Measurement of the Dynamical Dipolar Coupling in a Pair of Magnetic Nanodisks Using a Ferromagnetic Resonance Force Microscope

We perform a spectroscopic study of the collective spin-wave dynamics occurring in a pair of magnetic nanodisks coupled by the magnetodipolar interaction. We take advantage of the stray field gradient produced by the magnetic tip of a ferromagnetic resonance force microscope to continuously tune and detune the relative resonance frequencies between two adjacent nano-objects. This reveals the anticrossing and hybridization of the spin-wave modes in the pair. At the exact tuning, the measured frequency splitting between the binding and antibinding modes corresponds to the strength of the dynamical dipolar coupling $\Omega$. This accurate ferromagnetic resonance force microscope determination of $\Omega$ is measured versus the separation between the nanodisks. It agrees quantitatively with calculations of the expected dynamical magnetodipolar interaction in our sample.

Figure 1

(a) Schematic of the f-MRFM setup. An Fe sphere glued at the apex of a soft cantilever is scanned laterally above different pairs of Fe-V disks excited by a microwave field. Inset: SEM image of the tip. (b) Density plot of the f-MRFM signal versus the displacement $x$ of the sphere above an isolated disk. Inset: SEM image of the $2R=600\mathrm{nm}$ Fe-V disk placed below the microwave antenna.

Figure 2

(a) Density plot of the experimental f-MRFM spectra versus the displacement $x$ of the sphere above a pair of 600 nm Fe-V disks separated by $s=200\mathrm{nm}$. (b) Predicted behavior by micromagnetic simulations. The upper mode (blue) corresponds to the antibinding mode ($A$), while the lower (red) shows the binding mode ($B$). Insets: Simulated precession profiles in each disk for modes ($A$) and ($B$) at the anticrossing. The dashed lines would be the individual modes of each disk in the absence of dynamical coupling.

Figure 3

(a) Variation of the amplitude in arbitrary units of the binding ($B$, red) and antibinding ($A$, blue) resonances versus the lateral position of the probe for the two disks separated by 200 nm. The solid lines correspond to the behavior following from Eq. (4) (see text). (b) Linewidth of the binding mode for the same pair at the tuning position ($x=0$). (c) Linewidth of the single disk. Solid red lines are fit by Lorentzian curves.

Figure 4

Coupling strength versus the separation $s$ between two disks. The plot compares the experimental findings to the predicted amplitude of the magnetodipolar interaction either analytically (continuous line) or by micromagnetic simulations (small dots, dashed line is a guide to the eye).