Corrections to the extrinsic spin Hall conductivity of bulk n-doped GaAs: electron-electron interactions, multiple-impurity scattering, and impurity compensation

The extrinsic theory of the spin Hall effect developed by Engel, Halperin, and Rashba for bulk n-doped GaAs (Phys. Rev. Lett. 95, 166605 (2005)) predicts, without adjustable parameters, a dc spin Hall conductivity of magnitude comparable to the Kerr-rotation experiment of Kato et al. (edge spin accumulation detected through the rotation of reflected polarized light), but of opposite sign, as the difference of two comparable opposite-sign terms (skew scattering and side jump) computed in a single-particle Boltzmann framework valid only for $k_{\mathrm{F}}\ell\gg 1$ while the experiment has $k_{\mathrm{F}}\ell\approx 2$. The open problem is to establish quantitatively, for degenerate bulk n-GaAs, how (i) dynamical electron-electron interactions, (ii) multiple-impurity scattering and vertex corrections beyond the lowest-order Boltzmann/ladder treatment, and (iii) impurity compensation by coexisting attractive and repulsive charged impurity centers modify the transport skewness, the side-jump term, and the net extrinsic spin Hall conductivity including its sign, and to fix experimentally the absolute sign of the dc spin Hall conductivity so that extrinsic and intrinsic contributions can be definitively separated.

Background

The spin Hall effect (SHE) is the generation of a transverse spin current, and of spin accumulation near the sample edges, by an ordinary charge current flowing through a material with spin-orbit (SO) coupling; it was proposed by Dyakonov and Perel and by Hirsch as an electrical source of spin currents, and was first observed in semiconductors in 2004 by Kato et al., who mapped the edge spin accumulation in n-doped GaAs by scanning Kerr-rotation microscopy (measuring the local spin polarization through the rotation of the polarization axis of light reflected from the sample). Two broad microscopic mechanisms are distinguished. Extrinsic mechanisms arise from SO coupling in the electron-impurity scattering itself: Mott skew scattering, in which the spin-dependent part of the scattering cross section deflects spin-up and spin-down electrons asymmetrically, and the side-jump mechanism, in which the scattered wave packet is laterally displaced during each collision (a lateral displacement of order the SO length, which accumulates over the scattering rate). Intrinsic mechanisms arise instead from the band structure of the perfect crystal, where Berry curvature in momentum space gives anomalous transverse velocities (the clean-limit analog of the anomalous Hall effect (AHE), the Hall-like response of spin-polarized carriers without an external magnetic field). All of these questions inherit the long-studied subtleties of AHE theory: spin is not conserved in the presence of SO coupling, the definition of spin current is disputed, and the decomposition into mechanisms depends on the disorder approximation used.

For conduction-band electrons in a direct-gap cubic semiconductor such as GaAs, band mixing with p-type valence bands (8x8 Kane model) produces an effective SO term in the impurity potential, $\lambda\boldsymbol{\sigma}\cdot(\mathbf{k}\times\nabla V)$, with $\lambda = 5.3\,\mathrm{\AA}^2$ for GaAs — six orders of magnitude larger than the vacuum (Thomas) value. Engel, Halperin, and Rashba solved a spin-dependent Boltzmann equation with this SO impurity term for a 3D electron gas (the impurity potential treated in the second Born approximation, i.e. to second order in perturbation theory, and the current vertex dressed only by the standard ladder diagrams), obtaining two contributions: a skew-scattering term proportional to the longitudinal conductivity through a transport skewness $\gamma$ (a weighted angular average of the Sherman function, which measures the polarization produced by scattering), and a $\tau$-independent side-jump term. For a screened attractive Coulomb impurity model with the density $n = 3\times 10^{16}\,\mathrm{cm}^{-3}$ of the Kato et al. experiment, they found $\gamma\approx 1/900$, $\sigma^{\mathrm{SS}}_{\mathrm{SH}}\approx +1.7$ and $\sigma^{\mathrm{SJ}}_{\mathrm{SH}}\approx -0.8\,\Omega^{-1}\mathrm{m}^{-1}/|e|$, a net extrinsic $\sigma^{\mathrm{SH}}\approx 0.9\,\Omega^{-1}\mathrm{m}^{-1}/|e|$ against a measured magnitude $|\sigma^{\mathrm{SH}}_{\mathrm{expt}}|\approx 0.5\,\Omega^{-1}\mathrm{m}^{-1}/|e|$, while the intrinsic contribution from the Dresselhaus spin splitting (the spin-orbit splitting of the conduction band that exists in bulk GaAs because its zincblende crystal lacks inversion symmetry, whose part cubic in momentum drives a clean-limit spin current) was estimated two orders of magnitude smaller. The agreement in magnitude supports an extrinsic origin of the observed SHE, but the authors flagged essential caveats: the sign of their result is opposite to that quoted in the experiment while the absolute experimental sign was itself uncertain; the net value is the small difference of two larger opposite-sign terms, so the sign is sensitive to effects omitted from the theory; the Boltzmann expansion parameter $1/(k_{\mathrm{F}}\ell)$ ($k_{\mathrm{F}}$ the Fermi wave vector, $\ell$ the mean free path) is not small ($k_{\mathrm{F}}\ell\approx 2$, about 4 when extracted from the measured conductivity); and repulsive impurity centers (residual acceptors, i.e. compensation in an n-type sample) reverse the sign of $\gamma$ and can partially cancel the effect. The calculation is a single-particle one, so dynamical electron-electron interactions and multiple-scattering and vertex corrections beyond the lowest-order treatment are absent by construction.

Problem Statement

For bulk, degenerate (Fermi energy well above $k_{\mathrm{B}}T$) n-doped GaAs in the regime of the original spin Hall measurements (electron density of order $3\times 10^{16}\,\mathrm{cm}^{-3}$, $k_{\mathrm{F}}\ell\approx 2$–4), establish quantitatively how (i) dynamical electron-electron interactions, (ii) multiple-impurity scattering and vertex corrections beyond the lowest-order treatment used by Engel, Halperin, and Rashba (Boltzmann equation with the impurity potential in the second Born approximation and the current vertex dressed only by ladder diagrams), and (iii) impurity compensation — coexistence of attractive (donor) and repulsive (acceptor-like) charged scattering centers, which individually produce skewnesses $\gamma$ of opposite sign — modify the transport skewness $\gamma$, the side-jump contribution, and the net extrinsic spin Hall conductivity $\sigma^{\mathrm{SH}}$, including the sign of each contribution and of the total. In parallel, determine experimentally the absolute sign of the dc spin Hall conductivity in this degenerate regime — resolving the discrepancy between the calculated sign of the extrinsic $\sigma^{\mathrm{SH}}\approx 0.9\,\Omega^{-1}\mathrm{m}^{-1}/|e|$ (from skew-scattering $\approx +1.7$ and side-jump $\approx -0.8\,\Omega^{-1}\mathrm{m}^{-1}/|e|$ terms) and the measured sign — and test the predictions on samples with larger $k_{\mathrm{F}}\ell$ or systematically varied impurity types, including compensated samples, so that the extrinsic and intrinsic (Dresselhaus/Berry-curvature) contributions to the spin Hall effect in this system can be definitively separated. The three analytic items (i)–(iii) and the two experimental items (determining the absolute sign of the dc spin Hall conductivity, and testing the predictions on samples with larger $k_{\mathrm{F}}\ell$ or systematically varied impurity types) are posed jointly, each addressing a distinct sub-question.

The analytic target is a controlled quantitative calculation that states its approximation scheme and a quantified accuracy sufficient to fix the sign of the net extrinsic $\sigma^{\mathrm{SH}}$ and to compare its magnitude with the measured value; the net $\sigma^{\mathrm{SH}}$ is the combination of the skew-scattering and side-jump contributions. The two experimental items do not require newly fabricated samples and can be addressed with existing samples and measurements.

References

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