Role of nonequilibrium conduction electrons on the magnetization dynamics of ferromagnets in the $s$-$d$ model

In the literature on the $s$-$d$ model for the magnetization dynamics in systems without and with spin-polarized transport currents, it is generally assumed that the adiabatic part of the $s$-electron magnetization is everywhere parallel to the $d$-electron magnetization. We present model calculations within the ab initio spin-density functional theory which suggest that this assumption is not valid for systems with strongly noncollinear magnetization configurations.

Figure 1

Top: The angles ${\vartheta }_{\nu }$ with $\nu =s$, $p$, and $d$ which the ${\mathbf{M}}_{\nu }$ of the central Co atom form with the $z$ direction, as function of the prescribed angle $\vartheta$ for the total magnetic moment. Please note that we present $\pi -{\vartheta }_{\mathrm{s},\mathrm{p}}$ and ${\vartheta }_{\mathrm{d}}$. Thus the alignment of ${\mathbf{M}}_{\mathrm{s}}$ and ${\mathbf{M}}_{\mathrm{p}}$ is antiparallel with ${\mathbf{M}}_{\mathrm{d}}$ for $\vartheta =0$. Bottom: The magnitudes $M_{\nu }=\mid {\mathbf{M}}_{\nu }\mid$ of the central Co atom as function of the same prescribed angle $\vartheta$.