Theory has predicted that the damping of magnetization dynamics may be anisotropic; i.e., it may depend on the momentary orientation e͑t͒ of the magnetization in the crystal. In the present Brief Report it is shown that in general this anisotropy is averaged out at least to some extent for the special trajectory e͑t͒ of the magnetization vector in a ferromagnetic-resonance ͑FMR͒ experiment. In principle it may be that there is no anisotropy of the FMR linewidth although the damping is strongly anisotropic and although this anisotropy will be essential in a more complicated trajectory e͑t͒ of the magnetization vector.In recent years there has been a great research activity on fast magnetization dynamics for micro-and nanosized magnets because of their potential use in advanced information storage and data processing devices, with special emphasis on domain-wall dynamics ͑in particular for nanowires 1 ͒, on magnetization reversal in nanomagnets, 1 and on vortex dynamics. 2 Thereby, most theoretical investigations were based on the Gilbert equation of motion 3 for the magnetization M͑r , t͒,Here, ␥ is the gyromagnetic ratio; H eff is the effective field composed of the external field H, the exchange field H ex , the anisotropy field H ani , and the dipolar field H dip ; and ␣ is the scalar damping constant. Equation ͑1͒ is the simplest conceivable equation which describes precession of M͑r , t͒ around H eff ͑first term on the right-hand side͒ as well as damping ͑second term͒. Of course the question arises ͑see, e.g., Refs. 4-6͒ as to whether this equation really encompasses all the relevant physics for the time scale of the above discussed experiments, which is the near-adiabatic time scale 4 between several picoseconds and nanoseconds. For instance, Gilbert 3,6 himself suggested a generalization of his equation of motion, Eq. ͑1͒, with the local damping term and the scalar constant ␣ replaced by a nonlocal damping term ͑which relates the dynamics of the magnetization at site r to the dynamics of the magnetization at all other sites rЈ͒ with a damping matrix ␣͑r , rЈ͒. A damping of the dynamics described by a damping matrix was suggested also by Baryakhtar et al. 7 ͑and Safonov 8 ͒, who already mentioned that the replacement of the damping scalar by a damping matrix might have an influence on the FMR linewidth. Finally, such a generalized equation for the atomic magnetic moments M R = M R e R at the atomic sites R with magnitude M R and orientation e R could be derived 9 by a combination of the ab initio electron theory with the phenomenological breathing Fermi-surface model of Kamberský 10 for the magnetization dynamics close to the adiabatic limit, ͑2͒In Eq. ͑2͒ the effective field H eff,R depends on the magnetic configuration ͕e R Љ ͖ of the whole system. This is not surprising and is accepted by everybody because also the field H eff ͑r͒ of Gilbert equation ͑1͒ depends on the magnetization field M͑r͒ of the whole system via the exchange field and the dipolar field. Furthermore, the damping matrices ␣ R,R Ј depe...
Based on the breathing Fermi-surface model of Gilbert damping and on the Elliott-Yafet relation for the spin-relaxation time, a relation is established between the conductivitylike contribution to the Gilbert damping ␣ at low temperatures and the demagnetization time M for ultrafast laser-induced demagnetization at low laser fluences. Thereby it is assumed that, respectively, the same types of spin-dependent electron-scattering processes are relevant for ␣ and M . The relation contains information on the properties of single-electron states which are calculated by the ab initio electron theory. The predicted value for ␣ / M is in good agreement with the experimental value.
An equation of motion for the magnetization dynamics of systems with collinear or noncollinear magnetization is derived by a combination of the breathing Fermi surface model with a variant of the ab initio density functional electron theory given by the magnetic force theorem. The equation corresponds to a Gilbert equation with the constant Gilbert damping scalar α replaced by a nonlocal damping matrix , which depends on the momentary orientation of all atomic magnetic moments in the system. For collinear situations this corresponds to an anisotropy of the damping because it depends on the orientation of the magnetization in the crystal, and for systems with atomic-scale noncollinearity such as extremely narrow domain walls or vortices the nonlocality is essential. The range of validity of the theory is discussed, and the predictions are compared with experimental observations. In particular, it is outlined how the prediction of anisotropic damping can be tested by ferromagnetic resonance experiments.
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