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

We study quantum interference effects on the transition strength for strongly localized electrons hopping on two-dimensional (2D) square and three-dimensional (3D) cubic lattices in the presence of a magnetic field. 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. By analytically summing paths which allow backward excursions in the forward-scattering direction, we find that the interference patterns between the dominant winding paths are not drastically different from those between the directed paths. 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 2 compared to that at zero field. The periodicity in the flux of the MC is found to be equal to the superconducting flux quantum hc/2e. 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 the applied field B. At very small fields, for two sites diagonally separated a distance r, we find that the MC behaves as rB in quasi-1D systems, r^3/2B in 2D with B=(0,0,B), and rB (r^3/2B) in 3D with B parallel (perpendicular) to the (1,1,1) direction. Furthermore, for a 3D sample, the effect on the low-flux MC due to the randomness of the angles between the hopping direction and the orientation of B is examined analytically. © 1996 The American Physical Society.