Mode- and Size-Dependent Landau-Lifshitz Damping in Magnetic Nanostructures: Evidence for Nonlocal Damping

We demonstrate a strong dependence of the effective damping on the nanomagnet size and the particular spin-wave mode that can be explained by the theory of intralayer transverse-spin pumping. The effective Landau-Lifshitz damping is measured optically in individual, isolated nanomagnets as small as 100 nm. The measurements are accomplished by use of a novel heterodyne magneto-optical microwave microscope with unprecedented sensitivity. Experimental data reveal multiple standing spin-wave modes that we identify by use of micromagnetic modeling as having either localized or delocalized character, described generically as end and center modes. The damping parameter of the two modes depends on both the size of the nanomagnet as well as the particular spin-wave mode that is excited, with values that are enhanced by as much as 40% relative to that measured for an extended film. Contrary to expectations based on the ad hoc consideration of lithography-induced edge damage, the damping for the end mode decreases as the size of the nanomagnet decreases. The data agree with the theory for damping caused by the flow of intralayer transverse spin currents driven by the magnetization curvature. These results have serious implications for the performance of nanoscale spintronic devices such as spin-torque-transfer magnetic random access memory.

Figure 1

Measured resonance fields for a 100 nm nanomagnet: The center mode (red circles) has the lowest resonance field followed by the end mode 1 (green squares) and end mode 2 (blue triangles). The solid lines are fits to Eq. (1). The inset shows a spectrum obtained at 13.2 GHz. The solid (red) line is a fit to Eq. (1) in the Supplemental Material [26].

Figure 2

Linewidths for the center modes (a) and (c) and end modes (b) and (d) for a 200 nm and a 100 nm nanomagnet. The insets show the mode profile along the long axis of the ellipsoid, as determined by micromagnetic simulations. The horizontal red line indicates zero amplitude.

Figure 3

(a) Experimental damping data: we plot the dependence of $\alpha$ on nanomagnet size. The black circles (red triangles) are the average values of the end modes (center mode). $\alpha$ for the $20\times 20{\mathrm{mm}}^2$ square is marked with a blue bar, where the width of the bar indicates the measurement precision. (b) Intralayer spin-pumping model: the red circles are the fitted values of $\alpha$ for the end mode and the black circles for the center mode. ${\tau }_{\mathrm{sc}}$ was the only fitting parameter. (c) Edge-enhanced damping model: the red circles are $\alpha$ for the end mode and the black circles for the center mode.