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relatively far away ( W = 8 µm ) the physics is better cap- tured by a model employing PBC like ours. However, the possible role of boundary scattering deserves further attention. 

It is interesting to note the similarities and differences between this simple model for the voltage controlled spin- precession signal and the usual model for the Hanle sig- nal [14] 

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The cosine functions in Eqs. 5 can be written as cos(2 αk$_{0}$t/ ℏ ) where t is the transit time L/v$_{x}$ = m $^{∗}$L/ ℏ $_{k$_{0}$}$√$_{1}$ − s $^{2}$, showing that 2 αk$_{0}$ in our problem plays the role that gµ$_{B}$B plays in the Hanle precession signal. Since we assume ballistic rather than diffusive transport we have a different weighting function for dif- ferent transit times. But the most important difference is that Hanle signals are typically observed within tens of Gauss around B = 0 while the experiment we are analyz- ing has α varying between 8 × 10 − 12 and 12 × 10 − $^{12}$eV- m with k$_{0}$ = 4 . 1 × 10 $^{8}$m − 1 corresponding to values of | g | B close to ∼ 140T as noted in [1]. At such high val- ues of | g | B , the Hanle signal is usually reduced essen- tially to zero because of the spread in the transit time ‘ t ’ caused by diffusive transport. One would expect the same in the present case, were it not for ballistic trans- port. By contrast, the Hanle signal around B = 0 is relatively robust and it would be interesting to look for an analogous voltage-controlled signal in shorter struc- tures or perhaps in structures where the RSO, α ( V$_{G}$ ), can be tuned through α = 0 [15]. 

The simple model here makes no prediction about the amplitude B , but it does suggest that the peak-to-peak amplitude of the oscillation in V$_{X}$ should be 3 $_{π/}$√$_{2}$$_{πθ}$$_{L}$ times the spin-valve signal V$_{Y}$ . Using m ∗ = 0 . 05 m$_{0}$ , α ≃ 10 − $^{11}$eV-m, L = 1 . 65 µ m, this suggests V$_{Y}$ = 1 . 2 V$_{X}$ (p- p). Experimentally, the p-p oscillatory signal ∼ 6 µ V which equals V$_{X}$ (p-p) / 2(since the experiment measures the parallel-antiparallel difference we have defined as V$_{X}$ ) but V$_{Y}$ is only ∼ 6 µ V. Possible reasons for the discrep- ancy are discussed at the end of this paper, but here we would like to note that we expect a further reduction in the amplitude of the oscillatory component due to the ex- tension of the injecting and the detecting contacts along x giving rise to a spread in the values of θ$_{L}$ in Eqs. 5a and 5b. We can write 

where C$_{i}$ and C$_{d}$ are numbers less than one represent- ing the averaging effects of the injecting and detecting contacts respectively and θ$_{i}$ , θ$_{d}$ are the additional phase- shifts introduced by the injecting and detecting contacts respectively in addition to θ$_{0}$ . To estimate θ$_{i}$ , θ$_{d}$ or C$_{i}$ , C$_{d}$ we need to know (1) the spatial uniformity of the injecting and detecting contacts, (2) how the elec- tronic wavefunction evolves under the contacts, and (3) 

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how the RSO α ( V$_{G}$ ) varies under the contacts. Re- garding point 1 we assume the contacts to be uniform and the NEGF model described next should account for point 2 within this assumption. However, point 3 re- quires a careful treatment beyond the scope of this pa- per. Here we simply note that Eq. 9 describes the shape of the oscillatory V$_{X}$ ( V$_{g}$ ) quite well with the following choice: θ$_{0}$ = 2 m $^{∗}$α ( V$_{G}$ ) L$_{CH}$/ ℏ $^{2}$, where L$_{CH}$ = 1 . 65 µ m which is the experimental center-to-center distance be- tween contacts and θ$_{i,d}$ = m $^{∗}$α ( V$_{G}$ = 0) L$_{Ci,d}$/ ℏ 2 where L$_{Ci}$ = 0 . 2 µ m and L$_{Cd}$ = 0 . 25 µ m equal to half the con- tact widths. The result from Eq. 9 also matches that from the NEGF model (see Fig. 1(d)) to be described next in shape and amplitude if we use C$_{i,d}$ = sin( θ$_{i,d}$ ) /θ$_{i,d}$ which can be justified if the electronic wavefunction is assumed to remain constant under each contact. 

## III. QUANTITATIVE NEGF BASED MODEL 

One way to make the results from the quantum transport model quantitative is to use the NEGF-based method described in detail in [16]. The inputs to this model are the Hamiltonian [ H ] and the self-energy ma- trices [Σ] (Fig 2). For H we use a discrete version of the one used in Section II (Eq. 2), as described in [16] assuming PBC along y as discussed above. We ne- glect all scattering processes since both the mean free path and the spin-coherence length are believed to be longer than the longitudinal dimensions at low temper- atures. To understand the signal decay at higher tem- peratures will require a consideration of both momen- tum and spin-relaxation processes, but we leave this for future work. The self-energies for the FM contacts (Σ$_{2}$,Σ$_{3}$) have the form − ( i/ 2) γ [ I + P$_{C}$⃗ σ. ˆ n ] where the polarization, P$_{C}$ = ( G$_{M}$ G$_{m}$ ) / ( G$_{M}$ + G$_{m}$ ) and ˆ n is the unit vector in the direction of the magnet. The constant γ = π ( G$_{M}$ + G$_{m}$ ) ℏ $^{3}$/e$^{2}$m ∗ is chosen to give a tunneling conductance equal to the experimental value . The NM contacts (Σ$_{1}$,Σ$_{4}$) are represented similarly with P$_{C}$ = 0. Finally, the long extended regions outside the channel at two ends (see Fig. 1(a)) are represented by two semi-infinite contacts whose coupling is given by, Σ$_{L}$$_{(}$$_{R}$$_{)}$ = τ$_{L}$$_{(}$$_{R}$$_{)}$g$_{S}$τ † L ( R ) where τ is the spin-dependent cou- pling matrix between the contact and the channel and g$_{S}$ is the surface Green’s function. The transmission func- tions are calculated from the NEGF model and contacts 3,4,L and R are treated as voltage probes with zero cur- rent (following the approach introduced by Buttiker, see section 9.4, in [17]). Note that although we are not in- cluding scattering processes explicitly, the voltage probes introduce an effective spin-scattering that reduces the signal. Indeed both V$_{X}$ and V$_{Y}$ increase significantly if we remove the end regions represented by Σ$_{L}$ and Σ$_{R}$. 

For contacts 1 and 2 we adjust the applied potential difference ( µ$_{1}$ − µ$_{2}$ ) to obtain a current level equal to the experimental value. The voltage signal µ$_{3}$$_{P}$ is obtained from the difference between the µ$_{3}$$_{P}$ measured with par- 

