The spin accumulation due to the spin current induced by the perpendicular temperature gradient (the spin Nernst effect) is studied in a two-dimensional electron system (2DES) with spin-orbit interaction by employing the Boltzmann equation. The considered 2DES is confined within a symmetric quantum well with δ doping at the center of the well. A symmetry consideration leads to the spin-orbit interaction which is diagonal in the spin component perpendicular to the 2DES. As origins of the spin current, the skew scattering and the side jump are considered at each impurity on the center plane of the well. It is shown that, for repulsive impurity potentials, the spin-Nernst coefficient changes its sign at the impurity density where contributions from the skew scattering and the side jump cancel each other out. This is in contrast to the spin Hall effect in which the sign change of the coefficient occurs for attractive impurity potentials.
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