  Spin Hall effects are a collection of phenomena, resulting from spin-orbit
coupling, in which an electrical current flowing through a sample can lead to
spin transport in a perpendicular direction and spin accumulation at lateral
boundaries. These effects, which do not require an applied magnetic field, can
originate in a variety of intrinsic and extrinsic spin-orbit coupling
mechanisms and depend on geometry, dimension, impurity scattering, and carrier
density of the system--making the analysis of these effects a diverse field of
research. In this article, we give an overview of the theoretical background of
the spin Hall effects and summarize some of the most important results. First,
we explain effective spin-orbit Hamiltonians, how they arise from band
structure, and how they can be understood from symmetry considerations;
including intrinsic coupling due to bulk inversion or structure asymmetry or
due to strain, and extrinsic coupling due to impurities. This leads to
different mechanisms of spin transport: spin precession, skew scattering, and
side jump. Then we present the kinetic (Boltzmann) equations, which describe
the spin-dependent distribution function of charge carriers, and the diffusion
equation for spin polarization density. Next, we define the notion of spin
currents and discuss their relation to spin polarization. Finally, we explain
the electrically induced spin effects; namely, spin polarization and currents
in bulk and near boundaries (the focus of most current theoretical research
efforts), and spin injection, as well as effects in mesoscopic systems and in
edge states.
