudinal magnetoresistance (LMR), explaining why certain anisotropic spectra produce LMR while others do not.

Contribution

It derives a necessary and sufficient condition on Fermi surface anisotropy for non-zero LMR, highlighting the importance of non-separable spectra and specific velocity-momentum relationships.

Findings

Anisotropic but separable spectra do not produce LMR.

Non-separable spectra with certain velocity-momentum dependencies do produce LMR.

Lattice types and hopping terms influence the Fermi surface shape relevant for LMR.

Abstract

Since the Lorentz force is perpendicular to the magnetic field, it should not affect the motion of a charge along the field. This argument seems to imply absence of longitudinal magnetoresistance (LMR) which is, however, observed in many materials and reproduced by standard semiclassical transport theory applied to particular metals. We derive a necessary and sufficient condition on the shape of the Fermi surface for non-zero LMR. Although an anisotropic spectrum is a pre-requisite for LMR, not all types of anisotropy can give rise to the effect: a spectrum should not be separable in any sense. More precisely, the combination $k_{\rho}v_{\phi}/v_{\rho}$, where $k_\rho$ is the radial component of the momentum in a cylindrical system with the z-axis along the magnetic field and $v_{\rho} (v_{\phi}$) is the radial (tangential) component of the velocity, should depend on the momentum along the field. For some lattice types, this condition is satisfied already at the level of nearest-neighbor hopping; for others, the required non-separabality occurs only if next-to-nearest-neighbor hopping is taken into account.