Spin Hall Drag in Electronic Bilayers

We predict a new effect in electronic bilayers: spin Hall drag. The effect consists of the generation of spin accumulation across one layer by an electric current along the other layer. It arises from the combined action of spin-orbit and Coulomb interactions. Our theoretical analysis, based on the Boltzmann equation formalism, identifies two main contributions to the spin Hall drag resistivity: the side-jump contribution, which dominates at low temperature, going as $T^2$, and the skew-scattering contribution, which is proportional to $T^3$. The induced spin accumulation, while generally quite small, should be observable in optical rotation experiments.

Figure 1

(a) In ordinary Coulomb drag the current $j_{1x}$ in layer 1 induces, via interlayer Coulomb interaction, an electrochemical potential gradient $E_{2x}$ in layer 2. (b) In the spin Hall effect the current $j_x$ in a single layer induces, via spin-orbit interaction, a spin-dependent electrochemical potential gradient $E_y(\uparrow )=-E_y(\downarrow )$ causing electrons of opposite spin orientation to accumulate on opposite edges. (c) In Spin Hall drag the current $j_{1x}$ in layer 1 induces, via a combination of interlayer Coulomb interaction and spin-orbit interaction, electrochemical potential gradients $E_{2x}$ along layer 2, and $E_{2y}(\sigma )$ across layer 2.

Figure 2

Side-jump contribution to the spin Hall drag resistivity vs temperature $T/T_F$. The calculation includes dynamical screening, static exchange-correlation, and quantum well width effects along the lines of Ref. [7]. The width of the quantum well is 18 nm and the distance between the centers of the wells is 28 nm. The solid, dashed, and dotted lines correspond to the electron sheet densities of $18\times {10}^{14}{\mathrm{m}}^{-2}$, $3.8\times {10}^{14}{\mathrm{m}}^{-2}$ and $2\times {10}^{14}{\mathrm{m}}^{-2}$. The inset shows the ratio of the skew-scattering resistivity, evaluated from Eq. (14) with $W^{ee,a}$, to the side-jump resistivity. The value of $W^{ee,a}$ is chosen so that this ratio is 1 at $T=T_F$. The linear increase at low temperatures illustrates the $T^3$ behavior of skew-scattering resistivity, in contrast to the usual $T^2$ dependence of side-jump drag.