CORRIGENDUM: Tunable negligible-loss energy transfer between dipolar-coupled magnetic disks by stimulated vortex gyration

A wide variety of coupled harmonic oscillators exist in nature. Coupling between different oscillators allows for the possibility of mutual energy transfer between them and the information-signal propagation. Low-energy input signals and their transport with negligible energy loss are the key technological factors in the design of information-signal processing devices. Here, utilizing the concept of coupled oscillators, we experimentally demonstrated a robust new mechanism for energy transfer between spatially separated dipolar-coupled magnetic disks - stimulated vortex gyration. Direct experimental evidence was obtained by a state-of-the-art experimental time-resolved soft X-ray microscopy probe. The rate of energy transfer from one disk to the other was deduced from the two normal modes' frequency splitting caused by dipolar interaction. This mechanism provides the advantages of tunable energy transfer rates, low-power input signals and negligible energy loss in the case of negligible intrinsic damping. Coupled vortex-state disks might be implemented in applications for information-signal processing.

direction of the incident X-rays. Local field pulses of ~3 mT strength (corresponding to a current density of 7.0×10 6 A/cm 2 ), 90 ns length and 2.5 ns rise-and-fall time were applied along the x axis on which the two disks in each pair were positioned. The electronic excitation pulses were synchronized to the X-ray pulses at ~3 MHz frequency. To obtain time-resolved x-ray images, the current pulses were delayed from 0 to 200 ns in time steps of 0.78 ns or 0.83 ns (eight times per cycle period of vortex gyration, at a frequency of 160 MHz or 150 MHz). In order to achieve adequate XMCD contrasts, ten individual images that had been measured at identical delay times were accumulated. The structural contrast was normalized to an image obtained under a saturation field.

B. Classical coupled oscillator model
In one-dimensional (1D) coupled harmonic oscillators with the corresponding linear restoring force, the coupled motions of both oscillators can be expressed by a force balance S1a and S1b. In Fig. S1c, the total energy of the system is constant and the energy of each oscillator is transferred repeatedly between the two oscillators for the case of no damping.
For the case of damped oscillation, the force balance equation can be rewritten as . Thus, the displacements are given as Calculations of the energy and displacement of the oscillators with damping are shown in Fig. S1d. Comparing the cases of no damping with certain levels of damping (see Figs. S1b and S1d), it was found that some damping intrinsically results in mismatch between the nodes of one oscillator and the anti-nodes of the other.

C. Micromagnetic simulation procedure and results
We performed micromagnetic simulations of coupled vortex gyrations under free relaxation for several pairs of Py disks of different center-to-center distances d int /(2R). We To mimic the experimental conditions, we displaced the initial vortex-core position of disk 1 to −37.5 nm in the y axis, by applying a static field of + 15 mT along the x axis and then allowing it to relax. This displaced vortex core showed a CCW spiral motion with decreasing amplitude (Fig. S2b, right). In contrast, the vortex core initially positioned at the center of disk 2 gyrated in a CW sense with increasing amplitude (Fig. S2b, left). Figure  is known to be caused by dipolar interaction between the two separated vortex oscillators. In isolated single-vortex-state disks, the eigenfrequency is calculated to be 580 MHz, which is close to the average over those higher and lower frequencies. In

D. Energy attenuation
The energy attenuation during vortex-gyration-mediated signal transfer can be represented by the ratio of the maximum amplitude of oscillation of disk 2 to that of disk 1.
On the basis of the analytical forms of We additionally examined the propagation of stimulated vortex gyrations in 1D arrays consisting of two more disks (i.e. ten disks) made of Py and NiMnSb, during free relaxation. In the simulations, we used the same means of first-disk excitation as in the main paper, but employed much smaller disks [2R = 201, L = 6 nm and d int /(2R) ~ 1.04] to reduce the computation time. Figure S4 shows the model geometry as well as the displacements of the vortex-core position vectors of the individual disks (up to the 5th disk) in the chains as a function of time. For such longer chains, the attenuation of energy transfer cannot be estimated simply by the parameter /   , as explained above. Further study is necessary.
Here, we sought to interpret the attenuation of energy transfer through chains qualitatively. As can be seen, from the first to the second disk, the ratio of the maximum amplitude |X i | of the oscillation of disk (i) to |X i-1 | of disk (i-1) was relatively low, that is, |X 2 |/|X 1 | ~ 0.61 (0.69) for Py (NiMnSb). However, from the second disk to the third, and from the third to the fourth, and so on, the values of |X i |/|X i-1 | for Py (NiMnSb) increased to the 0.87~0.88 (0.93~0.97) range. The relatively small value of |X 2 |/|X 1 | compared with those of |X i |/|X i-1 | resulted from the fact that the stored energy in the first disk, upon the commencement of free relaxation, dissipates through not only the second disk but also the third one. Comparing the Py and NiMnSb chains, the smaller damping constant resulted in a more significantly reduced attenuation effect. All of the above results confirm that information signal transfer by stimulated vortex gyration exhibits low attenuation; further, they show that low energy dissipation can be achieved with low-damping-constant materials such as NiMnSb. The results suggest also that logic functions based on stimulated gyrations in coupled disks can be conducted with signal transfer only through the nearest (first) and second-nearest neighboring disks.

E. Normal-mode representation
As noted in Section C, the observed coupled vortex gyrations can be interpreted in terms of a coupled harmonic oscillator system. The vortex cores move in the plane of the disks. To understand the two different modes existing in coupled vortex oscillators, complicated vortex-core oscillations in real space can be represented, on the basis of the normal-mode coordinates (N1 X = x 1 +x 2 , N1 Y = y 1 -y 2 ) and (N2 X = x 1 -x 2 , N2 Y = y 1 +y 2 ), by coordinate transformations. In this normal-mode coordinate representation, the two normal modes have their own mode frequencies, which, with damping, act as corresponding uncoupled oscillators. Figure S5 is a normal-mode representation of the experimentally observed coupled vortex gyrations under free relaxation for the case d int /(2R) = 1.10.
Compared with the case of d int /(2R) = 1.05, the interaction, as discussed in the main text, is relatively weak. Thus, the frequency splitting for d int /(2R) = 1.10 is smaller than that for 1.05.
In the frequency spectra, the frequencies for both normal modes are close to each other, and thus, due to the limited experimental resolution in the present case, were not resolved.

F. Frequency splitting and energy transfer rate
A beating pattern in a simple coupled harmonic oscillator contains information on the frequency splitting caused by coupling between the individual oscillators. Thus, the frequency splitting can be determined by fitting the modulation envelopes of the vortex-core displacements in both disks (Fig. S2e) to the corresponding envelope functions, here cos( / 2) t  and sin( / 2) t  in disks 1 and 2, respectively, with the damping term exp( ) t

G. Tunable energy transfer by varying interdistance
Since a dipolar interaction between separated magnetic disks changes with their interdistance, the frequency splitting caused by the interaction also varies with d int /(2R).

H. Relative polarization configuration dependence
We performed micromagnetic simulations for two cases, the same and different vortex-core polarization, with the same geometry noted in Fig. S2, and d int /(2R) = 1.05. As shown in Fig. S8, the vortex-core gyrations were distinctly different; the energy transfer rate for the same polarization was two times slower ( ex  = 10.6 ns) than that ( ex  = 4.9 ns) for the opposite polarization.

I. Oersted field distribution generated by current flow
To generate the exciting Oersted fields, a pulse current was applied along the Cu stripline. Based on Ampere's law with the value of current (I), flowing through the Cu stripline, calculated by using Ohm's Law I = V out /Z with the measured output voltage V out = 400 mV and the impedance of the used oscilloscope of Z = 50 Ω, we obtained the local Oersted field distributions. S5 Figure S9 shows the calculation results (the blue dashed line through the centers of the two disks). The Oersted field (~ 3 mT) is concentrated mostly on disk 1 located under the Cu stripline, whereas the field is negligible at the position of disk 2.
Referencing a sample that had no Py disk beneath the long Cu electrode, we were able to confirm that the local field did not excite either disk far from the stripline electrode.

Supplementary Figures
Supplementary Figure