United States

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.&lt;br/&gt;DW-MTJ devices are nonvolatile memory-in-logic elements that can be integrated into an artificial neural network&lt;sup&gt;1&lt;/sup&gt;. 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 computing&lt;sup&gt;1,2,3&lt;/sup&gt;. However, experimental work has focused on binary DW-MTJs&lt;sup&gt;4&lt;/sup&gt;, 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 device&lt;sup&gt;5&lt;/sup&gt;.&lt;br/&gt;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



2022 Joint MMM-INTERMAG

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

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.

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

technical paper

Materials with helical crystal structures have been suggested to be a condensed matter analogue of solenoids, where a current flow produces inductive magnetic fields [1]. At the microscopic scale, a finite spin accumulation can be generated in strong spin-orbit coupling and low symmetry systems via the Edelstein effect [2,3]. Recently, chiral non-magnetic materials have received a lot of attention due to their low symmetries and complex radial spin textures, that could be potentially exploited for spintronic device applications [4,5,6]. However, a straightforward magnetotransport study on the relationship between chirality, spin texture and current-induced spin polarization is still missing. Here, we link one-to-one the direction of the spin polarization, induced by a chirality-dependent Edelstein effect, to the handedness of chiral tellurium (Te) nanowires (NWs), through simple angle-dependent magnetotransport experiments. Transmission electron microscopy (TEM) analysis and theoretical calculations corroborate the transport data. Te NWs show a large, chirality-dependent non-reciprocal charge transport when the electrical current and the magnetic field applied are collinear to the chiral axis of the crystal. By applying a gate voltage, we can also manipulate the magnetotransport response, acting on the carrier density and modifying the spin texture at the Fermi surface. The efficiency of the charge-to-spin interconversion process is one order of magnitude larger than what has been reported so far. Our results are a step forward towards the development of chiral spintronics.

![](https://assets.underline.io/uploads/markdown_image/1/image/cb46e659c3ceca2ac8d3e113ec8dbb71.jpg)

Figure 1: Sketch of the chiral structure of right- and left-handed Te NWs where the current-induced spin polarization depends on the crystal handedness

![](https://assets.underline.io/uploads/markdown_image/1/image/6de41c2cd5dcdb4672e53a8c0641cd6e.jpg)

Figure 2: On the leftt, sketch representing the Edelstein effect mechanism. The shift in kz of the Fermi contours lead to the formation of spin accumulations oriented in opposite directions according to the chirality-dependent radial spin texture of Te. On the right, current-dependent angular dependences of the magnetoresistance Rdiff= [R(Iz) - R(-Iz)]/2 measured at 9 T and 10 K for two Te NWs with opposites handedness. The signals show specular features between left- and right-handed NWs.




Chirality-dependent Edelstein effect in elemental Tellurium nanowires

Electrically conductive metals are typically good transmitters or absorbers of pure spin currents. Here, we report evidence that an interface of two conductive elemental metals – namely, Cu and Cr – constitutes a good spin-current reflector. Our study was initially motivated by the question of whether the elemental antiferromagnet Cr acts as a spin transmitter (akin to electrically insulating NiO [1]) or a spin absorber (akin to metallic IrMn [2]). To this end, we have performed ferromagnetic resonance (FMR) spin pumping measurements on spin-valve-like stacks of NiFe(10)/Cu(5)/Cr(tCr)/(Co)Fe(4), where the FMR-driven NiFe layer pumps an AC pure spin current. Any spin current transmitted through the Cu/Cr spacer would be absorbed by the (Co)Fe sink, generating additional Gilbert damping, △α, for the NiFe source. Thus, △α is proportional to how much a spin current is transmitted or absorbed across the Cr interlayer. In Figure 1, the sharp decline of △α with increasing Cr thickness indicates a remarkable trend: The introduction of Cr suppresses spin transmission and absorption – i.e., the spin current is largely reflected by the Cu/Cr spacer. However, our complementary element-selective X-ray FMR measurements confirm that the spin current is transmitted through the Cr spacer in NiFe/Cr/CoFe. We therefore identify the Cu/Cr interface, rather than the Cr bulk, as the source of the strong spin reflection. Figure 2 shows that △α varies only weakly with decreasing temperature (increasing antiferromagnetic order in Cr), which suggests that antiferromagnetism in Cr is not responsible for the spin-reflective behavior. Our finding points to the possibility that other electrically conductive metals can form effective spin reflectors, thereby offering new avenues in the fundamental studies and engineering of pure spin currents across interfaces.


![](https://assets.underline.io/uploads/markdown_image/1/image/e7884bfa3e9da4b8fbfbd64553b2d860.jpg)


[Fig. 1] △α vs. Cr layer thickness tCr in NiFe/Cu/Cr(tCr)/(Co)Fe measured at room temperature. Both the epitaxial and polycrystalline series show decaying with tCr, due to spin reflection at the Cu/Cr interface.

![](https://assets.underline.io/uploads/markdown_image/1/image/dc5e03a64fe9cbcdaa9a866583d3cb89.jpg)


[Fig. 2] △α vs. temperature, showing that spin transport is affected only weakly even as Cr develops stronger antiferromagnetic order at lower temperatures.

Spin-Reflective Metallic Cu/Cr Interface

Asymmetric charge transport at the interface of two materials with dissimilar electrical properties, such as metal/semiconductor and p-n junctions, is the fundamental aspect behind modern diode and transistor technology. Therefore, it is interesting to study pure-spin-current transport at the interface of two materials with different spin transport properties. Here we demonstrate asymmetric transport of pure-spin-currents across an interface of dissimilar non-magnetic materials, Au/Pt. In this work we present a study of spin transport by spin-pumping from either Py (Ni80Fe20) or Co in Py/Au/Pt/Co and Py/Au/Pt structures. The thickness of Pt was varied from 0 to 3 nm and the magnetic damping was measured by ferromagnetic resonance. The magnetic damping due to spin-pumping of Py in Py/Au/Pt follows the characteristic thickness dependence of ferromagnet/normal-metal. The typical ferromagnet/normal-metal/spin-sink thickness dependence is reproduced by spin-pumping from Py in the Py/Au/Pt/Co structure. Here the Co layer acts as a spin-sink. However, pumping from Co while Py is the spin-sink in the Py/Au/Pt/Co structure is very uncharacteristic of a spin-sink structure. Instead, this data suggests that very little spin-current makes it to the Py layer, i.e. a reflection of spin-current at the Au/Pt interface. The data could not be self consistently fit with the continuity of spin-accumulation and continuity of spin-current boundary conditions at the Au/Pt interface [1,2]. Including spin-memory-loss also failed to fit the data [3]. We proposed a new boundary condition that assumes that the reflected and transmitted spin-currents at the Au/Pt interface are proportional to the established spin chemical potentials on either side of the interface. This model can self consistently fit all the measured data. It suggests that spin-pumping from Py results in a ~30% reflection and ~70 transmissions of the spin-current across the Au/Pt interface. However, spin pumping from Co results in ~70% reflection and 30% transmission of spin-current at the Au/Pt interface.

Nonreciprocal transport of pure spin current at the Au/Pt interface

Spin-related magnetoresistance (MR) from versatile condensed matter systems can be employed as the main mechanism to detect the orientation of magnetization in next generation magnetoresistive random-access memories (MRAM). Recently, novel MR effects such as spin Hall MR (SMR) [1,2] and spin-dependent unidirectional MR (SD-UMR) [3,4] are extensively studied due to the correlation between MR and spin-orbit interactions in both classical and emergent materials systems. In our work, we disentangle the underlying mechanisms of the UMR in W/CoFeB magnetic heterostructures by systematic current-dependent and field-dependent MR measurements at room temperature. As shown in Fig.1, the current-dependent UMR of W/CoFeB heterostructures can be roughly divided into three regimes. In the first and second regimes, UMR increases linearly and then nonlinearly to the applied current density J as J reaches ~5×1010 A/m2, which originates from the SD-UMR [Fig.2(a)] and spin-flip UMR (SF-UMR) [Fig.2 (b)] [4]. In the third regime, as J reaches ~ 1011 A/m2, the UMR ratio will further saturate at 0.36%, which is larger than the typical UMR magnitudes reported previously in metallic system. With the aid of numerical simulations, this large UMR can be attributed to the extra contribution from SMR, which stems from the tilting of magnetization M caused by the competition between the spin Hall effect induced spin torques (STs) from W layer and the external field Hy. This tilting behavior is illustratively shown in Fig.2 (c). Consequently, we define this UMR as the spin-transfer torque UMR (STT-UMR). The damping like-ST (DL-ST) efficiency of W/CoFeB heterostructures can be also roughly estimated by using both the experimental UMR data and the simulation results. Our finding therefore verifies the correlation between the STs and this large room-temperature UMR, thereby providing an alternative approach to determine DL-ST efficiency.

![](https://assets.underline.io/uploads/markdown_image/1/image/40f566960248469705c79873e22caf42.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/7ec5345f8d8c1cc8ce22f8a88abc6e0c.jpg)

Observation of Large Spin Torques-Related Unidirectional Magnetoresistance in Metallic Heterostructures

Observation of Large Spin Torques-Related Unidirectional Magnetoresistance in Metallic Heterostructures

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.

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

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Chirality-dependent Edelstein effect in elemental Tellurium nanowires

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

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Chirality-dependent Edelstein effect in elemental Tellurium nanowires

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

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