  We study the Nernst effect and the spin Nernst effect, that a longitudinal
thermal gradient induces a transverse voltage and a spin current. A mesoscopic
four-terminal cross-bar device having the Rashba spin-orbit interaction (SOI)
under a perpendicular magnetic field is considered. For zero SOI, the Nernst
coefficient peaks when the Fermi level crosses the Landau Levels. In the
presence of the SOI, the Nernst peaks split, and the spin Nernst effect appears
and exhibits a series of oscillatory structures. The larger SOI is or the
weaker magnetic field is, the more pronounced the spin Nernst effect is. The
results also show that the Nernst and spin Nernst coefficients are sensitive to
the detailed characteristics of the sample and the contacts. In addition, the
Nernst effect is found to survive in strong disorder than the spin Nernst
effect does.
