United States

Spintronic diodes (STDs) are emerging as a technology for the realization of high-performance microwave detectors [1]. In this paper, we use their rectification properties to perform the analog multiplication between any two numbers. For this purpose, we introduce the concept of degree of rectification (DOR) and we confirm its applicability with micromagnetic and experimental data. The proposed method is suitable for highly integrated implementations and has promising perspectives for neural networks applications [2].&lt;br/&gt;The multiplication can be obtained by measuring the response of a parabolic phenomenon to three well-determined inputs [3][4][5]. Therefore, we analyze the DOR of spintronic diodes, which is the output &lt;i&gt;dc&lt;/i&gt; voltage measured for a variable input &lt;i&gt;dc&lt;/i&gt; current in injection-locked devices. This &lt;i&gt;V&lt;sub&gt;dc&lt;/sub&gt;/I&lt;sub&gt;dc&lt;/sub&gt;&lt;/i&gt; curve can be well approximated to a parabola as shown in Fig. 1, the experimental data are taken from [6].&lt;br/&gt;The voltage can be approximated by the following equation [6]:&lt;br/&gt;&lt;i&gt;V&lt;sub&gt;dc&lt;/sub&gt;=[(R&lt;sub&gt;AP&lt;/sub&gt;-R&lt;sub&gt;P&lt;/sub&gt; )/4] I&lt;sub&gt;ac,MAX &lt;/sub&gt;&lt;/i&gt;cos&lt;i&gt;(φ(I&lt;sub&gt;dc&lt;/sub&gt;)) √p &lt;/i&gt;(1)&lt;br/&gt;where &lt;i&gt;R&lt;sub&gt;AP&lt;/sub&gt;&lt;/i&gt; and &lt;i&gt;R&lt;sub&gt;P&lt;/sub&gt;&lt;/i&gt; are the device resistances in the antiparallel and parallel state, respectively, &lt;i&gt;p&lt;/i&gt; is the power of the magnetoresistive signal, &lt;i&gt;φ&lt;/i&gt;&lt;i&gt;(I&lt;sub&gt;dc&lt;/sub&gt;)&lt;/i&gt; is the intrinsic phase shift between the &lt;i&gt;ac&lt;/i&gt; current and the magnetoresistive signal, and the cosine can be well approximated with a parable using Taylor’s second-order expansion. Moreover, Eq. 1 can be fully characterized knowing the physical parameters of the device and the values of &lt;i&gt;I&lt;sub&gt;dc,MAX&lt;/sub&gt;&lt;/i&gt;, &lt;i&gt;V&lt;sub&gt;dc,MAX&lt;/sub&gt;&lt;/i&gt; (both represented in Fig. 1) and the second derivative of &lt;i&gt;V&lt;sub&gt;dc&lt;/sub&gt;&lt;/i&gt; with respect to &lt;i&gt;I&lt;sub&gt;dc&lt;/sub&gt;&lt;/i&gt; evaluated in &lt;i&gt;I&lt;sub&gt;dc&lt;/sub&gt;=I&lt;sub&gt;dc,MAX&lt;/sub&gt;&lt;/i&gt;.&lt;br/&gt;Fig. 2 shows the application of the presented method for 200 cases with random inputs, the red line shows the ideal result; the correlation coefficient found is &lt;i&gt;r&lt;/i&gt;=99.83%.&lt;br/&gt;We will show how the proposed method can be used to implement the convolution for neural networks applications with excellent results for image recognition.

![](https://assets.underline.io/uploads/markdown_image/1/image/8be5d570eb2198667334d746191b22a6.jpg)

Fig. 1: Output dc voltage depending on the input dc current (data taken from [4]) for an injection-locked diode (blue circles) and parabolic fit (red line).

![](https://assets.underline.io/uploads/markdown_image/1/image/ef2d86778d75be9a7a2f863250048857.jpg)

Fig. 2: 200 multiplications obtained with the proposed method (black circles) compared with the ideal result (red line).

2022 Joint MMM-INTERMAG

Analog multiplication implemented with spin-torque diodes for neural networks applications

The 15th Joint MMM-INTERMAG Conference (2022 Joint) is sponsored jointly by AIP Publishing and the IEEE Magnetics Society. Members of the international scientific and engineering communities interested in recent developments in fundamental and applied magnetism are invited to attend and contribute to the technical sessions. The technical program will include invited and contributed papers in oral and poster sessions, invited symposia, a plenary session, and an evening session, with about 1500 presentations overall. This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

The 2022 Joint Conference will offer both in-person and prerecorded on-demand content.

For tickets to the event, please register at the following link: https://magnetism.org/register

Please register for the Joint MMM-Intermag conference to join the event.

This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

Spintronic diodes (STDs) are emerging as a technology for the realization of high-performance microwave detectors [1]. In this paper, we use their rectification properties to perform the analog multiplication between any two numbers. For this purpose, we introduce the concept of degree of rectification (DOR) and we confirm its applicability with micromagnetic and experimental data. The proposed method is suitable for highly integrated implementations and has promising perspectives for neural networks applications [2]. The multiplication can be obtained by measuring the response of a parabolic phenomenon to three well-determined inputs [3][4][5]. Therefore, we analyze the DOR of spintronic diodes, which is the output dc voltage measured for a variable input dc current in injection-locked devices. This Vdc/Idc curve can be well approximated to a parabola as shown in Fig. 1, the experimental data are taken from [6]. The voltage can be approximated by the following equation [6]: Vdc=[(RAP-RP )/4] Iac,MAX cos(φ(Idc)) √p (1) where RAP and RP are the device resistances in the antiparallel and parallel state, respectively, p is the power of the magnetoresistive signal, φ(Idc) is the intrinsic phase shift between the ac current and the magnetoresistive signal, and the cosine can be well approximated with a parable using Taylor’s second-order expansion. Moreover, Eq. 1 can be fully characterized knowing the physical parameters of the device and the values of Idc,MAX, Vdc,MAX (both represented in Fig. 1) and the second derivative of Vdc with respect to Idc evaluated in Idc=Idc,MAX. Fig. 2 shows the application of the presented method for 200 cases with random inputs, the red line shows the ideal result; the correlation coefficient found is r=99.83%. We will show how the proposed method can be used to implement the convolution for neural networks applications with excellent results for image recognition.

![](https://assets.underline.io/uploads/markdown_image/1/image/8be5d570eb2198667334d746191b22a6.jpg)

Fig. 1: Output dc voltage depending on the input dc current (data taken from [4]) for an injection-locked diode (blue circles) and parabolic fit (red line).

![](https://assets.underline.io/uploads/markdown_image/1/image/ef2d86778d75be9a7a2f863250048857.jpg)

Fig. 2: 200 multiplications obtained with the proposed method (black circles) compared with the ideal result (red line).

technical paper

Topological magnetic semimetals, like Co3Sn2S2 and Co2MnGa, display exotic transport properties, such as large intrinsic anomalous (AHE) due to uncompensated Berry curvature. The highly symmetric XPt3 compounds display anti-crossing gapped nodal lines, a driving mechanism in the intrinsic Berry curvature Hall effects. Uniquely, these compounds contain two sets of gapped nodal lines that harmoniously dominate the Berry curvature in this complex multi-band system. We calculate a maximum AHE of 1965 S/cm in the CrPt3 by a state-of-the-art first principle electronic structure. We have grown high-quality thin films by magnetron sputtering and measured a robust AHE of 1750 S/cm for different sputtering growth conditions. Additionally, the cubic films display a hard magnetic axis along [111] direction. The facile and scalable fabrication of these materials makes them ideal candidates for integration into topological devices [1].

![](https://assets.underline.io/uploads/markdown_image/1/image/4135773548df436a10a909902db4a303.jpg)

Hard magnet topological semimetals in XPt3 compounds with the harmony of Berry curvature

Despite having published no papers related to magnetism prior to 1996, In 1996 Prof. Buhrman initiated a new project in his group to explore phenomena relevant to proposed MRAM technologies. Over the next 25 years, his group would make several seminal contributions to this field, and in this talk, I will present a retrospective on the impact Bob Buhrman’s group had on MRAM during this timeframe. Twenty-five years ago magnetism played a key role in HDD and tape storage technologies, but magnetism was not exploited in memories that could be integrated with planar semiconductor processing. Today, embedded MRAM is offered at all of the leading foundries, and standalone MRAM memory chips are available at capacities up to 1 Gbit. STT-MRAM is the basis of these of these memories. Many research groups have played key roles in the commercialization of this technology, but Prof. Buhrman’s group was responsible for significant advances including the first demonstration of STT switching in lithographically patterned bits [1] and early demonstrations of STT switching in magnetic tunnel junctions [2]. There is considerable effort today on using STT-MRAM as a replacement for embedded SRAM, but it is difficult for STT-MRAM to meet the endurance and speed requirements for these applications. Three-terminal SOT-MRAM technology is a potential alternative to STT-MRAM for such applications. Here again, Prof. Buhrman’s group was at the forefront of this research [3], and has subsequently made many important contributions to the understanding of SOT physics.

The Impact of the Buhrman Group on MRAM over the Past 25 Years

The processing of artificial neural networks (ANNs) is limited by the memory wall in von Neumann architectures, driving the need for hardware ANNs realized by neuromorphic devices [1]. Previously proposed multi-weight synaptic devices often suffer from a nonlinear and asymmetric update response, limiting their viability for supervised learning [2]. Domain wall-magnetic tunnel junction (DW-MTJ) devices driven by spin transfer torque (STT) and spin orbit torque (SOT) have shown promise in neuromorphic applications, realizing leaky integrate-and-fire neurons in simulation [3-8]. Simulations also show that DW-MTJs can implement synaptic spike-timing dependent plasticity [9]. A type of DW-MTJ with multiple MTJs was demonstrated experimentally [10], and we have recent results on experimental DW-MTJ synapses that are submitted separately for consideration in the conference. Most prior work on DW-MTJ synapses in ANNs do not consider thermal effects and process variation. We evaluate notched DW-MTJ designs shown in Fig. 1a and account for these effects using MuMax3 for device simulation and CrossSim for network simulation [11-13]. Figure 1b shows that excellent linearity, high symmetry, and low thermal noise are maintained at 300 K. High training accuracy was obtained on the Fashion-MNIST task, shown in Fig. 2 [14]. Notably, the greater stochasticity of SOT-driven synapses counters the discretizing effect of notches, boosting accuracy to near ideal. We also implemented short term potentiation for network pruning using shape anisotropy. These results propose a magnetic synapse with tunable plasticity, superior backpropagation performance, and fast updates, a foundational step toward ANNs with fully spintronic matrix operations. SNL is managed and operated by NTESS under DOE NNSA contract DE-NA0003525

![](https://assets.underline.io/uploads/markdown_image/1/image/a44eed1961abbd5be85c2b76d154f44f.jpg)

a) Device diagram b) Conductance G vs dG statistics of a DW-MTJ at 300 K for STT and SOT propagation. Heatmaps are generated from micromagnetic simulations. The CDF represents the probability that a given conductance update is less than the average dG.

![](https://assets.underline.io/uploads/markdown_image/1/image/8b8eb9cbd24d31e6dd1da9e5c0cf1410.jpg)

Validation accuracy of a two-layer perceptron of STT and SOT DW-MTJ synapses at 300 K assuming (a) continuous levels and (b) 32 notches and periodic carry.

Multifunctional Design of Domain Wall-Magnetic Tunnel Junction Artificial Synapses for Accurate Training of Neural Networks

Reservoir computing (RC) is a machine learning paradigm where computational power is obtained from the intrinsic memory and nonlinearity of a dynamic system. Traditionally, the dynamic system is provided by a recurrent neural network, though many recent studies show this network can be substituted with a physical system. Within the field of magnetism, many systems have been proposed as candidates for RC, such as magnetic tunnel junctions (MTJs) [1], skyrmion textures [2], superparamagnetic arrays [3], and even single domain walls [4]. However, aside from MTJs, few of these proposed systems have been experimentally realised. Recently, we showed that single-layer arrays of interconnected permalloy nanorings exhibit emergent domain wall dynamics suitable for computation [5]. Here, we demonstrate RC in materio using a device containing the aforementioned arrays. Data is encoded to the array by modulating the amplitude of a rotating magnetic field. Using two-point resistance measurements, we show that we can extract information on the state of the array through anisotropic magnetoresistance (AMR). These measurements exhibit nonlinearities between applied field and the measured magnetic state of the system, as well as hysteretic memory. Next, we show how we can control the system’s dynamics through scaling of the applied magnetic field, allowing a range of different transfer functions between input and measured magnetic state. These different transfer functions can be exploited to provide improved computational performance on signal, speech, and writing processing tasks. Through optimised scaling of the applied fields, the different computational demands of each task are met on a single device. We believe that the computational versatility, relative simplicity of manufacture, as well as the wide range of available parameter space to further explore makes the ring arrays an exciting platform for neuromorphic computation.

![](https://assets.underline.io/uploads/markdown_image/1/image/2dd48ac941e88dbd5d2d7ca08ddcf6a4.jpg)

Measured AMR response for 4 rotations of 30 Oe applied field, followed by 4 rotations of 20 Oe applied field, showing both nonlinear and hysteretic behaviour.

![](https://assets.underline.io/uploads/markdown_image/1/image/7848690af170509ddac1fd479d8a859f.jpg)

Micrograph of patterned nanoring array and electric contacts taken from RAITH Voyager electron beam lithography system.

Experimental Demonstration of Reservoir Computation using Emergent Domain Wall Dynamics in a Patterned Magnetic Substrate

Emulation of neurobiological information processing by artificial neural networks enables new frontiers in efficiency when performing data-intensive and real-time artificial intelligence tasks1,2. In this work, we experimentally demonstrate a multi-weight (MW) artificial synapse using a domain wall-magnetic tunnel junction (DW-MTJ) with a straight, notched geometry, depicted in Fig. 1(a). While previous simulations show that DW-MTJ synapses have the necessary characteristics for neuromorphic computing, experimental work has focused on binary DW-MTJs3, and no experimental works have closely studied the behavior of the MW-DW-MTJ synapse, in contrast with works studying multi-MTJ devices4. The heavy metal/ferromagnet/oxide trilayer stack3 is patterned into a DW track with a single, long output MTJ centered on top of the track. To achieve MW synaptic switching, five notches are fabricated on the track (n1-5). Fig. 1(b) shows the MTJ resistance (RMTJ) vs. out-of-plane magnetic field for a synapse with DW track width w = 350 nm, showing the maximum and minimum RMTJ. To study synaptic switching behaviors, voltage V is applied between the IN and CLK terminals (50 ns pulses of increasing amplitude, with step size 0.1 V) to move the DW from notch to notch via current driven via spin orbit torque. Fig. 2(a) shows RMTJ vs. V for 10 cycles, showing five resistances each corresponding to the DW in one of the notches. From the spacing dimensions of the notches, we estimate RMTJ when the DW is pinned at each notch, as shown in Fig. 2(b), which are close to the resistances observed in Fig. 2(a). We will show that repeated single-amplitude V pulses also move the DW notch to notch but ramping up the amplitude allows us to observe all switches without tuning the amplitude precisely. We will show how the cycle-to-cycle stochasticity may aid certain online learning tasks when V can be applied repeatably over time to set the synapse weight, allowing for stochastic outer product updates. This work informs essential design parameters using DW-based synapses for future neuromorphic computing.

![](https://assets.underline.io/uploads/markdown_image/1/image/0326a4371e8db824bf485edae008119b.jpg)

Fig. 1. (a) DW-MTJ synapse. (b) Field loop.

![](https://assets.underline.io/uploads/markdown_image/1/image/25feb81acfb52d72151ddba93b51f603.jpg)

Fig. 2. (a) MW switching for 10 cycles. (b) Expected vs measured RMTJ when the DW is pinned at each notch.

Multi-Weight Artificial Synapses with Straight Notched Geometry for Neuromorphic Computing

FeRh is widely studied because of its novel temperature-dependent antiferromagnetic (AFM) to ferromagnetic (FM) phase transition. This AFM-FM phase transition, which is accompanied by a significant change in resistivity, occurs at a critical temperature that can be fine-tuned over a wide range through substitutional doping, strain, and patterning.[1,2] Moreover, the temperature dependence of the transition provides a means to manipulate the state via Joule heating. Recent reports, based on ultrafast pump-probe measurements, show the AFM-FM transition occurs on a sub-picosecond timescale, thus devices operating in switching applications have the potential to operate in excess of 100’s GHz provided adequate thermal dissipation is achieved. In this work we demonstrate high-speed switching of FeRh wires, giving rise to a dynamic memristive device based on a metamagnetic transition. The thermally-induced AFM-FM transition was evaluated using two-terminal devices consisting of an FeRh wire and Ti/Au contacts. We identified geometrical dependencies in the AFM-FM transition temperature, which scaled with both current density and wire length. Pulsed I-V measurements were used to investigate the dynamic Joule heating effects, including the device switching speed and resulting power switching losses accompanying the AFM-FM transition. The lower bound of our device switching time, measured to be near 300 nanoseconds, was limited by measurement equipment limitations, not the material system. The performance of this rudimentary device is comparable to other phase change memory technologies with more intricate device architectures. FeRh could be the basis for a very fast, phase-change approach to future computing.

High-Speed Switching of FeRh Memristors

The development of efficient processing of unstructured information necessitates research into neuromorphic computing. In this work, we report domain wall-magnetic tunnel junction (DW-MTJ) artificial synaptic devices with a trapezoidal geometry that are uniquely suited for neuromorphic computation. DW-MTJ devices are nonvolatile memory-in-logic elements that can be integrated into an artificial neural network1. The bottom ferromagnetic layer of the MTJ is a DW track, and the magnetization of that layer is determined by the position of the DW within the track. The device can act as a programmable resistor based on the location of the DW. Simulations have shown that this high level of resistance state control and stability could allow for advanced functionalities for neuromorphic computing1,2,3. However, experimental work has focused on binary DW-MTJs4, and almost no experimental work has been done to understand the behavior of the multi-weight (MW) DW-MTJ synapse, with one paper showing results on a multi-MTJ device5. Here, we present a DW-MTJ based MW synapse that is directional, has stable and repeatable resistance states, and uses a single MTJ. The MTJ was lengthened and pinning notches were fabricated along the DW track. This way the DW could rest under the MTJ, opening more resistance states. A trapezoidal DW track was patterned for directional switching, shown in Fig. 1. As the DW is moved from IN to CLK using spin orbit torque switching, the voltage amplitude required to propagate the DW increases as the track widens, which results in MW switching. Going CLK to IN does not exhibit MW switching since depinning from the right-most notch requires the maximum voltage, shown in Fig. 2. We will show how this asymmetry in writing and reset is useful for offline learning to bring the synapse to a resistance state using a particular voltage, and then to reset the device efficiently. We will compare the behavior of the trapezoidal DW-MTJ synapse to the straight DW-MTJ synapse presented separately.

![](https://assets.underline.io/uploads/markdown_image/1/image/d8758b10df5900a61b218c67399668ce.jpg)

Fig. 1: Trapezoidal Synapse. Left) SEM image. Right) Schematic with DW track in blue, MTJ in red, electrodes in green.

![](https://assets.underline.io/uploads/markdown_image/1/image/00ff215da248f99f97ffb70a113e6586.jpg)

Fig. 2: Three consecutive directional write then reset cycles

Multi-Weight Directional Artificial Synapses with Trapezoidal Geometry for Neuromorphic Computing

CoFeB/MgO-based p-MTJs are the de facto standard for current STT-MRAM [1-7]. A Co/Pt-based Ru-synthetic antiferromagnetic (SyF) coupling reference layer is widely used in p-MTJs to compensate the stray magnetic field from the reference layer owing to its high thermal stability [4]. However, to scale down MTJs, a higher stable reference layer is required [8]. To improve the reference layer stability, an Ir-SyF coupling layer has been proposed [9]. To utilize the Ir coupling layer in the Co/Pt-based reference layer, one needs to understand the relationship between MTJ properties and sputtering conditions. In this study, we have investigated the effect of the sputtering conditions on the magnetic and transport properties and also compared them with those of Ru. We prepared two types of stacks with Co/Pt-based Ir- and Ru-SyF coupling layers, as shown in Figs. 1-2. Ir was deposited by two sputtering conditions, denoted as Condition-A and Condition-B. The stacks were annealed at 400°C for 1 h. The magnetic and transport properties of the MTJ stacks were measured using VSM and CIPT method, respectively. The stack with Condition-A exhibits a larger exchange coupling field Hex than that with Condition-B (Fig.1). In the EDX line analysis (Fig.2), Ir profile for Condition-A is wider than that for Condition-B, resulting in lower Hex for Condition-B. A possible reason for the different intermixing degree of Ir and Co seems to be that the atoms for Condition-A recoiled from the Ir sputtering target have a higher energy than the atoms for Condition-A[10]. The stacks with the Ir-SyF show lower TMR ratio (Fig.2), resulting from the Pt diffusion into the MgO barrier. In contrast, the stacks with Ru show high TMR ratio because the Pt diffusion is suppressed. These results indicate that the suppression of Ir and Pt diffusion in the reference layer is crucial to achieving robust MTJs that can withstand annealing. This work was supported by CIES’s Industrial Affiliation on the STT MRAM program, CIES Consortium, JST-OPERA, and NEDO.

![](https://assets.underline.io/uploads/markdown_image/1/image/8b97ecf7501066e33eae609224bcb3c4.jpg)

Fig. 1 Coupling layer thickness dependence of Hex for SyF.

![](https://assets.underline.io/uploads/markdown_image/1/image/fdf4e7680e9287faf2ebfc8521f25e40.jpg)

Fig. 2 EDX depth profiles of the stacks with (a) Ir for Condition-B, (b) Ir for Condition-A , and (c) Ru.

Influence of sputtering conditions on the magnetic properties of Co/Pt-based iridium-synthetic antiferromagnetic coupling reference layer

As one of the next-generation memory technologies, magnetic random access memory (MRAM) is of great interest because of its non-volatility, high access speed, large integration density and low power consumption. Current-induced spin-orbit torque (SOT) magnetization switching has been proposed for improving the writing performance of MRAM. Meanwhile, it has been demonstrated that the Co-based full Heusler alloys show potential for increasing the magnetoresistance ratio and achieving efficient reading due to their relatively large spin polarization1. Therefore, achieving the SOT magnetization switching in full Huesler alloys will be promising for optimizing both the reading and writing performance of MRAM devices. Here, we report a successful full SOT magnetization switching in a perpendicularly magnetized full Heusler alloy Co2FeSi2 by using Pd as a spin current generating layer. To eliminate the heating effect, a large writing pulse current with a pulse width of 0.1 ms (red columns in Fig. 1) was applied to switch the magnetization, and a small reading current of 1 mA (blue columns in Fig. 1) was applied for 0.5 s to read the Hall resistance. As shown in Fig. 2, with the assistance of the external magnetic field of 500 Oe, the magnetization can be fully switched with a switching current density of 3.7×107 A cm-2, which is in the same order of magnitude as that required in the conventional heavy metal (HM)/ferromagnet system even though the Pd shows a relatively smaller spin Hall angle than that of HM3. By using harmonic Hall measurements, the damping-like and field-like effective fields per unit current density are estimated to be 56.9 (10–7 Oe A–1 cm2) and 39.8 (10–7 Oe A–1 cm2), respectively. Our findings will advance the development of SOT-MRAM devices with both better reading and writing performance. This work was partly supported by Grants-in-Aid for Scientific Research (Nos. 18H03860, 20H05650, and 20F20366), CREST Program of JST (JPMJCR1777), Spintronics Research Network of Japan (Spin-RNJ), and Grant-in-Aid for JSPS Fellows (20F20366).

![](https://assets.underline.io/uploads/markdown_image/1/image/3b1aaee3c5eb41883e2f839f28df7ddf.jpg)


Fig. 1 Pulse current train used for the SOT switching measurements.

![](https://assets.underline.io/uploads/markdown_image/1/image/2dd84c170180b84b3de155838d52a263.jpg)

Fig. 2 SOT switching in the MgO/Co2FeSi/Pd heterostructure at room temperature.

Spin-orbit torque magnetization switching in a perpendicularly magnetized full Heusler alloy Co2FeSi

The rhythmic spiking behavior observed in biological neurons [1] has inspired oscillator based implementations of neural networks [2]. We theoretically investigate the use of synchronized spin-torque nano-oscillators (STNOs) to create an associative memory encoded with synaptic weights as time delays. STNOs frequency lock to external AC drives if the external drive frequency is close to the natural frequency of the STNO [3] but the relative phase varies with the details. Fig. 1 shows the implementation of a complex Hopfield network in which the relative phases between the synchronized STNO oscillations encode quasi-continuously varying information, such as the pixel values of a grayscale image. The synaptic feedback matrix implements a complex weight matrix by taking the AC output voltages from the STNO array and re-injecting weighted and time-delayed superpositions of AC currents back into the STNO array. The feedback matrix is set using one of several training methods. Then, retrieval of a stored pattern in the associative memory proceeds as shown in Fig. 2. The feedback is disconnected and the individual oscillators are injection locked to an array of external AC sources that encode the query image. A schematic evolution of the relative phases of the individual oscillators, starting from a random state, during this preparation phase is plotted in Fig. 2a. Then, the feedback connections are engaged and the network evolves towards one of the patterns stored in the synaptic weights as shown in Fig. 2b. We design CMOS circuitry to implement the synapses and use a compact model for the STNO dynamics to simulate the circuit with SPICE. These simulations give energy efficiencies and parametric restrictions on the properties of the STNOs and the CMOS circuitry needed for implementing large associative memories.

![](https://assets.underline.io/uploads/markdown_image/1/image/a51918677f3e133fdef9e7b4b0ee3a96.jpg)

Schematic of the complex Hopfield network implemented using STNO neurons

![](https://assets.underline.io/uploads/markdown_image/1/image/a751556a04383cb4f92620da4a94de34.jpg)

Steps involved in a typical pattern retrieval operation. (a) The phases of the oscillators are prepared to encode the query image by driving the oscillators with an array of external inputs. (b) The oscillators evolve towards one of the stored images as the feedback is turned on.



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