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 $\alpha$ in three typical device ferromagnets, Ni$_{ 79}$Fe$_{ 21}$, Co, and Co$_{ 40}$Fe$_{ 40}$B$_{ 20}$, and demonstrate for the first time the presence of a $k^2$ term as $\Delta\alpha=\Delta\alpha_0+A_{ k}\cdot k^2$ 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 as measured by conventional (interlayer) spin pumping.
Categories: cond-mat.mtrl-sci