Frequency multiplication by collective nanoscale spin-wave dynamics

Frequency multiplication is a process in modern electronics in which harmonics of the input frequency are generated in nonlinear electronic circuits. Devices based on the propagation and interaction of spin waves are a promising alternative to conventional electronics. The characteristic frequency of these excitations is in the gigahertz (GHz) range and devices are not readily interfaced with conventional electronics. Here, we locally probe the magnetic excitations in a soft magnetic material by optical methods and show that megahertz-range excitation frequencies cause switching effects on the micrometer scale, leading to phase-locked spin-wave emission in the GHz range. Indeed, the frequency multiplication process inside the magnetic medium covers six octaves and opens exciting perspectives for spintronic applications, such as all-magnetic mixers or on-chip GHz sources. Description Magnetism hits the high notes The generation and propagation of magnetic excitations such as magnons and spin waves in ferromagnetic thin films provides a platform for the development of spin-based device technology. Koerner et al. report measurements on the magnetization dynamics of a nickel–iron film excited coherently by microwave magnetic fields from a coplanar wave guide. Using nitrogen vacancy center–based magnetometry and a time-resolved magneto-optical Kerr effect, the authors show that low excitation frequencies and low bias fields in the range of only a few milli-tesla results in the generation of magnons emitted at higher frequency. Extending over 60 harmonics of the excitation frequency, such upconversion of magnetic excitation frequencies should prove useful for spintronics applications. —ISO Magnetic excitations in a thin ferromagnetic film can be upconverted from megahertz to gigahertz frequencies.

Frequency multiplication is a process in modern electronics in which harmonics of the input frequency are generated in nonlinear electronic circuits. Devices based on the propagation and interaction of spin waves are a promising alternative to conventional electronics. The characteristic frequency of these excitations is in the gigahertz (GHz) range and devices are not readily interfaced with conventional electronics. Here, we locally probe the magnetic excitations in a soft magnetic material by optical methods and show that megahertz-range excitation frequencies cause switching effects on the micrometer scale, leading to phase-locked spin-wave emission in the GHz range. Indeed, the frequency multiplication process inside the magnetic medium covers six octaves and opens exciting perspectives for spintronic applications, such as all-magnetic mixers or on-chip GHz sources. F requency multiplication in modern electronics is usually achieved in nonlinear electronic circuits or transmission lines. In recent years, spin-based electronics has emerged as a promising extension to conventional electronics, providing new functionalities such as nonvolatile memory (1,2). In such spin-based devices, the spin transport and spin dynamics in the gigahertz (GHz) frequency range are inherently coupled, as demonstrated by the spin torque effect. For many applications, the possibility of frequency up-conversion from megahertz (MHz) frequencies using the nonlinearity of the magnetization dynamics would be desirable but has been restricted typically to a few harmonics only (3)(4)(5). Nonperturbative frequency multiplication processes over several octaves were only observed at optical frequencies in response to extremely intense laser pulses (6). In most cases, nonlinear behavior either emerges when an external driving force exceeds a certain threshold or is caused by the intrinsic nonlinearity of the equation of motion. In the case of spin waves, major attention has been directed to the threshold behavior with respect to the radio frequency (rf) pumping field, showing that large excitation amplitudes drive nonlinear spin-wave modes (7,8). These nonlinear excitations are utilized in spintronic devices such as memory applications (9,10) and spin transfer torque-driven nanooscillators (11)(12)(13).
We explore the nonlinear response of a soft ferromagnet at very small magnetic fields. Spin waves are probed by using diamond nitrogen vacancy (NV) center nanoscale magnetometry (14)(15)(16)(17)(18) and time-resolved magneto-optic Kerr microscopy (MOKE) (19). In addition to spin waves predicted by a generalized theory of spin-wave turbulence oscillating at 3/2 of the driving frequency (20), we observe a series of spin-wave excitations precessing at up to the 60th harmonic of the pumping frequency. This phenomenon can be understood as the result of dynamic, periodic, and synchronized switching of magnetic texture.
In our experiments, a ferromagnetic (FM) Ni 80 Fe 20 layer is deposited on top of an Au coplanar waveguide (CPW) and covered with nanodiamonds containing NV centers (14,17,21) (Fig. 1A). The rf magnetic fields generated by the CPW excite the electron spin resonance (ESR) in diamond NV centers as well as magnetization dynamics in the FM layer. In NV center magnetometry, the ESR of the NV centers is optically detected by monitoring their photoluminescence (PL), i.e., optical detection of magnetic resonance (ODMR) (22,23), as explained in section M2 of materials and methods. The measured PL intensity modulation due to microwave excitation recorded from the NV center ensemble is shown as a function of static bias field and the rf excitation frequency f rf in Fig. 1B. Whenever applied magnetic field and rf frequency coincide with the ESR condition, a change in the PL signal is observed. In addition to the expected ESR signal of the individual NV centers (dotted lines in Fig. 1B), we find a number of additional features in the ODMR signal caused by the proximity of the FM layer. These may be characterized as follows: (I) a continuous resonance at and below the ferromagnetic resonance (FMR) field, (II) a strong enhancement at 2/3 of the NV centers resonance frequency, and (III) a series of resonance lines at low bias fields. The occurrence of the FMR in the ODMR signal (I) has been observed before (15)(16)(17)24), and the effect is attributed to dipolar stray fields generated by spin waves acting on the NV centers (16). Feature (II) occurs at a frequency of 2/3 f ESR , i.e., at 1.9 GHz. This feature is attributed to nonlinear spinwave excitations as predicted for small bias fields by Bauer et al. (20) (for further details, see fig. S1). For magnetic fields below 4 mT, a series of narrow resonance lines whose spacing scales down with decreasing frequency (Fig. 2) (feature III) is observed. All of these resonances are very sharp replicas of the ESR cone of the NV centers (around 2.87 GHz) occurring at the nth fraction of the ESR frequency, where n is an integer. The observed Zeeman splitting of these replicas is proportional to the bias field and scales inversely with n. In the inset of Fig. 2, more than 60 replicas of the ESR are observed with a signal amplitude comparable to that of the main ESR signal of the NV ensemble and only weakly decaying with increasing n. The replicated ESR lines in the PL signal are a consequence of the presence of the FM layer ( fig. S2), and their origin must be attributed to its dynamic response. Because the NV centers are sensitive only to a narrow frequency range around f ESR , the observed 1/n-spaced sharp lines can be considered a fingerprint of a frequency comb, which induces a signal whenever a comb line coincides with f ESR . For example, at an excitation frequency of f rf = 287 MHz, the 10th comb line of a frequency comb matches f ESR , leading to the observed signal ( Fig. 2; n = 10).
The observation of such a frequency comb is not at all expected and deserves close attention.
To investigate the spin-wave modes responsible for the frequency comb, we directly map the dynamic response at a specific harmonic of the excitation using super-Nyquist-sampling MOKE (SNS-MOKE) (25). By addressing the individual harmonics of the frequency comb (expected from the NV center results), we observe phase-stable spin-wave patterns. In Fig. 3, spatially resolved measurements of magnitude and phase of the dynamic magnetization are shown that represent the 1st, 10th, and 20th harmonic. Here, an almost homogeneous response at the fundamental f rf and an increasingly fragmented spatial modulation in the micrometer and submicrometer range is found for higher harmonics (all harmonics are shown in fig. S5). The frequency comb generation is most pronounced at magnetic bias fields below 2 mT.
To understand the physical mechanism responsible for this peculiar behavior, we perform micromagnetic simulations. The most simple case, a periodically excited homogeneous magnetic layer, does not reproduce the experiments, as only a few odd harmonics of the excitation frequency with rapidly decaying amplitude are observed for the zcomponent of the magnetization, as shown for f rf = 191 MHz in Fig. 4A (black curve). This result is not surprising because, based on the equation of motion of the magnetization and its intrinsic nonlinearity, one indeed expects just a few odd harmonics with rapidly decreasing amplitude. However, in the experiments, both even and odd multiples of the excitation frequency occur, extending up to the 60th harmonic. Therefore, a crucial aspect must be missing in the discussion.
Because the frequency comb is only observed at very small fields, where the magnetization may not be fully saturated, we map the magnetic texture of the sample close to zero magnetic field using static MOKE ( fig. S8). At very small fields (below 1 mT), we find that the magnetization of the Ni 80 Fe 20 stripe shows a magnetic ripple structure. This is a well-known effect that occurs in soft magnetic materials such as Ni 80 Fe 20 , and it is caused by the interplay between internal magnetic fieldse.g., due to strain or polycrystalline grain structure-and stray field energy, leading to a buckling structure of the magnetization known as a ripple pattern in extended layers or concertina patterns in stripes (26). When the simulation includes the magnetic stray fields from the edges and a small randomly oriented anisotropy field (reflecting the polycrystalline sample structure), the micromagnetic simulation well reproduces the observed static ripple pattern; fig. S9A). When this texture is used, the simulated frequency spectrum (Fig. 4A)  and even harmonics with orders-of-magnitude larger amplitude. Analyzing this simulation in detail, we find that local switching events of the transverse magnetization components of the ripple pattern cause the emission of spin waves ( fig. S9C and movie S1). Next, to simplify the discussion, we only consider a single step in magnetization density perpendicular to the applied field (Fig. 4C). Here, the stray field energy due the magnetic surface charges is minimized by a spontaneous tilt of the magnetization at the boundary. Two energetically equivalent states (transverse component up or down) are possible, as illustrated in Fig. 4, D and E. In this situation, a low-frequency (MHz range) rf excitation can cause switching of the magnetization between the two states at the excitation frequency. To demonstrate that this simple model indeed fully captures the essential physical processes, we simulate its dynamics. A corresponding snapshot of the magnetization dynamics during this process is shown in Fig. 4B.
Because of the open boundary conditions (materials and methods M4), the switching process starts at one edge and propagates along the boundary (Fig. 4B, and movie S3). As shown in fig. S11, the spin waves are emitted from the fast-propagating switching front. Here, the transverse magnetization component rapidly changes from +0.4 M S to −0.4 M S on a time scale of less than 100 ps, where M S is the saturation magnetization. Because the switching front moves at a speed of 4 km/s, and thus three to four times faster than the spin-wave phase velocity (in the GHz range), the spin waves pile up and form a shock front analogous to a supersonic cone moving across the sample. This phenomenon is known as the spin-wave Cherenkov effect (27). We conclude that in the actual sample, with a spatial distribution of switching regions, the emitted spin-wave shock fronts from one region can directly trigger the switching in the neighboring regions and thereby synchronize the emitted spin waves. This effect is known as spin wave-assisted switching (28).
Overall, this process leads to the observed coherent standing spin-wave amplitude pattern obtained by SNS-MOKE measurements at all harmonics phase locked to the MHzrange driving field. This behavior is observed in simulations and experiments. An example of such behavior is shown in a phase-resolved fashion for the 18th harmonic in movie S2. The result explains both the observation of ESR replicas in ODMR measurements and the higher harmonics directly observed by SNS-MOKE. However, in contrast to ODMR, SNS-MOKE is only sensitive to fully repeatable coherent dynamics. We emphasize that the observed behavior is extremely robust. ODMR and SNS-MOKE measurements show that the frequency comb is generated at static magnetic fields of up to 2 mT and with an exceptionally broad range of driving frequencies, from 50 MHz up to 1 GHz (Fig. 2). The range of generated spin waves is potentially even wider, but owing to limitations of our experimental setup, is not accessible. The simulations show that the frequency comb generation is largely independent of the details of the magnetic texture (compare figs. S14 and S13) and easily extends to above 25 GHz (see fig. S12).
Our results demonstrate that a simple layer of ferromagnetic material (in our case Ni 80 Fe 20 ) can be used to generate a frequency comb spanning six octaves. Using time-resolved Kerr microscopy, we correlate this frequencymultiplying behavior to the emergence of a collective dynamic phase in which the magnetization of the ferromagnet partially switches in a coherent and synchronized fashion due to 3 of 5 spin-wave coupling effects. As we demonstrate by magneto-optic imaging (Fig. 3), a standing spin-wave pattern establishes itself at all harmonics of the excitation frequency within the spin-wave band. The excited spin waves preferentially follow a linear dispersion ( fig.  S6). This behavior is expected as it allows for phase coherence of all spin waves between the localized emission centers, resulting in a standing spin-wave pattern. The coherent nature of the high harmonic response also implies that the line width of the individual comb lines is determined by the signal source used for pumping and is not related to spin-wave damping.
From a fundamental point of view, the effect of phase-stable pattern formation at many harmonics as discussed in this Report represents a self-synchronization phenomenon. Such effects occur in a large variety of nonlinear physical systems. In the present case, the rf switching events lead to spin-wave emission that mutually synchronizes the switching events and leads to the formation of a phase-stable time-periodic spin-wave pattern.
The presented possibility of frequency multiplication under continuous-wave conditions within the magnetic medium itself opens exciting perspectives for spin-based applications operating in the GHz range while conveniently being controlled with MHz input signals. In doing so, specific harmonic frequencies may be selected either by introducing a periodicity in the magnetic structure, resonantly enhancing the desired magnon mode, or by the initial excitation of all harmonics with subsequent separation. Here, the different k-vectors may be exploited, e.g., by coupling the spin waves into a tailored waveguide (29) or by using nanoscale narrow-band Fabry-Pérot spin-wave filters, which we recently demonstrated (19).