Variable-range hopping of spin polarons: Magnetoresistance in a modified Mott regime

We analize electrical conductivity controlled by hopping of bound spin polarons in disordered solids with wide distributions of electron energies and polaron shifts (barriers). By means of percolation theory and Monte Carlo simulations we have shown that in such materials at low temperatures, when hopping occurs in the vicinity of the Fermi level, a hard polaron gap does not manifest itself in the transport properties. This happens because as temperature decreases the hopping polaron trades the decreasing electron and polaron barriers for increasing hopping distance. As a result, in the absence of the Coulomb correlation effects, in this variable-range variable-barrier hopping regime, the electrical resistivity, $\ensuremath{\rho},$ as a function of temperature, T, obeys a nonactivation law: $\mathrm{ln}(\ensuremath{\rho}/{\ensuremath{\rho}}_{0})=(\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{T}{/T)}^{p}$ with $p=2/(d+2),$ where d is the dimensionality of the system. It differs from the standard Mott law for which $p=1/(d+1).$ Also, we studied the effects of upper and lower boundaries in the polaron shift distribution on hopping conduction, which may result in a partial reentrance of the hard polaron gap. We discuss possible applications to the problem of giant negative magnetoresistance in dilute magnetic semiconductors and nanocomposites where for paramanetic materials $p=3/(d+2).$