Magnetoconductance anisotropy and interference effects in variable-range hopping.

We investigate the magneto-conductance (MC) anisotropy in the variable range hopping regime, caused by quantum interference effects in three dimensions. When no spin-orbit scattering is included, there is an increase in the localization length (as in two dimensions), producing a large positive MC. By contrast, with spin-orbit scattering present, there is no change in the localization length, and only a small increase in the overall tunneling amplitude. The numerical data for small magnetic fields B, and hopping lengths t, can be collapsed by using scaling variables B⊥t 3/2, and B‖t in the perpendicular and parallel field orientations respectively. This is in agreement with the flux through a ‘cigar’–shaped region with a diffusive transverse dimension proportional to √ t. If a single hop dominates the conductivity of the sample, this leads to a characteristic orientational ‘finger print’ for the MC anisotropy. However, we estimate that many hops contribute to conductivity of typical