Mikhail Raikh, raikh@physics.utah.edu
Co-authors: R. C. Roundy
Spin dynamics of a diffusively moving electron in a random hyperfine field
We study the dynamics, hSz (t)i, of the average spin of electron hopping over sites which host random hyperfine
magnetic fields. If the typical waiting time for a hop is τ and the typical magnetic fields is bs0 , then the
typical spin-precession angle on a given site is δφ ∼ b0 τ 1. Then the Markovian theory predicts that
the spin, initially oriented along the z-axis decays, on average, as hSz (t)i = exp(−t/τs ), where τs =
1/b20 τ is the spin-relaxation time. We find that in low dimensions, d = 1, 2, the decay, hSz (t)i, is nonexponential 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, δφ, of spin during these visits add up. As a result, the Markovian description
does not apply. For one-dimensional diffusion of electron over sites, the average, hSz (t)i, is the universal
2/3
function of t3/2 /τ 1/2 τs , so that the characteristic decay time is τ 1/3 τs is much shorter than τs . Moreover,
when the random magnetic fields are located in the (x, y) plane, the decay of hSz (t)i to zero is preceded by a
reversal of hSz (t)i to the value hSz i = −0.16 at intermediate times. We develop an analytical self-consistent
description of the spin dynamics which explains this reversal. Another consequence of self-intersections of the
random-walk trajectories is that, in all dimensions, the average, hSz (t)i, becomes sensitive to a weak external
magnetic field directed along z. Our analytical predictions are complemented by the numerical simulations of
hSz (t)i.