Coulomb corrections to the extrinsic spin-Hall effect of a two-dimensional electron gas

We develop the microscopic theory of the extrinsic spin-Hall conductivity of a two-dimensional electron gas, including skew-scattering, side-jump, and Coulomb interaction effects. We find that while the spin-Hall conductivity connected with the side jump is independent of the strength of electron-electron interactions, the skew-scattering term is reduced by the spin-Coulomb drag, so the total spin current and the total spin-Hall conductivity are reduced for typical experimental mobilities. Further, we predict that in paramagnetic systems the spin-Coulomb drag reduces the spin accumulations in two different ways: (i) directly through the reduction of the skew-scattering contribution, and (ii) indirectly through the reduction of the spin diffusion length. Explicit expressions for the various contributions to the spin-Hall conductivity are obtained using an exactly solvable model of the skew scattering.

Figure 1

Antisymmetric scattering rate $W_{\mathit{int}}^a={\int }_0^{2\pi }d\theta W^a(ϵ,\theta ){\sin }^2\theta$ in units of $n_{i}h∕m^2\mathcal{A}$ [see Eq. (A22)] as a function of $k^2$ for a model circular well attractive potential $V_0=-5\mathrm{meV}$ and radius $a=9.45\mathrm{nm}$ (described in the Appendix). We choose the parameters typical for the experimental 2DEG confined in ${\mathrm{Al}}_{0.1}{\mathrm{Ga}}_{0.9}\mathrm{As}$ quantum well, i.e., density of electrons and impurities $n_{2D}=n_i=2.0\times {10}^{12}{\mathrm{cm}}^{-2}$, $m=0.074{\mathrm{m}}_e$, and mobility $\overline{\mu }=0.1{\mathrm{m}}^2∕\mathrm{Vs}$. The effective spin-orbit coupling $\alpha \hslash =4.4{\mathrm{\AA }}^2$ in accordance with (Ref. 36).

Figure 2

Spin Hall conductivity as a function of temperature. ${\sigma }_{yx}^{\mathit{sj}}$ (open squares), ${\sigma }_{yx}^{\mathit{ss}}$ (open and closed circles) are the side-jump and skew-scattering contributions, respectively, to the conductivity in the absence of electron-electron interactions, and ${\sigma }_{yx}^{\mathit{tot}}$ is the total spin conductivity when electron-electron interactions are taken into account. We choose the parameters typical for the experimental 2DEG confined in ${\mathrm{Al}}_{0.1}{\mathrm{Ga}}_{0.9}\mathrm{As}$ quantum well, i.e., density of electrons and impurities $n_{2D}=n_i=2.0\times {10}^{12}{\mathrm{cm}}^{-2}$, $m=0.074{\mathrm{m}}_e$, and two sets of mobilities and relaxations times: $\overline{\mu }=0.1{\mathrm{m}}^2∕\mathrm{Vs}$, $\tau =4\times {10}^{-5}\mathrm{ns}$, ${\tau }_{\mathit{ss}}=0.02\mathrm{ns}$, and $\overline{\mu }=1{\mathrm{m}}^2∕\mathrm{Vs}$ $\tau =4\times {10}^{-4}\mathrm{ns}$, ${\tau }_{\mathit{ss}}=0.2\mathrm{ns}$. The effective spin-orbit coupling $\alpha \hslash =4.4{\mathrm{\AA }}^2$ in accordance with Ref. 36. We used the model potential (see Appendix) where an effective impurity radius $a=9.45\mathrm{nm}$, the height of attractive impurity potential $V_0=-5\mathrm{meV}$ for $\overline{\mu }=0.1{\mathrm{m}}^2∕\mathrm{Vs}$, and $V_0=-1.6\mathrm{meV}$ for $\overline{\mu }=1{\mathrm{m}}^2∕\mathrm{Vs}$.