Nonlinear Spin Waves Research Articles

Nonlinear Spin-wave Theory for Dynamical Spin Correlations in the Kitaev Model at Finite Temperatures

Nonlinear Spin-wave Theory for Dynamical Spin Correlations in the Kitaev Model at Finite Temperatures

Investigation of a Nonlinear XNOR Logic Gate Based on an Induced Nonlinear Phase Shift of Spin Waves

Introduction. Recent years have seen a growing interest in studying the nonlinear properties of spin waves. Nonlinear phenomena, such as envelope solitons, nonlinear frequency shifts of intense spin waves, and etc., have attracted particular attention. However, a number of important issues remain to be underexplored, including the problem of induced nonlinear phase shift of spin waves. The relevance of this problem is related to the need to develop spin-wave logic gates that could be controlled by changing the spin wave phase.Aim. To study a nonlinear XNOR logic gate whose operation is based on the induced nonlinear phase shift of a spin wave.Materials and methods. An original theory is used to simulate the frequency response of a nonlinear XNOR logic gate. The operating principle of the nonlinear XNOR logic gate is substantiated. The possibility of implementing the nonlinear XNOR logic gate in a circuit similar to a spin-wave Mach-Zehnder interferometer is experimentally demonstrated.Results. An experimental study of the induced nonlinear phase shift of operating signals incident on identical nonlinear spin-wave phase shifters located in the arms of the logic gate was carried out. It is shown that an increase in the pump signal power up to 60 mW, supplied to nonlinear phase shifters, changes the induced nonlinear phase shift of the operating signal by more than 180°. Hence, nonlinear phase shifters can be used for constructing spin-wave logic gates. In addition, the operating principle of a spin-wave logic gate was experimentally studied. It is shown that the XNOR logical function is implemented in the low-frequency part of the device’s frequency response characteristic.Conclusion. Numerical simulation of the characteristics of a nonlinear XNOR logic gate based on the Mach-Zehnder interferometer circuit was carried out. It is shown that its logical functions are implemented due to the effect of an induced nonlinear phase shift of spin waves in nonlinear phase shifters located in different arms of the logic gate.

Theory of propagation of nonlinear spin wave through an antiferromagnetic magnonic crystal with four-sublattice interfaces

The purpose of the research is constructing an analytical model of a nonlinear spin wave propagating through a single antiferromagnet/antiferromagnet boundary as well as in one-dimensional antiferromagnetic magnonic crystal comprised of two sorts of antiferromagnets (AFM). Both AFMs that comprise the magnonic crystal are assumed to be two-sublattice uniaxial ones. The Landau-Lifshitz equations have been used in the sigma model with account for the exchange bias between magnetic sublattices of both AFMs, the magnetic anisotropy, the magnetic dipole–dipole interaction and the Dzyaloshynskyi-Moriya interaction. As a result, the allowed discrete sets of frequencies and velocities for the considered spin wave are obtained. In the process of investigation, the boundary conditions for the Néel vector on the interface between two AFMs are derived with the exchange bias between magnetic sublattices of both AFMs taken into account. These boundary conditions are, in particular, applied for both fully uncompensated interface and fully compensated one for the illustration purposes. Analysis of the results show that the considered wave is reflectionless (thus, energy losses on reflection on the interfaces are absent), phase-coherent and possess a number of parameters that can be considered as degrees of freedom for encoding information. These findings open up new possibilities of digital data processing utilizing nonlinear spin waves propagating through antiferromagnetic magnonic crystal.

Experimental Demonstration of High‐Performance Physical Reservoir Computing with Nonlinear Interfered Spin Wave Multidetection

Physical reservoir computing, which is a promising method for the implementation of highly efficient artificial intelligence devices, requires a physical system with nonlinearity, fading memory, and the ability to map in high dimensions. Although it is expected that spin wave interference can perform as highly efficient reservoir computing in some micromagnetic simulations, there has been no experimental verification to date. Herein, reservoir computing is demonstrated that utilizes multidetected nonlinear spin wave interference in an yttrium‐iron‐garnet single crystal. The subject computing system achieves excellent performance when used for hand‐written digit recognition, second‐order nonlinear dynamical tasks, and nonlinear autoregressive moving average (NARMA). It is of particular note that normalized mean square errors for NARMA2 and second‐order nonlinear dynamical tasks are 1.81 × 10−2 and 8.37 × 10−5, respectively, which are the lowest figures for any experimental physical reservoir so far reported. Said high performance is achieved with higher nonlinearity and the large memory capacity of interfered spin wave multidetection.

Broadband microwave detection using electron spins in a hybrid diamond-magnet sensor chip

Quantum sensing has developed into a main branch of quantum science and technology. It aims at measuring physical quantities with high resolution, sensitivity, and dynamic range. Electron spins in diamond are powerful magnetic field sensors, but their sensitivity in the microwave regime is limited to a narrow band around their resonance frequency. Here, we realize broadband microwave detection using spins in diamond interfaced with a thin-film magnet. A pump field locally converts target microwave signals to the sensor-spin frequency via the non-linear spin-wave dynamics of the magnet. Two complementary conversion protocols enable sensing and high-fidelity spin control over a gigahertz bandwidth, allowing characterization of the spin-wave band at multiple gigahertz above the sensor-spin frequency. The pump-tunable, hybrid diamond-magnet sensor chip opens the way for spin-based gigahertz material characterizations at small magnetic bias fields.

Spin motive force induced by parametric excitation

Spin motive force generated by parametrically excited magnetization dynamics is numerically investigated. We calculate spin motive force in a permalloy disk under an ac magnetic field with twice the ferromagnetic resonance frequency parallel to the static magnetic field based on the Landau–Lifshitz–Gilbert equation. We found that large spin motive force originating from standing spin waves driven by parametric excitation appears in the system. The observed time dependence of the voltage shows a dc voltage with an ac component oscillating with twice of the resonance frequency. The estimated amplitude of the voltage due to the spin motive force is ∼μV. We also investigate spin motive force driven by different modes of standing spin waves. Our numerical results extend the way to generate spin motive force by making use of the magnetization dynamics with the steep spatial modulation created by nonlinear spin waves excitation, without a non-uniform magnetization structure such as a conventional magnetic domain wall and a vortex.

Film-penetrating transducers applicable to on-chip reservoir computing with spin waves

We have proposed a spin-wave transducer structure named film-penetrating transducers (FPTs). FPTs penetrate an on-chip magnetic film for a spin-wave transmission medium and allow flexible spatial arrangements of many exciters/detectors due to their zero-dimensional feature. We constructed four device models with different spatial arrangements of FPT/conventional exciters using a 10-nm-thick ferrimagnetic garnet film with a central FPT detector. We performed numerical experiments that combine electromagnetics with micromagnetics including thermal noise at 300 K. We evaluated important device features of FPTs, such as the signal-to-noise ratios (SNRs), input/output signal transmission efficiencies, and nonlinear phenomena of spin waves. We applied in-phase sinusoidal input currents with various amplitudes and frequencies and altered the damping strengths near the film boundaries. We obtained sufficient SNRs for the practical use of FPTs and revealed that FPTs have both higher transmission efficiencies and nonlinear strengths than conventional antennas, as the input frequency approaches the ferromagnetic resonance frequency of the film. Moreover, we observed and analyzed various nonlinear phenomena of spin waves, including beats in the time-domain waveform, components of integer harmonic frequencies, wide-range scatterings of inter-harmonic frequencies, and frequency doubling in spin precession. These characteristics probably originate from various device effects: FPTs effectively excite dipolar spin waves with large-angle precession, propagating spin waves reflect from the film boundaries, and spin waves dynamically and nonlinearly interfere with each other. This study demonstrated that FPTs have promising features for both their applications to reservoir computing and the studies on the physics of nonlinear and space-varying spin waves.

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.

Microscopic nonlinear magnonic phase shifters based on ultrathin films of a magnetic insulator

Since magnonics takes advantage of not only the amplitude of spin waves but also their phase, tunable phase shifters are key elements for the implementation of magnonic circuits. Therefore, one of the major challenges in nano-magnonics is to find a physical mechanism to manipulate the spin-wave phase practically in simple and miniature devices. In this work, we experimentally demonstrate that intrinsic magnetic nonlinearities allow the implementation of efficient microscopic tunable phase shifters, where the phase is controlled by wave intensity. In the proposed devices, we achieve the tunability of the phase shift of more than 360° by a microwave power of few milliwatts over a propagation distance of about 10 μm. We show that the figure of merit of the demonstrated phase shifters is close to that of macroscopic devices based on alternative technologies. Our results also indicate that the ability to control the phase shift is primarily limited by nonlinear spin-wave damping and can be significantly improved by suppressing this effect. Our findings are important for the further development of integrated nano-magnonics for beyond-Moore computing.

Gap solitons in heterostructure magnonic crystal/semiconductor

The main features of the nonlinear spin wave propagating in the magnonic crystal/semiconductor (SC) layered heterostructure are studied. We demonstrate the possibility of double electric and magnetic control of the gap solitons parameters (their number, threshold of formation, velocity), bistability of spin waves, switching powers and nonlinear band gap shift in investigated structure. Our results show the possibility of nonlinear magnonics and SC electronics integration based on the proposed structure.

Possibility of Raising the Frequency of RF Limiters Using the Nonlinear Spin Wave–Electromagnetic Interaction in Ferrites With High Saturation Magnetization

Physics of the spin wave&#x2013;electromagnetic (EM) interaction is investigated for RF limiter applications, by looking at the magnetization <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> and its deviation <inline-formula> <tex-math notation="LaTeX">$\Delta M$ </tex-math></inline-formula> through equations of motion. Exchange effects are examined, and a complete magnetodynamics equation is developed and looked at for both the fundamental- and second-harmonic frequencies. Nonlinear conversion of energy occurs between the first and second harmonics. It is shown here that the usual operating frequencies for RF limiters, in the range of 1.5&#x2013;15-GHz frequencies, can be lifted up into the much higher frequencies of the 40&#x2013;60-GHz range by a new formula developed from fundamental spin wave&#x2013;EM interactions in magnetic material characterized by a magnetization <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula>. The treatment is self-contained in which all of the information needed to understand the magnetics aspect for RF applications is provided.

Nonlinear signal processing with magnonic superlattice with two periods

In this work, we provide investigations of nonlinear spin wave propagation in magnonic superlattice. Magnonic superlattice is a ferromagnetic film loaded on a dielectric ceramic substrate with the copper grating with two periods. It is shown that a superlattice with two periods allows the formation of narrower bandgaps as compared to magnonic crystal with one period. Nonlinear effects, such as the generation of spin wave excitations at half-frequency, caused by first-order parametric processes, and a nonlinear shift of the bandgap, caused by a change in the longitudinal component of magnetization with an increase in the input signal power, are revealed. For the experimental study, the Brillouin light scattering spectroscopy was used, and the theoretical model was built using the transfer matrix method.

Stability of ordered and disordered phases in the Heisenberg-Kitaev model in a magnetic field

The S = 1/2 Kitaev honeycomb model has attracted significant attention as an exactly solvable example with a quantum spin liquid ground state. In an properly oriented external magnetic field, the system exhibits chiral Majorana edge modes with an associated quantized thermal Hall conductance, and a distinct spin-disordered phase emerges at intermediate field strengths, below the polarized phase. However, since material realizations of Kitaev magnetism invariably display competing exchange interactions, the stability of these exotic phases with respect to additional couplings is a key issue. Here, we report a 24-site exact diagonalization study of the Heisenberg-Kitaev model in a magnetic field applied in the [001] and [111] directions. By mapping the full phase diagram of the model and contrasting the results to recent nonlinear spin-wave calculations, we show that both methods agree well, thus establishing that quantum corrections substantially modify the classical phase diagram. Furthermore, we find that, in a [111] field, the intermediate-field spin-disordered phase is remarkably stable to Heisenberg interactions and may potentially end in a novel quantum tricritical point.

Nanoscale neural network using non-linear spin-wave interference

We demonstrate the design of a neural network hardware, where all neuromorphic computing functions, including signal routing and nonlinear activation are performed by spin-wave propagation and interference. Weights and interconnections of the network are realized by a magnetic-field pattern that is applied on the spin-wave propagating substrate and scatters the spin waves. The interference of the scattered waves creates a mapping between the wave sources and detectors. Training the neural network is equivalent to finding the field pattern that realizes the desired input-output mapping. A custom-built micromagnetic solver, based on the Pytorch machine learning framework, is used to inverse-design the scatterer. We show that the behavior of spin waves transitions from linear to nonlinear interference at high intensities and that its computational power greatly increases in the nonlinear regime. We envision small-scale, compact and low-power neural networks that perform their entire function in the spin-wave domain.

Effect of probing signal on the output signals spectrum of nonlinear spin waves in a cross based on waveguides of iron-yttrium garnet film

Subject. A change in the spectrum of spin waves (SW) in a magnetic cross is investigated when two signals pass through it: a pump signal and a probe signal. Objective. Detection of specific features in formation of the spectra of the output signals of SW in the multiport structure based on a yttrium iron garnet (YIG) film in the case of excitation of two magnetostatic surface waves (MSSW) simultaneously by the input antenna, where the first, with power higher than the first-order parametric instability threshold is the pump, and the second one is a probe. Methods. The experiments were performed for a cross structure from YIG film in the form of two orthogonal waveguides with the SW wire antennas placed at the ends of the waveguides, where one of the antennas on the transversely magnetized waveguide was considered as the input. Result. It was found that by choosing the probing signal frequency, one can significantly (by 10 dB) change the relative signal levels for the satellite waves at the output antennas, which are secondary MSSWs with some new frequencies and appear in the output signals spectrum as a result of the thresholdless processes of merging of parametric spin waves generated by MSSW pumping. In this case the secondary MSSWs frequencies can differ at the output antennas located on orthogonal waveguides. Discussion. The discovered effect is associated with the nonreciprocal nature of propagation of both the pumping wave and the waves generated at parametric instability condition in the structure.

Characterization of nonlinear spin-wave interference by reservoir-computing metrics

We study the computational potential of a spin-wave (SW) substrate by applying two metrics known from reservoir computing. At low intensities, SW scatterers can perform linear operations, while at higher intensities, nonlinear phenomena dominate, possibly enabling high-function, general-purpose computing. The transition between the linear and nonlinear regimes can be quantified by the intensity-dependent kernel rank (KR) and generalization rank (GR). The KR and GR metrics prove that the SW substrate displays the nonlinearities required for computing and give recipes for device designs that utilize nonlinearity.

Semi-classical quantisation of magnetic solitons in the anisotropic Heisenberg quantum chain

Using the algebro-geometric approach, we study the structure of semi-classical eigenstates in a weakly-anisotropic quantum Heisenberg spin chain. We outline how classical nonlinear spin waves governed by the anisotropic Landau--Lifshitz equation arise as coherent macroscopic low-energy fluctuations of the ferromagnetic ground state. Special emphasis is devoted to the simplest types of solutions, describing precessional motion and elliptic magnetisation waves. The internal magnon structure of classical spin waves is resolved by performing the semi-classical quantisation using the Riemann--Hilbert problem approach. We present an expression for the overlap of two semi-classical eigenstates and discuss how correlation functions at the semi-classical level arise from classical phase-space averaging.

Nonlinear spin wave switches in layered structure based on magnonic crystals

The features of spin wave processes in a layered structure based on two magnonic crystals (MC-1 and MC-2) separated by a dielectric layer have been studied theoretically and experimentally at different levels of input power. The theoretical model has been developed for the cases of magnetic field oriented in-plane and out-of-plane to MC surfaces in order to identify the mechanisms of nonlinear interactions. Two types of nonlinear switching are implemented in the structure MC-1/MC-2 with increasing input power. Depending on parameters of the structure the input signal can exit from three (“double nonlinear switching”) or two (“single nonlinear switching”) outputs. This feature allows to consider MC-1/MC-2 structure as a basic element in a multifunctional device for the spatial separation of signals with different power levels.

Generation of Dark Multisoliton Complexes in a Magnonic Ring Resonator with Dispersion Management and Competing Nonlinear Spin-Wave Interactions

Multisoliton complexes have been generated in an active ring resonator with an L-shaped magnonic microwaveguide and a saturated amplifier. It has been shown that an irregular microwaveguide simultaneously supports both the propagation of magnetostatic spin waves with different (normal and anomalous) dispersion and competition between three- and four-wave spin-wave interactions. Generated multisoliton complexes consist of quasiperiodic sequences of two dark parametric pulses containing “soliton trains” consisting of three dark ultrashort four-wave envelope solitons. It has been found that quasiperiodic sequences of multisoliton complexes generated by the active ring resonator with a regular magnonic microwaveguide consist of one dark parametric pulse, which also contains three dark four-wave solitons. However, in contrast to the preceding case, four-wave solitons, having a longer duration, do not form soliton trains inside a dark parametric pulse.

Towards experimental observation of parametrically squeezed states of microwave magnons in yttrium iron garnet films

We demonstrate theoretically, and confirm experimentally, that nonlinear spin waves excited in thin yttrium iron garnet films are good candidates for squeezing vacuum quantum noise. The experimental demonstration is in the form of a measurement of spin-wave induced modulation instability (IMI) conducted in the classical regime. The experiment evidences strong phase locking of an idler wave parametrically generated in the film with a deterministic small-signal wave launched into the film from an external source. The theory predicts that the same behaviour will be observed for vacuum quantum noise, resulting in squeezing of the noise.