−3 −2 −1 0 1 2 3 0 0.1 0.2 0.3 V$_{G}$ [V] V (norm.) Approx. analytical Numerical integration Approx. analytical Numerical integration FM−FM, y FM−FM, x (c) Simple model −3 −2 −1 0 1 2 3 0 1 2 3 4 5 6 V$_{G}$ [V] V [ V] FM−FM, y FM−FM, x (d) NEGF based model 2 We define the spin-voltage V$_{X}$ (or V$_{Y}$ ) as the differ- ence between the voltages measured in the parallel and the anti-parallel configurations by X- (or Y-) directed in- jecting and detecting magnets. This is expected to be twice that measured in the parallel configuration using a setup like the one shown in Fig. 1(a) [10] which is ex- actly how the oscillatory signals V$_{X}$ in [1] are measured, although the spin-valve signal V$_{Y}$ is measured the same way our V$_{Y}$ is defined. It is shown in Appendix A (sup- plementary information) that for a point injector located at x = 0 and a point detector located at x = L , as- suming ballistic transport, the voltage signals for X- and Y-directed magnets can be written as ( B : constant) where s ≡ k$_{Y}$ /k$_{0}$ = ℏ k$_{Y}$ $_{/}$√$_{2}$$_{m}$ $^{∗}$E and θ$_{L}$ = 2 m $^{∗}$αL/ ℏ $^{2}$. The contributions from different E , k$_{Y}$ all act ‘in parallel’ giving a voltage equal to the average. At low tempera- tures we can average the contributions from all transverse wave-vectors k$_{Y}$ over the Fermi circle ( E = E$_{F}$ ) to write Interestingly, the results obtained from the integration in Eq. 6 looks almost exactly like the single cosine re- sult in Eq. 1 that describes the experimental observa- tions. This can be understood by noting that the argu- ment θ$_{L}$$_{/}$√$_{1}$ − s 2 has a stationary point at s = 0 [11] and we can use the method of stationary phase to write approximately, lower signs respectively. For small α we can write ap- proximately As shown in Appendix B (supplementary information) these approximations describe the results from the exact integration quite well for θ$_{L}$ ≳ 2 π which is true for the range of α and L involved in the experiment. This should help answer some of the concerns raised in a recent com- ment [12]. Let us note that the simple results presented above are made possible by our assumption of periodic boundary conditions (PBC) in the y-direction making k$_{Y}$ a ’good quantum number’ like E . Most of the prior work, on this topic [13], uses hardwall boundary condition (HBC) which does not seem to permit the simple decoupling of different transverse wavevectors ( k$_{Y}$ ) due to non-trivial ‘boundary scattering’. We have checked numerically that the use of HBC does not change the conclusions described above in a significant way although some details are dif- ferent. Furthermore, one could argue that since the ac- tual boundaries in the experimental structure of [1] are V$_{X}$$_{0}$ ( E, k$_{Y}$ ) = B { s 2 + ( 1 − s 2 ) cos ( θ$_{L}$ √$_{1}$ − s 2 )} (5a) V$_{Y}$$_{0}$ ( E, k$_{Y}$ ) = B ( 1 − s 2 ) + s $^{2}$cos θ$_{L}$ √$_{1}$ − s 2 (5b) V$_{X}$$_{(}$$_{Y}$$_{)}$ = ∫ + k$_{0}$ − k$_{0}$ dk$_{y}$ 2 πk$_{0}$ V$_{X}$$_{0(}$$_{Y}$$_{0)}$ ( E$_{F}$ , k$_{Y}$ ) (6) V$_{X}$ ≃ B 3 π + B √$_{2}$$_{πθ}$$_{L}$ cos( θ$_{L}$ + π 4 ) (7a) V$_{Y}$ ≃ 2 B 3 π (7b) k$_{X}$$_{−}$ − k$_{X}$$_{+}$ ≈ 2 m $^{∗}$α ℏ 2 k$_{0}$ √ k 2 0 − k 2 Y (4) with k$_{0}$ ≡ √$_{2}$$_{m}$ $^{∗}$E/ ℏ . FIG. 1: (a) Schematic structure and (b) experimentlly ob- served non-local signal in [1]. Calculated non-local signal for the structure in (a) using: (c) simple analytical model (section II) and (d) using NEGF based model (section III). Parameters: P C = 6 . 8%= G$_{M}$ − G$_{m}$ G$_{M}$ + G$_{m}$ and G M + G m = 4 × 10 $^{10}$/m$^{2}$/ Ω ( G M ( m ) being the contact conductance per unit area for majority (minority) spins), carrier density n S = 2 . 7 × 10 $^{16}$m − $^{2}$, FM contact lengths L Ci = 0 . 2 µ m, L Cd = 0 . 25 µ m, FM contact spacing L CH = 1 . 65 µ m, width W = 8 µ m, effec- tive mass m ∗ = 0 . 05 m$_{0}$ .