We propose a mechanism of spin Hall effect in two-dimensional electron gas with spatially random Rashba spin-orbit interaction. The calculations based on the Kubo formalism and kinetic equation show that in contrast to the constant spin-orbit coupling, spin Hall conductivity in the random spin-orbit field is not totally suppressed by the potential impurity scattering. Therefore, the intrinsic spin Hall effect exists being, however, nonuniversal.

We calculate the efficiency of infrared optical spin injection in single-layer graphene with Rashba spin-orbit coupling and for the in-plane magnetic field. The injection rate in the photon frequency range corresponding to the Rashba splitting is shown to be proportional to the ratio of the Zeeman and Rashba splittings. As a result, a large spin polarization can be controllably achieved for experimentally available values of the spin-orbit coupling and in magnetic fields below 10 T, without using ferromagnetic contacts.

We analyze theoretically the optical generation of a spin-polarized charge current (photogalvanic effect) and spin polarization in graphene with Rashba spin-orbit coupling. An external magnetic field is applied in the graphene plane, which plays a crucial role in the mechanism of current generation. We predict a highly efficient resonant photogalvanic effect in a narrow frequency range that is determined by the magnetic field. A relatively less efficient photogalvanic effect appears in a broader frequency range, determined by the electron concentration and spin-orbit coupling strength.

Thermoelectric effects in graphene are considered theoretically with particular attention paid to the role of resonant scattering on impurities. Using the $T$-matrix method we calculate the impurity resonant states and the momentum relaxation time due to scattering on impurities. The Boltzmann kinetic equation is used to determine the thermoelectric coefficients. It is shown that the resonant impurity states near the Fermi level give rise to a resonant enhancement of the Seebeck coefficient and figure of merit $ZT$. The Wiedemann-Franz ratio deviates from that known for ordinary metals, where this ratio is constant and equal to the Lorentz number. This deviation appears for small chemical potentials and in the vicinity of the resonant states. In the limit of a constant relaxation time, this ratio has been calculated analytically for $\mu =0$.

We predict two spin-dependent transport phenomena in two-dimensional electron systems, which are induced by a spatially fluctuating Rashba spin-orbit interaction. When the electron gas is magnetized, the random Rashba interaction leads to the anomalous Hall effect. An example of such a system is a narrow-gap magnetic semiconductor-based symmetric quantum well. We show that the anomalous Hall conductivity reveals a strongly nonlinear dependence on the magnetization, decreasing exponentially at large spin density. We also show that electron scattering from a fluctuating Rashba field in a two-dimensional nonmagnetic electron system leads to a negative magnetoresistance arising solely due to spin-dependent effects.

We consider theoretically a wide graphene ribbon that is attached on both ends to electronic reservoirs, which generally have different temperatures. The graphene ribbon is assumed to be deposited on a substrate that leads to a spin-orbit coupling of the Rashba type. We calculate the thermally induced charge current in the ballistic transport regime as well as the thermoelectric voltage (Seebeck effect). Apart from this, we also consider thermally induced spin current and spin polarization of the graphene ribbon. The spin currents are shown to have generally two components; one parallel to the temperature gradient and the other perpendicular to this gradient. The latter corresponds to the spin current due to the spin Nernst effect. Additionally, we also consider the heat current between the reservoirs due to transfer of electrons.