Spatiotemporal spin fluctuations caused by spin-orbit-coupled Brownian motion

We develop a theory of thermal fluctuations of spin density emerging in a two-dimensional electron gas. The spin fluctuations probed at spatially separated spots of the sample are correlated due to Brownian motion of electrons and spin-orbit coupling. We calculate the spatiotemporal correlation functions of the spin density for both ballistic and diffusive transport of electrons and analyze them for different types of spin-orbit interaction, including the isotropic Rashba model and the persistent spin helix regime. The measurement of spatial spin fluctuations provides direct access to the parameters of spin-orbit coupling and spin transport in conditions close to thermal equilibrium.

Figure 1

Probe-probe measurements of the spatial correlations of spin fluctuations in an electron gas. Fluctuations of the spin density at points ${\mathit{r}}_1$ and ${\mathit{r}}_2$ are correlated due to Brownian motion of electrons. The sign of the correlation function, positive or negative, depends on the distance ${\mathit{r}}_1-{\mathit{r}}_2$ and the frequency $\mathbf{\Omega }\left(\mathit{p}\right)$ of spin precession in the effective magnetic field which acts upon the electron spins when electrons walk.

Figure 2

Diagrammatic representation of the contributions to the spin susceptibility which corresponds to the series equation (5).

Figure 3

Spin-density correlation functions $K_{\alpha \beta }(\mathit{r},t)$ for the Rashba spin-orbit interaction and coordinate difference $\mathit{r}\parallel x$ between the points where the spin fluctuations are probed. Maps are calculated for ${\mathrm{\Omega }}_R\tau =5$, which is close to the ballistic transport of electrons between the points.

Figure 4

Spin-density correlation functions $K_{\alpha \beta }(\mathit{r},t)$ for the Rashba spin-orbit interaction and coordinate difference $\mathit{r}\parallel x$ between the points where the spin fluctuations are probed. Maps are calculated for ${\mathrm{\Omega }}_R\tau =0.2$, which corresponds to diffusive transport of electrons between the points.

Figure 5

(a)–(d) Correlation function of the out-of-plane spin fluctuations $K_{zz}$ in real space in a (001)-grown QW. Maps are calculated for the delay time $t=20/\left({\mathrm{\Omega }}_R^2\tau \right), {\mathrm{\Omega }}_R\tau =0.2$, and different ratios of the Dresselhaus and Rashba terms. (e) Distributions of the effective magnetic field on the Fermi circle for different ratios of the Dresselhaus and Rashba terms.