Transverse spin diffusion in ferromagnets

We discuss the dissipative diffusion-type term of the form $\mathbf{m}\times {\nabla }^2{\partial }_t\mathbf{m}$ 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 $s\text{−}d$ 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.

Figure 1

In this RPA summation for the magnon propagator [Eq. (64)], the magnon self-energy is provided by the $s$-electron spin-response bubble [Eq. (41)]. Straight double lines denote disorder-averaged $s$-electron Green’s functions and dashed lines describe vertex ladder corrections. Each bubble contains ladder corrections to the vertex function. Wavy lines are the $d$-electron-spin propagators.

Figure 2

A schematic view of the superlattice considered in the text: an $\text{F}\mid \text{N}$ bilayer is repeated along the $x$ axis, with either ferromagnetic or antiferromagnetic alignment of the consecutive magnetic layers. The system is translationally invariant along the two remaining axes.