Extrinsically vs. intrinsically driven spin Hall effect in disordered mesoscopic multiterminal bars

We show that pure spin Hall current, flowing out of a four-terminal two-dimensional electron gas (2DEG) within inversion asymmetric heterostructure, contains contributions from both the extrinsic mechanisms (spin-orbit-dependent scattering off impurities) and the intrinsic ones (due to the Rashba coupling). The extrinsic contribution vanishes in the weakly and strongly disordered limits, and the intrinsic one dominates in the quasiballistic limit. However, in the crossover transport regime the spin Hall conductance is not simply reducible to either of the two mechanisms, which can be relevant for interpretation of experiments on dirty 2DEGs (Sih V. et al., Nature Phys., 1 (2005) 31). We also predict sample-to-sample mesoscopic fluctuations, and even sign change, of both the spin Hall conductance and the applied voltages to transverse terminals (ensuring that spin Hall current outflowing through them is pure with no accompanied net charge current) in phase-coherent bars. Copyright c © EPLA, 2007 Introduction. – Recent experimental discovery of the spin Hall effect in three-dimensional n-type semiconductors [1] and two-dimensional hole [2] or electron gases [3] provides deep insight into the relativistic effects in solids which manifest through the spin-orbit (SO) couplings [4]. The flow of conventional unpolarized charge current in the longitudinal direction through systems governed by SO interactions can lead to spin flux in the transverse direction which deposits non-equilibrium spin Hall accumulation on the lateral sample edges, as observed in experiments [1–3]. Moreover, anticipated experiments should demonstrate flow of spin into the transverse electrodes attached at those edges [5], thereby offering a semiconductor analog of the Stern-Gerlach device as a source of spin currents or flying spin qubits where spatial separation of spin-↑ and spin-↓ electrons does not require any external magnetic fields. However, the effect observed in electron systems [1,3] is rather small, and its magnitude cannot be tuned easily since it is determined by material parameters which govern the SO-dependent scattering off impurities [6–8] deflecting a beam of spin-↑ (spin-↓) electrons predominantly to the right (left). In contrast to such extrinsically (i.e., impurity) induced spin Hall effect, recent theories have argued for several orders of magnitude larger spin Hall currents due to the intrinsic SO couplings capable of spin-splitting the energy bands and inducing transverse SO forces in bulk semiconductors in the clean limit [9,10], as well as in ballistic mesoscopic multiterminal nanostructures made of such materials [5,11]. Such mechanisms were invoked to explain [12] observation of two orders of magnitude larger spin Hall accumulation in 2D hole gases [2], whose magnitude can still be affected by the effects of the disorder [13]. Moreover, in infinite 2DEGs with linear in momentum SO coupling any non-vanishing spin-independent disorder completely suppresses the spin Hall effect [14,15], while in open finite-size 2DEG structures attached to external current and voltage probes increasing disorder can only gradually diminish its signatures [11,15,16]. Despite these advances, the intrinsic vs. extrinsic debate on the origin of the detected signatures of the spin Hall effect persists [17,18], closely mimicking several decades old discourse [19] on the Karplus-Luttinger intrinsic (due to anomalous velocity, interpreted in modern language via the Berry phase associated with electronic band structure) vs. extrinsic (i.e., skew-scattering+ sidejump) explanations of the anomalous Hall effect in ferromagnetic systems. Even though much of this controversy might be resolved through a unified theoretical treatment of different mechanisms [19], surprisingly enough, no theory has been constructed to describe spin Hall currents in the transition from quasiballistic to dirty transport