Two-Dimensional Electron Transport in the Presence of the Rashba Effect

Patrick Bruno and Yonko Millev
Spin-orbit coupling is an intrinsic relativistic effect and as such it is truly ubiquitous, though usually small. Quantum mechanically, the spin-orbit interaction breaks symmetries, so that neither spin nor orbital momentum are conserved separately. The consequences of such symmetry breaking in the solid-state context have been studied group-theoretically and can be rather spectacular. Thus, in bulk crystalline semiconductors lacking inversion symmetry there arises an effective spin-orbit interaction which is of third order in the momentum of the conduction-band electrons. A similar, yet distinct effect arises in asymmetric two-dimensional quantum wells where it is known as the Rashba effect which is linear in momentum and whose amplitude  is proportional to the spatial average of the electric field across the well. In practice, high-quality semiconductor (III-V or II-VI) heterostructures have been designed where the electric potential landscape at the interface provides for a narrow quantum well where carriers are confined to move only within a plane parallel to the stacking of the heterostructure (Fig. 1). The relevant parameters of the 2d electron gas can be controlled simply and effectively by the gate voltage applied to the heterostructure. In addition, rather perfect structures can be grown nowadays. With the help of modulation doping, huge mobilities of the carriers of the order of tens of m²/Vs< are rather usual nowadays. Accordingly, the mean-free path in such systems is very large and of the order of micrometers.

The Rashba effect is of considerable interest in the context of the vigorously developing field of spin electronics (spintronics, spin electronics). Here, one hopes to make use of the spin of the conduction electrons because of the following attractive properties. The electron spins can store information in the form of spin polarization. This information can be transported. Finally, the state of spin polarization can be detected optically or electronically. Physically, the Rashba effect amounts to the action of an effective magnetic field $$B_eff on each of the moving carriers.$$B_eff lies within the plane of the $2d$ electron gas and is perpendicular to the instantaneous wavevector $$k. It lifts the spin degeneracy and gives rise to two energy dispersion branches (Fig. 2). Not less importantly, $$B_eff gives the proper quantization axis for the spin of the carriers and leads to a precession of the spin vector. This results in a spatial modulation of the net spin polarisation of the current which can be controlled by the gate voltage and, hence, implemented in spintronic devices. From the theoretical point of view, one encounters a novel situation even if only spin-independent scatterers are present in the system, because the two spin channels of conduction are intermixed as a result of the randomising effect of the potential scattering on the local quantization axis.

To study the effect of this new type of magnetic inhomogeneity, we put to action Kubo's linear response theory. A key issue is the understanding that one has to deal with spin accumulation in the course of the carriers' drift under the application of a small driving field. This is closely analogous to, but not so well understood as, the usual charge accumulation. Accordingly, one has to study the relevant spin-polarised conductivity as incorporating the linear response to the effective spin-dependent driving field.
This and related questions, as e.g. the examination of properly generalised Einstein relations for diffusivity and conductivity when both of them are spin-polarised due to the Rashba (alias: spin-orbit) effect, are currently examined within this project. It is part of the wholesome plan that the quantitative predictions and their interpretation be used to help design properly real spintronic devices or to indicate reliable ways of realization of some recently proposed devices which, at least for now, bear the touch of Gedanken experiments for lack of sufficient interpretational and quantitative basis.