Inverse spin Hall effect and anomalous Hall effect in a two-dimensional electron gas

We study the coupled dynamics of spin and charge currents in a two-dimensional electron gas in the transport diffusive regime. For systems with inversion symmetry there are established relations between the spin Hall effect, the anomalous Hall effect and the inverse spin Hall effect. However, in two-dimensional electron gases of semiconductors like GaAs, inversion symmetry is broken so that the standard arguments do not apply. We demonstrate that in the presence of a Rashba type of spin-orbit coupling (broken structural inversion symmetry) the anomalous Hall effect, the spin Hall and inverse spin Hall effect are substantially different effects. Furthermore, we discuss the inverse spin Hall effect for a two-dimensional electron gas with Rashba and Dresselhaus spin-orbit coupling; our results agree with a recent experiment. Copyright c © EPLA, 2010 Despite the anomalous and spin Hall effect being closely related, their histories are rather different. The anomalous Hall effect was experimentally discovered [1] almost at the same time as the ordinary Hall effect, while the spin Hall effect, first predicted in 1971 [2,3] and several times recently [4–7], has been experimentally seen only in the last few years [8–12]. This is not surprising as the anomalous Hall effect entails the measurement of currents and voltages which is well established experimentally, whereas the spin Hall effect requires the detection of a spin current, which has to be done in an indirect way; for a review see for example [13,14] and [15,16]. Since the anomalous and spin Hall effect have the same physical origin, namely the spin-orbit interaction which couples charge and spin degrees of freedom, their dependence on various physical parameters is expected to share similar trends. Depending on whether the spin-orbit coupling is intrinsic in the band structure or appears due to coupling to impurities one speaks about intrinsic or extrinsic mechanisms. The interplay of intrinsic and extrinsic mechanisms is non-trivial. For instance the intrinsic Rashba type of spin-orbit coupling in a two-dimensional electron gas suppresses drastically (a)E-mail: [email protected] the extrinsic (skew-scattering) contribution to the spin Hall conductivity [17–20]. It is the purpose of this paper to develop a similar analysis for the anomalous Hall effect and for the inverse spin Hall effect. We will start with a phenomenological discussion of charge and spin currents in a metal or semiconductor with diffusive charge carrier dynamics. We will study in detail the two-dimensional Rashba model including extrinsic skew-scattering. In the anomalous Hall conductivity we find an unexpected anomaly in the magnetic field dependence. Our analysis of the inverse spin Hall effect for a system with Rashba and Dresselhaus spin-orbit coupling is consistent with the experimental results of [21]. As a starting point we consider a system with spin-orbit coupling, where a spin-polarized current in the x-direction generates a small current δjy into the transverse direction with δjy↑ = 2γjx↑, δjy↓ =−2γjx↓. (1) Clearly, from these equations we can conclude that: a) a charge current generates a transverse spin current, δjy↑ − δjy↓ = 2γ(jx↑ + jx↓) (spin Hall effect), b) a spin current generates a transverse charge current, δjy↑ + δjy↓ = 2γ(jx↑ − jx↓). If the spin current is due to an electric field in a spin-polarized medium like a ferromagnet this is called the anomalous Hall effect. If it is instead due