Spin Hall and spin Nernst effects: temperature dependence

We have considered temperature dependence of spin Hall and spin Nernst effect in two-dimensional electron gas with spin-orbit interaction of Rashba type [arXiv:1510.03080]. In our considerations we have employed the approach based on the Matsubara Green functions. The formalism used in the case of electric field as a driving force was subsequently adopted to the situation of a spin current driven by a temperature gradient. To achieve this, we have used the concept of an auxiliary vector field. Such a description gives the possibility to consider all mechanisms leading to the spin Hall and spin Nernst effect on equal footing and also their behavior at finite temperatures. Both spin Hall and spin Nernst conductivities were calculated in the approximation including the vertex correction. The total spin Hall conductivity, including vertex correction, has been shown to vanish exactly in the whole temperature range. Thus, our results extend the earlier ones to an arbitrary temperatures. In turn, the total spin Nernst conductivity remains finite when the vertex corrections are included. Using the Ioffe-Regel localization criterion, we have also estimated the range of parameters where the calculated results for the spin Hall and spin Nernst conductivities are applicable.