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jumps at a step. Around an extreme value, any func- tion is almost symmetry, thus one only can get a low value of N$_{s}$ . While at both sides of the extreme value, Δ T$_{2}$$_{,}$$_{3}$ monotonically increases or decreases, we can get a local maximum magnitude of N$_{s}$ . Now why Δ T$_{2}$$_{,}$$_{3}$ has an extreme value at a peak of S for a small V$_{R}$ . Due to the Rashba SOI, each eigen-energy band splits into two sub-bands with opposite spin directions. These two sub-bands degenerate at k$_{y}$ = 0, and the lower sub-band has two valleys below this degenerate point. The two sub-bands are very close to each other when V$_{R}$ is small. If the lower sub-band of high level (for example, E$_{1}$$_{,k}$$_{y}$ ) has the similar spin direction with the upper sub-band of low level energy (for example, E$_{0}$$_{,k}$$_{y}$ ), Δ T$_{2}$$_{,}$$_{3}$ continually increases when E$_{F}$ goes from the upper band of E$_{0}$$_{,k}$$_{y}$ to the two valleys of lower sub-band of E$_{1}$$_{,k}$$_{y}$ , and than rapidly decreases when E$_{F}$ goes through the degenerate point ( k$_{y}$ = 0) of E$_{1}$$_{,k}$$_{y}$ , thus we get a peak in Δ T$_{2}$$_{,}$$_{3}$ ; otherwise we get a valley in Δ T$_{2}$$_{,}$$_{3}$ . 

When V$_{R}$ is very big, we can see the external peaks for both N$_{s}$ and S . For example V$_{R}$ = 0 . 45 in Fig.2(d), close to the 2 nd and 3 rd main peaks of S , we can see a very sharp sub-peak of N$_{s}$ . In fact, these are also the small peaks of S , though not very big. This is because for these two band (see Fig.3(c)), the two valleys of the lower sub-band is far from the degenerate point at k$_{y}$ = 0, the two channel of these two sub-bands is separated. Thus we can see two peaks. At this time, the change of spin transmission coefficients can be roughly thought as the change of transmission coefficients, thus we can see N$_{s}$ peaks at the S ’s peak. 

In Fig.4, we show the spin Nernst coefficient N$_{s}$ and 

the seebeck coefficient S versus the Rashba SOI V$_{R}$ in the clean system ( W = 0) for E$_{F}$ = − 3. The seebeck coefficient S decreases and maintains for a small value for quite a while before shows another peak. This is because increasing V$_{R}$ moves the energy bands and makes them go through the fermi energy. It should be mentioned that we found the negative seebeck coefficient S (Fig. 2d), which means a longitudinal current occurs in the opposite direction of the temperature gradient Δ T . This is due to the boundary conditions V$_{R}$ = 0 at leads-2,4. As a compare, we also show S for a uniform system, i.e. leads-2,4 having the same strength of V$_{R}$ as in the sample. For this situation, the seebeck coefficient S is no longer negative. In fact, when the Rashba SOI is absent in the leads-2,4, an interface between V$_{R}$ = 0 and V$_{R}$ ̸ = 0 ocurrs $^{9}$, this interface causes additional scattering for an incident electron. In some special case like E$_{F}$ = − 3, this may make the electrons below E$_{F}$ easier to transport than the electrons above E$_{F}$ , thus a negative S . 

Finally we discuss the disorder effect on the spin Nernst effect. Fig.5 shows N$_{s}$ versus disorder strength W for dif- ferent Fermi energies. The calculations are averaged over 500 disorder configurations. Around W < 1 . 7, N$_{s}$ shows an oscillatory structure. N$_{s}$ changes sign with increas- ing of the disorder strength (see E$_{F}$ = − 3 . 6 and − 3 . 0 in Fig.5). It is interesting to see that, comparing to a clean system ( W = 0), N$_{s}$ can be unexpectedly increased by disorder W . This is because the disorder changes the os- cillating structure of N$_{s}$ (see Fig.5b). As expected, the disorder decreases the strength of oscillating, however, it also shifts the peak positions of N$_{s}$ . It is possible to have a peak in N$_{s}$ at finite disorder while it is almost zero initially at clean limit. In Fig.5a, around W = 1 ∼ 1 . 5, for the Fermi level E$_{F}$ = − 3 . 8, − 2 . 2, we can see that N$_{s}$ is up to about three times of N$_{s}$ at W = 0. The behav- ior of N$_{s}$ v.s. W is very apparent when the Fermi level E$_{F}$ is close to the bottom of energy band ( E$_{F}$ = − 4). For E$_{F}$ = − 3 . 8, N$_{s}$ is much bigger than those at other Fermi levels, and we can see a very remarkable peak at about W = 1 . 4. For E$_{F}$ = − 2 . 2, N$_{s}$ begins from − 0 . 005, changes its sign at about W ∼ 1 . 25 and than increases, again reaches to 0 . 005 at about W ∼ 1 . 75. With a very big disorder, N$_{s}$ should go to zero as system enters into an insulating regime. We find that the zero of N$_{s}$ occurs at W = 3 for V$_{R}$ = 0 . 05 . This is roughly independent of the locations of the Fermi energy. 

In summary, in the absence of a perpendicular mag- netic field, the interplay between the spin Nernst ef- fect and the seebeck effect is investigated in a two- dimensional cross-bar with a spin-orbit interaction. The spin Nernst effect exhibits an oscillatory structure for a wide range of the Rashba SOI. With a large Rashba SOI, the N$_{s}$ oscillation has a peak when the seebeck coeffi- cient possesses one. However, the inverse condition is not always satisfied, namely, the seebeck coefficient can be almost zero while N$_{s}$ has a peak. The disorder ef- fect on the spin Nernst effects is also studied. We find that disorder can enhance N$_{s}$ up to three times for some 

