Spin Hall phenomenology of magnetic dynamics

Physical Review B - Condensed Matter and Materials Physics

Volumn 90, Issue 1, 2014, Pages

  Tserkovnyak, Yaroslav ^a   Bender, Scott A ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84905037571     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.90.014428     Document Type: Article

References (44)

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19. More precisely, it is only the symmetric part of (Equation presented) that should be identified with a generalized Gilbert damping. Indeed, Onsager reciprocity requires (Equation presented), while the dissipative (i.e., time-reversal symmetry breaking) character dictates (Equation presented), which together lead to (Equation presented). The antisymmetric component of (Equation presented), on the other hand, contributes to the effective, matrix-valued gyromagnetic ratio.
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22. When the ferromagnet is insulating, (Equation presented) so defined describes the conversion between 3D current density in the normal metal and the spin-current density absorbed by the ferromagnetic insulator. In a simple limit of weak spin-orbit interaction at the interface, such (Equation presented) may correspond to the bulk spin Hall angle of the normal metal. When the thickness (Equation presented) is larger than the spin-relaxation length (Equation presented) in the normal metal, it is natural to expect (Equation presented) defined by Eq. (13), as well as (Equation presented), to be essentially thickness (Equation presented) independent.
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27. When the ferromagnet is metallic, furthermore, (Equation presented), (Equation presented), and (Equation presented) may also depend on its thickness (Equation presented) when the ferromagnet is thinner than its spin-relaxation length (Equation presented).
28. It would, however, be interesting to experimentally study the dependence of these two coefficients on the layer thicknesses as well as the substrate and cap materials, in the ultrathin limit.
29. We remark, however, that Eqs. (9) and (10) can effectively produce an anisotropic Gilbert damping even if the original (Equation presented) is scalar: solving Eq. (10) for (Equation presented) in the limit (Equation presented) and (Equation presented), for example, and substituting the resultant current into Eq. (9) gives the torque (Equation presented), where (Equation presented) (setting, for simplicity, (Equation presented)), of which the dissipative term (Equation presented) contributes to magnetic damping (while the Hall term (Equation presented) effectively makes the gyromagnetic ratio anisotropic). An anisotropic and (Equation presented)-dependent correction to the resistivity tensor (Equation presented) can similarly be constructed, for example, in the limit (Equation presented).
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