Side-jumps in the spin-Hall effect: construction of the Boltzmann collision integral

We present a systematic derivation of the side-jump contribution to the spin-Hall current in systems without band structure spin-orbit interactions, focusing on the construction of the collision integral for the Boltzmann equation. Starting from the quantum Liouville equation for the density operator we derive an equation describing the dynamics of the density matrix in the first Born approximation and to first order in the driving electric field. Elastic scattering requires conservation of the total energy, including the spin-orbit interaction energy with the electric field: this results in a first correction to the customary collision integral found in the Born approximation. A second correction is due to the change in the carrier position during collisions. It stems from the part of the density matrix off-diagonal in wave vector. The two corrections to the collision integral add up and are responsible for the total side-jump contribution to the spin-Hall current. The spin-orbit-induced correction to the velocity operator also contains terms diagonal and off-diagonal in momentum space, which together involve the total force acting on the system. This force is explicitly shown to vanish (on the average) in the steady state: thus the total contribution to the spin-Hall current due to the additional terms in the velocity operator is zero.Comment: Added references, expanded discussion, revised introductio

Experimental observation of the spin-Hall effect in a two dimensional spin-orbit coupled semiconductor system

We report the experimental observation of the spin-Hall effect in a two-dimensional (2D) hole system with Rashba spin-orbit coupling. The 2D hole layer is a part of a p-n junction light-emitting diode with a specially designed co-planar geometry which allows an angle-resolved polarization detection at opposite edges of the 2D hole system. In equilibrium the angular momenta of the Rashba split heavy hole states lie in the plane of the 2D layer. When an electric field is applied across the hole channel a non zero out-of-plane component of the angular momentum is detected whose sign depends on the sign of the electric field and is opposite for the two edges. Microscopic quantum transport calculations show only a weak effect of disorder suggesting that the clean limit spin-Hall conductance description (intrinsic spin-Hall effect) might apply to our system.Comment: 4 pages, 3 figures, paper based on work presented at the Gordon Research Conference on Magnetic Nano-structures (August 2004) and Oxford Kobe Seminar on Spintronics (September 2004); accepted for publication in Physical Review Letters December 200

Anomalous Rashba spin splitting in two-dimensional hole systems

It has long been assumed that the inversion asymmetry-induced Rashba spin splitting in two-dimensional (2D) systems at zero magnetic field is proportional to the electric field that characterizes the inversion asymmetry of the confining potential. Here we demonstrate, both theoretically and experimentally, that 2D heavy hole systems in accumulation layer-like single heterostructures show the opposite behavior, i.e., a decreasing, but nonzero electric field results in an increasing Rashba coefficient.Comment: 4 pages, 3 figure

Anomalous Spin Polarization of GaAs Two-Dimensional Hole Systems

We report measurements and calculations of the spin-subband depopulation, induced by a parallel magnetic field, of dilute GaAs two-dimensional (2D) hole systems. The results reveal that the shape of the confining potential dramatically affects the values of in-plane magnetic field at which the upper spin subband is depopulated. Most surprisingly, unlike 2D electron systems, the carrier-carrier interaction in 2D hole systems does not significantly enhance the spin susceptibility. We interpret our findings using a multipole expansion of the spin density matrix, and suggest that the suppression of the enhancement is related to the holes' band structure and effective spin j=3/2.Comment: 6 pages, 4 figures, substantially extended discussion of result

Invariant expansion for the trigonal band structure of graphene

We present a symmetry analysis of the trigonal band structure in graphene, elucidating the transformational properties of the underlying basis functions and the crucial role of time-reversal invariance. Group theory is used to derive an invariant expansion of the Hamiltonian for electron states near the K points of the graphene Brillouin zone. Besides yielding the characteristic k-linear dispersion and higher-order corrections to it, this approach enables the systematic incorporation of all terms arising from external electric and magnetic fields, strain, and spin-orbit coupling up to any desired order. Several new contributions are found, in addition to reproducing results obtained previously within tight-binding calculations. Physical ramifications of these new terms are discussed.Comment: 10 pages, 1 figure; expanded version with more details and additional result

Anomalous magneto-oscillations in two-dimensional systems

The frequencies of Shubnikov-de Haas oscillations have long been used to measure the unequal population of spin-split two-dimensional subbands in inversion asymmetric systems. We report self-consistent numerical calculations and experimental results which indicate that these oscillations are not simply related to the zero-magnetic-field spin-subband densities.Comment: 4 pages, 3 figures; changed content (clarifications

The Effect of Spin Splitting on the Metallic Behavior of a Two-Dimensional System

Experiments on a constant-density two-dimensional hole system in a GaAs quantum well reveal that the metallic behavior observed in the zero-magnetic-field temperature dependence of the resistivity depends on the symmetry of the confinement potential and the resulting spin-splitting of the valence band

Lateral spin-orbit interaction and spin polarization in quantum point contacts

We study ballistic transport through semiconductor quantum point contact systems under different confinement geometries and applied fields. In particular, we investigate how the {\em lateral} spin-orbit coupling, introduced by asymmetric lateral confinement potentials, affects the spin polarization of the current. We find that even in the absence of external magnetic fields, a variable {\em non-zero spin polarization} can be obtained by controlling the asymmetric shape of the confinement potential. These results suggest a new approach to produce spin polarized electron sources and we study the dependence of this phenomenon on structural parameters and applied magnetic fields. This asymmetry-induced polarization provides also a plausible explanation of our recent observations of a 0.5 conductance plateau (in units of $2e^2/h$) in quantum point contacts made on InAs quantum-well structures. Although our estimates of the required spin-orbit interaction strength in these systems do not support this explanation, they likely play a role in the effects enhanced by electron-electron interactions.Comment: Summited to PRB (2009