An alternative approach to the calculation of the scattering-induced magnetoresistance in the regime of variable-range-hopping conductivity is developed. Under the assumption that the direct tunneling and the various single-scattering tunneling paths give the main contribution to the amplitude of a hop, the exact averaging over the positions and energies of scatterers of the logarithm of the resistance is performed. It is shown that in the strong-scattering limit the magnetoresistance is universal in the sense that it does not depend on the scattering strength. The transition from negative magnetoresistance in low magnetic fields, caused by the suppression of the destructive interference, to positive magnetoresistance in strong enough fields, caused by the orbital shrinkage of the impurity wave functions, is traced. The theory developed is applied to the system of two parallel impurity planes in a magnetic field parallel to the planes. The oscillations of the hopping resistance with magnetic field in such a system are studied quantitatively.
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Raǐkh et al. (1993) studied this question.