An auto-oscillator driven by a harmonic signal at about twice its free-running frequency is characterized by a bistable phase dynamics where the two states are separated by π radians. This phase bistability enables an oscillator to emulate a single Ising spin, providing a fundamental building block for the oscillator-based Ising machines (OIM). At the same time, a driving signal close to the oscillator free-running frequency locks the oscillator's phase at a single value, playing the role of a magnetic field bias in ensembles of real spins. We introduce a universal theory of phase auto-oscillators driven by a biharmonic signal (having frequency components close to single and double of the free-running oscillator frequency) with noise; with it, we show how deterministic phase locking and stochastic phase slips can be continuously tuned by varying the relative amplitudes and frequencies of the driving components. Using, as an example, a spin-torque nano-oscillator, we numerically validate this theory by implementing a deterministic Ising machine paradigm, a probabilistic one, and dual-mode operation of the two. This demonstration introduces the concept of adaptive Ising machines (AIM), a unified oscillator-based architecture that dynamically combines both regimes within the same hardware platform by properly tuning the amplitudes of the biharmonic driving relative to the noise strength. Benchmarking on different classes of combinatorial optimization problems, the AIM exhibits complementary performance compared to OIMs and probabilistic Ising machines, with adaptability to the specific problem class. This Letter introduces the first OIM capable of transitioning between deterministic and probabilistic computation taking advantage of a proper design of the trade-off between the strength of phase-locking of an auto-oscillator to a biharmonic external driving and noise, opening a path toward scalable, CMOS-compatible hardware for hybrid optimization and inference.

We present the design and numerical simulation of a spiking neuron capable of on-chip machine learning. Built within the CMOS+X framework, the spiking neuron consists of an NMOS transistor combined with a magnetic tunnel junction (MTJ). This NMOS+MTJ unit, when simulated in the industry-standard circuit simulation software LTspice, reproduces multiple functions of a biological neuron, including threshold spiking, latency, refractory periods, synaptic integration, inhibition, and adaptation. These behaviors arise from the intrinsic magnetization dynamics of the MTJ and do not require any additional control circuitry. By interconnecting the NMOS+MTJ neurons, we construct a model of an analog multilayer network that learns through spike-timing-dependent weight updates derived from a gradient-descent rule, with both training and inference modeled in the analog domain. The simulated CMOS+X network achieves reliable spike propagation and successful training on a nonlinear task, indicating a feasible path toward compact, low-power, in-memory neuromorphic hardware for edge applications.

Abstract Magnonic frequency combs (MFCs) offer a promising route to compact, energy-efficient platforms for on-chip coherent microwave signal generation and processing. Conventional on-chip comb generation typically relies on nonlinear resonators supporting equidistant, low-loss resonances driven by a monochromatic signal, resulting in fixed comb spacing. Here we introduce and experimentally demonstrate a distinct mechanism for ultra-broadband MFC generation using a highly nonlinear miniaturized magnonic resonator. The small resonator volume, combined with a slow-wave transducer, drives the system deep into the bistable regime where parametric excitation of propagating spin waves facilitates comb formation. Our approach yields over 350 comb lines spanning a 450 MHz bandwidth, with spacing continuously tunable via a two-tone external drive, representing an order-of-magnitude enhancement over prior reports at relatively low power. The platform is ultra-compact, scalable, and highly tunable, establishing a distinct frequency comb paradigm with transformative opportunities in microwave signal processing, neuromorphic computing, and precision sensing.

We introduce a universal theory of phase auto-oscillators driven by a bi harmonic signal (having frequency components close to single and double of the free-running oscillator frequency) with noise. With it, we show how deterministic phase locking and stochastic phase slips can be continuously tuned by varying the relative amplitudes and frequencies of the driving components. Using, as an example, a spin-torque nano-oscillator, we numerically validate this theory by implementing a deterministic Ising machine paradigm, a probabilistic one, and dual-mode operation of the two. This demonstration introduces the concept of adaptive Ising machines (AIM), a unified oscillator-based architecture that dynamically combines both regimes within the same hardware platform by properly tuning the amplitudes of the bi-harmonic driving relative to the noise strength. Benchmarking on different classes of combinatorial optimization problems, the AIM exhibits complementary performance compared to oscillator based Ising machines and probabilistic Ising machines, with adaptability to the specific problem class. This work introduces the first OIM capable of transitioning between deterministic and probabilistic computation taking advantage of a proper design of the trade-off between the strength of phase-locking of an auto-oscillator to a bi harmonic external driving and noise, opening a path toward scalable, CMOS compatible hardware for hybrid optimization and inference.

Spin-transfer-torque and spin-orbit-torque nanooscillators (STNOs) are fundamentally nonlinear devices interesting for various applications in spintronics and microwave micro- and nanoelectronics. While the simplest, single-mode, generation regime of STNO is well-studied, the more complex regimes involving the generation of two or more spin wave modes simultaneously, which were observed in laboratory experiments, were not systematically analyzed in theory, so far. In this work, we analyze a two-mode generation regime in an STNO using a general model, which accounts for the inter-mode interactions, inevitably present in any STNO: mode competition for common (shared) pumping energy and, also, self- and cross-mode nonlinear frequency shift and nonlinear damping. It is shown that the key parameter governing the two-mode generation regime is the cross-mode coupling, which can be simplistically understood as being proportional to the spatial overlap of the profiles of interacting modes. In addition to the self-oscillation regime, the phase-locking of the self-generated modes to an external periodic force is studied, showing in some cases nontrivial dynamics.

In novel, beyond Von Neumann, computational approaches the use of magnons (or quanta of spin waves) is particularly promising due to the small intrinsic energies of individual magnons $(\mu \mathrm{V})$, the possibility of using phase, in addition to magnitude, as a state variable, and the possibility to control the magnon dispersion properties in a magnetic sample by varying the direction and magnitude of the bias magnetic field [1].

Nonreciprocity of propagation of surface acoustic waves (SAWs) in the microwave frequency band can be achieved using the magnetoelastic interaction of SAWs with spin waves (SWs) propagating in magnetic heterostructures. Recent works have shown that the ultimate isolation of a counterpropagating hybridized SAW/SW is achieved in heterostructures consisting of a synthetic antiferromagnet-a ferromagnetic (FM) bilayer with antiferromagnetic Ruderman-Kittel-Kasuya-Yosida interlayer coupling-placed on top of a piezoelectric acoustic waveguide. In this work, we study in detail a more practical and technologically simpler system based on an FM bilayer, where layers are coupled by only dipole-dipole interaction, and having noncollinear magnetizations of the FM layers. A weak in-plane anisotropy with noncollinear easy axes in the layers is shown to be the only essential factor for the realization of strongly nonreciprocal propagation of a hybridized SAW/SW. We formulate requirements for the relative orientation of the layer's magnetizations and wave propagation direction necessary to realize an efficient SAW isolator, and demonstrate examples of SAW transmission characteristics which prove the possibility of achieving an isolation exceeding 50 dB for a submillimeter-long FM bilayer with insertion losses of just a few decibels more than those of a pure SAW device. In addition to relative fabrication simplicity, the proposed magnetoelastic heterostructure exhibits a reasonable robustness in respect to deviations in the anisotropy axes and/or bias field directions-an important benefit for device mass production.

This paper proposes a novel spiking artificial neuron design based on a combined spin valve/magnetic tunnel junction (SV/MTJ). Traditional hardware used in artificial intelligence and machine learning faces significant challenges related to high power consumption and scalability. To address these challenges, spintronic neurons, which can mimic biologically inspired neural behaviors, offer a promising solution. We present a model of an SV/MTJ-based neuron which uses technologies that have been successfully integrated with CMOS in commercially available applications. The operational dynamics of the neuron are derived analytically through the Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation, demonstrating its ability to replicate key spiking characteristics of biological neurons, such as response latency and refractive behavior. Simulation results indicate that the proposed neuron design can operate on a timescale of about 1 ns, without any bias current, and with power consumption as low as 50 uW.

Recent advances in spintronics resulted in the development of a new class of radiation-resistant nano-sized microwave devices - spin-torque nano-oscillators (STNO). To use these novel nano-scale devices in wireless communications system as either microwave sources or detectors it is necessary to develop antennas coupled to STNO and providing efficient radiation and reception of microwave radiation. We demonstrate that it is possible to design antennas of a sub-wavelength size that have sufficiently high efficiency to be successfully used in spintronic communication devices. A coplanar antenna has the best performance characteristics, because its impedance could be easily matched with the impedance of nano-scale spintronic devices. We developed prototype spintronic devices with matched coplanar antennas (oscillators and radar detectors) which could be embedded into armor, thereby improving the survivability of the antennas as well as reducing the visual signature of antennas on military vehicles.

We analyze the performance of an active terahertz (THz)-frequency signal detector based on an antiferromagnetic tunnel junction (ATJ) Pt/Ir0.2Mn0.8/MgO/Pt in a wide temperature range T = 4.2-300 K. Assuming that the geometric parameters of the ATJ depend on temperature due to the standard thermal expansion, and using usual thermal dependencies for the ATJ resistance-area product and tunneling anisotropic magnetoresistance (TAMR) ratio, we show that the output dc voltage of the detector operating in the frequency range of 0.1-1 THz increases by 50%-70% as the temperature decreases from 300 to 4.2 K. In addition, the cooling of the detector results in a substantial reduction of the low-frequency Johnson-Nyquist noise, and a corresponding increase of the detector signal-to-noise ratio (SNR) (up to 20), as well as in the reduction of the minimum detectable power of the detector (to 1 pW or less). The detector characteristics reach their optimal values near the temperature of T similar or equal to 10 K and frequency of f approximate to 0.15 THz due to the interplay between the inertial properties of the detector, and its impedance-matching conditions. The developed formalism can be used for the optimization of practical parameters of ATJ-based spintronic devices.

Spintronic devices offer a promising avenue for the development of nanoscale, energy-efficient artificial neurons for neuromorphic computing. It has previously been shown that with antiferromagnetic (AFM) oscillators, ultra-fast spiking artificial neurons can be made that mimic many unique features of biological neurons. In this work, we train an artificial neural network of AFM neurons to perform pattern recognition. A simple machine learning algorithm called spike pattern association neuron (SPAN), which relies on the temporal position of neuron spikes, is used during training. In under a microsecond of physical time, the AFM neural network is trained to recognize symbols composed from a grid by producing a spike within a specified time window. We further achieve multi-symbol recognition with the addition of an output layer to suppress undesirable spikes. Through the utilization of AFM neurons and the SPAN algorithm, we create a neural network capable of high-accuracy recognition with overall power consumption on the order of picojoules.

Time-multiplexed Coherent Ising Machines (CIMs) have demonstrated promising results in rapidly solving large-scale combinatorial problems. However, CIMs remain relatively large and power-demanding. Here, we demonstrate a spinwave-based Ising machine (SWIM) that due to the low spinwave group velocity allows for sufficient miniaturization and reduced power consumption. The SWIM is implemented using a 10-mm-long 5- μ m-thick Yttrium Iron Garnet film with off-the-shelf microwave components and can support an 8-spin MAX-CUT problem and solve it in less than 4 μ s consuming only 7 μ J. As the SWIM minimizes its energy, we observe that the spin states can demonstrate both uniform and domain-propagation-like switching. The developed SWIM has the potential for substantial further miniaturization with reduction of power consumption, scalability in the number of supported spins, increase of operational speed, and may become a versatile platform for commercially feasible high-performance solvers of combinatorial optimization problems.

Spiking artificial neurons emulate the voltage spikes of biological neurons and constitute the building blocks of a new class of energy efficient, neuromorphic computing systems. Antiferromagnetic materials can, in theory, be used to construct spiking artificial neurons. When configured as a neuron, the magnetization in antiferromagnetic materials has an effective inertia that gives them intrinsic characteristics that closely resemble biological neurons, in contrast with conventional artificial spiking neurons. It is shown here that antiferromagnetic neurons have a spike duration on the order of picoseconds, a power consumption of about 10−3 pJ per synaptic operation, and built-in features that directly resemble biological neurons, including response latency, refraction, and inhibition. It is also demonstrated that antiferromagnetic neurons interconnected into physical neural networks can perform unidirectional data processing even for passive symmetrical interconnects. The flexibility of antiferromagnetic neurons is illustrated by simulations of simple neuromorphic circuits realizing Boolean logic gates and controllable memory loops.

Spin-wave amplification techniques are key to the realization of magnon-based computing concepts. We introduce a novel mechanism to amplify spin waves in magnonic nanostructures. Using the technique of rapid cooling, we create a nonequilibrium state in excess of high-energy magnons and demonstrate the stimulated amplification of an externally seeded, propagating spin wave. Using an extended kinetic model, we qualitatively show that the amplification is mediated by an effective energy flux of high energy magnons into the low energy propagating mode, driven by a nonequilibrium magnon distribution.

We present a spinwave-based time-multiplexed Ising Machine (SWIM) where artificial Ising spins are formed with ns-long spinwave RF pulses propagating in a delay line based on $5-\mu m$ thick Yttrium Iron Garnet (YIG) film. The phase of artificial Ising spins is binarized using an off-the-shelf phase-sensitive microwave amplifier and the coupling between spins is implemented with microwave delay cables. Thanks to the very low spinwave group velocity, the 7-mm long YIG delay line can host an 8-spin MAX-CUT combinatorial optimization problem and solve it in less than 4 $\mu s$ while consuming only 7 $\mu J$ .

Lack of nonreciprocity-in particular, nonreciprocity of phase accumulation-is one of the major drawbacks of microwave solid-state acoustic devices, which has prevented the development of acous-tic isolators and circulators. Here we report the observation of the phase nonreciprocity of hybridized surface acoustic waves (SAWs) and spin waves in a magnetoelastic heterostructure. Our system consists of a Fe-Ga-B/Al2O3/Fe-Ga-B multilayer on top of a LiNbO3 crystal. Maximum values of the observed nonreciprocal phase accumulation easily exceed pi radians over a broad range of field conditions, which is necessary for the development of an effective circulator. In addition, under the application of bias magnetic field, the structure demonstrates tunable giant nonreciprocity of propagation losses with isolation as high as 48 dB, necessary for the development of isolators. Theoretical calculations provide an insight into the observed phenomena and demonstrate a pathway for further improvement of nonreciprocal SAW devices based on magnetoelastic coupling.

The advantage of an ultrafast frequency-tunability of spin-torque nano-oscillators (STNOs) that have a large (>100 MHz) relaxation frequency of amplitude fluctuations is exploited to realize ultrafast wide-band time-resolved spectral analysis at nanosecond time scale with a frequency resolution limited only by the "bandwidth" theorem. The demonstration is performed with an STNO generating in the 9 GHz frequency range and comprised of a perpendicular polarizer and a perpendicularly and uniformly magnetized "free" layer. It is shown that such a uniform-state STNO-based spectrum analyzer can efficiently perform spectral analysis of frequency-agile signals with rapidly varying frequency components.

We present a theory of a detector of terahertz-frequency signals based on an antiferromagnetic (AFM) crystal. The conversion of a THz-frequency electromagnetic signal into the DC voltage is realized using the inverse spin Hall effect in an antiferromagnet/heavy metal bilayer. An additional bias DC magnetic field can be used to tune the antiferromagnetic resonance frequency. We show that if a uniaxial AFM is used, the detection of linearly polarized signals is possible only for a non-zero DC magnetic field, while circularly polarized signals can be detected in a zero DC magnetic field. In contrast, a detector based on a biaxial AFM can be used without a bias DC magnetic field for the rectification of both linearly and circularly polarized signals. The sensitivity of a proposed AFM detector can be increased by increasing the magnitude of the bias magnetic field, or by by decreasing the thickness of the AFM layer. We believe that the presented results will be useful for the practical development of tunable, sensitive and portable spintronic detectors of THz-frequency signals based of the antiferromagnetic resonance (AFMR).

We study the collective dynamics of two distant magnets coherently coupled by acoustic phonons that are transmitted through a nonmagnetic spacer. By tuning the ferromagnetic resonances of the two magnets to an acoustic resonance of the intermediate, we control a coherent three-level system. We show that the parity of the phonon mode governs the indirect coupling between the magnets: the resonances with odd (even) phonon modes correspond to out-of-phase (in-phase) lattice displacements at the interfaces, leading to bright (dark) states in response to uniform microwave magnetic fields, respectively. The experimental sample is a trilayer garnet consisting of two thin magnetic films epitaxially grown on both sides of a half-millimeter-thick nonmagnetic single crystal. In spite of the relatively weak magnetoelastic interaction, the long lifetimes of the magnon and phonon modes are the key to unveil strong coupling over a macroscopic distance, establishing the value of garnets as a platform to study multipartite hybridization processes at microwave frequencies.

We demonstrate a spin-wave-based time-multiplexed Ising Machine (SWIM), implemented using a 5 μm thick Yttrium Iron Garnet (YIG) film and off-the-shelf microwave components. The artificial Ising spins consist of 34–68 ns long 3.125 GHz spinwave RF pulses with their phase binarized using a phase-sensitive microwave amplifier. Thanks to the very low spinwave group velocity, the 7 mm long YIG waveguide can host an 8-spin MAX-CUT problem and solve it in less than 4 μs while consuming only 7 μJ. Using a real-time oscilloscope, we follow the temporal evolution of each spin as the SWIM minimizes its energy and find both uniform and domain-propagation-like switching of the spin state. The SWIM has the potential for substantial further miniaturization, scalability, speed, and reduced power consumption, and may become a versatile platform for commercially feasible optimization problem solvers with high performance.