Radio-Frequency Multiply-and-Accumulate Operations with Spintronic Synapses

Nathan Leroux, Danijela Marković, Erwann Martin, Teodora Petrisor, Damien Querlioz, Alice Mizrahi, and Julie Grollier

Phys. Rev. Applied 15, 034067 – Published 23 March 2021

Abstract

Exploiting the physics of nanoelectronic devices is a major lead for implementing compact, fast, and energy-efficient artificial intelligence. In this work, we propose a strategy in this direction, where assemblies of spintronic resonators used as artificial synapses can classify analogue radio-frequency signals directly without digitalization. The resonators convert the radio-frequency input signals into direct voltages through the spin-diode effect. In the process, they multiply the input signals by a synaptic weight, which depends on their resonance frequency. We demonstrate through physical simulations with parameters extracted from experimental devices that frequency-multiplexed assemblies of resonators implement the cornerstone operation of artificial neural networks, multiply and accumulate (mac), directly on microwave inputs. The results show that, even with a nonideal realistic model, the outputs obtained with our architecture remain comparable to that of a traditional mac operation. Using a conventional machine-learning framework augmented with equations describing the physics of spintronic resonators, we train a single-layer neural network to classify radio-frequency signals encoding 8 × 8 pixel handwritten-digit pictures. The spintronic neural network recognizes the digits with an accuracy of 99.96%, equivalent to purely software neural networks. This mac implementation offers a promising solution for fast low-power radio-frequency classification applications and another building block for spintronic deep neural networks.

Received 13 November 2020
Revised 23 February 2021
Accepted 2 March 2021

DOI:https://doi.org/10.1103/PhysRevApplied.15.034067

Physics Subject Headings (PhySH)

Magnetization dynamics Radio frequency techniques Spin transfer torque Synapses

Artificial neural networks Coupled oscillators Diodes

Microwave techniques

Nonlinear DynamicsNetworksCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Nathan Leroux ^1,*, Danijela Marković¹, Erwann Martin², Teodora Petrisor², Damien Querlioz³, Alice Mizrahi¹, and Julie Grollier¹

¹Unité Mixte de Physique, CNRS, Thales, Université Paris-Saclay, 91767 Palaiseau, France
²Thales Research and Technology, 91767 Palaiseau, France
³Université Paris-Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies, 91120 Palaiseau, France

^*nathan.leroux@cnrs-thales.fr

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Vol. 15, Iss. 3 — March 2021

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Images

Figure 1
(a) Multiply-and-accumulate operation: neural signals P₁, P₂, P₃, and P₄ are multiplied by different synaptic weights W_ji and summed. (b) Multiply-and-accumulate operation with different radio-frequency signals sent simultaneously in two chains of resonators: each resonator rectifies mostly one of the input signals, hence multiplying it by a weight. Chain voltages are the sum of all their resonator voltages.
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Figure 2
Analytical calculations based on the realistic spin-diode model described by Eqs. (2) and (6). (a) Spin-diode rectified voltage of a spintronic resonator with resonance frequency $f^{res} = 200 MHz$ versus the frequency of an input radio-frequency signal for microwave powers between 10 and 50 µW (in color scale). (b) Circles are the spin-diode voltage of a spintronic resonator with different resonance frequencies versus the microwave power of the rf signal at 200 MHz. Solid lines are linear fits. (c) Spin-diode voltage of a chain of four spintronic resonators wired head-to-tail with resonance frequencies $f^{res} = 200 Hz$ , 204.0 MHz, 208.2 MHz, and 212.4 MHz, versus the frequency of a radio-frequency signal with a power of 50 µW. (d) Calculations for the same chain with four different radio-frequency signals for 6561 different combinations of microwave powers for the radio-frequency signals (5, 10, and 15 µW) and different resonance frequencies for the resonators. Scatter dots are the voltages of the calculations with nonlinear resonators plotted against the voltages of the calculations with ideal linear resonators. Red solid line corresponds to calculated voltages with ideal linear resonators plotted against themselves. Root-mean-square deviation between scatter dots and the red solid line is 0.58 µV and the correlation is 99.98%.
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Figure 3
(a) Horizontal component of a spintronic resonator magnetization with four different radio-frequency signals simulated with an ordinary differential equation solver. (b) Theoretical model $m_{x}^{'} (t) = \sum_{i}^{N} \sqrt{p_{i}^{' res}} \cos (ψ^{rf} - ω_{i}^{rf} t)$ . (c) ODE simulations (blue solid line) and theoretical model (black dashed line). (d) Simulation of four different radio-frequency signals sent through a chain of four different spintronic resonators for 6561 different combinations of microwave powers for the radio-frequency signals (5, 10, and 15 µW) and different resonance frequencies for the resonators. Scatter dots are the voltages of the ODE simulations plotted against the voltages of the calculations with theoretical model. Simulations are averaged over 20 repetitions, each with a random initialization for the rf signal phases. Red solid line corresponds to voltages of simulations realized with the theoretical model plotted against themselves. Root-mean-square deviation between the scatter dots and the red solid line is 1.72 µV and the correlation is 99.94%.
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Figure 4
(a) Classical neural-network architecture to solve the digits dataset. From left to right: 8 × 8 pixel input images, 64 × 1 flattened input layer, synaptic layer connecting the input to 10 outputs, and a comparison of outputs with the targets. (b) Equivalent radio-frequency spintronic-synapse-based neural-network architecture. From left to right: 8 × 8 pixel input images and 64 × 1 flattened input layer; each input is encoded in the microwave power of a radio-frequency signal with a different frequency. All 64 signals are summed and sent to 10 chains of 64 resonators wired in series head to tail. Each resonator rectifies its matching frequency signal, thus applying a synaptic weight to it. Output voltages are compared with targets. (c) Analytical simulations of the spin-diode voltage of a chain of 64 spintronic resonators wired head to tail versus the frequency of a rf signal with a power of 50 µW. First resonator has a resonance frequency of $f_{0}^{res} = 200 MHz$ and others are arranged following Eq. (A1) of the Appendix. (d) Percentage of successful classifications versus number of epochs. Black (purple) corresponds to the results for the training (test) set for the resonator neural network and green (blue) is for the training (test) set for the equivalent regular neural-network software. Lines (dashed lines for neural-network software) represent the mean success rates and the shading indicates standard deviations. Success rate reaches 99.96% for both for the neural-network software and resonator-based neural network for training and test sets.
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Figure 5
Percentage of successful classifications of a single-layer resonator network on the MNIST dataset versus magnetic damping of the material used for spintronic resonators (log₁₀ scale). In blue (red) are plotted the results for the test (training) set. Results are averaged over 10 repetitions and the error bar corresponds to the root-mean-square deviation.
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Figure 6
Magnetization-normalized oscillation power versus frequency of a rf signal for three spintronic resonators of different frequencies following Eq. (A1).
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