The effect of spin-orbit scattering on the hopping transport of strongly localized electrons in the variable-range hopping regime is investigated analytically in both two and three dimensions. In particular, we focus on the behavior of the magnetoconductance, which is a manifestation of the quantum interference between different electron forward-scattering paths associated with a critical hop. Treating these paths independently, we obtain closed-form results for the magnetoconductance for quasi-one- and two-dimensional systems in the limit of strong spin-orbit scattering. The magnetoconductance is positive with reduced magnitude and identical periodicity compared to that without spin-orbit scattering. Moreover, a simple formula for the field strength that first saturates the magnetoconductance is provided and can directly be compared with experiments. The magnetoconductance for three-dimensional systems in a weak magnetic field of arbitrary orientations is also obtained analytically, which shows magnetoconductance anisotropy and thus enables us to determine the direction of the critical hop. Indeed, our results for the magnetoconductance which varies according to the orientation of the applied field can be potentially used as a theoretical basis for further experimental tests of the quantum interference effects for electrons in the strongly localized regime.

• Received 8 June 1998

DOI:https://doi.org/10.1103/PhysRevB.58.13544