United States

Hybrid magnonic-electric computing architectures are promising for ultra-low power data processing [1,2]. To realize these computing systems, scalable magnetoelectric
transducers are required that efficiently convert signals between the electric and magnetic domain. Several transducer types have been proposed of which voltage-based concepts are
expected to have the best scaling characteristics [3]. A promising voltage-based transducer type consists of a piezoelectric/magnetostrictive bilayer [4]. When a voltage is applied to this
transducer, dynamic strain is induced in the bilayer via the piezoelectric effect. This dynamic strain then interacts with the magnetization via the magnetostrictive effect and a coupling
between the electric and magnetic domain is obtained. Although this transducer type is very promising, its fundamental understanding as well as energy efficiency estimates are currently
lacking.
In this work, we develop a theoretical model of the displacement and magnetization dynamics in a resonator consisting of a piezoelectric/magnetostrictive bilayer. The model considers
both elastic as well as magnetic resonances and captures the magnetoelastic effect via an effective stiffness. Via this approach, both the elastic and magnetic power flow inside the device
can be determined. A comprehensive analysis was conducted of the dimensional as well as material parameter influence on the elastic and magnetic power dissipation (see Fig. 1).
Furthermore, the power calculation also results in magnetic transducer efficiency estimates. The results show that magnetic excitation efficiencies up to 90% can be achieved for nanometer
thick magnetostrictive layers (Fig. 2). Finally, a figure of merit is determined which allows to select the optimal piezoelectric-magnetostrictive material combination.&lt;br&gt;

**References:**&lt;br&gt;

1. A. Khitun and K. L. Wang, J. Appl. Phys. 110, 034306 (2011).&lt;br&gt;
2. B. Dieny, I. L. Prejbeanu, K. Garello, P. Gambardella, P. Freitas, R. Lehndorff, W. Raberg, U. Ebels, S. O. Demokritov, J. Akerman, et al., Nature Electron, 3 (2020).&lt;br&gt;
3. F. Vanderveken, V. Tyberkevych, G. Talmelli, B. Soree, F. Ciubotaru, and C. Adelmann, Sci. Rep. 12, 3796 (2022).&lt;br&gt;
4. A. Khitun, D. E. Nikonov, and K. L. Wang, J. Appl. Phys. 106, 123909 (2009).&lt;br&gt;

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/0867d6ce1e1696b44c9b050ecec6782e.png)&lt;br&gt;

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MMM 2022

Power transfer in magnetoelectric resonators

Hybrid magnonic-electric computing architectures are promising for ultra-low power data processing [1,2]. To realize these computing systems, scalable magnetoelectric
transducers are required that efficiently convert signals between the electric and magnetic domain. Several transducer types have been proposed of which voltage-based concepts are
expected to have the best scaling characteristics [3]. A promising voltage-based transducer type consists of a piezoelectric/magnetostrictive bilayer [4]. When a voltage is applied to this
transducer, dynamic strain is induced in the bilayer via the piezoelectric effect. This dynamic strain then interacts with the magnetization via the magnetostrictive effect and a coupling
between the electric and magnetic domain is obtained. Although this transducer type is very promising, its fundamental understanding as well as energy efficiency estimates are currently
lacking.
In this work, we develop a theoretical model of the displacement and magnetization dynamics in a resonator consisting of a piezoelectric/magnetostrictive bilayer. The model considers
both elastic as well as magnetic resonances and captures the magnetoelastic effect via an effective stiffness. Via this approach, both the elastic and magnetic power flow inside the device
can be determined. A comprehensive analysis was conducted of the dimensional as well as material parameter influence on the elastic and magnetic power dissipation (see Fig. 1).
Furthermore, the power calculation also results in magnetic transducer efficiency estimates. The results show that magnetic excitation efficiencies up to 90% can be achieved for nanometer
thick magnetostrictive layers (Fig. 2). Finally, a figure of merit is determined which allows to select the optimal piezoelectric-magnetostrictive material combination. 

**References:** 

1. A. Khitun and K. L. Wang, J. Appl. Phys. 110, 034306 (2011). 
2. B. Dieny, I. L. Prejbeanu, K. Garello, P. Gambardella, P. Freitas, R. Lehndorff, W. Raberg, U. Ebels, S. O. Demokritov, J. Akerman, et al., Nature Electron, 3 (2020). 
3. F. Vanderveken, V. Tyberkevych, G. Talmelli, B. Soree, F. Ciubotaru, and C. Adelmann, Sci. Rep. 12, 3796 (2022). 
4. A. Khitun, D. E. Nikonov, and K. L. Wang, J. Appl. Phys. 106, 123909 (2009). 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/0867d6ce1e1696b44c9b050ecec6782e.png) 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/5a2191baf82121f314207356220e5bb0.png)

technical paper

#### Welcome to the 67th Annual Conference on Magnetism and Magnetic Materials, MMM 2022!

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Hydrogen is one of the most promising species for fast magneto-ionic control in iron triad metals. (1,2)
In this case, a voltage, which is applied to the magnetic metal via a solid or liquid electrolyte, (3) triggers proton reduction and subsequent hydrogen ad- and absorption. The composition of the electrolyte may have a great influence on these processes, since the absorption of hydrogen in metals is highly dependent on the pH values, (4) and moreover, additives to electrolyte can improve hydrogen permeability and diffusivity in ferromagnetic metals. (5) 
However, this knowledge has not been applied and tested for magneto-ionic systems yet. Thus, the optimization of electrolyte composition and the use of additives to promote hydrogen ad- or absorption may be a promising pathway to increase the efficiency of magneto-ionic systems.
In this work, we study the influence of alkaline electrolyte composition on the H-based magneto-ionic effect in electrodeposited nanocrystalline Ni films. In Ni, H is expected to significantly modulate the magnetization. (6) 
The voltage-triggered changes in magnetic properties are analyzed by in situ Anomalous Hall effect and in situ SQUID measurements in aqueous electrolytes with different concentrations of KOH (0.5, 2 and 5 M), as well as in 0.5 M KOH with thiourea additive.
H-based magneto-ionic control is clearly demonstrated by a voltage-controllable decrease and increase of saturation magnetization of the Ni films upon hydrogen absorption and desorption, respectively. We find that an increase in electrolyte pH and the addition of thiourea improves the efficiency of the H-based magneto-ionic effect. Results of in situ Raman spectroscopy indicate that the enhancement of H-absorption by thiourea relates to the formation of a Ni(OH)2 layer on the Ni surface.
The obtained results should stimulate further research in the field toward the use of multicomponent electrolytes and the optimization of underlying chemical and electrochemical reaction mechanisms to improve magneto-ionic efficiency. 

*This research was funded by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant number 861145.* 

**References:** 
(1) Tan, A. J. et al. Magneto-ionic control of magnetism using a solid-state proton pump. Nat. Mater. 18, 35–41 (2019). 
(2) Huang, M. et al. Voltage control of ferrimagnetic order and voltage-assisted writing of ferrimagnetic spin textures. Nat. Nanotechnol. (2021) doi:10.1038/s41565-021-00940-1. 
(3) M. Nichterwitz, S. Honnali, M. Kutuzau, S. Guo, J. Zehner, K. Nielsch, and K. Leistner. Advances in magneto-ionic materials and perspectives for their application. APL Materials 9, 030903 (2021). 
(4) Fujimoto, N., Sawada, T., Tada, E. & Nishikata, A. Effect of pH on Hydrogen Absorption into Steel in Neutral and Alkaline Solutions. Mater. Trans. 58, 211–217 (2017). 
(5) Latanision, R. M. & Kurkela, M. Hydrogen Permeability and Diffusivity in Nickel and Ni-Base Alloys. CORROSION 39, 174–181 (1983). 
(6) León, A. et al. Magnetic effects of interstitial hydrogen in nickel. Journal of Magnetism and Magnetic Materials 421, 7–12 (2017).

Influence of alkaline electrolyte composition on H based magneto

Magneto-ionics is an emerging approach to controlling the magnetic properties of materials via voltage-driven ion motion. Nitrogen-based magneto-ionics in transition metal nitride thin films (CoN & FeN) can induce reversible ON-OFF transitions of ferromagnetic states at faster rates and lower threshold voltages than oxygen-based magneto-ionics (1,2), attributed to the comparably smaller energy barrier for ion diffusion and the lower electronegativity of nitrogen when compared with transition metal oxide thin films. Furthermore, this effect largely relies on the strength and penetration of the induced electric field into the target material, the amount of generated ion transport pathways, and the ionic mobility inside the magnetic media. Optimizing all these factors in a simple way is a huge challenge, although highly desirable for technological applications. Here we demonstrate that introduction of suitable transition metal elements in binary nitride compounds can drastically boost magneto-ionics. More specifically, we show the multiple benefits that moderate substitution of Co by Mn brings to the magneto-ionic performance of CoN-based heterostructures. With 10% substitution of Co by Mn in the thin film composition, a transformation from nanocrystalline to amorphous-like structures, as well as from metallic to semiconducting behaviors has been observed. The former increases N ion transport channels, and the latter significantly extends the electric field effect which is normally limited to a few Å in metals. In this way, Mn incorporation leads to a 6.7-fold enhancement of the saturation magnetization (MS) and improved toggling speeds and cyclability. In addition, Ab initio calculations reveal a lower formation energy barrier for Mn-N compared to Co-N, which allows a fundamental understanding of the crucial role of Mn addition on the voltage-driven magnetic effects. These results constitute an important step forward towards enhanced voltage control of magnetism via electric-field-driven ion motion (3). 

**References:** 
(1) de Rojas, J. et al. Voltage-driven motion of nitrogen ions: a new paradigm for magneto-ionics. Nature Communications 11, 5871 (2020). 
(2) de Rojas, J. et al. Magneto-Ionics in Single-Layer Transition Metal Nitrides. ACS Applied Materials & Interfaces 13, 30826–30834 (2021). 
(3) Zhengwei Tan et al. From binary to ternary transition metal nitrides: a boost towards nitrogen magneto-ionics. Submitted (2021).
 

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Boosting nitrogen magneto ionics: From binary to ternary transition metal nitrides

There are mainly two types of exchange interactions in a multiple-spins system. First being the symmetric exchange, which is directly responsible for the existence of ferromagnetism (FM) and antiferromagnetism (AFM) in which the spins collectively form either parallel or antiparallel configurations. On the other hand, the antisymmetric exchange, or the Dzyaloshinskii-Moriya interaction (DMI) [1-3] has gained significant attention lately. For the majority of cases, DMI is an intralayer effect, i.e. the nonlinear spin configurations are confined within a single magnetic layer. However, pioneering predictions [4] have recently expanded the possibility to an interlayer case, where the spins are from two distinct FM layers. We hereby demonstrate an interlayer DMI in a CoFeB/Pt/Co/Pt multilayer where CoFeB has in-plane anisotropy (IMA), while Co has perpendicular magnetic anisotropy (PMA). The wedge sputtering provides sufficient symmetry breaking for the interlayer DMI to manifest (Fig.1 (a)). A field sweep protocol involving the out of plane external field Hz sweep while applying an in-plane field Hφ with various φ orientation is used to capture the effective field experienced by the PMA layer. The resultant hysteresis loops of the Co layer are substantially shifted (Fig.1 (b)) and follow a sinusoidal form with respect to the φ direction (Fig.1 (d)). The importance of in-plane symmetry breaking is confirmed by comparing results with control samples without symmetry breaking (Fig.1 (c)). The reciprocal effect experienced by the IMA layer is captured by magneto-optical Kerr effect experiments (Fig.1 (d)). Moreover, it is shown that current-induced field-free switching of a PMA layer is achievable utilizing the DMI induced effective field and the switching polarity can be controlled by the interlayer DMI landscape (Fig.2). 
**References:** [1] Dzyaloshinsky, A thermodynamic theory of ‘weak’ ferromagnetism of antiferromagnetics, J. Phys. Chem. Sol. 4, 241 (1957). 
[2] T. Moriya, Anisotropic Superexchange Interaction and Weak Ferromagnetism, Phys. Rev. 120, 91 (1960). 
[3] A. Fert and P. M. Levy, Role of Anisotropic Exchange Interactions in Determining the Properties of Spin-Glasses, Phys. Rev. Lett. 44, 1538 (1980). 
[4] E. Y. Vedmedenko, P. Riego, J. A. Arregi, and A. Berger, Interlayer Dzyaloshinskii-Moriya Interactions, Phys. Rev. Lett. 122, 257202 (2019). 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/e96cf0a6ee880782704d00f31d5c561c.png) 
Fig.1 (a) Sketch of the magnetic multilayers, external fields, coordinate system. Interlayer DMI’s manifestation on both IMA and PMA hysteresis loops presented in (b) and (d), respectively. Comparison between samples without symmetry breaking presented in (c). 
![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/8f09170333a6d60da6ba318c9fd2d8ae.png) 
Fig.2 Interlayer DMI landscape demonstrating field free switching and the influence of the device angle on the switching polarity.

Growth dependent Interlayer Chiral Exchange and Field

We propose to directly and quantum-coherently couple a superconducting transmon qubit to magnons — the quanta of the collective spin excitations, in a nearby magnetic particle. The magnet’s stray field couples to the qubit via a superconducting interference device (SQUID) leading to strong and tunable interactions. More specifically, we predict a
resonant magnon-qubit exchange coupling, similar to the one studied in 3D cavity setups [1], and a novel magnon-qubit nonlinear interaction that is similar to the radiation-pressure
coupling in optomechanics [2]. We show that the latter can be resonantly enhanced by dynamically driving the flux in the SQUID, and demonstrate a quantum control scheme to generate
magnon-qubit entanglement and magnonic Schrödinger cat states with high fidelity. Our results [3] enrich the quantum control toolbox in magnonic devices and open new possibilities for
constructing quantum magnonic networks on a chip. 
**References** [1] Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, Science Vol. 349, p. 405 (2015).
[2] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. Vol. 86, p. 1391 (2014).
[3] M. Kounalakis, G. E. W. Bauer, Y. M. Blanter, arXiv:2203.11893 (2022). 
![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/cfe8e04007f7b0cbf30588e906f93677.png) 
Fig. 1. Proposed circuit architecture.

Analog quantum control of magnonic cat states on a

Engineering coupling between systems is essential for many phenomena of quantum physics and technology, especially in long distance without field overlap. We construct a long-distance cavity magnonic system to realize indirect coupling mediated by travelling photons. A high-quality dielectric resonator (cavity photon mode) and an Yttrium Iron Garnet sphere (YIG, magnon mode) are placed along the transmission line with a longitudinal distance of over 1 metre to avoid direct coupling. By tuning the longitudinal distance given by phase and the YIG sphere's transverse position, we realized indirect coupling in transmission: in the reciprocal case with the YIG sphere in the center of the transmission line, we have coherent and dissipative couplings at the longitudinal phase Î¦=(2n+1)Ï€ and Î¦=(2n+2)Ï€; while in the nonreciprocal case with the YIG sphere on the side of the transmission line, we have coherent and dissipative couplings for the left- and right-going travelling photons at the longitudinal phase Î¦=(2n+0.5)Ï€ and Î¦=(2n+1.5)Ï€. The coupled-mode theory modified by a higher-order coupling term is adopted to explain our results and enhances our understanding of long-distance coupling and nonreciprocity. Our illustration of switching long-distance coherent and dissipative couplings, gives access to designing nonreciprocal links and constructing advanced remote-control strategies among spatially separated systems to build networks.

Travelling Photons Mediated Coherent and Dissipative Couplings in Long

We predict how design aspects of the HAMR system such as media properties, thermal profile, and the geometry of the magneto-resistive (MR) reader affect user density through micromagnetic simulation [1]. It is found that maximum user density is achievable by optimizing the track pitch. Fig 1. shows dependence on track pitch for a small grain pitch, fly height, and thermal spot size: Fig 2. shows corresponding data for large values. User density increases when the value of shield-to-shield spacing (SSS) of the reader is reduced owing to improved resolution, even though SNR often decreases [2]. Moreover, the impact of the shield-to-shield spacing is much stronger in case of small grain size compared to larger grain size because the limit to resolution becomes the transition broadening induced by the grain pitch. It is also evident that higher full width at half maximum (FWHM) of the temperature profile reduces user density owing mostly to increased track pitch, but also owing to increased erase after write. Transition curvature in combination with shingled recording is predicted to yield asymmetric transitions with respect to the track center [3]. Therefore, to adequately match the curvature of these asymmetrical curved transitions, we employed a rotated read head rotation: a single rotated head ( θopt = 350 ) results in a 9% improvement in user density over a single non-rotated head at 5.8 nm grain pitch, 15 nm reader width, 30 nm FWHM, 16 nm SSS, and 15 nm track pitch. Finally, we also demonstrate that media Tc variation impacts density more severely at high densities, but anisotropy field variation is the more significant effect at lower densities. 
**References:** [1] R. H. Victora and P.-W. Huang, IEEE Trans. Magn., vol. 49, no. 2, pp. 751–757, (2013) 
[2] Z. Liu, Y. Jiao, and R. H. Victora, Appl. Phys. Lett., vol. 108, no. 23, p. 232402, (2016) 
[3] W.-H. Hsu and R. H. Victora, Appl. Phys. Lett., vol. 118, no. 7, p. 72406, (2021) 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/c2cf57679726d155490bab4700d33d04.png) 
Fig. 1. User density as a function of track pitch for different shield-to-shield spacing (SSS) of ECC (FePt) media for optimal bit length = 9nm and reader width =15nm. 
![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/6461352a8cf41ad94217f5df9c4355ea.png) 
Fig. 2. User density profile as a function of track pitch for different shield-to-shield spacing (SSS) of ECC (FePt) media for optimal bit length = 9nm and reader width =15nm.

Dependence of User Density on System and Media Parameters for Shingled Heat Assisted Magnetic Recording (HAMR)

The resource criticality of rare-earth (RE) elements on the global market has triggered significant research into more resource-efficient alternative hard-magnetic materials[1]. The utilization of the RE-lean ThMn12 materials system in combination with the abundant RE element Ce is a promising strategy. One of the main challenges for the Ce-based hard-magnetic materials is the formation of detrimental Laves phases next to the ThMn12-type compound CeFe11Ti[2]. In this contribution, we present an ab initio-based approach to modify the stability of these phases in the Ce-Fe-Ti system by additions of 3d and 4d-elements. We start with Cu and Ga substitutions and employ state-of-the-art approaches for vibrational, electronic, and magnetic entropy contributions[3,4] to calculate the Helmholtz free energy F(T,V) for all relevant phases. The results are used to provide two fundamental methodological insights. One of them is our new modeling concept of partial decomposition, which considers the enrichment of the added solutes in phases that would at the considered temperature not be stable in a full decomposition. In the case of Cu, this effect suppresses the formation of CeFe11Ti, whereas in the case of Ga this ensures the presence of a small phase fraction of CeFe11Ti. The second conclusion is the dominant impact of 0 K formation enthalpies on the solute-enhances phase stability compared to finite temperature entropy terms. Based on this a screening approach, considering the substitution of all 3d and 4d-elements, is developed. We show that substituted elements with more than a half-filled 3d-shell or with less than a half-filled 4d-shell mainly reduce the formation temperature of the 1:12 phase. According to these thermodynamic considerations, especially Zn and Tc turn out to be promising substitution candidates. Therefore, more detailed finite temperature ab initio evaluations of the free energies have been performed for these elements. The difference between the screening determination of the critical temperature and a fully temperature-dependent determination of free energies is within the error bar of the other approximations. This demonstrates the robustness and efficiency of our developed screening approach. 
**References:** [1] O. Gutfleisch, et al.: Magnetic materials and devices for the 21st century: Stronger, lighter, and more energy efficient. Advanced Materials, 23, (2011) 821. 
[2] H. I. Sözen, et al.: Ab initio phase stabilities of Ce-based hard magnetic materials and comparison with experimental phase diagrams. Physical Review Materials, 3 (2019) 084407. 
[3] H. I. Sözen, et al.: Impact of magnetism on the phase stability modeling for rare-earth based hard magnetic materials. CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry, 68 (2020) 101731. 
[4] H. I. Sözen, et al.: Ab initio phase stabilities of rare-earth lean Nd-based hard magnets. Journal of Magnetism and Magnetic Materials, 559 (2022) 169529. 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/b3c664197ef98be7d78fc805536f1286.png) 
Figure 1

Ab initio based Thermodynamic Modeling for Sustainable Hard Magnetic Materials

There has been much interest in the layered material MoS2 Waals interactions because some of the intrinsic defects are magnetic, including a vacancy on an S site and Mo
on an S site[1]. After implantation with Gd ions XRD, XSP, Raman scattering indicated the presence of S vacancies and conductive behaviour at room temperature [2]. We report on
detailed investigation of a film deposited with Gd implantation, 1×1015 ions/cm3, over a temperature range 5<T<300K in fields up to 70kOe in order to clarify the contribution of the native
defects produced as a result of radiation damage and that from the Gd ions. Below 80K hysteresis loops were analysed in terms of two components, Curie behaviour from small
independent clusters and larger clusters that saturated in 70kOe as shown in raw data in Fig 1 and the saturating part in Fig 2. respectively. In all cases the magnetisation per Gd ion vastly
exceeded the ~7mB expected for Gd3+. The fall in the saturated part of the magnetisation, 2,700 mB/Gd at 5K to 40 mB/Gd at 150K, was also apparent in the paramagnetic part where the
Curie constant dropped from 18,000 mB/Gd at 40K to 1,200 mB/Gd at 150K. This indicated that the moments arising from isolated the native defects are unstable at high temperature.
Very different behaviour was seen between 150K and 300K where the saturation magnetisation remained constant at 40 mB/Gd. Moreover the magnetisation per Gd fell only slightly as the
implantation dose of Gd was raised to 5×1015 ions/cm3, leading to a magnetisation of ~340emu/cm3 in a 40nm layer with a coercive field of 400Oe. This behaviour has not been seen with
other rare earth ions [3] and indicates a specially strong interaction between the Gd ions and the native defects such that some native defects are enabled to retain their moments at 300K
and provide exchange between the Gd ions. 

**References:** 

HK Komsa and A. V. Krasheninnikov Phys Rev B 91, 125304 (2015)
M. Zhang, M. Ying et al
X. Ding, X. Cui Adv. Quantum Technol. 2000093 (2020) 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/5c0ab74d0f4b20209df911d627ed4560.png)

Exceptionally Large Magnetism in Gd Implanted MoS2 Due to Both Radiation Damage and Gd3+ Ions

The combination of the two key effects of spintronics, magnetization dynamics and magneto-resistive effects, allows the realization of nano-neurons and nano-synapses with high computational capabilities. However, multilayer spintronic neural networks have never been realized with these nanodevices because there is no way to connect them that can work on a large scale. I will show that it is possible to exploit the two key effects of spintronics to connect together magnetic tunnel junctions that implement both neurons and synapses of a multilayer network. The magnetic tunnel junctions perform neural operations on DC signals and output the result as RF, operate synaptic operations on RF signals and output the result as DC thus giving rise to multilayer networks based on successive, clean, and fast conversions from RF to DC and from DC to RF. I will give a proof of concept with a two-layer neural network composed of nine interconnected magnetic tunnel junctions functioning as both synapses and neurons and experimentally demonstrate its ability to solve non-linear tasks with high performance. I will show that with junctions downscaled to 20 nm, such a network would consume 10fJ per synaptic operation and 100fJ per neuronal operation, several orders of magnitude less than current software neural network implementations. Finally, I will show through physical simulations that these networks, which can process DC data, can also natively classify RF inputs, achieving state-of-the-art drone identification from their RF transmissions. This study lays the foundation for deep spintronic neural networks at the nanoscale.

Series of synaptic and neuronal operations in a multilayer magnetic tunnel junction network through RF to

The parallels between associative learning observed in the brain and spontaneous synchronization observed in weakly coupled oscillators have led to proposals for artificial associative memories based on synchronized oscillators [1,2]. Here, we consider spin-torque oscillator arrays to form a frequency synchronized oscillator network [1]. Spin-torque oscillators are capable of frequency locking to external AC drives, provided the external drive frequencies are close to their natural frequencies [3]. Information is encoded in the coupling network while the relative phases between the frequency-synchronized oscillators encode the current state of the network. There are several possible topologies to implement the coupling networks [2] and the couplings themselves can be implemented either using magnetostatic and exchange interactions [3], or more easily though electrical circuitry. 

Here, we consider an all-to-all pairwise-connected feedback network, as shown in Fig. 1 with each connection representing a complex weight [4]. These complex weights consist of time-delay and scaling networks implemented using memristor-augmented CMOS circuitry. When connected in feedback, the relative phases of the oscillators stabilize into one of the states corresponding an encoded vector in the feedback network. We demonstrate storage and retrieval of twelve 16×12 images using 192 spin-torque oscillators. The pixel values of the images considered are drawn from a saturated color wheel which naturally maps to the phase of the oscillator corresponding to that pixel. A schematic of the phase evolution during a typical retrieval process is shown in Fig. 2. Starting from a corrupt state, the oscillator phases evolve in the presence of feedback so that the final image corresponds to one of the stored images that closely resembles the corrupt image. For the oscillators and circuitry considered, the image retrieval process when presented with an image with 5 % root mean square deviations from the ideal image requires approximately 5 μs and consumes roughly 130 nJ. These simulations show that for the network to function as desired, the resonant frequencies of the oscillators are required to have fractional spread lower 0.01 %. 

**References:** 

[1] N. Prasad, P. Mukim, A. Madhavan, and M. D. Stiles. arXiv preprint arXiv:2112.03358 (2022). 
[2] D. E. Nikonov, G. Csaba, W. Porod, et al. IEEE J. Explor. Solid-State Comput. Devices Circuits 1, 85 (2015). 
[3] S. Kaka, M. R. Pufall, W. H. Rippard, et al. Nature 437, 389 (2005). 
[4] S. Jankowski, A. Lozowski, and J. M. Zurada. IEEE Trans. Neural Netw. 7, 6 (1996). 

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