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Volumn 80, Issue 18, 2009, Pages

Hydrodynamic theory of coupled current and magnetization dynamics in spin-textured ferromagnets

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EID: 73249136465     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.80.184411     Document Type: Article
Times cited : (42)

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    • Coupling of this exchange energy to the electronic density fluctuations can also be treated systematically. However, since it does not lead to any significant effects, while at the same time unnecessarily complicating our discussion, we will disregard it in the following.
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    • In the more general two-band model of spin-up and spin-down electrons, we would have to develop a two-fluid effective theory, with spin-flip scattering between the two fluid components. The phenomenology simplifies, however, reducing formally to the halfmetallic case, in the long-wavelength low-frequency limit: ω τsf 1 and k λsd 1, where ω, k are the characteristic frequency, wave number of the magnetohydrodynamics and τsf, λsd τsf are the spin-flip time, spin-diffusion length. In this limit, it may be possible to describe the hydrodynamic state of the system by the spin-texture field, the charge-density distribution, and the charge-current field. If any out-of-equilibrium spin imbalance decays sufficiently fast, therefore, we only need to retain a one-fluid description for the charge flows. The key phenomenological modification is then to introduce a material-dependent dimensionless "spin-polarization" parameter p, such that q→pq in the following equations of motion. Namely, the effective charge that couples the electronic particle-number flux densities j with the spin-texture gauge field is renormalized by p. While in the halfmetallic limit p=1 and in normal metals p=0, we may expect some intermediate value in realistic multiple-band ferromagnets with fast spin relaxation.
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    • To make contact with Ref., define the canonical current as J= □C p□dl, for an arbitrary closed curve C. Its equation of motion is given by ∂t J=q □C dl□ (v×b). If the curve C coincides with a quasi-1D wire, then vdl and we recover the reactive equation of Ref.: ∂t J=0.
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    • Similarly to q that should generally be viewed as a material-dependent phenomenological parameters, the electron mass me from now on is also an effective parameter, which is not necessarily identical with the free-electron mass.
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    • We are not including the ordinary hydrodynamic viscosity in our treatment.
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