Phase-Binarized Spin Hall Nano-Oscillator Arrays: Towards Spin Hall Ising Machines

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Phase-Binarized Spin Hall Nano-Oscillator Arrays: Towards Spin Hall Ising Machines

Afshin Houshang, Mohammad Zahedinejad, Shreyas Muralidhar, Jakub Chęciński, Roman Khymyn, Mona Rajabali, Himanshu Fulara, Ahmad A. Awad, Mykola Dvornik, and Johan Åkerman

Phys. Rev. Applied 17, 014003 – Published 3 January 2022

Abstract

Ising machines (IMs) are physical systems designed to find solutions to combinatorial optimization (CO) problems mapped onto the IM via the coupling strengths between its binary spins. Using its intrinsic dynamics and different annealing schemes, the IM relaxes over time to its lowest-energy state, which is the solution to the CO problem. IMs have been implemented on different platforms, and interacting nonlinear oscillators are particularly promising candidates. Here we demonstrate a pathway towards an oscillator-based IM using arrays of nanoconstriction spin Hall nano-oscillators (SHNOs). We show how SHNOs can be readily phase binarized and how their resulting microwave power corresponds to well-defined global phase states. To distinguish between degenerate states, we use phase-resolved Brillouin-light-scattering microscopy and directly observe the individual phase of each nanoconstriction. Micromagnetic simulations corroborate our experiments and confirm that our proposed IM platform can solve CO problems, showcased by how the phase states of a $2 \times 2$ SHNO array are solutions to a modified max-cut problem. Compared with the commercially available D-Wave ${Advantage}^{TM}$ , our architecture holds significant promise for faster sampling, substantially reduced power consumption, and a dramatically smaller footprint.

Received 17 August 2021
Revised 29 October 2021
Accepted 9 December 2021
Corrected 25 January 2022

DOI:https://doi.org/10.1103/PhysRevApplied.17.014003

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by Bibsam.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Adiabatic quantum optimization Quantum computation Spin Hall effect Spintronics

Ising model Kuramoto model Magnetization measurements Micromagnetic modeling

Condensed Matter, Materials & Applied Physics

Corrections

25 January 2022

Correction: The author order was set incorrectly and has been fixed.

Authors & Affiliations

Afshin Houshang^1,†, Mohammad Zahedinejad^1,†, Shreyas Muralidhar ¹, Jakub Chęciński ^1,2, Roman Khymyn¹, Mona Rajabali¹, Himanshu Fulara ¹, Ahmad A. Awad ¹, Mykola Dvornik ³, and Johan Åkerman ^1,*

¹Physics Department, University of Gothenburg, Gothenburg 412 96, Sweden
²AGH University of Science and Technology, Institute of Electronics, Al. Mickiewicza 30, Kraków 30-059, Poland
³NanOsc AB, Electrum 229, Kista 164 40, Sweden

^*johan.akerman@physics.gu.se
^†These authors contributed equally to this work.

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Vol. 17, Iss. 1 — January 2022

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Images

Figure 1
Injection locking and principle of phase-binarized spin Hall nano-oscillator (SHNO) arrays. (a) Scanning-electron-microscope image of a $2 \times 2$ SHNO array fabricated from a Pt(5 nm)/ ${Ni}_{80} {Fe}_{20}$ (3 nm) bilayer (Hf dusting layer not shown). The width of each SHNO is $w$ = 120 nm, and their separation (pitch) is $p$ = 300 nm. $H_{IP}$ shows the in-plane direction of the external magnetic field, $H$ . (b) Schematic illustration of the SHNO array in (a), with red arrows showing the local magnetization vector and springs representing their couplings. (c) Ordinary first-harmonic injection locking ( $f_{inj} \sim f_{SHNO}$ ), where all SHNOs are locked in phase to the external source. (d) Second-harmonic injection locking (SHIL at $f_{inj} \sim 2 f_{SHNO}$ ) allows the oscillators to acquire one of two possible phase states separated by $180^{\circ}$ , giving rise to phase binarization.
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Figure 2
Experimentally demonstrated phase binarization. PSD of a $1 \times 2$ SHNO array (a) vs injection power $P_{IL}$ at a fixed injected frequency $f_{IL} = 18.69$ GHz, and (b) vs $f_{IL}$ at a fixed $P_{IL} = 0$ dB. In (a), the array starts off in a synchronized state (“Synced state”), while in (b) the SHNOs are not synchronized. Under SHIL, the linewidth decreases, while the power shows intermittent fluctuations between a high- and a low-power state. (c),(d) show the same trend for $2 \times 2$ arrays, where three different energy levels can be distinguished in the SHIL state. PP refers to peak power.
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Figure 3
Phase-resolved BLS microscopy of individual SHNOs. (a) SEM image of the $1 \times 2$ SHNO array under study; the green arrow shows the direction of the BLS line scan. SHIL with $P_{IL} = 0$ dBm and $f_{IL} = 15.6$ GHz is used. (b)–(e) show the spin wave (SW) intensity profile of the SHNOs for the phase-resolved BLS signal at $90^{\circ}$ and $270^{\circ}$ . The SHNOs in (b),(c) are energized with a $180^{\circ}$ phase difference ( $⟨ ↑↓ ⟩$ ) when $2 f_{SHNO} < f_{IL}$ , and in (d),(e) with a $0^{\circ}$ phase difference ( $⟨ ↑↑ ⟩$ ) when $2 f_{SHNO} ≃ f_{IL}$ . (f) SEM image of a $2 \times 2$ SHNO array with corresponding BLS line-scan directions. Phase-resolved BLS line scans are shown in (g)–(j) for $P_{IL} = + 4$ dBm, corresponding to the $⟨ ↓↑↑↓ ⟩$ phase state, while (k)–(n) show the $⟨ ↓↓↓↓ ⟩$ state obtained at $P_{IL} = + 3$ dBm. (o)–(r) BLS counts for the least stable phase state $⟨ ↓↓↑↓ ⟩$ at $P_{IL} = - 2$ dBm, measured by sweeping the phase ( $φ$ ) of the BLS.
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Figure 4
Micromagnetic simulations of a $2 \times 2$ SHNO Ising machine. (a),(b) Spatial maps of the FFT amplitude and phase, respectively, obtained for a 7.8-GHz mode under a field $H = 5980$ Oe. Phase maps are also shown for $H$ increased to 5990 Oe (c) and to 6020 Oe (d), where the change in the Ising-problem parameters results in different phase patterns. The modified max-cut problems corresponding to fields of 5980 Oe (e) and 6000 Oe (f) are also shown. By reading out the phase patterns of the SHNO network, optimal solutions to these problems are obtained, as illustrated in (g),(h).
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