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allel contacts and µ$_{3}$$_{AP}$ measured with anti-parallel con- tacts. We use a contact conductance of G$_{M}$ + G$_{m}$ = 4 × 10 $^{10}$Ω − $^{1}$m − 2 based on the experimental parameters in [1] and a P$_{C}$ = ( G$_{M}$ − G$_{m}$ ) / ( G$_{M}$ + G$_{m}$ ) = 0 . 068 to match the spin-valve signal, V$_{Y}$ . Fig. 1(d) shows the nu- merical results obtained using a channel with α ( V$_{G}$ ) of length L$_{CH}$ = 1 . 65 µ m and contacts with fixed α ( V$_{G}$ = 0) of lengths L$_{Ci}$ = 0 . 2 µ m and L$_{Cd}$ = 0 . 25 µ m. The os- cillatory signal matches the experimental observation in shape but the amplitude is smaller. One possibility is that the P$_{C}$ we use has been calibrated for the spin- valve signals obtained with Y-directed magnets. The same magnets when forced into the X-direction for the oscillatory signals may have a higher effective P$_{C}$ espe- cially since no anti-parallel measurements are involved. However, to account fully for the discrepancy a signifi- cant increase in P$_{C}$ would be needed and other sources of discrepancy should be investigated. 

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[6] See for example (a) M. Johnson and R. H. Silsbee, Phys. Rev. B 37 , 5312 (1988) and (b) F. J. Jedema, M. S. Nijboer, A. T. Filip and B. J. van Wees, Phys. Rev. B 67 , 085319 (2003). 

[7] M. G. Pala, M. Governale, J. K¨ onig, and U. Z¨ ulicke, Europhys. Lett. 65 , 850 (2004). 

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In summary, we have presented (1) a straightforward extension of the Datta-Das theory [2] to include the angular spectrum of electrons and the extended con- tacts, and (2) a more elaborate model that treats the actual non-local experimental structure using a NEGF based approach widely used in nanoelectronics. The sim- pler theory provides a number of insights and is well- supported by the more elaborate model, identifying sev- eral features that deserve further investigation. Specifi- cally it seems that while the experimental oscillation pe- riod shows good agreement with theory, the amplitude relative to the spin-valve signal V$_{Y}$ is larger showing es- sentially no reduction expected from averaging over the angular spectrum and the extended contacts. Possible reasons for this discrepancy deserve carefull attention. On the theoretical side, it is possible that the contribu- tion from high k$_{Y}$ components are suppressed because they have shorter effective spin coherence lengths and a purely ballistic theory misses this aspect. Another pos- sibility is that the effective polarization P$_{C}$ is lower in the antiparallel case which affects only V$_{Y}$ and not V$_{X}$ . The structure and nature of the injecting and detecting contacts also require careful consederation. 

This work is supported by the office of Naval Research under Grant No. N0014-06-1-2005 and the Network for Computational Nanotechnology (NCN). 

J. Nitta, T Akazaki, and H. Takayanagi, Phys. Rev. Lett 89 , 046801 (2003). 

[10] S. Takahashi and S. Maekawa, Phys. Rev. B 67 , 052409 (2003). 

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[12] S. Bandyopadhyay, arXiv:cond-matt/0911.0210 (2009). 

[13] See for example (a) B. K. Nicoli` c and S. Souma, Phys. Rev. B 71 , 195328 (2005) and (b) J. S. Jeong and H. W. Lee, Phys. Rev. B, 74 , 195311 (2006). 

[14] See for example (a) M. Johnson and R. H. Silsbee, Phys. Rev. Lett. 55 1790 (1985), (b) F. J. Jedema, H. B. Heer- sche, A. T. Filip, J. J. A. Baselmans and B. J. van Wees, Nature 416 , 713 (2002) and (c) Lou X. et. al., Phys. Rev. Lett. 96 , 176603 (2006). 

[15] J. D. Koralek, C. P. Weber, J. Orenstein, B. A. Bernevig, S.-C. Zhang, S. Mack. and D. D. Awschalom, Nature 458 , 610 (2009). 

[16] S. Datta, Nanoelectronics: A unified view Chapter 1, vol. I, the Oxford Handbook on Nanoscience and Nanotech- 

## IV. DISCUSSION 

## V. ACKNOWLEDGEMENT 

