Kinetic equations for hopping transport and spin relaxation in a random magnetic field

We derive the kinetic equations for a hopping transport that take into account an electron spin and the possibility of double occupation. In the Ohmic regime, the equations are reduced to the generalized Miller-Abrahams resistor network. We apply these equations to the problem of the magnetic moment relaxation due to the interaction with the random hyperfine fields. It is shown that in a wide range of parameters the relaxation rate is governed by the hops with the similar rates as spin precession frequency. It is demonstrated that at the large time scale spin relaxation is nonexponential. We argue that the nonexponential relaxation of the magnetic moment is related to the spin of electrons in the slow-relaxing traps. Interestingly, the traps can significantly influence the spin relaxation in the infinite conducting cluster at large times.