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Macroscopic-scale nanostructures, situated at the interface of nanostructures and bulk materials, hold significant promise in the realm of thermoelectric materials. Nanostructuring presents a compelling avenue for enhancing material thermoelectric performance as well as unlocking intriguing nanoscale phenomena, including spin-dependent thermoelectric effects. This is achieved while preserving high power output capabilities and ease of measurements related to the overall macroscopic dimensions. Within this framework, the recently developed three-dimensional interconnected nanowire and nanotube networks, integrated into a flexible polymer membrane, emerge as promising candidates for macroscopic nanostructures. The flexibility of these composites also paves the way for advances in the burgeoning field of flexible thermoelectrics. In this study, we demonstrate that the three-dimensional nanowire networks made of ferromagnetic metals maintain the intrinsic bulk thermoelectric power of their bulk constituent even for a diameter reduced to approximately 23 nm. Furthermore, we showcase the pioneering magneto-thermoelectric measurements of three-dimensional interconnected nickel nanotube networks. These macroscopic materials, comprising interconnected nanotubes, enable the development of large-area devices that exhibit efficient thermoelectric performance, while their nanoscale tubular structures provide distinctive magneto-transport properties. This research represents a significant step toward harnessing the potential of macroscopic nanostructured materials in the field of thermoelectrics., Comment: 6 pages, 3 figures

Three-dimensional interconnected nanowire networks have recently attracted notable attention for the fabrication of new devices for energy harvesting/storage, sensing, catalysis, magnetic and spintronic applications and for the design of new hardware neuromorphic computing architectures. However, the complex branching of these nanowire networks makes it challenging to investigate these 3D nanostructured systems theoretically. Here, we present a theoretical description and simulations of the geometric properties of these 3D interconnected nanowire networks with selected characteristics. Our analysis reveals that the nanowire segment length between two crossing zones follows an exponential distribution. This suggests that shorter nanowire segments have a more pronounced influence on the nanowire network properties compared to their longer counterparts. Moreover, our observations reveal a homogeneous distribution in the smallest distance between the cores of two crossing nanowires. The results are highly reproducible and unaffected by changes in the nanowire network characteristics. The density of crossing zones and interconnected nanowire segments are found to vary as the square of the nanowire density multiplied by their diameter, further multiplied by a factor dependent on the packing factor. Finally, densities of interconnected segments up to 10$^{13}$ cm$^{-2}$ can be achieved for 22-$\mu$m-thick nanowire networks with high packing factors. This has important implications for neuromorphic computing applications, suggesting that the realization of 10$^{14}$ interconnections, which corresponds to the approximate number of synaptic connections in the human brain, is achievable with a nanowire network of about 10 cm$^{2}$., Comment: 6 pages, 3 figures

Micromagnetic simulations are used to study a spin-torque vortex oscillator excited by an out-of-plane dc current. The vortex core gyration amplitude is confined between two orbits due to periodical vortex core polarity reversals. The upper limit corresponds to the orbit where the vortex core reaches its critical velocity triggering the first polarity reversal which is immediately followed by a second one. After this double polarity reversal, the vortex core is on a smaller orbit that defines the lower limit of the vortex core gyration amplitude. This double reversal process is a periodic phenomenon and its frequency as well as the upper and lower limits of the vortex core gyration are controlled by the input current density while the vortex chirality determines the onset of this confinement regime. In this non-linear regime, the vortex core never reaches a stable orbit and thus, it may be of interest for neuromorphic application, for example as a leaky integrate-and-fire neuron., Comment: 6 pages, 4 figures

We present a demonstration of image classification using an echo-state network (ESN) relying on a single simulated spintronic nanostructure known as the vortex-based spin-torque oscillator (STVO) delayed in time. We employ an ultrafast data-driven simulation framework called the data-driven Thiele equation approach (DD-TEA) to simulate the STVO dynamics. This allows us to avoid the challenges associated with repeated experimental manipulation of such a nanostructured system. We showcase the versatility of our solution by successfully applying it to solve classification challenges with the MNIST, EMNIST-letters and Fashion MNIST datasets. Through our simulations, we determine that within an ESN with numerous learnable parameters the results obtained using the STVO dynamics as an activation function are comparable to the ones obtained with other conventional nonlinear activation functions like the reLU and the sigmoid. While achieving state-of-the-art accuracy levels on the MNIST dataset, our model's performance on EMNIST-letters and Fashion MNIST is lower due to the relative simplicity of the system architecture and the increased complexity of the tasks. We expect that the DD-TEA framework will enable the exploration of deeper architectures, ultimately leading to improved classification accuracy., Comment: 6 pages, 4 figures

In this study, we developed a quantitative description of the dynamics of spin-torque vortex nano-oscillators (STVOs) through an unconventional model based on the combination of the Thiele equation approach (TEA) and data from micromagnetic simulations (MMS). Solving the STVO dynamics with our analytical model allows to accelerate the simulations by 9 orders of magnitude compared to MMS while reaching the same level of accuracy. Here, we showcase our model by simulating a STVO-based neural network for solving a classification task. We assess its performance with respect to the input signal current intensity and the level of noise that might affect such a system. Our approach is promising for accelerating the design of STVO-based neuromorphic computing devices while decreasing drastically its computational cost., Comment: Presented in ISCS2023

The spin-diode effect is studied both experimentally and with our original semi-analytical method. The latter is based on an improved version of the Thiele equation approach (TEA) that we combine to micromagnetic simulation data to accurately model the non-linear dynamics of spin-torque vortex oscillator (STVO). This original method, called data-driven Thiele equation approach (DD-TEA), absorbs the difference between the analytical model and micromagnetic simulations to provide a both ultra-fast and quantitative model. The DD-TEA model predictions also agree very well with the experimental data. The reversal of the spin-diode effect with the chirality of the vortex, the impact of the input current and the origin of a variation at half of the STVO frequency are presented as well as the ability of the model to reproduce the experimental behavior. Finally, the spin-diode effect and its simulation using the DD-TEA model are discussed as a promising perspective in the framework of neuromorphic computing., Comment: 2 pages, 2 figures

We design a neural network based on a single spin-torque vortex nano-oscillator (STVO) multiplexed in time. The behavior of the STVO is simulated with an improved ultra-fast and quantitative model based on the Thiele equation approach. Different mathematical and numerical adaptations are brought to the model in order to increase the accuracy and the speed of the simulations. We demonstrate the high added value and adaptability of such a neural network through the resolution of three standard machine learning tasks in the framework of reservoir computing. The first one is a task of waveform (sines and squares) classification. We show the ability of the system to effectively classify waveforms with high accuracy and low root-mean-square error thanks to the intrinsic short-term memory of the device. Given the high throughput of the simulations, two innovative parametric studies on the intensity of the input signal and the level of noise in the system are performed to demonstrate the value of our new models. The efficiency of our system is then tested during a speech recognition task on the TI-46 dataset and shows the agreement between the new models and the corresponding experimental measurements. Finally, we use our STVO-based neural network to perform image recognition on the MNIST dataset. State-of-the-art performances are demonstrated, and the interest of using the STVO dynamics as an activation function is highlighted. These results support and facilitate the future development of neuromorphic STVO-based hardware for energy-efficient machine learning., Comment: 11 pages, 8 figures

Due to their memristive properties nanowire networks are very promising for neuromorphic computing applications. Indeed, the resistance of such systems can evolve with the input voltage or current as it confers a synaptic behaviour to the device. Here, we propose a network of silver nanowires (Ag-NWs) which are grown in a nanopourous membrane with interconnected nanopores by electrodeposition. This bottom-up approach fabrication method gives a conducting network with a 3D architecture and a high density of Ag-NWs. The resulting 3D interconnected Ag-NW network exhibits a high initial resistance as well as a memristive behavior. It is expected to arise from the creation and the destruction of conducting silver filaments inside the Ag-NW network. Moreover, after several cycles of measurement, the resistance of the network switches from a high resistance regime, in the GOhm range, with a tunnel conduction to a low resistance regime, in the kOhm range., Comment: 8 pages, 5 figures

The dynamics of vortex based spin-torque nano-oscillators is investigated theoretically. Starting from a fully analytical model based on the Thiele equation approach, fine-tuned data-driven corrections are carried out to the gyrotropic and damping terms. These adjustments, based on micromagnetic simulation results, allow to quantitatively model the response of such oscillators to any dc current within the range of the vortex stability. Both, the transient and the steady-state regimes are accurately predicted under the proposed data-driven Thiele equation approach. Furthermore, the computation time required to solve the dynamics of such system is reduced by about six orders of magnitude compared to the most powerful micromagnetic simulations. This major breakthrough opens the path for unprecedented high-throughput simulations of spin-torque vortex oscillators submitted to long-duration input signals, for example in neuromorphic computing applications., Comment: 6 pages, 4 figures

Understanding the dynamics of magnetic vortices has emerged as an important challenge regarding the recent development of spin-torque vortex oscillators. Either micromagnetic simulations or the analytical Thiele equation approach are typically used to study such systems theoretically. This work focuses on the precise description of the restoring forces exerted on the vortex when it is displaced from equilibrium. In particular, the stiffness parameters related to a modification of the magnetic potential energy terms are investigated. A method is proposed to extract exchange, magnetostatic and Zeeman stiffness expressions from micromagnetic simulations. These expressions are then compared to state-of-the-art analytical derivations. Furthermore, it is shown that the stiffness parameters depend not only on the vortex core position but also on the injected current density. This phenomenon is not predicted by commonly used analytical ans\"atze. We show that these findings result from a deformation of the theoretical magnetic texture caused by the current induced Amp\`ere-Oersted field., Comment: 10 pages, 6 figures

Deep learning has an increasing impact to assist research, allowing, for example, the discovery of novel materials. Until now, however, these artificial intelligence techniques have fallen short of discovering the full differential equation of an experimental physical system. Here we show that a dynamical neural network, trained on a minimal amount of data, can predict the behavior of spintronic devices with high accuracy and an extremely efficient simulation time, compared to the micromagnetic simulations that are usually employed to model them. For this purpose, we re-frame the formalism of Neural Ordinary Differential Equations (ODEs) to the constraints of spintronics: few measured outputs, multiple inputs and internal parameters. We demonstrate with Spin-Neural ODEs an acceleration factor over 200 compared to micromagnetic simulations for a complex problem -- the simulation of a reservoir computer made of magnetic skyrmions (20 minutes compared to three days). In a second realization, we show that we can predict the noisy response of experimental spintronic nano-oscillators to varying inputs after training Spin-Neural ODEs on five milliseconds of their measured response to different excitations. Spin-Neural ODE is a disruptive tool for developing spintronic applications in complement to micromagnetic simulations, which are time-consuming and cannot fit experiments when noise or imperfections are present. Spin-Neural ODE can also be generalized to other electronic devices involving dynamics., Comment: 16 pages, 4 figures

Arrays of spin-torque nano-oscillators are promising for broadband microwave signal detection and processing, as well as for neuromorphic computing. In many of these applications, the oscillators should be engineered to have equally-spaced frequencies and equal sensitivity to microwave inputs. Here we design spin-torque nano-oscillator arrays with these rules and estimate their optimum size for a given sensitivity, as well as the frequency range that they cover. For this purpose, we explore analytically and numerically conditions to obtain vortex spin-torque nano-oscillators with equally-spaced gyrotropic oscillation frequencies and having all similar synchronization bandwidths to input microwave signals. We show that arrays of hundreds of oscillators covering ranges of several hundred MHz can be built taking into account nanofabrication constraints.

Spin-related effects in thermoelectricity can be used to design more efficient refrigerators and offer novel promising applications for the harvesting of thermal energy. The key challenge is to design structural and compositional magnetic material systems with sufficiently high efficiency and power output for transforming thermal energy into electric energy and vice versa. Here, the fabrication of large-area 3D interconnected Co/Cu nanowire networks is demonstrated, thereby enabling the controlled Peltier cooling of macroscopic electronic components with an external magnetic field. The flexible, macroscopic devices overcome inherent limitations of nanoscale magnetic structures due to insufficient power generation capability that limits the heat management applications. From properly designed experiments, large spin-dependent Seebeck and Peltier coefficients of $-9.4$ $\mu$V/K and $-2.8$ mV at room temperature, respectively. The resulting power factor of Co/Cu nanowire networks at room temperature ($\sim7.5$ mW/K$^2$m) is larger than those of state of the art thermoelectric materials, such as BiTe alloys and the magneto-power factor ratio reaches about 100\% over a wide temperature range. Validation of magnetic control of heat flow achieved by taking advantage of the spin-dependent thermoelectric properties of flexible macroscopic nanowire networks lay the groundwork to design shapeable thermoelectric coolers exploiting the spin degree of freedom., Comment: 11 pages, 7 figures

The reservoir computing neural network architecture is widely used to test hardware systems for neuromorphic computing. One of the preferred tasks for bench-marking such devices is automatic speech recognition. However, this task requires acoustic transformations from sound waveforms with varying amplitudes to frequency domain maps that can be seen as feature extraction techniques. Depending on the conversion method, these may obscure the contribution of the neuromorphic hardware to the overall speech recognition performance. Here, we quantify and separate the contributions of the acoustic transformations and the neuromorphic hardware to the speech recognition success rate. We show that the non-linearity in the acoustic transformation plays a critical role in feature extraction. We compute the gain in word success rate provided by a reservoir computing device compared to the acoustic transformation only, and show that it is an appropriate benchmark for comparing different hardware. Finally, we experimentally and numerically quantify the impact of the different acoustic transformations for neuromorphic hardware based on magnetic nano-oscillators., Comment: 13 pages, 5 figures

Spin-torque nano-oscillators can emulate neurons at the nanoscale. Recent works show that the non-linearity of their oscillation amplitude can be leveraged to achieve waveform classification for an input signal encoded in the amplitude of the input voltage. Here we show that the frequency and the phase of the oscillator can also be used to recognize waveforms. For this purpose, we phase-lock the oscillator to the input waveform, which carries information in its modulated frequency. In this way we considerably decrease amplitude, phase and frequency noise. We show that this method allows classifying sine and square waveforms with an accuracy above 99% when decoding the output from the oscillator amplitude, phase or frequency. We find that recognition rates are directly related to the noise and non-linearity of each variable. These results prove that spin-torque nano-oscillators offer an interesting platform to implement different computing schemes leveraging their rich dynamical features.

Substantial evidence indicates that the brain uses principles of non-linear dynamics in neural processes, providing inspiration for computing with nanoelectronic devices. However, training neural networks composed of dynamical nanodevices requires finely controlling and tuning their coupled oscillations. In this work, we show that the outstanding tunability of spintronic nano-oscillators can solve this challenge. We successfully train a hardware network of four spin-torque nano-oscillators to recognize spoken vowels by tuning their frequencies according to an automatic real-time learning rule. We show that the high experimental recognition rates stem from the high frequency tunability of the oscillators and their mutual coupling. Our results demonstrate that non-trivial pattern classification tasks can be achieved with small hardware neural networks by endowing them with non-linear dynamical features: here, oscillations and synchronization. This demonstration is a milestone for spintronics-based neuromorphic computing.

In this paper, we propose to control the strength of phase-locking between two dipolarly coupled vortex based spin-torque nano-oscillators by placing an intermediate oscillator between them. We show through micromagnetic simulations that the strength of phase-locking can be largely tuned by a slight variation of current in the intermediate oscillator. We develop simplified numerical simulations based on analytical expressions of the vortex core trajectories that will be useful for investigating large arrays of densely packed spin-torque oscillators interacting through their stray fields., Comment: 4 pages, 4 figures

The topologically protected magnetic spin configurations known as skyrmions offer promising applications due to their stability, mobility and localization. In this work, we emphasize how to leverage the thermally driven dynamics of an ensemble of such particles to perform computing tasks. We propose a device employing a skyrmion gas to reshuffle a random signal into an uncorrelated copy of itself. This is demonstrated by modelling the ensemble dynamics in a collective coordinate approach where skyrmion-skyrmion and skyrmion-boundary interactions are accounted for phenomenologically. Our numerical results are used to develop a proof-of-concept for an energy efficient ($\sim\mu\mathrm{W}$) device with a low area imprint ($\sim\mu\mathrm{m}^2$). Whereas its immediate application to stochastic computing circuit designs will be made apparent, we argue that its basic functionality, reminiscent of an integrate-and-fire neuron, qualifies it as a novel bio-inspired building block., Comment: 41 pages, 20 figures

Neurons in the brain behave as non-linear oscillators, which develop rhythmic activity and interact to process information. Taking inspiration from this behavior to realize high density, low power neuromorphic computing will require huge numbers of nanoscale non-linear oscillators. Indeed, a simple estimation indicates that, in order to fit a hundred million oscillators organized in a two-dimensional array inside a chip the size of a thumb, their lateral dimensions must be smaller than one micrometer. However, despite multiple theoretical proposals, there is no proof of concept today of neuromorphic computing with nano-oscillators. Indeed, nanoscale devices tend to be noisy and to lack the stability required to process data in a reliable way. Here, we show experimentally that a nanoscale spintronic oscillator can achieve spoken digit recognition with accuracies similar to state of the art neural networks. We pinpoint the regime of magnetization dynamics leading to highest performance. These results, combined with the exceptional ability of these spintronic oscillators to interact together, their long lifetime, and low energy consumption, open the path to fast, parallel, on-chip computation based on networks of oscillators.

Phase coupling between auto-oscillators is central for achieving coherent responses such as synchronization. Here we present an experimental approach to probe it in the case of two dipolarly coupled spin-torque vortex nano-oscillators using an external microwave field. By phase-locking one oscillator to the microwave field, we observe frequency pulling on the second oscillator. From coupled Thiele equations we show analytically that this frequency pulling results from concerted actions of dipolar and microwave couplings. The analysis allows us to determine the amplitude of dipolar coupling and the phase lag of each oscillator, yielding important information for the implementation of large oscillator networks.

By investigating thoroughly the tunable behavior of coupled modes, we highlight how it provides new means to handle the properties of spin transfer nano-oscillators. We first demonstrate that the main features of the microwave signal associated with coupled vortex dynamics i.e. frequency, spectral coherence, critical current, mode localization, depends drastically on the relative vortex core polarities. Secondly we report a large reduction of the nonlinear linewidth broadening obtained by changing the effective damping through the control of the core configuration. Such a level of control on the nonlinear behavior reinforces our choice to exploit the microwave properties of collective modes for applications of spintronic devices in novel generation of integrated telecommunication devices.