Ab initio Calculation of the Intrinsic Spin Hall Effect in Semiconductors

Relativistic band theoretical calculations reveal that intrinsic spin Hall conductivity in hole-doped archetypical semiconductors Ge, GaAs, and AlAs is large [$\sim 100(\hslash /e)(\Omega \mathrm{cm}{)}^{-1}$], showing the possibility of a spin Hall effect beyond the four-band Luttinger Hamiltonian. The calculated orbital-angular-momentum (orbital) Hall conductivity is one order of magnitude smaller, indicating no cancellation between the spin and orbital Hall effects in bulk semiconductors. Furthermore, it is found that the spin Hall effect can be strongly manipulated by strains, and that the ac spin Hall conductivity is large in pure as well as doped semiconductors.

Figure 1

Relativistic band structure of GaAs: (a) Energy bands and (b) density of states (DOS) ($\mathrm{\text{states}}/\mathrm{eV}/\mathrm{\text{cell}}$). The zero energy is at the top of the valence bands. The dotted lines just below the top of the valence bands denote the Fermi levels for the hole concentration $n_h=0.025$, 0.05, 0.1, and $0.2e/\mathrm{\text{cell}}$, respectively.

Figure 2

Calculated (a) spin (${\sigma }_{xy}^{sH}$) and (b) orbital (${\sigma }_{xy}^{oH}$) Hall conductivity as a function of hole concentration ($n_h$) for Si, Ge, GaAs, and AlAs. (c) Calculated spin Hall conductivity for $n_h=0.05e/\mathrm{\text{cell}}$ vs the top valence band spin-orbit splitting (${\Delta }_{\mathrm{so}}$). The dashed line denotes the parabola fitted to the calculated data. The $x$ is the spin-orbit coupling scaling factor [25]: $x=1$ means full spin-orbit coupling and $x=0$ means no spin-orbit coupling. $n_h=0.1e/\mathrm{\text{cell}}$ is equivalent to $n_h$ of $2.5\times {10}^{21}$, $2.2\times {10}^{21}$, $2.2\times {10}^{21}$, and $2.3\times {10}^{21}{\mathrm{cm}}^{-3}$ for Si, Ge, GaAs, and AlAs, respectively.

Figure 3

Calculated spin Hall conductivity (${\sigma }_{xy}^{sH}$) (squares) and energy splitting (diamonds) of heavy-hole and light-hole bands at $\Gamma$ of GaAs as a function of uniaxial strain at the hole concentration $n_h=0.1(e/\mathrm{\text{cell}})$.

Figure 4

Calculated spin (${\sigma }_{xy}^{sH}$) and orbital (${\sigma }_{xy}^{oH}$) Hall conductivity of GaAs as a function of frequency at the hole concentrations $n_h=0.0$, $0.1(e/\mathrm{\text{cell}})$, respectively.