Introduction 1 .1 Spin-Orbit Coupling in Solid-State Physics 1 .2 Spin-Orbit Coupling in Quasi-Two-Dimensional Systems. .. . 1 .3 Overview References 2 Band Structure of Semiconductors 2 .1 Bulk Band Structure and k • p Method 2 .2 The Envelope Function Approximation 2 .3 Band Structure in the Presence of Strain 2 .4 The Paramagnetic Interaction in Semimagnetic Semiconductors 2 .5 Theory of Invariants References 3 The Extended Kane Model 21 3 .1 General Symmetry Considerations 21 3 .2 Invariant Decomposition for the Point Group Td 3 .3 Invariant Expansion for the Extended Kane Model 23 3 .4 The Spin-Orbit Gap Ao 26 3 .5 Kane Model and Luttinger Hamiltonian 3 .6 Symmetry Hierarchies 29 References 33 4 Electron and Hole States in Quasi-Two-Dimensional Systems 35 4 .1 The Envelope Function Approximation for Quasi-Two-Dimensional Systems 35 4 .1 .1 Envelope Functions 36 4 .1,2 Boundary Conditions 36 4 .1 .3 Unphysical Solutions 37 4 .1 .4 General Solution of the EFA Hamiltonian Based an a Quadrature Method 39 4 .1 .5 Electron and Hole States for Different Crystallographic Growth Directions 41 4 .2 Density of States of a Two-Dimensional System 41

In this review, we describe and analyze a mesoscale simulation method for fluid flow, which was introduced by Malevanets and Kapral in 1999, and is now called multi-particle collision dynamics (MPC) or stochastic rotation dynamics (SRD). The method consists of alternating streaming and collision steps in an ensemble of point particles. The multi-particle collisions are performed by grouping particles in collision cells, and mass, momentum, and energy are locally conserved. This simulation technique captures both full hydrodynamic interactions and thermal fluctuations. The first part of the review begins with a description of several widely used MPC algorithms and then discusses important features of the original SRD algorithm and frequently used variations. Two complementary approaches for deriving the hydrodynamic equations and evaluating the transport coefficients are reviewed. It is then shown how MPC algorithms can be generalized to model non-ideal fluids, and binary mixtures with a consolute point. The importance of angular-momentum conservation for systems like phase-separated liquids with different viscosities is discussed. The second part of the review describes a number of recent applications of MPC algorithms to study colloid and polymer dynamics, the behavior of vesicles and cells in hydrodynamic flows, and the dynamics of viscoelastic fluids.

In two-dimensional (2D) hole systems the inversion asymmetry induced spin splitting differs remarkably from its familiar counterpart in the conduction band. While the so-called Rashba spin splitting of electron states increases linearly with in-plane wave vector k_|| the spin splitting of heavy hole states can be of third order in k_|| so that spin splitting becomes negligible in the limit of small 2D hole densities. We discuss consequences of this behavior in the context of recent arguments on the origin of the metal-insulator transition observed in 2D systems.Comment: 4 pages, 2 figure

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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

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The Rashba effect, discovered in 1959, continues to supply fertile ground for fundamental research and applications. It provided the basis for the proposal of the spin transistor by Datta and Das in 1990, which has largely inspired the broad and dynamic field of spintronics. More recent developments include new materials for the Rashba effect such as metal surfaces, interfaces and bulk materials. It has also given rise to new phenomena such as spin currents and the spin Hall effect, including its quantized version, which has led to the very active field of topological insulators. The Rashba effect plays a crucial role in yet more exotic fields of physics such as the search for Majorana fermions at semiconductorsuperconductor interfaces and the interaction of ultracold atomic Bose and Fermi gases. Advances in our understanding of Rashba-type spin-orbit couplings, both qualitatively and quantitatively, can be obtained in many different ways. This focus issue brings together the wide range of research activities on Rashba physics to further promote the development of our physical pictures and concepts in this field. The present Editorial gives a brief account on the history of the Rashba effect including material that was previously not easily accessible before summarizing the key results of the present focus issue as a guidance to the reader.

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.For many years, it was widely accepted that there can be no metallic phase in a disordered two-dimensional (2D) carrier system. This was due to the scaling arguments of Abrahams et al. [1], and the support of subsequent experiments [2]. In the last few years, however, experiments on high quality 2D systems have provided us with reason to re-visit the question of whether or not a metallic phase in a 2D system can exist [3][4][5][6][7][8][9]. Early temperature dependence data by Kravchenko et al., from high-mobility silicon metal-oxide-semiconductor field effect transistors (MOSFETs), showed a drop in resistivity as the temperature (T ) was reduced below 2 K. This metallic behavior is the opposite of the expected insulating behavior in which the resistivity should become infinite as T approaches zero. In addition, the behavior not only was metallic in a certain electron density range, but also scaled with a single parameter as the density was reduced and the sample became insulating, suggesting a true metal-insulator phase transition [3].Since these experiments, the metallic behavior has been observed in Si MOSFETS [3,4], SiGe quantum wells [5,6], GaAs/AlGaAs heterostructures [7,8], and AlAs quantum wells [9], demonstrating that there are still some unsolved puzzles in the fundamental nature of 2D carrier systems. Multiple mechanisms including electronelectron interaction [10], spin-splitting [11], and temperature dependence of traps [12] have been proposed as causes for the metallic behavior, but no clear model has emerged which fully describes this sizeable body of experimental data. The experiments reported here add to our understanding by demonstrating a correlation between the zero-magnetic-field spin-splitting and the metallic behavior.Spin-splitting of carriers in a 2D system at zero magnetic field is caused by the spin-orbit interaction and by an inversion asymmetry of the potential in which the carriers move [13]. The energy bands are split into two spinsubbands, which have different populations because their energies at any non-zero k are slightly different. The existence of these spin-subbands has been well established both experimentally and theoretically [13][14][15][16][17][18].Our experiments are performed on high-mobility 2D hole systems in GaAs quantum wells (QWs), chosen because they have a large inter-carrier separation r s [19], have already shown metallic behavior [7,8], and exhibit a large and tunable spin-splitting [17]. In GaAs, the spin-splitting arises from the inversion asymmetries of the zincblende crystal structure and of the potential used to confine the electrons to two dimensions. The asymmetry of the crystal structure is fixed, but the asymmetry of the confining potential, and therefo...