Hopping conduction in disordered systems in the presence of a magnetic field: Linearized rate equation and magnetoresistance

Abstract

Rate equations are derived, in the linear approximation with respect to the electric field, for hopping transport in disordered systems in the presence of a magnetic field H. The resulting system of linear equations is in general not equivalent to a random resistor network. By means of an effective-medium approximation, an analytical expression for the magnetoresistance \ensuremath{\Delta}\ensuremath{\sigma}(H) has been obtained for R hopping. In the case of strong coupling with phonons, \ensuremath{\Delta}\ensuremath{\sigma}(H) is independent of the temperature T and it is proportional to ${\mathit{H}}^{2}$ in the region of weak magnetic fields. For weak coupling with phonons, we find that \ensuremath{\Delta}\ensuremath{\sigma}(H)\ensuremath{\sim}${\mathit{H}}^{2}$/T. $DELTA sigma (H)--- shows a monotonic behavior in a wide range of magnitude of the magnetic field. It does not exhibit an oscillating contribution. The sign of \ensuremath{\Delta}\ensuremath{\sigma}(H) depends on the character of the conductivity (electronlike or holelike). \ensuremath{\Delta}\ensuremath{\sigma}(H) approaches zero at some point if one varies the chemical potential.