Analytical results on quantum interference and magnetoconductance for strongly localized electrons in a magnetic field: Exact summation of forward-scattering paths

TL;DR

This paper derives exact analytical expressions for quantum interference effects on magnetoconductance in strongly localized electrons on 2D and 3D lattices, revealing how magnetic fields influence electron tunneling and conductance.

Contribution

It provides the first exact summation of forward-scattering paths for tunneling probabilities and magnetoconductance in localized electron systems in any dimension.

Findings

Positive magnetoconductance observed with increasing magnetic field.

Magnetoconductance doubles at a certain field strength in 2D.

Interference patterns in 3D depend on magnetic field orientation.

Abstract

We study quantum interference effects on the transition strength for strongly localized electrons hopping on 2D square and 3D cubic lattices in the presence of a magnetic field B. These effects arise from the interference between phase factors associated with different electron paths connecting two distinct sites. For electrons confined on a square lattice, with and without disorder, we obtain closed-form expressions for the tunneling probability, which determines the conductivity, between two arbitrary sites by exactly summing the corresponding phase factors of all forward-scattering paths connecting them. An analytic field-dependent expression, valid in any dimension, for the magnetoconductance (MC) is derived. A positive MC is clearly observed when turning on the magnetic field. In 2D, when the strength of B reaches a certain value, which is inversely proportional to twice the hopping length, the MC is increased by a factor of two compared to that at zero field. We also investigate transport on the much less-studied and experimentally important 3D cubic lattice case, where it is shown how the interference patterns and the small-field behavior of the MC vary according to the orientation of B. The effect on the low-flux MC due to the randomness of the angles between the hopping direction and the orientation of B is also examined analytically.