Transverse spin-orbit force in the spin Hall effect in ballistic semiconductor wires

Abstract

We introduce the spin and momentum dependent force operator which is defined by the Hamiltonian of a clean semiconductor quantum wire with homogeneous Rashba spin-orbit (SO) coupling attached to two ideal (i.e., free of spin and charge interactions) leads. Its expectation value in the spin-polarized electronic wave packet injected through the leads explains why the center of the packet gets deflected in the transverse direction. Moreover, the corresponding spin density will be dragged along the transverse direction to generate an out-of-plane spin accumulation of opposite signs on the lateral edges of the wire, as expected in the phenomenology of the spin Hall effect, when spin-↑ and spin-↓ polarized packets (mimicking the injection of conventional unpolarized charge current) propagate simultaneously through the wire. We also demonstrate that spin coherence of the injected spin-polarized wave packet will gradually diminish (thereby diminishing the "force") along the SO coupled wire due to the entanglement of spin and orbital degrees of freedom of a single electron, even in the absence of any impurity scattering. The classical Hall effect 1 is one of the most widely known phenomena of condensed matter physics because it represents manifestation of the fundamental concepts of classical electrodynamics-such as the Lorentz forcein a complicated solid state environment. A perpendicular magnetic field B exerts the Lorentz force F = qv × B on current I flowing longitudinally through metallic or semiconductor wire, thereby separating charges in the transverse direction. The charges then accumulate on the lateral edges of the wire to produce a transverse "Hall voltage" in the direction qI × B. Thus, Hall-effect measurements reveal the nature of the current carriers.Recent optical detection 2,3 of the accumulation of spin-↑ and spin-↓ electrons on the opposite lateral edges of current carrying semiconductor wires opens new realm of the spin Hall effect. This phenomenon occurs in the absence of any external magnetic fields. Instead, it requires the presence of SO couplings, which are tiny relativistic corrections that can, nevertheless, be much stronger in semiconductors than in vacuum.4 Besides deepening our fundamental understanding of the role of SO couplings in solids, 4,5 the spin Hall effect offers new opportunities in the design of all-electrical semiconductor spintronic devices that do not require ferromagnetic elements or cumbersome-to-control external magnetic fields. 5While experimental detection of the strong signatures of the spin Hall effect brings to an end decades of theoretical speculation for its existence, it is still unclear what spin-dependent forces are responsible for the observed spin separation in different semiconductor systems. One potential mechanism-asymmetric scattering of spin-↑ and spin-↓ electrons off impurities with SO interactionwas invoked in the 1970s to predict the emergence of pure (i.e., not accompanied by charge transport) spin current, in the transverse direction to the flow of lon...

Imaging mesoscopic spin Hall flow: Spatial distribution of local spin currents and spin densities in and out of multiterminal spin-orbit coupled semiconductor nanostructures

“…However, this expectation value (i.e., the SO "force") oscillates along the sample due to the precession of the deflected spin in the effective Rashba magnetic field which is nearly parallel to the y-axis because of the transverse confinement effects. 52 In ballistic strongly coupled SO structures such α 2 -dependent SO "force", which oscillates on the mesoscale set by the spin precession length L SO = π 2 /2m * α = πt o a/2t SO (on which the spin precesses by an angle π, i.e., the state |↑ evolves into |↓ ), will lead to a change of the sign 15 of spin Hall current as a function of the system size L/L SO . Also, since the mesoscopic spin Hall effect sensitively depends on the measurement geometry, 15 the sign of the transverse spin Hall current can change when non-ideal leads 17 are attached to the sample.…”

Spin-dependent Klein tunneling in graphene: Role of Rashba spin-orbit coupling

“…5 exhibit opposite lateral preference of the ↑ and ↓ electron spins, which is an intrinsic spin-Hall mechanism due to the Rashba SOC. In a semiconductor two-dimensional electron gas (i.e., a continuous system rather than discrete as in the TBM), such an intrinsic spin-Hall deflection of opposite S z electrons can be easily explained by the concept of a spin-orbit force based on the Heisenberg equation of motion, 44,45 …”

Universal Spin-Hall Conductance Fluctuations in Two Dimensions

“…It has also been shown that any presence of weak disorder destroys this spin-Hall effect in the large sample limit 5,6 . On the other hand, numerical studies have provided evidence that for mesoscopic samples, spin-Hall conductance can survive weak disorder 7,8,9 .…”

“…It has also been shown that any presence of weak disorder destroys this spin-Hall effect in the large sample limit 5,6 . On the other hand, numerical studies have provided evidence that for mesoscopic samples, spin-Hall conductance can survive weak disorder 7,8,9 .One of the most striking quantum transport features in mesoscopic regime is the universal charge conductance fluctuation (UCF) 10,11,12 : quantum interference gives rise to the sample-to-sample fluctuation of charge conductance of order e 2 /h, independent of the details of the disorder, Fermi energy, and the sample size as long as transport is in the coherent diffusive regime characterized by the relation between relevant length scales, l < L < ξ. Here L is the linear sample size, l the elastic mean free path and ξ the phase coherence length.…”