Inhomogeneous Gilbert damping from impurities and electron-electron interactions

We present a unified theory of magnetic damping in itinerant electron ferromagnets at order $q^2$ including electron-electron interactions and disorder scattering. We show that the Gilbert damping coefficient can be expressed in terms of the spin conductivity, leading to a Matthiessen-type formula in which disorder and interaction contributions are additive. In a weak ferromagnet regime, electron-electron interactions lead to a strong enhancement of the Gilbert damping.

Figure 1

(Color online) The Gilbert damping $\alpha$ as a function of the disorder scattering rate $1/\tau$. The red (solid) line shows the Gilbert damping for polarization $p=0.1$ in the presence of the $e\text{−}e$ and disorder scattering, while the dashed line does not include the $e\text{−}e$ scattering. The blue (dotted) and black (dash-dotted) lines show Gilbert damping for $p=0.5$ and $p=0.99$, respectively. We took $q=0.1k_F$, $T=54\text{K}$, ${\omega }_0=E_F\left[{\left(1+p\right)}^{2/3}-{\left(1-p\right)}^{2/3}\right]$, $M_0=\gamma pn/2$, $m_{\ast }=m_e$, $n=1.4\times {10}^{21}{\text{cm}}^{-3}$, $r_s=5$, and $a_{\ast }=2a_0$.