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Ultra-fast spectrum analysis concept based on rapidly tuned spintronic nano-oscillators has been under development for the last few years and has already demonstrated promising results. Here, we demonstrate an ultra-fast microwave spectrum analyzer based on a chain of five mutually synchronized nano-constriction spin Hall nano-oscillators (SHNOs). As mutual synchronization affords the chain a much improved signal quality, with linewidths well below 1 MHz at close to a 10 GHz operating frequency, we observe an order of magnitude better frequency resolution bandwidth compared to previously reported spectral analysis based on single magnetic tunnel junction based spin torque nano-oscillators. The high-frequency operation and ability to synchronize long SHNO chains and large arrays make SHNOs ideal candidates for ultra-fast microwave spectral analysis., Comment: 5 Figures

Ising machines (IM) have recently been proposed as unconventional hardware-based computation accelerators for solving NP-hard problems. In this work, we present a model for a time-multiplexed IM based on the nonlinear oscillations in a delay line-based resonator and numerically study the effects that the circuit parameters, specifically the compression gain $\beta_r$ and frequency nonlinearity $\beta_i$, have on the IM solutions. We find that the likelihood of reaching the global minimum -- the global minimum probability (GMP) -- is the highest for a certain range of $\beta_r$ and $\beta_i$ located near the edge of the synchronization region of the oscillators. The optimal range remains unchanged for all tested coupling topologies and network connections. We also observe a sharp transition line in the ($\beta_i, \beta_r$) space above which the GMP falls to zero. In all cases, small variations in the natural frequency of the oscillators do not modify the results, allowing us to extend this model to realistic systems., Comment: 7 pages, 5 figures; supplementary material 2 pages

As nascent nonlinear oscillators, nano-constriction spin Hall nano-oscillators (SHNOs) represent a promising potential for integration into more complicated systems such as neural networks, magnetic field sensors, and radio frequency (RF) signal classification, their tunable high-frequency operating regime, easy synchronization, and CMOS compatibility can streamline the process. To implement SHNOs in any of these networks, the electrical features of a single device are needed before designing the signal detection CMOS circuitry. This study centers on presenting an empirical electrical model of the SHNO based on a comprehensive characterization of the output impedance of a single SHNO, and its available output power in the range of 2-10 GHz at various bias currents., Comment: 6 pages

The ever increasing demand for computational power combined with the predicted plateau for the miniaturization of existing silicon-based technologies has made the search for low power alternatives an industrial and scientifically engaging problem. In this work, we explore spintronics-based Ising machines as hardware computation accelerators. We start by presenting the physical platforms on which this emerging field is being developed, the different control schemes and the type of algorithms and problems on which these machines outperform conventional computers. We then benchmark these technologies and provide an outlook for future developments and use-cases that can help them get a running start for integration into the next generation of computing devices.

This chapter reviews the state of the art in mutually synchronized spin-torque and spin Hall nano-oscillator (STNO and SHNO) arrays. After briefly introducing the underlying physics, we discuss different nano-oscillator implementations and their functional properties with respect to frequency range, output power, phase noise, and modulation rates. We then introduce the concepts and the theory of mutual synchronization and discuss the possible coupling mechanisms in spintronic nano-oscillators, such as dipolar, electrical, and spin-wave coupling. We review the experimental literature on mutually synchronized STNOs and SHNOs in one- and two-dimensional arrays and discuss ways to increase the number of mutually synchronized nano-oscillators. Finally, the potential for applications ranging from microwave signal sources/detectors and ultrafast spectrum analyzers to neuromorphic computing elements and Ising machines is discussed together with the specific electronic circuitry that has been designed so far to harness this potential., Comment: Book Chapter (Dynamical Phenomena in Spintronics/nano-magnetism)

Time-multiplexed Spinwave Ising Machines (SWIMs) have unveiled a route towards miniaturized, low-cost, and low-power solvers of combinatorial optimization problems. While the number of supported spins is limited by the nonlinearity of the spinwave dispersion, other collective excitations, such as surface acoustic waves (SAWs), offer a linear dispersion. Here, we demonstrate an all-to-all, fully FPGA reprogrammable, 50-spin surface acoustic wave-based Ising machine (SAWIM), using a 50-mm-long Lithium Niobate SAW delay line, off-the-shelf microwave components, and a low-cost FPGA. The SAWIM can solve any 50-spin MAX-CUT problem, with arbitrary coupling matrices, in less than 340 $\mu$s consuming only 0.62 mJ, corresponding to close to 3000 solutions per second and a figure of merit of 1610 solutions/W/s. We compare the SAWIM computational results with those of a 100-spin optical Coherent Ising machine and find a higher probability of solution. Moreover, we demonstrate that there is an optimum overall coupling strength between spins at which the probability of the exact solution reaches 100%. The SAWIM illustrates the general merits of solid state wave-based time-multiplexed Ising machines in the microwave domain as versatile platforms for commercially feasible high-performance solvers of combinatorial optimization problems.

A spinwave Ising machine (SWIM) is a newly proposed type of time-multiplexed hardware solver for combinatorial optimization that employs feedback coupling and phase sensitive amplification to map an Ising Hamiltonian into phase-binarized propagating spin-wave RF pulses in an Yttrium-Iron-Garnet (YIG) film. In this work, we increase the mathematical complexity of the SWIM by adding a global Zeeman term to a 4-spin MAX-CUT Hamiltonian using a continuous external electrical signal with the same frequency as the spin pulses and phase locked with with one of the two possible states. We are able to induce ferromagnetic ordering in both directions of the spin states despite antiferromagnetic pairwise coupling. Embedding a planar antiferromagnetic spin system in a magnetic field has been proven to increase the complexity of the graph associated to its Hamiltonian and thus this straightforward implementation helps explore higher degrees of complexity in this evolving solver., Comment: Index Terms: combinatorial optimization problems, Ising machines, spinwaves, unconventional computing, physical computing, spinwaves

Nano-constriction based spin Hall nano-oscillators (SHNOs) are at the forefront of spintronics research for emerging technological applications such as oscillator-based neuromorphic computing and Ising Machines. However, their miniaturization to the sub-50 nm width regime results in poor scaling of the threshold current. Here, we show that current shunting through the Si substrate is the origin of this problem and study how different seed layers can mitigate it. We find that an ultra-thin Al$_{2}$O$_{3}$ seed layer and SiN (200 nm) coated p-Si substrates provide the best improvement, enabling us to scale down the SHNO width to a truly nanoscopic dimension of 10 nm, operating at threshold currents below 30 $\mu$A. In addition, the combination of electrical insulation and high thermal conductivity of the Al$_{2}$O$_{3}$ seed will offer the best conditions for large SHNO arrays, avoiding any significant temperature gradients within the array. Our state-of-the-art ultra-low operational current SHNOs hence pave an energy-efficient route to scale oscillator-based computing to large dynamical neural networks of linear chains or two-dimensional arrays., Comment: 10 pages, 6 figures

The reduction of phase noise in electronic systems is of utmost importance in modern communication and signal processing applications and requires an understanding of the underlying physical processes. Here, we systematically study the phase noise in mutually synchronized chains of nano-constriction spin Hall nano-oscillators (SHNOs). We find that longer chains have improved phase noise figures at low offset frequencies (1/f noise), where chains of two and ten mutually synchronized SHNOs have 2.8 and 6.2 dB lower phase noise than single SHNOs. This is close to the theoretical values of 3 and 10 dB, and the deviation is ascribed to process variations between nano-constrictions. However, at higher offset frequencies (thermal noise), the phase noise unexpectedly increases with chain length, which we ascribe to process variations, a higher operating temperature in the long chains at the same drive current and phase delays in the coupling between nano-constrictions.

We demonstrate a spin-wave-based time-multiplexed Ising Machine (SWIM), implemented using a 5 $\mu$m 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 $\mu$s while consuming only 7 $\mu$J. 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.

The main limitation in order to exploit spin torque nano-oscillators (STNOs) in various potential applications is their large phase noise. In this work, we demonstrate its efficient reduction by a highly reconfigurable, compact, specifically on-chip designed PLL based on custom integrated circuits. First, we thoroughly study the parameter space of the PLL+STNO system experimentally. Second, we present a theory which describes the locking of a STNO to an external signal in a general sense. In our developed theory, we do not restrict ourselves to the case of a perfect phase locking but also consider phase slips and the corresponding low offset frequency $1/f^2$ noise, so far the main drawback in such systems. Combining experiment and theory allows us to reveal complex parameter dependences of the system's phase noise. The results provide an important step for the optimization of noise properties and thus leverage the exploitation of STNOs in prospective real applications.

The advantage of an ultra-fast frequency-tunability of spin-torque nano-oscillators (STNOs) that have large (> 100MHz) relaxation frequency of amplitude fluctuations is exploited to realize ultra-fast wide-band time-resolved spectral analysis at nanosecond time scale with the 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 study the collective dynamics of two distant magnets coherently coupled by acoustic phonons that are transmitted through an intercalated crystal. By tuning the ferromagnetic resonances of the two magnets to an acoustic resonance of the crystal, we control a coherent three levels system. We show that the parity of the phonon mode governs the nature of the indirect coupling between the magnets: the resonances with odd / even phonon modes correspond to out-of-phase / in-phase lattice displacements at the magnets, leading to bright / dark states in response to uniform microwave magnetic fields, respectively. The sample is a tri-layer garnet consisting of two thin magnetic layers epitaxially grown on both sides of a half-millimeter thick non-magnetic single crystal. In spite of the relatively weak magneto-elastic interaction, the long lifetimes of the magnon and phonon modes in the sample are the key to unveil this long range strong coupling. This demonstrates that garnets are a great platform to study multi-partite hybridization process between magnon and phonons at microwave frequencies., Comment: 8 pages, 5 figures

We report ferromagnetic resonance in the normal configuration of an electrically insulating magnetic bilayer consisting of two yttrium iron garnet (YIG) films epitaxially grown on both sides of a 0.5-mm-thick nonmagnetic gadolinium gallium garnet (GGG) slab. An interference pattern is observed and it is explained as the strong coupling of the magnetization dynamics of the two YIG layers either in phase or out of phase by the standing transverse sound waves, which are excited through a magnetoelastic interaction. This coherent mediation of angular momentum by circularly polarized phonons through a nonmagnetic material over macroscopic distances can be useful for future information technologies.