Interference mechanism of magnetoresistance in variable-range hopping conduction: The effect of paramagnetic electron spins and continuous spectrum of scatterer energies

Despite the fact that the problem of the interference mechanism of magnetoresistance in semiconductors with hopping conductivity has been widely discussed, most existing studies focused on the model of spinless electrons. This model can be justified only when all electron spins are frozen. However, there is always an admixture of free spins in the semiconductor. This study presents the theory of interference contribution to magnetoresistance that explicitly includes effects of both frozen and free electron spins. We consider the cases of small and large number of scatterers in the hopping event. For the case of a large number of scatterers an approach is used that takes into account the dispersion of the scatterer energies. It is shown that with the presence of both frozen and free spins, the negative magnetoresistance is proportional to the ratio of the frozen spins. Its temperature dependence in the case of the large number of scatterers is obtained to be weaker than in other models that neglect the variance of scatterer energies. Also, there is an additional spin-ordering positive magnetoresistance mechanism that is proportional to the ratio of free spins and has a strong temperature dependence. It is shown that the latter mechanism can dominate at low temperatures and can lead to a suppression of negative magnetoresistance. The results provide an explanation of the experimental observations of suppression of negative magnetoresistance with temperature decrease.