Imaging and phase-locking of non-linear spin waves

Non-linear processes are a key feature in the emerging field of spin-wave based information processing and allow to convert uniform spin-wave excitations into propagating modes at different frequencies. Recently, the existence of non-linear magnons at half-integer multiples of the driving frequency has been predicted for Ni80Fe20 at low bias fields. However, it is an open question under which conditions such non-linear spin waves emerge coherently and how they may be used in device structures. Usually non-linear processes are explored in the small modulation regime and result in the well known three and four magnon scattering processes. Here we demonstrate and image a class of spin waves oscillating at half-integer harmonics that have only recently been proposed for the strong modulation regime. The direct imaging of these parametrically generated magnons in Ni80Fe20 elements allows to visualize their wave vectors. In addition, we demonstrate the presence of two degenerate phase states that may be selected by external phase-locking. These results open new possibilities for applications such as spin-wave sources, amplifiers and phase-encoded information processing with magnons.


Supplementary Figures:
S1. Response at f rf and 3/2 f rf for different sample geometries. S2. Threshold analysis of 3/2 f rf NLSWs. S3. Micromagnetic simulations of the field resolved spin-wave spectra analyzed at different frequencies with different spatial variations of the saturation magnetization. S4. Experimentally obtained signal amplitudes at different frequency components. S5. Wave vector analysis as a function of the rf-frequency. S6. Micromagnetic simulations of spatially-resolved NLSW excitations with different spatial variations of the saturation magnetization. S7. Influence of the in-plane field orientation on NLSW generation. S8. NLSW pattern in 5 µm × 4 µm element with seed frequency. S9. Micromagnetic simulation of non-linear spin-wave generation in different element geometries. S10. NLSW signal obtained at different seed frequencies. S11. Micromagnetic simulation of the phase-locking of NLSWs to an external seed frequency.

Fig. S1
| Response at f rf and 3/2 f rf for different sample geometries. We performed field swept measurement with increasing rf-power levels in different sample geometries to identify suitable conditions for coherent NLSW generation. Panels a, b and c show the ferromagnetic resonance spectra for an extended layer of 20 nm thick Ni 80 Fe 20 , a 30 µm × 15 µm ellipse and a 5 µm × 4 µm rectangular element, respectively. All measurements indicate the onset of non-linear processes by showing a phase shift and a saturation of the uniform mode. However, only in panel e a phase coherent non-linear response is detectable at 3/2 f rf for the 30 µm × 15 µm ellipse, while the other two geometries show no phase-stable NLSWs as shown in d and f. Moreover, we were able to obtain NLSW generation in all rectangular and elliptical structures with dimensions larger than 20 µm × 10 µm.  | Micromagnetic simulations of the field resolved spin-wave spectra analyzed at different frequencies with different spatial variations of the saturation magnetization. Rf-power dependent simulations of the field-resolved spin-wave spectra. (a)-(c) The imaginary part of the response is analyzed at the driving frequency which results in typical Lorentzian FMR line shapes for low-power driving amplitudes. For increasing rf-power levels the phase starts to shift and the mode saturates. From panel a to c the spatial variation of M s was increased from 0 % up to 10 % (the size of the grains is 100 nm). For larger variations of M s the FMR condition is shifted towards lower fields, as depicted in b and c (the dotted red line shows the low power FMR condition for 0 % variation). Panel d-e show the imaginary part of the 1/2 f rf response while g-i display the 3/2 f rf components. Note that the field range in which NLSW generation appears is broad. The experimentally obtained coherent part of the spectrum is on the order of 1 mT, while the incoherent part accessible by the simulation covers a field range of 3 mT (corresponding to the line width of the ferromagnetic resonance) with a symmetric distribution around the FMR condition. We compare the different signal amplitudes obtained at 1/2 f rf , 3/2 f rf and 5/2 f rf shown in a, b and c, respectively. The signal ratio taken from such measurements between 3/2 and 1/2 f rf is on the order of 6:1, while the ratio from 3/2 and 5/2 is on the order of 3:1. The wave vector components in the 1/2 and 3/2 f rf channel appear to be similar, while the 5/2 f rf signal contains higher-order wave vector components.   The response of the system is detected as a function of the external bias field for different in-plane field orientations at f rf , 1/2 f rf and 3/2 f rf , respectively. At ±90 • the uniform response in a is strongly suppressed while three-magnon scattering effects give rise to signal components at 1/2 f rf in b. This corresponds to the parallel pumping geometry. In the range between ±45 • the dominant non-linear effect becomes visible at 3/2 f rf in c. The obtained spin-wave pattern from spatially-resolved imaging at 3/2 f rf for different angular orientations is demonstrated in d and e. In d the external field's direction is changed by 32 • resulting in a rotation of the NLSW pattern obtained by performing a 2D-FFT. A change of −34 • leads to the pattern shown in b. As indicated by red circles, the symmetry of the pattern obtained for 0 • orientation changes and the spin waves tend to align along the direction with smaller k y components.  a and b, respectively. While panel a shows a mostly uniform signal across the entire structure the 3/2 f rf signal in b consists of a complex spin-wave pattern superimposed with a more uniform background due to the direct excitation at this frequency. In this element, the obtained NLSW pattern only becomes accessible due to the applied seed frequency component. By utilizing a 2D-FFT the wave vector components of this pattern can be accessed as presented in c. Note that for a better visibility of the NLSW we subtracted the direct excitation background.

Fig. S9
| Micromagnetic simulation of non-linear spin-wave generation in different element geometries. We performed micromagnetic simulations for a 5 µm × 4 µm rectangular and a 2 µm × 1 µm elliptical shaped element with magnetic properties of Ni 80 Fe 20 as shown in a and c, respectively. We analyzed the response of the system at f rf and 3/2 f rf . Above threshold condition a non-linear spin-wave pattern with the expected wave vector components becomes visible. Note that the resonance condition for small elements shifts significantly to larger frequencies.
Fig. S10 | NLSW signal obtained with different seed frequencies. The spatially-resolved NLSW signals oscillating at 3/2 f rf is recorded with different seed frequencies in the 20 µm × 10 µm ellipse. In panel a and d half-integer seed frequencies of 1/2 f rf and 5/2 f rf were used, respectively. These seed components result in a purely non-linear signal in the read-out channel since the detected NLSW signal at 3/2 f rf contains no additional background due to direct excitations as shown in b where a 3/2 f rf seed was used. In order to illustrate, that half-integer seed frequency signals are required, we show as an example f seed = 2 f rf in panel c. Here, no phase stability of the NLSW excitation is obtained. The seed power levels were set to 0 dBm at the sample.
Fig. S11 | Micromagnetic simulation of the phase-locking of NLSWs to an external seed frequency. In the presented micromagnetic simulations we investigate the influence of an additional seed frequency at 3/2 f rf with an amplitude of 0 dBm. We performed the simulations for a 5 µm × 4 µm rectangular shaped element with magnetic properties of Ni 80 Fe 20 and analyzed the non-linear response of the system as a function of the phase shift ∆θ. Panel a-c show spatially-resolved images of the NLSW signal's phase obtained at 3/2 f rf for three different phase shifts ∆θ. One clearly observes a reversal of the signal's sign from a to c. In analogy to our experiments, we analyzed the phase and the magnitude as a function of the phase shift ∆θ in e and f. Here, the responses at 1/2 f rf (green), 3/2 f rf (blue) and 5/2 f rf (violet) are shown, respectively. In panel d, the normalized magnitude shows the expected 180 • periodicity for all analyzed channels (scaling factors on the right). While the phase for the 3/2 f rf signal in e appears mostly linear, the phase values for both other NLSW channels indicate the presence of two distinct phase states. The mostly linear behavior of the phase at 3/2 f rf is due to a superposition of the NLSW signal at this frequency and the direct excitation at the seed frequency with similar wavev ector.