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In spin orbit torque MRAM, the magnetization of the free layer is erratically switched at the time of reading. We previously proposed a bi-directional read method as a solution to reduce this read disturbance [1]. We examined the size dependence of the effect with this method [2], and the threshold current became constant as the size increased in the short-side direction (Fig. 1(a)). In this study, including the elucidation of this behavior, we compared the size dependence of the movement during magnetization switching between conventional method and our method, and examined the effect of reducing the disturbance.&lt;br/&gt;Figures. 1 (c) to (f) are graphs of magnetization switching, and Fig. 2 shows the state during the magnetization switching in the short-side direction.&lt;br/&gt;In conventional method, when X was 30 to 60 nm, the magnetization was switched in the monodomain, and when X was 90 to 180 nm, a magnetic domain was formed once and the magnetization was switched due to the movement of the domain wall.&lt;br/&gt;Magnetization switching in both directions can suppress magnetization reversal because a spin current in the opposite direction is generated, but when the threshold current is exceeded, the magnetization state becomes unstable. When X is 30 to 60 nm, the monodomain is disrupted and magnetization is switched, and when X is 90 to 180 nm, the magnetization is switched by the movement of the domain wall from the switched magnetic domain.&lt;br/&gt;Therefore, the threshold current becomes constant while the size increases in the short-side direction because the magnetization is switched by the generation of the magnetic domain and movement of the domain wall. In the long-side direction, the magnetization is switched by the generation of magnetic domains, because the shape anisotropy strengthens the connection of magnetic domains and the current for switching one magnetic domain increases as the size increases.

![](https://assets.underline.io/uploads/markdown_image/1/image/350c1be3b15b035b5718734851aa7037.jpg)

Fig. 1(a) Comparison of conventional and our method threshold currents for different sizes.
(b) The magnetoresistive element.
(c) Time-dependent magnetization.

![](https://assets.underline.io/uploads/markdown_image/1/image/f49e4bba05ecc0c35966d9bfc9dcdafb.jpg)

Fig. 2 Magnetization with conventional method.
(a) Initial magnetization at 30 × 30 nm.
(b), (c) Magnetization switching at 30 × 30 nm.
(d), (e) Domain wall movement at 120 × 30 nm.

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Examination of Magnetization Switching Behavior by Bi-directional Read of Spin Orbit Torque MRAM

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INTERMAG 2021

In spin orbit torque MRAM, the magnetization of the free layer is erratically switched at the time of reading. We previously proposed a bi-directional read method as a solution to reduce this read disturbance [1]. We examined the size dependence of the effect with this method [2], and the threshold current became constant as the size increased in the short-side direction (Fig. 1(a)). In this study, including the elucidation of this behavior, we compared the size dependence of the movement during magnetization switching between conventional method and our method, and examined the effect of reducing the disturbance. Figures. 1 (c) to (f) are graphs of magnetization switching, and Fig. 2 shows the state during the magnetization switching in the short-side direction. In conventional method, when X was 30 to 60 nm, the magnetization was switched in the monodomain, and when X was 90 to 180 nm, a magnetic domain was formed once and the magnetization was switched due to the movement of the domain wall. Magnetization switching in both directions can suppress magnetization reversal because a spin current in the opposite direction is generated, but when the threshold current is exceeded, the magnetization state becomes unstable. When X is 30 to 60 nm, the monodomain is disrupted and magnetization is switched, and when X is 90 to 180 nm, the magnetization is switched by the movement of the domain wall from the switched magnetic domain. Therefore, the threshold current becomes constant while the size increases in the short-side direction because the magnetization is switched by the generation of the magnetic domain and movement of the domain wall. In the long-side direction, the magnetization is switched by the generation of magnetic domains, because the shape anisotropy strengthens the connection of magnetic domains and the current for switching one magnetic domain increases as the size increases.

![](https://assets.underline.io/uploads/markdown_image/1/image/350c1be3b15b035b5718734851aa7037.jpg)

Fig. 1(a) Comparison of conventional and our method threshold currents for different sizes.
(b) The magnetoresistive element.
(c) Time-dependent magnetization.

![](https://assets.underline.io/uploads/markdown_image/1/image/f49e4bba05ecc0c35966d9bfc9dcdafb.jpg)

Fig. 2 Magnetization with conventional method.
(a) Initial magnetization at 30 × 30 nm.
(b), (c) Magnetization switching at 30 × 30 nm.
(d), (e) Domain wall movement at 120 × 30 nm.

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Spin-orbit torques (SOTs) provide an electrical way to manipulate the magnetization of magnetic films, which is promising for their application in magnetic random-access memory (MRAM) with high speed and long endurance [1,2]. SOT is usually observed in magnetic-heavy-metal bilayers and casued by the spin Hall effect (SHE) in the heavy metal and as-induced spin current. Restricted by the symmetry of the SHE, the spin current is normally polarized in plane, which can only determinstically switch a magnetic film with in-plane magnetic anisotropy (IMA). To switch a magnetic film with perpendicular magnetic anisotropy (PMA), which is desired for high-density memory applications, an in-plane symmetry-breaking field is indispensable. The field can be provided directly by a sourcing current via the Ostered mechanism or from the exchange-bias or coupling effects [3]. Here, we demonstrate through both theory and experiments that an effective perpendicular field scaling with SOT can inherently result from a magnetic trilayer, which consists of a magnetic film with PMA, another magnetic film with IMA, and a spacer in between to mediate their coupling. More importantly, when the two magnetic films experience two SOTs with the same (opposite) polarity, the effective perpendicular field can favor (hamper) SOT switching of the perpendicular film. This special magentic trilayer can guide the design of efficient and versatile SOT MRAM devices. In our work, a macrospin model is employed to explain the generation of effective perpendicular field. And after a seris of experiments of SOT switching under different in-plane fields, the efficiency to produce the perpendicular field reaches 2.1 Oe/1010 Am-2[4].

![](https://assets.underline.io/uploads/markdown_image/1/image/130bf5ae1259f90b28ef46b38ebb0c93.jpg)

Fig.1. Physics scenario to create Hz1eff and predicted dependence of Mz of the perpendicular layer on applied torque.

![](https://assets.underline.io/uploads/markdown_image/1/image/75788d4c598295b4d3750d6ee5817573.jpg)

Fig. 2. Direct measurement of Hz1eff and dependence of SOT efficiency χ on applied field Hy.

Efficient Spin-Orbit-Torque Switching Assisted by an Effective Perpendicular Field in a Magnetic Trilayer

GdFeCo, a rare-earth/transition-metal composition, is a competitive candidate for practical applications on GMR devices [1,2], magneto-optical recording medium [3,4], and ultrafast magnetic switching [5,6]. Particularly, the ability to control magnetic coercivity (Hc) is one of the most important parameters for the advantage in magnetic data storage technologies. It is well known that the coercivity of the GdFeCo compound can be tuned by varying content of Gd or FeCo component [7]. Although these methods can effectively alter the magnetic properties of a sample, the changes are permanent, thus, one may expect a further increase in the flexible ability to manipulate Hc will be explored. Here, we reveal that the coercive field of GdFeCo adjacent to heavy metal Hf can be effectively modulated by electric current. The stack structure of Gd28.5(Fe80Co20)71.5 (7nm)/Hf(4nm)/MgO(4nm), as illustrated in Fig.1(a), is prepared by means of a RF magnetron co-sputtering, and then characterized by the X-Ray Fluorescence (XRF) spectroscopy technique and the Alternating Gradient Magnetometer (AGM). A 5 μm-width and 100 μm-length Hall bar is fabricated using standard photolithography, E-beam lithography, and Ar ion-beam etching technique. We perform a measurement by injecting a DC current along the Hall channel in the presence of an external magnetic field which is swept perpendicularly to the sample plane, as illustrated in Fig.1(b). Interestingly, we observe that in the early stage of measurement, the Hc reduces drastically while at a higher current level coercivity is able to recover back to its initial value, as shown in Fig. 2. Such dependence of Hc on electric current can be explained by the competing interplay of the spin-orbit torque (SOT) that arises from heavy metal layer Hf and the Joule heating effects. This finding is a supplement for researchers and engineers who are working on spintronics materials

![](https://assets.underline.io/uploads/markdown_image/1/image/08b9c854f08b49452e0f1b365d6455ba.jpg)

(a) Schematic illustrations of stacked structure for thin films; (b) Optical microscopy image of the Hall bar device and the schematic of the experimental setup for the magneto-transport measurement.


![](https://assets.underline.io/uploads/markdown_image/1/image/693fba0be8e46cdfc523beab4832c892.jpg)

Hc variation as a function of input DC obtained from the anomalous hall effect (AHE) measurements.

Manipulation of coercivity in GdFeCo/Hf heterostructures by an electrical current

The spin orbit torque is one of the most promising effects for inducing magnetization dynamics and switching in a thin ferromagnetic (FM) layer and is at the base of several advancements in the field of spintronic memory devices [1,2]. The effect is caused by the spin current electrically generated by the spin Hall effect in a adjacent heavy metal (HM) layer. The description of the spin orbit torque effects on the magnetization of FM requires the introduction of two torque components in the magnetization dynamic equation: the field-like (FL) and the damping-like (DL), both proportional to intensity of the spin Hall effect [3]. By projecting all the vectors into the plane perpendicular to the magnetization, one finds the DL in the direction of the projected spin current and the FL in the perpendicular direction. The harmonic Hall method permits a detailed characterization of the two components [4] and reveals that the FL is much smaller than the DL and that they both have a non obvious dependence upon the thicknesses of the layers. In the literature, the two torque terms are derived by the introduction of the spin mixing conductance of the interface, whose real and imaginary parts give rise to the two torque terms [3]. However the spin mixing conductance has a difficult physical interpretation and justification. We propose here an alternative approach which is based on the definition of the spin current at the interface of a ferromagnet. To derive the torque terms we allow for the presence of a spin current at the interface of a FM and therefore we extend the boundary conditions of micromagnetism, therefore replacing the classical Brown's boundary conditions. The analysis reveals that by using the continuity of the spin current at the interface between the FM and the HM, one is able to derive two torque components. As a result the ratio between the DL and the FL is proportional to the damping constant of the FM and depends on thicknesses of the two layers. In the present paper we will present the derivation and the properties of the DL and FL torque terms and we will compare the outcome with experimental results from the literature [5].

Spin currents at the interface and spin Hall torque

Spintronics devices based on spin-orbit torque (SOT) are promising for next generation information technology. Perpendicularly magnetized MnGa/Co2MnSi (CMS) bilayer [1], in which MnGa and CMS layers are antiferromagnetically coupled to each other, is attractive for a ferromagnetic layer of such devices because of the relatively large perpendicular magnetic anisotropy of MnGa and the half metallicity of CMS. Moreover, previous our study showed that SOT was enhanced by using a MnGa/CMS bilayer structure compared with that in a MnGa single layer structure [2]. Further enhancement of SOT is expected at or near the magnetization-compensation point of MnGa/CMS bilayer because the efficiency of SOT generation is inversely proportional to the magnetization of the ferromagnetic layer. In this work, we systematically investigated SOT in MnGa/CMS/Ta structures with various CMS thicknesses (tCMS) including the compensation point. A wedged structure consisting of, from the substrate side, MgO (10)/NiAl (5)/MnGa (2.5)/CMS (tCMS = 0-1.7)/Ta (5)/MgO (2) was deposited on an MgO(001) substrate (numbers in parentheses are nominal thicknesses in nm ). The wedged structure was processed into Hall devices with a 5-µm wide channel to investigate the SOT. We measured the effective magnetic field Heff arising from the interaction between SOT and magnetization in domain walls (DWs) created during the magnetization reversal under current I and in-plane magnetic field Hx along current direction. The magnitude of Heff almost linearly increases with increasing |Hx| at low |Hx| and saturates at large |Hx|. The saturated Heff per I (µ0Heff*/I, µ0 is the permeability in vacuum) is shown as a function of tCMS in Figure. The value of µ0Heff*/I is increased near the compensation point and reaches about six times that in the device without CMS. These results indicate that magnetization compensation is useful for enhancing the efficiency of SOT generation. This work was supported in part by Japan Society for the Promotion of Science KAKENHI (Grant No. 20H02174 and 20H02598), the Center for Spintronics Research Network.

![](https://assets.underline.io/uploads/markdown_image/1/image/3841f8a8ab9a9cb33f93fd31f98698af.jpg)

Fig. µ0Heff* per I as a function of tCMS. The black dotted lines at tCMS = 0.45 nm indicates the magnetization-compensation point.

Spin-orbit torque in structures with magnetization-compensated MnGa/Co2MnSi bilayer

Generation of spin-current and its transport across interfaces, as well as its interaction with magnetic moments, are the central themes of spintronic community to enable fast and efficient magnetization dynamics for data storage devices. There are two main mechanisms to produce the spin-current in non-magnetic materials: (a) spin Hall effect (SHE) in heavy metals such as Pt or Ta and (b) Rashba effect (RE) at the interfaces1. Irrespective of the origin, the spin-current exerts torques on adjacent ferromagnetic layer with two components: damping-like (DL) & field-like (FL)2. However, disentangling properly SHE from RE remains highly challenging. In this experimental study we identify the mechanisms of spin-transport in ultrathin magnetic multilayer whose thicknesses range from less to above the characteristic spin- decoherence length. To this aim, we quantify the different SOT contributions in series of samples Pt8|Cox|Al1.4|Pt3 and P8|Cox|Cu1.4|Pt3 (as a reference ) with x varying from 0.4 to 1.4 nm. We extract, in each case, a spin decoherence length to be around 1.1 nm. Surprisingly, SOT investigation in the Pt8|Cox|Al 1.4|Pt 3 reveals a dominating FL torque (HFL/HDL&gt;1), which is a signature of a strong Rashba interface in Al-based systems. As a comparison, we find that the Rashba interaction is small in Pt8|Cox|Cu1.4|Pt3 samples. Moreover, we determine the SOTs characteristics in Pt(x)/Co(0.9)/Al(5) by varying the Pt thickness (tPt). The bulk spin Hall angle of Pt(θSHE) and ratio of HFL and HDL as a function of tPt are shown in Figure 1(a) and (b). The increase in HFL/HDL in the range of Pt thickness where the bulk SHE of Pt started decreasing, strongly favors our argument of the presence of Rashba interface3, exhibiting thus the specific role of Al insertion layers to enhance spin-orbit effects. Financial support from the Agence Nationale de la Recherche, France, No. ANR-17-CE24-0025 (TOPSKY), the Horizon2020 Framework Programme of the European Commission under FET-Proactive Grant agreement No. 824123 (SKYTOP), FLAGERA SOGRaphMEM and the DARPA TEE programme through grant MIPR no. HR0011831554 are acknowledged

![](https://assets.underline.io/uploads/markdown_image/1/image/ae330d77631623209411e1c48a048ff1.jpg)

Figure 1(a) θSHE vs tPt in Ta(5)/Pt(x)/Co(0.9)/Al(5). (b) Ratio of HFL/HDL as a function of tPt.

Rashba Spin-Orbit Torque in Interface Engineered Ultrathin Metallic Multilayers

Ferromagnetic materials with high spin polarization are highly suitable materials for the spin transfer torque based magnetoresistive random access memories. In this regard, spin gapless semiconductor (SGS) materials such as quaternary Heusler Alloy CoFeMnSi (CFMS) are very promising candidates for the same because of their unique band structure [1]. However, all these exotic properties are susceptible to the microstructure of the grown film. Therefore, optimized growth of a SGS material is a prerequisite for its practical realization in spintronic-based devices. Here, in this study we have tailored the structural ordering in CFMS thin films (~60 nm) grown at different substrate temperatures (TS) from 150oC to 550oC range over Si(100) substrates using pulsed DC magnetron sputtering. All the grown films were polycrystalline in nature with an enhancement in structural ordering with Tc. Among all the film samples, the film grown at 550oC showed the optimal structure with L21 ordering. The magnetic measurements revealed that saturation magnetization (Ms) of the sample decreased with the measurement temperature with a value ~3.42 μB/f.u. at 5K to 3.11 μB/f.u. at 300 K. The Curie temperature (TC) of the sample was found to be 778±13K. The longitudinal resistivity measurement clearly exhibited a negative temperature coefficient of resistivity, indicating the semiconducting behavior of these CFMS films. The carrier concentration (n) and mobility (μ) were found to be nearly temperature-independent, revealing typical SGS characteristics. The room temperature n and μ values were found to be 3.9 ± 0.04 cm-3 and 88.83 ± 1.58 cm2/V.s, respectively. The temperature-dependent anomalous Hall effect measurements have also been carried out to identify the dominant scattering mechanisms occurring in these films. This measurement suggests the dominance of phonon-induced scattering over impurity-induced scattering.

![](https://assets.underline.io/uploads/markdown_image/1/image/36480f5387b1c5c03309ccafab5e5fa1.jpg)

Fig. 1: (a) GIXRD spectra of the samples grown at different TS, (b) longitudinal resistivity (and/or conductivity at the inset) vs. temperature curve, and (c) Hall resistivity vs. magnetic field recorded at different temperatures (5K to 250K) for A-550 film. Hall resistivity data at 300K is shown in the inset.

Investigation of Spin Gapless Semiconducting Behaviour in Quaternary CoFeMnSi Heusler Alloy Thin Films on Si (100)

Magnetic random access memories (MRAMs) driven by current-induced spin-orbit torques (SOTs) become a promising candidate of next generation nonvolatile memory due to the high reliability, ultrafast writing speed, and good compatibility with COMS circuits[1] Pursuing the efficient manipulation of SOT in SOT-MRAM, recent research has focused on sorts of materials with large spin hall angle, like Platinum, β-tungsten, and topological insulators, or on SOT-channels with the special structure like ultrathin insertion layer of Hf or Co in Pt[2-4]. Here we demonstrate a type of SOT channel with large spin hall angle (θSH) up to 0.62 and relatively low resistivity of about ~200μΩ*cm by oxygen and nitrogen periodically doped in tungsten. As shown in Fig.1(a), our sample stacks from bottom to top are Si/SiO sub./X(t)/CoFeB(1.9)/MgO(1.5)/Ta(1.5)(units in nm). The X layer refers the different SOT channels which include tungsten, oxygen doped tungsten(WO), nitrogen doped tungsten(WN), and oxygen and nitrogen periodically doped in tungsten(WO/WN). The resistivity shows that we obtain a larger β-phase window with oxygen or nitrogen or both two elements incorporation in Fig.1(b), which can provide a more stable deposition condition of β-tungsten. The spin hall angle measurement was carried out by spin-torque ferromagnetic resonance (ST-FMR). The voltage spectra Vmix measured by ST-FMR at frequency f = 5 ~ 10 GHz in (WO/WN) sample is displayed in Fig. 2(a). Fig. 2(b) shows the fitting of the measured curve at 7GHz by the Lorentzian function. By analyzing the Vmix spectra, we figure out that the damping constant is about 0.01(Fig. 2(c)). While the spin hall angle attains approximately 0.62 which implies its high efficiency to exert spin-orbit torques and a decline of writing current density in SOT-MRAM.

![](https://assets.underline.io/uploads/markdown_image/1/image/b4980a8338376f3596a1d98c92a7e303.jpg)

Fig.1 (a). Schematic of samples film stack; (b). The resistivity of samples.

![](https://assets.underline.io/uploads/markdown_image/1/image/b4757424a8a5ee8ec32d57606f9ab40c.jpg)

Fig2. (a). Voltage spectra Vmix of WN[WO/WN]4 measured by ST-FMR; (b). Fitting of Vmix at 7GHz The Vsym (Vasy) denotes the symmetric (antisymmetric) part of Vmix, respectively; (c) The linewidth Δ extracted from the fitting of Vmix; (d) The resonant frequency as the function of resonant field; (e) Spin hall angel results.

Large Spin-Current Conversion in Tungsten-based Materials

The realization of magnetization switching induced by in-plane current injection in heavy metal/ferromagnetic heterostructures has drawn increasing attention to spin-orbitronics, leading to the advent of spin-orbit torques magnetoresistive random access memories (SOT-MRAM). Compared to heavy metals, oxide 2D electron gases have emerged as alternative spin-orbitronics material systems, which benefit from an efficient spin-charge interconversion through the direct and inverse Edelstein effects. Recently, we have demonstrated an enhancement of the spin-to-charge conversion efficiency by two orders of magnitude in SrTiO3-based 2D gas compared to conventional heavy metals [1], along with a non-volatile electric-control of the spin-to-charge conversion [2]. While the sign and efficiency of the SOTs are fixed by the stack of materials in conventional SOT devices, achieving an electric-control of the mirror charge-to-spin conversion would be of great interest for developing reconfigurable SOT-MRAM and logic gates. Here we report electric-field control of spin orbit torques with electrical remanence in a perpendicular ferromagnet - SrTiO3 system. Non-volatile electric-field control of the sheet resistance is achieved with 1150% contrast, with two switchable and remanent high and low resistivity states of the 2D electron gas. Spin-orbit torques effective fields are further measured using second harmonic Hall methods. A remanent electric-field control of the SOT efficiency is demonstrated, with inversion of the sign of the SOT anti-damping-like effective field. These results are consistent with a combination of both intrinsic modulation of the SOT efficiency together with extrinsic modulation due to the non-volatile electric-control of the current injection in the 2D gas. The non-volatile control of the SOT effective field is further evidenced by reproducible inversion of the SOT torques after initializing with negative or positive voltage pulses of ±130 V, opening the way to reconfigurable spin-orbit torque memory and logic gate architectures.

![](https://assets.underline.io/uploads/markdown_image/1/image/c659e26f3f654d16f4c29d73b4605f6d.jpg)

Fig.1: (a) Structure of the sample and measurement geometry. (b) Electric-field control of spin orbit torques with electrical remanence.

Non-Volatile Electric-Field Control of Spin-Orbit Torques and Anomalous Hall Effect in Perpendicular Ferromagnet - SrTiO3 System

The magnetic tunnel junction (MTJ) has gained a lot of interest in neuromorphic computing owing to its tunable switching behavior. Particularly, the stochastic switching behavior of the MTJ provides an intrinsic capability to tackle computationally difficult problems in an energy efficient manner. At the architectural level, the potential associative ability of Boltzmann Machine (BM) allows it to make inference for the input patterns which is not included in the defined input set [1][2]. Motivated by the computational abilities of MTJ and associative abilities of Boltzmann Machine, in this work, we present the spin orbit torque (SOT) driven MTJ based Boltzmann Machine neural network for solving classification problems. We use a fixed learning approach to demonstrate the ability of the BM based topology. Stochastic MTJ neuron model is used in artificial neural network. The sigmoidal response of MTJ switching probability enables us to utilize the inherent stochastic device characteristic to be implemented as a stochastic unit in BM. Fig. 1(a) shows the sigmoidal switching response of the stochastic MTJ and Fig. 1(b) shows the magnetization dynamics of MTJ. We can clearly observe an SOT tunable stochastic switching behavior. The block diagram of the BM pattern classifier is shown in Fig. 2(a), where each neuron in first layer is connected to output neuron via a fixed weight. To ensure inhibitory connection, the output from each neuron acts as input to rest of the output layer. The average value of the output for all possible outcomes of neurons and for the particular case of specific output to be high (for O2 and O6) is shown in Fig. 2(b). Based on input pattern and synaptic configuration only one output has high value and a winner takes all (WTA) circuit will be able to select the desired output, signifying the proper classification problem. Finally, we conclude by discussing the applicability of these device models and architectures for various other unconventional tasks such as combinatorial optimization and probabilistic inference.

![](https://assets.underline.io/uploads/markdown_image/1/image/d6dec4c4276171978a280d8974d529be.jpg)

(a): Acceptance probability vs difference function in BM (b) Stochastic switching behavior of SOT driven MTJ.

![](https://assets.underline.io/uploads/markdown_image/1/image/4d6347c0a39efe2f8c22e4b90aa8a17a.jpg)

(a): MTJ based BM architecture. (b) Mean value of output neuron layer.

Spin Orbit Torque Driven Stochastic MTJ Neuron-based Boltzmann Machine for Pattern Classification Application

Oscillator-based data-classification schemes have been proposed recently using the Kuramoto model which predicts synchronization behaviour of coupled oscillators through a general framework neglecting underlying physics [1,2]. Here we propose hardware implementation of a Kuramoto-model-based data-classification scheme through an array of dipole-coupled uniform-mode spin Hall nano-oscillators (SHNOs) [2,3]. Using micromagnetic simulations on 'mumax3' software [4], which actually capture physics of SHNOs, we first study how synchronization range between two uniform-mode SHNOs (150 nm diameter) varies with physical distance between them (Fig. 1b). Thus we correlate the coupling constant in Kuramoto model (Fig. 1a) with dipole-coupling strength between two SHNOs in 'mumax3' (Fig. 1b) . Using this correlation, we generate a synchronization map for a two-input-two-output dipole-coupled uniform-mode SHNO system through micromagnetics (Fig. 2a). Red circles show frequency values of input oscillators (F1,F2) for which the two output oscillators are synchronized, and black circles show (F1,F2) for which they are not. These data points coincide with pink region showing sync. and grey region showing de-sync., obtained from Kuramoto model (Fig. 2a). Thus we establish here that synchronization behaviour of SHNOs obtained from physics-based modeling (micromagnetics) is consistent with that obtained from Kuramoto model, where underlying physics of SHNOs is ignored. This shows that a Kuramoto-model-based data classification scheme [2] can indeed be implemented on an array of SHNOs. Next we show, through micromagnetics, classification of data from a popular data set (Fisher's Iris [5]) using an array of uniform-mode SHNOs. While distinguishing flowers in Iris of Setosa type from Virginica type, output oscillators synchronize for Setosa and desynchronize for Virginica (Fig. 2b). The obtained accuracy is 98.67%

![](https://assets.underline.io/uploads/markdown_image/1/image/76425c03e78db7f1e75e35cd0fc9ffbe.jpg)

Fig. 1. For a two-oscillator system, coupling constant in Kuramoto model (a) is correlated with distance between two SHNOs in micromagnetics (b) using the synchronization-range values.

![](https://assets.underline.io/uploads/markdown_image/1/image/894a943d95a0240af6a1aeeb9c7fd828.jpg)

Fig. 2. Synchronization map of the two-input-two-output oscillator system in (a) (details in text) used for data classification from Iris set (b)

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Efficient Spin-Orbit-Torque Switching Assisted by an Effective Perpendicular Field in a Magnetic Trilayer

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