Introduction

Nonlinear phenomena are ubiquitous in nature, ranging from gravitational waves in the cosmos1 to spin waves (or magnons) in nanomagnets2,3. Among many intriguing nonlinear effects, harmonic generation is currently being intensively investigated. Because of appealing application prospects in information science and technology4, the higher harmonic generation of magnons has been pursued in a variety of magnetic systems5,6,7,8,9,10,11,12. A recent breakthrough is the demonstration of the harmonic generation up to the sixtieth harmonic in a ferromagnetic film at very weak magnetic fields, which was ascribed to the spin-wave Cherenkov effect on top of the ripple pattern of the magnetization13,14. However, the microscopic mechanism for the nonlinear spin-wave mixing is still elusive since the conceived spin-wave Cherenkov effect has not been directly observed in experiment14. Nevertheless, previous theory has suggested that frequency multiplication can be realized in topological spin textures like domain walls15, skyrmions16,17,18, and vortices19, because the large magnetization gradient can generate significant three-magnon processes that are indispensable for the formation of magnon frequency comb17. The extension of this mechanism to non-topological spin texture, like inhomogeneous edge magnetization, is still not clear.

The coherence of high-order spin waves generated by non-linear excitation is of vital importance if used as a source for coherent magnonic devices and information processing20,21. For instance, one may use the phase of the spin wave for information exchange and information processing without the Joule heating of charge transport22. For quantum information science, coherent spin waves provide an efficient route to realizing long-range quantum control of solid-state qubits (for example, color centers in diamond or SiC)23,24,25,26,27,28,29. Propagating spin wave-based hybrid quantum sensors significantly extend the frequency detection range30 and boost their sensitivity to magnetic signals31,32,33,34,35,36. Despite these promising applications, it is a non-trivial task to probe the coherent nonlinear spin dynamics of a geometrically confined magnetic layer, as the stray field decays fast with the distance from the thin film, and in-situ magnetic probes with good spatial and temporal resolution are demanded.

Here, we report on the harmonic generation in a ferromagnetic film detected by nitrogen-vacancy (NV) centers in nanodiamonds that serve as in-situ quantum sensors. We demonstrate the harmonic generation of magnons in ferromagnetic films in single or multiple magnetic domain states that can be controlled by an external magnetic field. The generated high-order spin waves drive the Rabi oscillations of the NV electron spin, which provides solid evidence of the high quality of their phase coherence. Analytical calculations and micromagnetic simulations show that nonlinear spin-wave excitations nucleate in magnetic textures at the device edges.

Results

Sample characterization and measurement setup

The device and measurement setup are shown in the left panel of Fig. 1a. A ferromagnetic film (permalloy (Py), Ni80Fe20) is deposited on top of the coplanar waveguide (CPW) made of a Pt(5)|Au(100)|Pt(5) (thicknesses in nanometers) film. A magnetic field applied during the deposition in a magnetron sputtering chamber imprints an easy axis on the Py film. The vibrating sample magnetometer (VSM) measurements in Fig. S4 (Please refer to the Supplementary Information at … (link) for technical details.) confirm the directions of easy and hard axes. Two types of devices were fabricated, in which the signal line (along the X axis) is parallel to the easy (device A) and hard axis (device B), respectively. Finally, we deposited NV-center containing nanoscale diamond crystals (average diameter of 100 nm). A microwave-frequency (rf) electrical current generates rf magnetic fields that drive the magnetization precession in the Py. At the resonance frequency (fESR) of the NV electron spin, the photoluminescence (PL) of the NV centers exhibits dips that enable optically detected magnetic resonance (ODMR)37,38. All the measurements we present were carried out at room temperature (25 ± 4°C) and a humidity of 30–40%.

Fig. 1: The measurement setup and measured ODMR spectra.
Fig. 1: The measurement setup and measured ODMR spectra.
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a Left panel: device for the detection of harmonic generation with nanodiamonds, emphasizing the “edges” of the magnetic film. The red arrow indicates the magnetic field (when applied). The red circle ‘S’ represents the microwave input, which is connected to the signal line. The blue arrow indicates the location of the magnetic edge domains along the Y direction (smaller ones along the X direction do not contribute to the high-order spin wave generation). Right panel: Kerr microscopy images (scale bar, 200 μm) for signal line S (X axis) parallel to the easy axis (upper picture) and hard axis (lower picture), respectively, at Bext = 0 mT after reducing the external magnetic field along X from a large value (~ 26 mT) to zero (white arrows indicate magnetization directions in the underlying domain); b, c ODMR spectra for the two devices at 0 mT and a non-magnetic reference. The driving rf power is 15.85 mW.

Harmonic generation in the magnetic films

Figure 1b and c shows the ODMR spectra without an applied magnetic field for the two devices. The magnetization was initially saturated along the X axis by a magnetic field of ~ 26 mT that was subsequently reduced to zero. Because of the in-plane magnetic anisotropy, device A (device B) with an easy (hard) axis along the X axis settles into a single (multiple) domain state, as shown by the magneto-optic Kerr microscopy (MOKE) image in the right panel of Fig. 1a (see also Fig. S5 (Please refer to the Supplementary Information at … (link) for technical details.)). We observe a clear frequency comb in the ODMR signal of both devices. The resonance dips appear at excitation frequencies of 2.87/n GHz (n is an integer), as reported previously14. The comb lines emerge when the NV resonance frequency fESR (= 2.87 GHz under zero field) is a multiple of the microwave frequency. At the first resonance dips (n = 1), the microwave field matches the ESR frequency of the NV spins, while all resonance dips with a frequency below 2.87 GHz have to be attributed to the ferromagnetic layer. A microwave with a frequency of 1.43 GHz, for example, causes a signal via the stray field of a second harmonic magnon. Further evidence is the absence of the resonance dips in a non-magnetic reference sample (dotted blue line in Fig. 1b, c). We observe similar spectra for a CoFeB layer (see Fig. S6 (Please refer to the Supplementary Information at … (link) for technical details.)), which shows the robustness of the phenomenon of harmonic generation of magnons in other magnetic materials. Importantly, the harmonic generation appears for both the single domain (see the top MOKE image of the right panel in Fig. 1a) and multidomain (see the bottom MOKE image of the right panel in Fig. 1a) states. Device A displays a stronger signal than device B under a zero magnetic field, as depicted in Fig. 1b, c. Below, we attribute this difference to the magnetic domain walls in device B that impede the propagation of spin waves generated at the edges and do not efficiently generate high-order harmonic spin waves15, even though domain walls, as nonuniform magnetic structures, are also considered as a source of high-order spin waves. The magnetic ripple domain thought to be essential in previous work14 is therefore not a necessary ingredient for the higher harmonic generation.

Magnetic field and microwave power dependence of harmonic generation

Next, we assess the effect of an applied magnetic field (Bext). Figure 2a shows the ODMR signals of device A when Bext is along the X axis (those of device B are referred to Fig. S7 (Please refer to the Supplementary Information at … (link) for technical details.)). All the magnetic fields are set by scanning a large ~ 26 mT (larger than saturation field) to the target value. The high-order harmonic gradually decreases and eventually disappears with increasing Bext. Moreover, the high-order harmonics of device A appear at a lower magnetic field than in device B, presumably because of the additional magnetic anisotropy, as argued below.

Fig. 2: The magnetic field strength and power dependence of the frequency comb.
Fig. 2: The magnetic field strength and power dependence of the frequency comb.
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a ODMR data for the device with signal line parallel to the easy X axis for Bext ∈ {0 – 4.0} mT. The driving rf power is 15.85 mW. b Power dependence of ODMR spectra under zero magnetic field. The driving rf powers are 1 mW (upper panel) and 10 mW (lower panel). c, ODMR contrasts of higher-order signals with n = 1, 5, 10, 15 and 20 as a function of power.

According to Fig. 2b, c, an increasing microwave power enhances the features of the frequency comb (see also Fig. S12 (Please refer to the Supplementary Information at … (link) for technical details.)). The direct excitation of the ESR and therefore the ODMR signal of the first resonance dip increases with power. The low-order resonance dips show a similar power dependence, implying their excitation does not require a clear threshold, in contrast to, e.g., their parametric excitation. At even higher powers, the low-order harmonics saturate while higher-order resonance dips emerge gradually, consistent with the threshold-less harmonic generation predicted by micromagnetic simulations of topological magnetic textures17. The observed frequency comb is therefore the result of a nonlinear confluence of highly excited low frequency magnons.

Micromagnetic Simulations

We interpret the experimental findings with the help of micromagnetic simulations of a two-dimensional Py film with an easy X axis using the Mumax3.0 code39. Decreasing the bias magnetic field from saturation to zero generates the initial magnetization displayed in Fig. 3a, where we have considered roughness along all edges of the device. A microwave field along Y with frequency f0 = 2.87/3 GHz then drives the magnetization dynamics. Figure 3b shows the fast Fourier transformation (FFT) spectrum of the magnetization dynamics at the orange point in the bulk of the film in Fig. 3a. It reveals a series of resonance lines at the frequencies nf0 as observed in the experiments. The stray fields at the edges are smaller than in the bulk, thereby causing a triangular-like potential that confines spin wave modes, also referred to as spin-wave well state40,41,42, with frequencies below the bulk magnon gap. Moreover, inhomogeneous spin textures near the edges can significantly enhance the nonlinear spin-wave mixing15,16. Therefore, the edge region efficiently generates higher harmonics. The harmonic generation disappears when there is no magnetic texture at the film edge, as shown in Fig. S15 (Please refer to the Supplementary Information at … (link) for technical details.). We hence conclude that the inhomogeneous dipolar fields at the edges allow the formation of non-uniform magnetic textures. In these edge states, the cancelation of torques stabilizes low-frequency in-gap edge-bound magnon modes that are essential for the observed harmonic generation of up to 55 orders (see Figs. S8 and S9 (Please refer to the Supplementary Information at … (link) for technical details.)). We also compute the dependences on the driving field amplitude. Spin-wave amplitude in Fig. S17 (Please refer to the Supplementary Information at … (link) for technical details.) increases with input power but saturates at a certain point. A larger ac field is required to reach saturation for high-order harmonics (0.5 mT for n = 1, 1.5 mT for n = 2, and 2.5 mT for n = 3), in agreement with the experimental results shown in Fig. 2c. As the harmonic generation effect is due to the nonlinear spin dynamics in a geometrically confined soft magnetic layer, it can be affected by the device geometry and damping constant of the permalloy. More details can be found in the Supplementary Materials (Please refer to the Supplementary Information at … (link) for technical details.).

Fig. 3: Micromagnetic simulations.
Fig. 3: Micromagnetic simulations.
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a The static magnetization at 0 T after a magnetic field scan from a saturation field along the +X axis to zero. The orange dot is the detection position. (The scale bar in Fig. 3a is 5 μm) (b) The FFT spectrum of the computed magnetization time series at the orange dot marked in Fig. 3a under a microwave field with frequency f0 = 2.87/3 GHz. The errors margins of the FFT spectrum are plotted as the gray region. c The dispersion relation of spin waves along the +X axis under a zero magnetic field. The green color and black curve are obtained from the simulations and Eq. (1), respectively. The blue curve depicts the dispersion of the surface spin waves47. d The mode profiles of spin waves with the frequencies f0, 2f0 and 3f0. f0 is the frequency of an external RF magnetic field far below the spin wave gap of the bulk material (Fig. 3c) that resonates with the low-frequency edge modes. Since it penetrates the entire sample, it also reactively (non-dissipatively) drives the bulk magnetization as seen in the upper panel of Fig. 3d. The middle panel of Fig. 3d illustrates that the waves with frequency 2f0 below the band gap decay away from the edge. Those at 3f0 belong to the continuous spin-wave spectrum above the band gap and may propagate, as shown in the lower panel of Fig. 3d.

The second resonance peak in Fig. 3b is much weaker than the others. To understand this behavior, we plot the spin-wave dispersion along the X axis parallel to the external field \({H}_{0}\hat{X}\). Micromagnetic simulations agree with the analytical formula (see black curve Fig. 3c)43,44,45:

$$\omega=\sqrt{\left({A}^{*}{k}^{2}+{\omega }_{H}+{\omega }_{K}\right)\left({A}^{*}{k}^{2}+{\omega }_{H}+{\omega }_{K}+{\omega }_{m}{F}_{p}\right)}$$
(1)

where A* = 2γA/MS in terms of gyromagnetic ratio γ, exchange constant A, and saturated magnetization MS. We also used ωH = γH0, ωK = 2γKu/MS with anisotropy constant Ku for an easy axis along \(\hat{X}\), ωm = γμ0MS, the dipole-dipole matrix element FP = 1+gkcos2θk+gk(1−gk)/(ωH + ωK + A*k2) with gk = 1−[1−exp(−kd)]/(kd)46, and Py film thickness d. The spin wave state with frequency 2f0 is a localized edge state in the magnon band gap, as shown in Fig. 3d. On the other hand, third and higher-order waves lie above the band gap and may propagate to the detection point deep in the film. It is noted that the dispersion of Damon-Eshbach (DE) magnons (propagating along the Y axis)47 is much steeper (see blue curve in Fig. 3c) and is thus irrelevant to our experimental finding.

The coherence of higher harmonic generation

Next, we assess the phase coherence of the multiplied spin waves by time-dependent measurements of the NV electron spin Rabi oscillations. As shown in Fig. 4a, a 3-μs green laser pulse initializes the NV spin to the ms = 0 states. Microwave pulses of different durations are then applied to manipulate NV spin state. A second laser reads out the final state through spin-dependent fluorescence. Figure 4c presents typical Rabi oscillations driven now by the second harmonic spin-wave (fESR/2 = 1.417 GHz) and different input powers. We also observed Rabi oscillation driven by the third harmonic spin wave (see Fig. 4d). The observed Rabi oscillations indicate that the high-order spin waves stay phase-coherent for at least a few hundred nanoseconds, which is an essential requirement for quantum control and related applications. The Rabi frequencies plotted as a function of the input microwave power (see Fig. 4e) illustrate the good efficiency of manipulating NV spin state by the higher-order AC stray fields as compared with Rabi oscillations of the same NV centers driven by the microwave directly (fESR = 2.834 GHz, Fig. 4a)33. Both coherent and incoherent (thermal noise) spin waves coexist in the NiFe film, and the latter deteriorates the performance of possible quantum devices. Nevertheless, the demonstration of robust phase-coherence of the multiplied spin waves and the enlarged frequency detection window implies that a hybrid sensor composed of a NiFe film and NV centers could be useful for wide-band magnetic frequency of signal detection. The observation of coherent Rabi oscillations up to the third harmonic but absence at higher ones gives us an indication under what conditions higher harmonic modes lose their phase coherence. One physical reason is that, as the order increases, the magnon-magnon or magnon-phonon interactions act as an effective thermal bath for the high-order magnon, which causes significant dephasing effect. If the spin wave dephasing by magnon-magnon and magnon-phonon interactions or Joule heating through eddy currents prevents observation of the higher-order Rabi oscillations, we should repeat the experiments at low temperatures.

Fig. 4: Phase coherence of higher spin-wave harmonics.
Fig. 4: Phase coherence of higher spin-wave harmonics.
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a Pulse sequences to measure Rabi oscillation frequencies and spin relaxation times (T1) of NV centers in diamonds. b Rabi oscillations driven by the 1st harmonic signal at different input powers. The applied microwave frequency is 2.834 GHz. c Rabi oscillations driven by the 2nd harmonic signal at different input powers. The applied microwave frequency is 1.417 GHz. d Rabi oscillations driven by the 3rd harmonic signal at different input powers. The applied microwave frequency is 0.938 GHz. e Power dependence of the Rabi frequency, n = 1 is the 1st, and n = 2 is the 2nd harmonic. f Spin relaxation signal of the NV centers in the nanodiamond in Fig. 4a–f.

The proximity of the ferromagnet increases the attenuation of the coherent Rabi oscillations. The observed spin relaxation time T1 = 1.7 ± 0.2 μs (Fig. 4f) of the NV centers on the NiFe film is about two orders of magnitude shorter than in bare nanodiamonds (~100 μs)48 because at the small bias magnetic fields (~ 6.4 mT), the magnetic noise of the NiFe film at the resonant frequency (2.768 GHz) stimulates the NV spin relaxation. The short T1 time also explains why the Rabi oscillations driven by high-order spin waves occur only for a rather narrow window of the input power. The NV spin polarization should not decay on the scale of multiple coherent oscillations, which sets a lower bound for the input power, while the strong magnetic noise, on the other hand, suppresses the signals at high input powers. Numerical simulation indicates that strong driving fields bring large-angle deformation of the equilibrium magnetization (see Fig. S18 (Please refer to the Supplementary Information at … (link) for technical details.)). Another possible reason is that higher intensities create additional magnetic textures (see simulation results in Fig. S19 (Please refer to the Supplementary Information at … (link) for technical details.)). Another option would be harmonic spin wave generation by topologically protected spin textures such as skyrmions and hopfions, with possible higher coherence of the emitted spin waves.

Discussion

In summary, we report magnonic harmonic generation in a soft ferromagnet as detected by diamond NV center spectroscopy. The observed Rabi oscillation is evidence of good phase coherence of higher-order spin waves in our devices. Since the driving frequency is well below the magnon band gap, the nonlinear spin-wave mixing induced by the nonuniform magnetic structures at the film edge is instrumental. We thus exclude the domain-switching as the dominating origin for explaining the observed high-harmonic spin-waves. The spin-texture-induced magnon frequency comb may help open the door for magnon-based information processing and be beneficial to wide-band magnetic signal detection.

Methods

Films growth and sample preparation

The multilayer films were deposited on Si/SiO2 wafers using magnetic sputtering systems with a base pressure of less than 8×10−7 Pa. To produce a uniform film, the sample must rotation at a rate of approximately π/5 rad/s during sputtering due to the tilted magnetron target. A permanent magnet attached to the sample holder induces a magnetic field of 0.18 mT, aligning the easy-axis or hard-axis of the magnetic films (NiFe and CoFeB) in a predetermined direction. The specific description of the devices and the relevant discussion about the two systems are shown in Table S2. The specific film growth conditions are shown in Table S1. The coplanar waveguides (CPWs) were patterned using ultraviolet (UV) lithography and the excess materials were subsequently removed through Ar ion milling. The detailed progress is depicted in Fig. S1. Figure S2 illustrates the structure of a device captured under optical microscopy.

Deposition of nanodiamonds

The nanodiamonds, which contain an ensemble of nitrogen-vacancy (NV) centers (Adámas Nanotechnologies, diameter ~100 nm, NV concentration ~3 ppm), are suspended in an isopropanol solution (5 μg/ml). The deposition of nanodiamonds was accomplished by transferring a drop of ~10 μl ND ethanol solution to the device with pipette. Nanodiamonds were deposited on the device surface after the volatilization of isopropanol.

NV magnetometry and Rabi oscillations

The nanodiamonds contain an ensemble of nitrogen-vacancy (NV) centers (Adámas Nanotechnologies, diameter ~100 nm, NV concentration ~3 ppm) and are deposited on the device described above. Measurements are performed in a home-built scanning confocal microscope setup, and the diagram of the ODMR system is shown in Fig. S3. Samples are illuminated by a green solid-state 532-nm laser (CNI, MLL-III-532-150mW) with a dedicated acousto-optic modulator (Gooch & Housego, 3350-199). The light is sent into a microscope frame (Olympus, IX73) equipped with dichroic mirrors and high-NA objectives (Olympus, MPlanFL N100X/0.9). The fluorescence signal is detected through a 650-nm long-pass filter (semrock) by an avalanche photodiode (Excelitas, APCM-700-10-FC). Scanning and positioning are realized by a 3-axis piezo-scanning stage (Physik Instrumente, E-727.3CDA). During the ODMR measurements, the signal generator (Rohde & Schwarz SMIQ06B, or Keysight N5183B) continuously sends microwaves (MW) into the device directly. The fluorescence counts of the NV centers are recorded as a function of the MW frequency. For the Rabi measurements, the MW signal is pulsed by an RF switch. A high-power MW amplifier (Mini-Circuits, ZHL-16W-43-s + ) is used in measuring Rabi oscillations driven by resonant microwave pulses. The amplifier is not used when measuring the Rabi oscillations driven by high-order spin waves.

Measurement protocols and data analysis steps

Each measurement was started by initially saturating the magnetization along the X axis using a magnetic field of ~ 26 mT, and subsequently reducing it to zero. The ODMR spectra were measured by recording the photoluminescence (PL) counts of NV centers while the applied microwave frequency was swept. The integration time of each microwave frequency is 1 ms, followed by a referenced count of the same duration (no MW). The measurement sequence was repeated 3000 times, and for each microwave frequency, the sum of signal counts was normalized by the sum of the referenced counts. To do so, we can eliminate quasi-static fluctuation of the laser power.

Model or data processing code

We carried the micromagnetic simulations out using the software package Mumax3.0 with RF field polarized along the y-direction. We discretized the film into cuboids of dimensions 10×10×20 nm3. We consider a ferromagnetic film with dimensions 20×10×0.02 μm3, which is saturated by a bias magnetic field. Then, by decreasing the bias field from the saturation to zero, an initial state is obtained, as shown in Fig. 3a. A microwave field along y direction with frequency f0 = 2.87/3 GHz is applied on the whole ferromagnetic film to drive the magnetization dynamics. By the fast Fourier transformation (FFT) of the dynamic magnetization component δmz(t) at the orange point in the bulk of the film in Fig. 3a, the spin-wave spectrum δmz(f) is obtained, as shown in Fig. 3b. The space distribution of spin-wave modes at resonance lines in Fig. 3d is extracted from the FFT spectrum of the dynamic magnetization over the whole film. The spin-wave dispersion in Fig. 3c is obtained by the FFT of the dynamic magnetization excited by a sinc-function field. Micromagnetic simulations are performed by using the software Mumax3.0. In Mumax3.0, the RF field along the y-direction can be easily applied via the code ‘B_ext = vector (0, h0sin(2πf0t), 0)’. In simulations, a cuboid with the size 10×10×20 nm3 was used to discretize the film sample. The present simulations do not account for defects, magnon-phonon coupling, or Joule heating that require further investigation.

Micromagnetic model parameters

Magnetic parameters used in micromagnetic simulations are as follows: the saturated magnetization MS = 8.6×105 A/m, exchange constant A = 13 pJ/m, uniaxial anisotropy constant Ku = 2.53×103 J/m3 with the easy axis along the X direction, and the damping parameter α = 0.01.