Spin relaxation of a diffusively moving carrier in a random hyperfine field

Relaxation, <Sz(t)>, of the average spin of a carrier in course of hops over sites hosting random hyperfine fields is studied theoretically. In low dimensions, d = 1, 2, the decay of average spin with time is non-exponential at all times. The origin of the effect is that for d = 1, 2 a typical random-walk trajectory exhibits numerous self-intersections. Multiple visits of the carrier to the same site accelerates the relaxation since the corresponding partial rotations of spin during these visits add up. Another consequence of self-intersections of the random-walk trajectories is that, in all dimensions, the average, <Sz(t)>, becomes sensitive to a weak magnetic field directed along z. Our analytical predictions are complemented by the numerical simulations of <Sz(t)>.