E. Medina, M. Kardar, and R. Rangel, Physical Review B 53, 7663 (1996)

Magnetoconductance anisotropy and interference effects in variable-range hopping

We investigate the magnetoconductance (MC) in the variable-range hopping regime, caused by quantum interference effects in three dimensions. We find that, in the absence of spin-orbit scattering, there is an increase in the localization length, 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 as in two dimensions. Orientational effects, of the sample with respect to an external magnetic field, can be considered in three dimensions, and we find the magnetoconductance anisotropy depends critically on the number of dominant hops in the sample and the magnetic field intensity. If a single hop (or a few) dominates the conductivity of the sample, this leads to a characteristic orientational "fingerprint" for the MC anisotropy that could be probed experimentally. Samples probed to date, however, exhibit a conductance dominated by many hops, and thus averaging over critical hop orientations renders the bulk sample isotropic. Anisotropy appears, however, for thin films, when the length of the hop is comparable to the thickness. The hops are then restricted to align with the sample plane, leading to different MC behaviors parallel and perpendicular to it, even after averaging over many hops. We predict, on the basis of the Nguyen-Spivak-Shklovskii model, the variations of such anisotropy with both the hop size and the magnetic field strength. An orientational bias of dominant hops produced by strong electric fields is suggested as an interesting probe of anisotropy effects due to interference mechanisms in the variable-range hopping regime.