Theory of coupled spin-charge transport due to spin-orbit interaction in inhomogeneous two-dimensional electron liquids

Spin-orbit interactions in two-dimensional electron liquids are responsible for many interesting transport phenomena in which particle currents are converted to spin polarizations and spin currents and vice versa. Prime examples are the spin Hall effect, the Edelstein effect, and their inverses. By similar mechanisms, it is also possible to partially convert an optically induced electron-hole density wave to a spin density wave and vice versa. In this paper, we present a unified theoretical treatment of these effects based on quantum kinetic equations that include not only the intrinsic spin-orbit coupling from the band structure of the host material, but also the spin-orbit coupling due to an external electric field and a random impurity potential. The drift-diffusion equations we derive in the diffusive regime are applicable to a broad variety of experimental situations, both homogeneous and nonhomogeneous, and include on equal footing “skew scattering” and “side jump” from electron-impurity collisions. As a demonstration of the strength and usefulness of the theory we apply it to the study of several effects of current experimental interest: the inverse Edelstein effect, the spin-current swapping effect, and the partial conversion of an electron-hole density wave to a spin density wave in a two-dimensional electron gas with Rashba and Dresselhaus spin-orbit couplings, subject to an electric field.

Figure 1

Diagrams for the impurity-averaged self-energy ${\Sigma }_0,{\Sigma }_1,{\Sigma }_2$, and ${\Sigma }_{\mathrm{EY}}$. The dashed line denotes the impurity averge and the vertices with open circles represent the impurity-induced spin-orbit coupling.

Figure 2

Magnitudes of spin Hall velocities due to skew scattering $\left(v^{\mathrm{ss}}\right)$, side jump $\left(v^{\mathrm{sj}}\right)$, and intrinsic mechanism $\left(v^{\mathrm{int}}\right)$ as functions of momentum relaxation time (lower axis) and mobility (upper axis) in 10-nm GaAs QW grown in (001) direction with electron density $n_e={10}^{12}{\text{cm}}^{-2}$. The electric field is taken to be $1\mathrm{kV}/\mathrm{cm}$. The results for the intrinsic mechanism are computed with two different values of Rashba coefficient.

Figure 3

Time evolution of the amplitude of the spin grating $S^z$ arising from an electron-hole grating of wave vector $q$ in the presence of an electric field perpendicular to the wave vector. Here only the extrinsic spin Hall effect (skew scattering and side-jump effect) is considered. The largest amplitude of the spin grating occurs in the central region of the plot (in red), and is maximized about $q=0.2$ $\mu {\mathrm{m}}^{-1}$. In this section, we take $\tau =1\mathrm{ps},D_a=20{\mathrm{cm}}^2/\mathrm{s}$, and $D_s=200{\mathrm{cm}}^2/\mathrm{s}$ unless otherwise specified. The electron-hole recombination rate $\Gamma =1{\mathrm{ns}}^{-1}$ is used.

Figure 4

(a) Time evolution of the amplitude of the spin grating $S^z$ arising from an electron-hole grating of wave vector $q$ (normalized by $q_0\simeq 3.5\mu {\mathrm{m}}^{-1})$ in the presence of an electric field perpendicular to the wave vector with $\alpha =-\beta$. (b)–(d) The corresponding spin profiles $S^y\text{−}{S}^z$ induced by collective spin Hall effect with different grating wave vectors around $q_0$. Four typical times are chosen for each case. For $q\ne q_0$, the profile at short time is a superposition of $S^{+}$ and $S^{-}$ with weights $q-q_0$ (large) and $q+q_0$ (small). At long time, only the $S^{+}$ mode $\left(\propto q-q_0\right)$ survives. (c) For $q=q_0$, the long-lived mode disappears and the long-time spin profile is qualitatively similar to the short-time state.

Figure 5

(a) Time evolution of the amplitude of the spin grating $S^z$ arising from an electron-hole grating of wave vector $q$ in the presence of an electric field perpendicular to the wave vector with $\alpha =\beta$. (b) The corresponding spin profiles $S^x\text{−}{S}^z$ induced by collective spin Hall effect at $q=q_0$. We observe that the helical modes $S^{+}$ and $S^{-}$ have the same lifetime and amplitude.

Figure 6

Time evolution of the amplitude of the spin grating $S^z$ arising from an electron-hole grating of wave vector $q$ in the presence of an electric field perpendicular to the wave vector without Rashba SOC.