Transverse spin diffusion in ferromagnets

Abstract

We discuss the dissipative diffusion-type term of the form $\mathbf{m}\ifmmode\times\else\texttimes\fi{}{\ensuremath{\nabla}}^{2}{\ensuremath{\partial}}_{t}\mathbf{m}$ in the phenomenological Landau-Lifshitz equation of ferromagnetic precession, which describes enhanced Gilbert damping of finite-momentum spin waves. This term arises physically from itinerant-electron spin flows through a perturbed ferromagnetic configuration and can be understood to originate in the ferromagnetic spin pumping in the continuum limit. We develop a general phenomenology as well as provide microscopic theory for the Stoner and $s\text{\ensuremath{-}}d$ models of ferromagnetism, taking into account disorder and electron-electron scattering. The latter is manifested in our problem through the Coulomb drag between the spin bands. The spin diffusion coefficient is identified with the transverse spin conductivity, in analogy with the Einstein relation in the kinetic theory.