L . (2.188)
We note that our results are compatible with the work of Duine [71]. From the same author, Ref. [72] also deals with the issue of spin-pumping due to a moving domain wall but uses a different approach than in Ref. [71]. However, the result of the pumped charge current in Ref. [72] shows the opposite sign of the β-term as compared to Ref. [71] and our result (2.188).
Let us briefly consider the kinetic equation for the force applied to this specific problem of a moving domain-wall. We straightforwardly obtain from Eq. (2.181), assuming a constant external magnetic field B~ext=Bexteˆz,
−wq2( ˙φ+αX˙
w) =−gµBBextwq2+a3
Sq2βjs , (2.189)
or
φ˙+αX˙
w =gµBBext− a3
Swβjs , (2.190)
which is one of the well known equations for the two collective coordinates representing chiralityφ and domain wall positionX [29, 66]. We again recover the resultjc =σs2
LgµBB by comparing the last expression with the pumped charge current in Eq. (2.187) and assuming α=β.
2.6. Conclusions and Outlook
We studied electronic transport in the presence of an inhomogeneous magnetization in fer-romagnetic conductors by deriving a quantum kinetic equation for the low energy dynamics including gradient corrections to the collision integral and in linear response. Our method allows us to obtain various results for charge- and spin-transport and the magnetization dynamics within one microscopic model, mathematical framework and compatible approxi-mations.
This equation has been first employed to calculate the domain-wall resistance for current perpendicular to wall (CPW) and current in-wall (CIW) geometries for a simple Bloch wall we found positive or negative values for the DWR. As opposed to previous studies of the DWR, dissipative spin-flip is taken into account and it gives rise to qualitatively different behavior, since in the absence of spin-flip processes, non-equilibrium spin-excitations do not relax, even infinitely far away from the domain wall which yields an additional potential drop at the domain-wall. Essentially, our treatment assumes that any spin-accumulation has decayed at the contact ends and thus, we neglect any finite size effects due to the contact geometry.
We have performed a detailed analysis of the domain-wall resistance in ferromagnetic conduc-tors and identified important mechanisms. Our results differ in various aspects from other
theories and we extend the description by including magnetic impurity scattering which leads to dissipative spin-flip scattering and spin-dephasing. The difference to other theories has been discussed at length and originates mostly from an inconsistent treatment in these works like disregarding the d-wave component, the neglect of spin-flip processes and different ap-proaches to include impurity scattering. While we use a fully microscopic approach for scat-tering, reflected in the full form of the collision integral (2.45) with gradient corrections and modification of the electronic structure properly taken into account, other works introduced momentum scattering rates phenomenologically.
We identified two distinct contributions to the DWR: the isotropic contribution that is always present in regions of non-vanishing magnetization gradient and an anisotropic term that contributes only when there is a magnetization gradient along the current flow. We found that transverse spin-dephasing contributes only to the anisotropic contribution, while the isotropic contribution does not depend on T2. Also, the anisotropic component is identical fors-dand itinerant Stoner models. In the limit ∆τ 1 mainly studied here, we found the general trend that the domain-wall resistance increases as we increase the magnetic impurity scattering strength.
In the limit of weak polarization, we find that the behavior of the DWR is dominated by the scattering asymmetry ofτ↑,↓, i.e. the difference of the momentum relaxation rate in the spin-up and down channel. In the opposite limit of strong polarization, the behavior of the DWR is dominated by the gradient corrections toτ↑. In the intermediate regime of P, it is difficult to isolate the various contributions to the DWR, especially since in our regime ∆τ 1 we need to include thed-wave moment of the spin-charge distribution function, which gives rise to complicated scattering, and is consistently described in terms of the complete collision integral with gradient correction included to second order in spatial magnetization gradients.
Other contributions arise from changes in the spectrum, the shift in the chemical potential, or local screening potentialδµ, and the reduction of the magnetic moment δ∆.
Contrary to thes-dmodel, we found that the Stoner limit has an overall stronger tendency to exhibit negative domain-wall resistance in the regime of small to intermediate polarizations P, which has its origin in the local reduction of the exchange field in areas of non-vanishing magnetization gradient. The situation is opposite in the limit of strong polarization, where the behavior is dominated by the magnetization gradient corrections to the scattering time τ↑, so that the DWR increases, eventually becoming positive again.
More complex domain-walls give rise to a transverse voltage due to the Berry curvature and we investigated this effect by calculating the transverse voltage drop induced by a charge current flowing through a vortex wall.
Employing our kinetic equation, we also have derived extended spin-charge continuity and diffusion equations which can be utilized to study magnetotransport and spin-transport in presence of an inhomogeneous magnetization in the mesoscopic limit. We calculated the full charge- and spin-current response due to the two sources of external perturbations: external electric field and a time-varying magnetization which least to charge- and spin-pumping. As application, we calculated the induced charge current by a moving domain wall. The results for the spin-current can then be directly used to obtain the spin-torque which affects the magnetization dynamics described in the framework of the Landau-Lifshitz-Gilbert equation, widely utilized to study for example domain-wall motion.
Finally, we have also illustrated the effects of the two dissipative terms, Gilbert damping and non-adiabatic spin-torque, which can be also understood in terms of friction between the electron system and the magnetization. On one hand, we saw that the Gilbert dampingα leads to a electron drag and consequently, a charge current due to a dynamic magnetization.
On the other hand, we directly calculated the pumped charge current (see Eq. (2.141)) with a contribution proportional toβ. In the end, both pictures should yield the same current, so this suggests a deep connection between the Gilbert damping constantα and the non-adiabatic coefficient β.
In summary, the kinetic equation derived in the first part of this work allows for a unified treatment of the complete response of a ferromagnetic conductor with an inhomogeneous and time-dependent magnetization structure in the regime where the magnetization m~ is smooth on the scale of the Fermi-wavelength λF and the precession length lprec. Aside from the thorough investigation of the domain wall resistance, we are able to confirm many findings published in literature using different methods and models, some of them of phenomenological nature, thus putting them on a solid ground. We also find new terms that have not been discussed before, in particular non-local corrections to various terms in the LLG equation that describe corrections relevant for strongly varying magnetization structures.