Microscopic theory for Doppler velocimetry of spin propagation in semiconductor quantum wells

Abstract

We provide a microscopic theory for the Doppler velocimetry of spin propagation in the presence of spatial inhomogeneity, driving electric field and the spin orbit coupling in semiconductor quantum wells in a wide range of temperature regime based on the kinetic spin Bloch equation. It is analytically shown that under an applied electric field, the spin density wave gains a time-dependent phase shift φ(t). Without the spin-orbit coupling, the phase shift increases linearly with time and is equivalent to a normal Doppler shift in optical measurements. Due to the joint effect of spin-orbit coupling and the applied electric field, the phase shift behaves differently at the early and the later stages. At the early stage, the phase shifts are the same with or without the spin-orbit coupling. While at the later stage, the phase shift deviates from the normal Doppler one when the spin-orbit coupling is present. The crossover time from the early normal Doppler behavior to the anomalous one at the later stage is inversely proportional to the spin diffusion coefficient, wave vector of the spin density wave and the spin-orbit coupling strength. In the high temperature regime, the crossover time becomes large as a result of the decreased spin diffusion coefficient. The analytical results capture all the quantitative features of the experimental results, while the full numerical calculations agree quantitatively well with the experimental data obtained from the Doppler velocimetry of spin propagation [Yang et al., Nat. Phys. 8, 153 (2012)]. We further predict that the coherent spin precession, originally thought to be broken down at high temperature, is robust up to the room temperature for narrow quantum wells. We point out that one has to carry out the experiments longer to see the effect of the coherent spin precession at higher temperature due to the larger crossover time.

Phase velocity of drifting spin wave packets in semiconductor two-dimensional electron gas

“…To overview the spin dynamics of a drifting spin packet, we start with an analytically solvable model with a SDD equation. 22) The time evolution of spin components under an in-plane electric field applied in the x direction [110] can be described by the following equation in Fourier space:…”

“…, , , i are determined by the SOI parameters and momentum-relaxation time. 22) The spins are transported with a constant drift velocity v d in the x direction. When the SOI strengths are sufficiently small and satisfy 22) we obtain an approximate solution for the z-component of the spin density in real space where A(x,t) is a Gaussian envelope with its width defined as…”