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33646600770
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note
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See Ref. 3. The differences in these two calculations are theirs is based on band structure and specular scattering at interface, ours is confined to a Boltzmann description for transport in the bulk of the layers with diffuse scattering at interfaces and assumes that transverse spin accumulation and current does exist in the ferromagnet's bulk. Referring to Fig. 3 and Eqs. (14) and (18) in the above referenced paper, one notes the absence of the component of the spin distribution function that is transverse to the magnetization (axis of spin quantization) in the ferromagnetic layers. Indeed, in a subsequent paper (Ref. 4) they have shown that the transverse component decays in a distance less than 1 nm for metals that fit the Stoner model. On the contrary, we are currently in the process of further justifying the assumption that a transverse component of the spin accumulation exists in the 3d transition-metal ferromagnets. Aside from these differences both approaches do find an amplification in the spin torque above that one anticipates on the basis of a nondiffusive transport calculation, i. e., we both find the angular momentum transferred from the spin current to the free magnetic layer far exceeds the bare portion of the transverse component of the spin current in the nonmagnetic layer adjacent to the free magnetic layer; M. D. Stiles (private communication).
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15
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33646605077
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note
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m, is also a vector in position space. To simplify the notation we limit ourselves to currents along the x axis which is perpendicular to the planes of the layers.
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16
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0001421957
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0. It is just this case that was discussed in a previous publication (Ref. 5).
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note
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We thank Albert Fert for pointing out this alternate route.
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23
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0141636577
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J.A. Katine, F.J. Albert, R.A. Buhrman, E.B. Myers, and D.C. Ralph, Phys. Rev. Lett. 84, 3149 (2000); F.J. Albert, J.A. Katine, R.A. Buhrman, and D.C. Ralph, Appl. Phys. Lett. 77, 3809 (2000); J. Grollier, V. Cros, A. Hamzic, J.M. George, H. Jaffres A. Fert, G. Faini, J. Ben Youssef, and H. Legall, ibid. 78, 3663 (2001).
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J.A. Katine, F.J. Albert, R.A. Buhrman, E.B. Myers, and D.C. Ralph, Phys. Rev. Lett. 84, 3149 (2000); F.J. Albert, J.A. Katine, R.A. Buhrman, and D.C. Ralph, Appl. Phys. Lett. 77, 3809 (2000); J. Grollier, V. Cros, A. Hamzic, J.M. George, H. Jaffres A. Fert, G. Faini, J. Ben Youssef, and H. Legall, ibid. 78, 3663 (2001).
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J.A. Katine, F.J. Albert, R.A. Buhrman, E.B. Myers, and D.C. Ralph, Phys. Rev. Lett. 84, 3149 (2000); F.J. Albert, J.A. Katine, R.A. Buhrman, and D.C. Ralph, Appl. Phys. Lett. 77, 3809 (2000); J. Grollier, V. Cros, A. Hamzic, J.M. George, H. Jaffres A. Fert, G. Faini, J. Ben Youssef, and H. Legall, ibid. 78, 3663 (2001).
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Ref. 2
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See Waintal, et al., Ref. 2.
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