Random walk approach to spin dynamics in a two-dimensional electron gas with spin-orbit coupling

We introduce and solve a semiclassical random walk (RW) model that describes the dynamics of spin polarization waves in zinc-blende semiconductor quantum wells. We derive the dispersion relations for these waves, including the Rashba, linear and cubic Dresselhaus spin-orbit interactions, as well as the effects of an electric field applied parallel to the spin polarization wave vector. In agreement with calculations based on quantum kinetic theory [P. Kleinert and V. V. Bryksin, Phys. Rev. B 76, 205326 (2007)], the RW approach predicts that spin waves acquire a phase velocity in the presence of the field that crosses zero at a nonzero wave vector, $q_0$. In addition, we show that the spin-wave decay rate is independent of field at $q_0$ but increases as ${\left(q-q_0\right)}^2$ for $q\ne q_0$. These predictions can be tested experimentally by suitable transient spin grating experiments.

Figure 1

(Color online) The dispersion relations for (a) the decay rate and (b) the rate of phase change of the SO enhanced mode in the SU(2) case. (a) The decay rate ${\gamma }_{-}\left(q\right)$ increases with the drift velocity $\left(\lambda \equiv v_d/v_F\right)$ but always vanishes at the resonant wave vector $q_0$. (b) The rate of phase change ${\dot{\varphi }}_{-}\left(q\right)$ is proportional to the drift velocity $v_d$ and it crosses zero at the resonant wave vector $q_0$.

Figure 2

(Color online) The space-time evolution of $S_z$ with a normalized $\delta$-function injection at $x=0$, $t=0$, and drift velocity $v_d=2D{q}_0$ in the SU(2) case. The spin polarization develops into a conserved stationary wave with a Gaussian wave packet.

Figure 3

(Color online) The dispersion relations for (a) the decay rate and (b) the rate of phase change of the SO enhanced mode in the linear-Dresselhaus-only case. The main features resemble those in the SU(2) case, both ${\gamma }_{-}\left(q\right)$ show a minimum and ${\dot{\varphi }}_{-}\left(q\right)$ vanishes at $q_0$, but the lifetime is finite in this case.

Figure 4

(Color online) The space-time evolution of $S_z$ in the linear-Dresselhaus-only case with the same initial condition and applied $E$ field as in the SU(2) case. The features are similar to those in the SU(2) case, except the envelope function decays exponentially.

Figure 5

(Color online) The the absolute value of the spin polarization integrated over position as a function of time. In the SU(2) case, ${\left|S_z\right|}^{tot}$ is conserved after an initial decay while in the linear-Dresselhaus-only case, ${\left|S_z\right|}^{tot}$ decays exponentially.

Figure 6

(Color online) The space-time images of the spin polarization in the (a) SU(2) and (b) linear-Dresselhaus-only cases, respectively.