2 0 0.1 (a) V$_{R}$ =0.02 N s  & S 0 0.1 (b) V$_{R}$ =0.05 −4 −3.5 −3 −2.5 0 0.1 (c) V$_{R}$ =0.1 E$_{F}$ N s  & S −3.5 −3 −2.5 0 0.1 (d) V$_{R}$ =0.45 E$_{F}$ FIG. 2: N s (red solid) and S (black dotted) vs. Fermi energy E F for different Rashba V R . The (scaled) transmission coef- ficient T $^{1}$,2 + T $^{1}$,3 (thin blue dashed) and spin transmission coefficient Δ T $^{2}$,3 (thin blue solid) are also shown. The other parameters are T = 0 . 01, and L = 19 a . FIG. 1: Schematic diagram of the four-terminal cross-bar sample. The area with SOI is marked by gray. A thermal gradient Δ T is applied between the longitudinal lead-1 and lead-3. where c † i $_{σ}$( c$_{i}$$_{σ}$ ) is the creation (annihilation) operator of electrons in the site i = ( n, m ) with spin σ , and δx and δy are the unit vectors along the x and y directions. ε$_{i}$ is the on-site energy, which is set to 0 everywhere for the clean system. When the center region is a disorder system, ε$_{i}$ is set by a uniform random distribution [-W/2,W/2]. Here t = ℏ $^{2}$/ (2 m $^{∗}$a $^{2}$) is the hopping matrix element with the lattice constant a , I is a two-dimensional identity matrix. The strength of Rashba SOI is represents by V$_{R}$ = α ℏ / 2 a , where α is the Rashba spin-orbital coupling. V$_{R}$ is set to zero in the lead-2 and lead-4. here Δ T$_{2}$$_{p}$ = T$_{2}$$_{↑}$$_{,p}$ − T$_{2}$$_{↓}$$_{,p}$ , and f is the zero order of Tay- lor expansion of the Fermi distribution function, it is the same for all four leads, f ( E ) = 1 / { exp[( E − E$_{F}$ ) /k$_{B}$ T ]+ 1 } . For the calculation of the longitudinal seebeck coeffi- cient S , we need the open boundary condition at lead-1,3, i.e. J$_{1}$$_{e}$ = J$_{3}$$_{e}$ = 0 to find the difference Δ V = V$_{1}$ − V$_{3}$ . Different from a quasi-one-dimensional 2-leads system $^{11}$, the extra leads-2,4 also affects the longitudinal seebeck coefficient S of the entire system. For example, with a perpendicular magnetic field B , the longitudinal see- beck coefficient S is affected by the bias in leads-2,4, V$_{2}$ and V$_{4}$ . However, without B , the sample’s symme- try increases from C$_{2}$ symmetry to D$_{2}$ symmetry, i.e., we have T$_{1}$$_{,}$$_{2}$ = T$_{1}$$_{,}$$_{4}$ , Here T$_{1}$$_{,}$$_{2}$ = T$_{1}$$_{↑}$$_{,}$$_{2}$ + T$_{1}$$_{↓}$$_{,}$$_{2}$ . After the Taylor expansion, we can get the longitudinal seebeck coefficient S ≡ − Δ V/ Δ T as: The equation above shows that, even with a higher sym- metry, the longitudinal seebeck coefficient S is still af- fected by the transport properties from lead-2 and lead- 4. S = 1 T ∫ dE ( T$_{1}$$_{,}$$_{2}$ + T$_{1}$$_{,}$$_{4}$ + 2 T$_{1}$$_{,}$$_{3}$ ) ( E − E$_{F}$ ) f (1 − f ) ∫ dE ( T$_{1}$$_{,}$$_{2}$ + T$_{1}$$_{,}$$_{4}$ + 2 T$_{1}$$_{,}$$_{3}$ ) f (1 − f ) . (4) where T$_{pσ,q}$ ( E ) is the transmission coefficient from the lead- q to the lead- p with spin σ and E is the energy of the incident electron. f$_{p}$ ( E ) is the electronic Fermi distribution function of the lead- p . The spin Hall current in lead-2 and lead-4 can be calcu- lated with the closed boundary condition in both lead-1,3 and lead-2,4, i.e. V$_{1}$ = V$_{3}$ = 0 and V$_{2}$ = V$_{4}$ = 0. From symmetry of the system, we know that J$_{2}$$_{s}$ = − J$_{4}$$_{s}$ $^{9}$. After the Taylor expansion, the spin Nernst coefficient N$_{s}$ ≡ J$_{2}$$_{s}$/ Δ T can be reduced to: N$_{s}$ = 1 4 π ∫ dE (Δ T$_{23}$ − Δ T$_{21}$ ) E − E$_{F}$ k$_{B}$ T 2 f (1 − f ) , (3) I$_{pσ}$ = 1 ℏ ∑ q = p ∫ dE T$_{pσ,q}$ ( E )[ f$_{p}$ ( E ) − f$_{q}$ ( E )] (2) Considering a small temperature gradient Δ T on the longitudinal lead-1,3, we can set the temperatures T$_{1}$ = T + Δ T / 2, T$_{3}$ = T − Δ T / 2, T$_{2}$ = T$_{4}$ = T . The charge current in lead- p can be written as J$_{pe}$ = e ( I$_{p}$$_{↑}$ + I$_{p}$$_{↓}$ ) and the spin current is J$_{ps}$ = ( ℏ / 2)( I$_{p}$$_{↑}$ − I$_{p}$$_{↓}$ ). Here I$_{pσ}$ is the particle current in lead- p with σ equals to ↑ or ↓ . I$_{pσ}$ can be obtained by the LB formula: 9,10 H = ∑ i σ ε$_{i}$c † i $_{σ}$c$_{i}$$_{σ}$ + ∑ i σσ ′ [ c † i + $_{δy,σ}$( − t I − iσ$_{x}$V$_{R}$ )$_{σσ}$ ′ c$_{i}$$_{σ}$ ′ + c † i + $_{δx,σ}$( − t I + iσ$_{y}$V$_{R}$ )$_{σσ}$ ′ c$_{i}$$_{σ}$ ′ + H.c. ] (1) with SOI can be written as: $^{10}$,