Interference mechanism of magnetoresistance in variable range hopping: the effect of paramagnetic electron spins

The work is aimed to modify the conventional approach to interference magnetoresistance for the situation when there is an admixture of free spins. We considered the system with arbitrary probability Pfree of electron spin to be free. We studied analytically and numerically both negative interference magnetoresistance and positive magnetoresistance due to spin ordering in magnetic field. We obtained the expression for magnetoresistance for two limiting cases corresponding to small and large number of scatterers in the hopping act. For both cases we obtained the explicit expressions for the dependence of the magnetoresistance on temperature and Pfree. Our results are in semiquantitative agreement with experiment.