Extendeds−dmodel for magnetization dynamics of strongly noncollinear configurations

Abstract

In the conventional s-d model of magnetization dynamics, it is assumed that for an adiabatic situation the s magnetization is ferromagnetically aligned to the d magnetization. For configurations with strong noncollinearity in the d magnetization, this no longer holds true for all cases. In the present paper it is shown that, as a consequence of this noncollinearity between the adiabatic s and d magnetizations, there arise several additional torques in the micromagnetic equation of motion for the d magnetization. The equation is solved for a Néel wall driven by an external field and/or a spin-polarized current, yielding a correction term to the result of the conventional theory. By calculations with the ab initio density-functional electron theory, the magnitude of the correction term is estimated. It is concluded that for 3d metals the effects of the additional torques are very small except possibly for atomic-scale noncollinearities or for the long-term trajectories of complicated magnetization configurations.The dynamics of noncollinear magnetization configurations driven by external magnetic fields or spin-polarized currents is often investigated by micromagnetic simulations with the Landau-Lifshitz-Gilbert ͑LLG͒ equation of motion. 1 The influence of a spin-polarized transport current thereby is treated by including two spin-torque terms ͑an adiabatic and a nonadiabatic spin torque͒ into the LLG equation. These two terms which originally have been introduced in a phenomenological manner 2 have been justified within the s-d model of Zhang and Li. 3 In the s-d model the electronic states are subdivided into "s" states and "d" states. The "s" states represent those electronic states ͑conduction electrons states͒ which dominate the electrical conductivity. For 3d ͑4f͒ metals these are the 4s and 4p ͑5d, 6s, and 6p͒ states which are strongly delocalized, i.e., itinerant. The magnetization related to these states is denoted by m͑r , t͒. The symbol "d" stands for the more localized states which are the 3d ͑4f͒ states and which contribute a magnetization denoted as M d ͑r , t͒ so that the total magnetization is given by M͑r,t͒ = M d ͑r,t͒ + m͑r,t͒. ͑1͒In the s-d model the dynamics of M d ͑r , t͒ is calculated in two steps. In the first step the response of m͑r , t͒ to a given M d ͑r , t͒ is calculated. As soon as m͑r , t͒ is not completely aligned to M d ͑r , t͒, a torque density will be exerted on M d ͑r , t͒ via the s-d exchange interaction, 3In Eq. ͑3͒ S is the effective spin quantum number which is related to the localized atomic moments, and J ex denotes the s-d exchange coupling. In the second step the dynamics of M d is determined from a LLG-type equation of motion,where ␥ is the gyromagnetic ratio, H eff,d is the effective field acting on M d , and ␣ denotes the Gilbert damping constant.In the conventional s-d model 3 m͑r , t͒ is not completely aligned to M d ͑r , t͒ because the conduction electrons experience spin-flip scattering processes ͓characterized by a spinflip scattering time sf , see Eq. ͑7͔͒ and t...

Constrained Density Functional Theory

“…These core equations have been widely used for determining the U parameter in LDA+ U , Anderson, and Hubbard models, − frequently in combination with the Hund’s rule exchange parameter J . − A closely related fixed spin moment (FSM) class of methods, which will not be covered in the present review, originated in tandem with the original CDFT work of Dederichs et al − Related use of CDFT for producing constrained magnetic configurations has been rather widespread, − and the ability to fix different spin orientations at distinct sites allows for ab initio spin dynamics with extension to relativistic spin dynamics . A survey of the results based upon CDFT finds that virial and Hellmann–Feynman theorems have been given for CDFT, and the theory has been generalized for application to the inverse Kohn–Sham problem .…”

Electron theory of fast and ultrafast dissipative magnetization dynamics

“…Rare-earth metals may be described in the same spirit by an s-f model where the 's-electrons' represent the 5d-6sp conduction electrons. In the literature there are several variants of the s-d model [10] (as examples we mention [23][24][25]) and a combination of the s-f model with an ab initio treatment of the 5d-6sp electrons is given in [26]. The second viewpoint is that in concentrated magnetic transition metals and alloys there is a considerable weight of the d-density of electronic states at the Fermi level and a strong hybridization of d-orbitals with sp-orbitals.…”

Phenomenological description of the nonlocal magnetization relaxation in magnonics, spintronics, and domain-wall dynamics

“…In this report, we demonstrate how the phenomenological magnetic relaxation term derived by Baryakhtar to explain the discrepancy between magnetic damping constants obtained from ferromagnetic resonance (FMR) and magnetic domain wall velocity measurements in dielectrics [20][21][22] can be applied to magnetic metallic samples. We show that the Landau-Lifshitz equation with Baryakhtar relaxation term (Landau-Lifshitz-Baryakhtar or simply LLBar equation) contains the Landau-Lifshitz-Gilbert (LLG) equation as a special case, while also naturally including the contribution from the nonlocal damping in the tensor form of Zhang and Zhang [23] and De Angeli [24]. The effects of the longitudinal relaxation and the anisotropic transverse relaxation on the magnetization dynamics excited by optical and magnetic field pulses, respectively, in continuous films and magnetic elements were discussed e.g.…”