Wavenumber dependent Gilbert damping in metallic ferromagnets

New terms to the dynamical equation of magnetization motion, associated with spin transport, have been reported over the past several years. Each newly identified term is thought to possess both a real and an imaginary effective field leading to fieldlike and dampinglike torques on magnetization. Here we show that three metallic ferromagnets possess an imaginary effective-field term which mirrors the well-known real effective-field term associated with exchange in spin waves. Using perpendicular standing spin wave resonance between 2-26 GHz, we evaluate the magnitude of the finite-wavenumber ($k$) dependent Gilbert damping of the uniform mode ($\alpha_u$) and the first spin wave mode ($\alpha_s$) in three typical ferromagnets, Ni$_{79}$Fe$_{21}$, Co, and Co$_{40}$Fe$_{40}$B$_{20}$. By taking the difference of $\alpha _s$ and $\alpha _u$ and excluding the eddy current damping $\alpha_E$ ($\Delta\alpha_k=\alpha_s-\alpha_u+\alpha_E$), we find the presence of a $k^2$ term, as $\Delta\alpha_k=\Delta\alpha_0+A_{k}\cdot k^2$ in all three metals. We interpret the new term as the continuum analog of spin pumping, predicted recently, and show that its magnitude, $A_{k}$=0.07-0.1 nm$^2$, is consistent with transverse spin relaxation lengths (1-3 nm) as measured by conventional spin pumping.