Demonstration of an energy-efficient Ising solver composed of Ovonic Threshold Switch (OTS)-based nano-oscillators (OTSNOs)

Demonstration of an energy-efficient Ising solver composed of Ovonic Threshold Switch (OTS)-based nano-oscillators (OTSNOs) | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Demonstration of an energy-efficient Ising solver composed of Ovonic Threshold Switch (OTS)-based nano-oscillators (OTSNOs) Young Woong Lee, Seon Jeong Kim, Jaewook Kim, Sangheon Kim, Jongkil Park, and 6 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3943189/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 22 May, 2024 Read the published version in Nano Convergence → Version 1 posted 4 You are reading this latest preprint version Abstract As there is an increasing need for an efficient solver of combinatorial optimization problems, much interest is paid to the Ising machine, which is a novel physics-driven computing system composed of coupled oscillators mimicking the dynamics of the system of coupled electronic spins. In this work, we propose an energy-efficient nano-oscillator, called OTSNO, which is composed of an Ovonic Threshold Switch (OTS) and an electrical resistor. We demonstrate that the OTSNO shows the synchronization behavior, an essential property for the realization of an Ising machine. Furthermore, we have discovered that the capacitive coupling is advantageous over the resistive coupling for the hardware implementation of an Ising solver by providing a larger margin of the variations of components. Finally, we implement an Ising machine composed of capacitively-coupled OTSNOs to demonstrate that the solution to a 14-node MaxCut problem can be obtained in 40 ms while consuming no more than 2.3 mJ of energy. Compared to a previous hardware implementation of the phase-transition nano-oscillator (PTNO)-based Ising machine, the OTSNO-based Ising machine in this work shows the performance of the increased speed by more than one order while consuming less energy by about an order. Ising solver Ovonic Threshold Switch (OTS) oscillator-based computing Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 1. Introduction Combinatorial optimization problems (COPs) are commonly found in everyday life, for example, schedule-planning, travel-planning, resource allocation, etc, and are solved in brute-force forms or rather sophisticated algorithmic forms. As COPs are expanding their application to logistics, routing in IC (integrated circuits) design, drug discovery, etc., there is a growing demand for an energy- and time-efficient solver for COPs since conventional computers with the von Neumann architecture are not good at solving these kinds of problems [ 1 ]. A promising alternative computing method is to use the so-called “compute-by-physics” strategy, where a COP is translated into a complex physical system which, ruled by the physics same as in the problem [ 2 – 4 ], evolves into the ground state leading to a solution to the problem. One example is the Ising machine (IM), a system composed of spins coupled by the exchange interaction, whose ground state naturally gives a solution to the MaxCut problem [ 5 ]. In practical implementations of IM, the Ising spins can be replaced with coupled oscillators with phase being quantized at a few values by the same physics governing the Ising spins. Prototypical large-scale IMs were demonstrated by using optical parametric oscillators (OPO) [ 6 , 7 ], showing excellent performance of Ising solvers compared to other competitors. Nevertheless, such OPO-based IM has a drawback in their scalability because they used the long fiber ring cavity requiring a lot of space. To resolve this scalability problem, scalable IMs have been demonstrated by using electronic oscillators such as ring oscillators (ROSCs) [ 8 , 9 ], the spin-torque nano-oscillator (STNO), and spin-Hall nano-oscillator (SHNO) based on the magnetic tunnel junction (MTJ) [ 10 – 12 ], and the phase-transition nano-oscillator (PTNO) based on Mott insulators [ 13 – 16 ]. However, due to scalability and/or energy efficiency issues with those oscillator devices, there is a need to find a more scalable and energy-efficient alternative for implementing a practical IM. In this work, we propose a novel frequency-tunable nano-oscillator based on the Ovonic threshold switch (OTS) as a solution to this problem. This OTS nano-oscillator (OTSNO) has a structure similar to that of the PTNO with the phase-change switching device replaced by an OTS device (see Fig. 1 a). Consisting of an amorphous chalcogenide sandwiched between the conducting electrodes, an OTS device shows reversible electrical switching [ 17 ], which is believed to be closely associated with the charging and discharging of the trap states inside the amorphous chalcogenide [ 18 – 26 ]. Such a switching mechanism endows the OTSNO with superior energy efficiency compared to the PTNO which requires Joule heating for the switching. In Fig. 1 a, note that an OTS device is identified by two characteristic voltage levels, the threshold voltage ( V th ) and the holding voltage ( V H ), at which the resistance of the OTS turns low ( R on ) and high ( R off ), respectively, with a very high R off / R on ratio (~ 10 6 ) [ 27 – 30 ]. For the 1OTS + 1FET structure to be an oscillator, the resistance of the FET ( R FET ) should have R on < < R FET < < R off . Under this condition, the oscillating behavior of the 1OTS + 1FET structure is easily understood considering the voltage-dividing relation between the OTS and the FET. When a voltage bias ( V bias > V th ) is applied to the 1OTS + 1FET structure with the OTS in its OFF state, most of V bias is dropped across the OTS because R FET < < R off , turning the OTS into ON state. On the contrary, with the OTS in its ON state, most of V bias is dropped across the FET because R on < < R FET , turning the OTS into OFF state and completing one cycle of oscillation. Since the aforementioned discharging period depends on R FET , the oscillation frequency can be controlled by the gate voltage ( V G ) of the FET. For applying the OTSNO to the IM, the coupling element, mostly a capacitor and/or a resistor, should be carefully selected such that the coupled OTSNOs have an appropriate phase relationship, in-phase (IP) or anti-phase (AP). We have performed a systematic circuit simulation study to examine the phase relationship between the coupled OTSNOs depending on the strength of the coupling element and the tolerance to the variations of the oscillator devices composing the IM. Finally, by using these findings, we have constructed an IM to successfully demonstrate the solution of a MaxCut problem with 14 nodes. Finally, we present a comparison of the OTSNO with other existing nano-oscillators, showing that the OTSNO is a promising candidate for developing large-scale IMs. 2. Results and Discussion Figure 1 b shows an optical microscope plane-view image (top) and a cross-sectional transmission electron microscope (TEM) image of an OTS device, which has a pore-type structure with a pore size ( d ) of 300 nm. The fabrication process of the OTS device is described in detail in the Experimental Section. Figure 1 c shows the characteristic current-voltage ( I - V ) curves of the device, which is nominally the same as Fig. 1 b. In ten repeated measurements, the OTS device shows the variation in V th within 2.5 ~ 3.0 V while V H is around 1 V with a negligible variation. In addition, R off and R on of OTS devices have been read around ~ 10 8 and ~ 10 2 Ω, respectively. Figure 1 d shows V out waveforms of a typical OTSNO with varying V G , where the OTSNO consists of an OTS device and a FET (n-MOSFET, LND150N3-G, Microchip Inc.). It is clearly shown that the natural frequency ( f nat ) of the OTSNO increases with V G as expected. To quantify f nat at each V G , the fast Fourier transform (FFT) is performed as shown in Fig. 1 e and f nat , defined as the frequency at the primary peak, is plotted as a function of V G in Fig. 1 f. It is observed that f nat is well defined and linearly proportional to V G being adjustable in the range of 0.5 ~ 2 MHz. These results clearly show that the oscillation in the OTSNO is composed of fundamental-frequency components and their harmonics while keeping other components negligible. Next, we investigate the synchronization behavior of coupled OTSNOs (see Fig. 2 a). Figure 2 b shows the output waveforms of two OTSNOs coupled by a capacitor. It is observed that the output waveforms of two oscillators look nearly the same with a certain degree of phase difference, demonstrating that two oscillators are synchronized to each other (for a detailed study of the synchronization behavior, see section S1 in Supplementary Information). In Fig. 2 c, V out2 is plotted as a function of V out1 , which presents the so-called “phase portrait” of a coupled-oscillator system. It shows a butterfly-shaped attractor curve, implying the AP relationship between the two oscillators. The attractor curve appears as a band implying the chaotic nature of the OTSNO [ 21 , 23 , 27 , 31 ], which is desirable for the application to the IM because it helps the system to find the solution of the network configuration with the global minimum energy [ 32 ]. To examine the phase stability of two coupled oscillators, we have performed a simulation study where the circuit parameters of the system are systematically varied. Especially, we have investigated two types of coupling - capacitive and resistive coupling - with varying their values, C C and R C , respectively. In the SPICE (Simulation Program with Integrated Circuit Emphasis) simulation, the OTS device is modeled as a voltage-controlled switch characterized by four parameters ( V th , V H , R on , R off ) in parallel connection with a parasitic capacitor. Figure 2 d and 2 e show the phase difference ( Δφ ) between oscillators #1 and #2 as a function of ( C C , R L ) and ( R C , R L ), respectively. Here, the parameters of both OTS devices are set to ( V th , V H , R on , R off )=(3.3 V, 0.7 V, 150 Ω, 10 MΩ). Δφ is calculated by 2*π*( T 2 - T 1 )* f sync after ~ 200 cycles, where T 2 - T 1 and f sync are the time difference between adjacent peaks of oscillator #1 and #2 and the oscillation frequency in the synchronized state, respectively. It is found that the AP relationship between two oscillators is more stable in the capacitive coupling with C C for AP relationship spanning almost two orders (2 ~ 200 pF) independent of R L while the range of R C for AP relationship is relatively narrow with a strong dependence on R L . For the practical application, an important aspect is the tolerance of the coupled oscillator system to the inevitable variation of the OTS device. Figure 2 f and 2 g show Δφ as a function of ΔV th (= V th2 - V th1 ) and ΔV H (= V H2 - V H1 ), where we have varied V th and V H of OTS #2 systematically while keeping those of OTS #1 fixed. It is shown that, in the case of the capacitive coupling, the AP relationship is robust against the V th -variation of up to ± 20%, which is significantly higher compared to resistive coupling where the limit is only ± 5%. The superior stability of the AP relationship in the C C -coupled oscillator system can be explained qualitatively by the ability of the coupling capacitor to store energy. In detail, if a voltage difference is generated between two oscillators, then it can be stored in the coupling capacitor. The stored energy is returned to the oscillators making them repel each other, stabilizing the AP relationship. In contrast, since the resistor can’t store energy in the R C -coupled oscillator system, the voltage difference between oscillators generates heat dissipation in R C . This makes the oscillators lose their energy destabilizing the AP relationship in the R C -coupled oscillator system. Another intriguing finding is that the phase relationship of the coupled OTSNOs in both cases is hardly sensitive to the variation of V H compared to the variation of V th . It is related to the asymmetry of the output waveform (see Fig. 1 d), which shows a much longer falling period compared to the rising period. Recalling the mechanism of the oscillating behavior of the OTSNO, it is easily found that the peak of the oscillation aligns with the ON-to-OFF transition of the OTS device whereas the valley point corresponds to the OFF-to-ON transition. Consequently, when the OTS reaches its peak, it transitions into a highly resistive state with R off ~ 10 MΩ. As mentioned above, since the OTS can be described by a parallel connection of a voltage-controlled switch and a parasitic capacitor, such a high R off results in a large RC delay, and consequently, the elongation of the falling period compared to the rising period. During the falling period, the parasitic capacitor is charged and the voltage across it runs into V th starting from V H . Since the charging time is mainly determined by V th , the variation in V H has little effect on the phase relationship of the coupled OTSNOs. The output waveforms and the phase portraits at points in various regions in Fig. 2 d and 2 e are presented in the Supplementary Information (Fig. S2). Based on these results, we have built an IM using OTSNOs and coupling capacitors ( C c =100 pF) as shown in Fig. 3 a and, as a benchmark test, have tried to solve a MaxCut problem with a Mobius ladder geometry composed of 14 nodes and cubic connections (see Fig. 3 b). We have also tried another geometry, which is presented in Fig. S3 in the Supplementary Information. Tlhe details of the measurement is described in the Experimental Section. A representative output waveforms of the 14 OTSNOs are presented in Fig. 3 c. We have employed both the second-harmonic injection locking (SHIL) and simulated annealing (SA) techniques [ 16 , 33 – 35 ] to improve the reliability of the solution by keeping the system from being stuck to a local minimum. In this technique, all the oscillators are driven by a DC voltage ( V dc ) and a modulational AC voltage ( V SH sin( ω SH t )), which locks the oscillation frequency to ω SH (= 2 ω sync ~ 2.75 MHz, where ω sync is the oscillation frequency in the synchronized state). In the right panel, which is the expansion of the waveforms in the range of 40 ~ 43 µs, it is shown that the odd-numbered OTSNOs oscillate with the nearly same phase while the even-numbered ones also show the synchronized behavior with a clearly distinguished phase. To quantify the phase difference between oscillators, we have calculated the amplitude of the in-phase component of each V out ( t ) by applying the low-pass filter (the cut-off frequency of 0.1 MHz) to the product of a reference wave (cos( ω SH t )) and V out ( t ). Such obtained in-phase amplitude (Amp in ) of the oscillators are plotted as a function of time in Fig. 3 d. It is clearly observed that the oscillators are separated into two groups with the opposite signs of Amp in implying the AP relationship between those two groups of oscillators. In addition, it shows that those two groups are clearly separated 35 µs after applying the bias. Considering that V SH gradually increases over 35 µs in the simulated annealing scheme, it indicates that the time-to-solution ( T sol ) is expected to be at most 35 µs. From repeated implementations of the same experiment, we have observed T sol is in the range of 35 ~ 40 µs. It is much shorter than T sol ~ 4 ms reported in a previous work on the hardware implementation of a PTNO-based Ising solver [ 16 ] for solving an 8-node MaxCut problem although, in a simulation study, T sol =30 µs was presented for solving a 100-node MaxCut problem with random cubic connections. The energy-to-solution ( E sol ), as another performance metric of the IM, is calculated by integrating the instantaneous power consumption ( P ( t ) = V ( t )× I ( t ), where \(V(t)={V_{dc}}+{V_{SH}}\sin (\omega t)\) applied to the system (refer to Fig. S3a) and \(I(t)=\sum\limits_{{i=1}}^{{14}} {{I_{i,out}}(t)}\) , the sum of the output currents ( \({I_{i,out}}(t)\) ) at each node) over T sol . P ( t ) is plotted as a function of time in Fig. 4 a, leading to an estimation of E sol around 2.3 µJ. It can not be compared with that of the PTNO-based IM as it was not known in the previous work. Instead, we have tried to compare the energy consumption in a single OTSNO and PTNO because of the negligible energy consumption in the coupling capacitor (see Fig. S4 in the Supplementary Information) as expected in the pure capacitive circuit. The energy consumption per cycle ( E cycle ) is estimated to be ~ 1.25 nJ/cycle (see Fig. 5 b) for a 500 nm ( d : diameter of the pore)-sized OTSNO and ~ 0.9 nJ/cycle for a 200 nm (the length of VO 2 channel)-sized PTNO, respectively. In the OTS device, E cycle is expected to be scaled with the dimension of the device because of the shrinkage of the switching volume. Therefore, we have investigated the dependence of E cycle of the OTSNO on the diameter of the pore as presented in Fig. 4 c (for the waveforms of V ( t ), I ( t ), and P ( t ) corresponding to OTSNOs with various pore sizes, see Fig. S5 in Supplementary Information). For comparison, E cycle of the PTNO [ 16 , 36 ] is also located in the same graph, clearly showing that the OTSNO consumes less energy by about an order than the PTNO with similar device dimensions. The difference is attributed to the difference in the switching mechanisms of the PTNO and the OTSNO. In detail, the PTNO is based on the phase transition of the Mott insulator, which requires heating of a part in the channel material up to the transition temperature. In contrast, the OTSNO is believed to be based on the filling and evacuation of the trap states in the OTS, the electronic process in nature, although there is still controversy about its switching mechanism [ 21 , 23 , 31 , 37 ]. In addition, Fig. 4 c shows that E cycle of the OTSNO scales as E cycle ~ d 1.81 with the diameter of the pore, being close to the expected relation of E cycle ~ d 2 for the case of the uniform current density in the switching material. Therefore, it seems to imply that the reduction in E cycle is attributed to the shrinkage of the switching volume inside the cylinder. As a result, E cycle of the OTSNO extrapolates to ~ 1 pJ/cycle at d = 10 nm, which enables the development of a highly efficient large-scale IM. Finally, we have performed a simulation study to investigate the scalability of the OTSNO-based IM. As a typical benchmark task, we have studied the Mobius ladder geometry composed of variable numbers of OTSNOs and connections (see Fig. 5 a). Figures 5 b ~ 5d show the representative examples of Amp in of oscillators with varying the number of nodes ( N ) from 22 to 102. The correctness of the obtained solution has been verified by comparing the energy of the solution with that of the ground state. From 20 repetitions of the simulation with varying the initial phases of each oscillator randomly, we have obtained the performance metrics, the success probability ( P success ) and T sol , as shown in Fig. 5 e. Note that, as N increases, T sol is linearly proportional to log( N ) with a slight decrease in P success . We have repeated similar simulations with varying the number of connections ( N c ) in the way of adding connections to the next nearest neighbors in order with a diagonal connection being fixed. Representative examples of Amp in of oscillators for various geometries are presented in the Supplementary Information (see Fig. S6). In Fig. 5 f, it is observed that T sol shows a sublinear dependence on N c while P success decreases slightly with increasing N c . It implies that, as N c increases, local minima are likely to be formed in the energy landscape and the oscillators are trapped in those local minima. Due to the intrinsic chaotic nature of the OTSNO device as mentioned in the discussion of Fig. 2 c, we expect that the problem of such local minima might be alleviated in the hardware implementation of the IM using OTSNOs. 3. Conclusion In summary, we have investigated the OTSNO as a candidate nano-oscillator with improved scalability for applications in Ising machines. We have demonstrated that the OTSNO shows robust oscillating behavior with a well-defined frequency, which can be controlled by the gate voltage applied to the FET. We have also demonstrated the synchronization between two coupled OTSNOs with their phase relation (in-phase or anti-phase) controlled by the strength of the coupling elements (electrical resistors or capacitors), which is systematically investigated leading to a conclusion that capacitive coupling provides the larger operation windows with respect to both the value of the coupling capacitance and the device variations of the OTS. Finally, we have implemented an Ising machine composed of capacitively-coupled OTSNOs and demonstrated that the solution to a 14-node MaxCut problem can be obtained in 35 µs while consuming no more than 2.3 µJ of energy. Moreover, it is shown that the OTSNO has superior energy scalability with consuming less energy by about an order compared with the PTNO based on the Mott insulator such as VO 2 . We have compared the characteristics of various nano-oscillators in the Supplementary Information (see “2. Supplementary Note: Comparison with other oscillators”). Based on this comparison and considering that the OTSNO provides an oscillator with a minimum size of 6F 2 with F down to ~ 14 nm [ 28 , 38 ], we believe that the OTSNO is highly promising for application to large-scale Ising machines. 4. Experimental Section Fabrication of the OTS OTS devices are fabricated with a pore-type structure, where the pore size ( d ) is defined by the electron beam lithography in the range of 100 ~ 500 nm. 60 nm-thick Sb x (GeSe) 1−x (SGS) layer is used as a switching material because it is found to have a lower threshold voltage ( V th ) compared to Ge 50 Se 50 . It is deposited by a co-sputtering technique (magnetron RF sputtering) using Ge, Sb, and GeSe 2 targets. Pt and TiN are used as the bottom and top electrodes, respectively. Characterization of the OTSNO To configure a frequency-adjustable OTSNO, an OTS device is connected in series to an n-MOSFET device (LND150N3-G, Microchip Inc.). The characteristics of OTS devices and OTSNOs have been investigated using an arbitrary function generator (AFG-3102, Tektronix), an oscilloscope (MSO-58, Tektronix), and a source-measure unit (2635B, Keithley). We have tested several OTSNO devices using OTS devices of various pore sizes and found similar behavior. The results presented in this paper are mainly obtained by using OTS devices with d = 500 nm. Implementation of the Ising solver The implementation of the Ising machine composed of OTSNOs including the measurement setup is shown in Fig. 3 a and Fig. S3. The OTS devices fabricated on the SiO 2 substrate are loaded onto a custom-made breakout board. Terminals of the OTS devices are connected to the power supply ( V dd ), R L , and C C on a specially-designed printed circuit board (PCB). We have used the second-harmonic injection locking (SHIL) technique to lock phases of OTSNO with an oscillation frequency of 2.75 MHz for the injection signal, twice the natural frequency of the single OTSNO. In addition, we have used the simulated annealing technique for the injection signal, whose amplitude gradually increases from 0 V to 2 V over 45 µs. V dc , R L , and C C are set to 4 V, 16 kΩ, and 100 pF, respectively. An arbitrary function generator (AFG-3102, Tektronix) has been used to bias the driving dc voltage and the SHIL ac voltage. To read the output waveforms of 14 OTSNOs by using an 8-channel oscilloscope (MSO-58, Tektronix), we have repeated the measurement runs two times. In the first and second runs, 14 OTSNOs have been read half-and-half relative to oscillator #1. SPICE simulation To perform SPICE simulations, we have used the LTSpice (Analog Device Inc.) software. The OTS device is modeled as a parallel connection between a voltage-controlled switch (VCS) and a parasitic capacitor (100 pF), the former of which is modeled to have hysteretic switching behavior mimicking the OTS with V th = 3.3 V, V H = 1 V, R on = 150 Ω, and R off = 10 6 Ω. Declarations Acknowledgments Not applicable. Author contributions S.L. designed and conceived the experiments with help from Y.W.L. and S.J.K., and S.K. Y.W.L., S.J.K., and S.K. fabricated OTS devices. Y.W.L., S.J.K., and S.K. performed the characterization of devices. S.L. and B.H.P. participated in the analysis of the synchronization. J.K., J.P., Y.J., G.W.H., and S.P. performed a circuit analysis of the behavior of the oscillator device and a simulation of a coupled oscillator network. All authors discussed the data and participated in the writing of the manuscript. Funding This work was supported by the Korea Institute of Science and Technology (KIST) through 2E32960 and by National Research Foundation program through NRF- 2021M3F3A2A03017782, 2021M3F3A2A01037738, 2021M3F3A2A01037814, 2022R1A2C3004135, and NRF-2021R1A2C3011450. B.H.P. was supported by Korea Basic Science Institute (National Research Facilities and Equipment Center) grant funded by the Ministry of Education. (Grant No. 2022R1A6C101A754). Availability of data and materials Not applicable. Competing interests The authors declare no competing financial interest. Author details 1 Center for Neuromorphic engineering, Korea Institute of Science and Technology, Seoul 02792, Rep. of Korea. 2 Department of Physics, Konkuk University, Seoul 05029, Rep. of Korea. 3 Core Facility Center for Quantum Characterization/Analysis of Two-Dimensional Materials & Heterostructures, Konkuk University, Seoul 05029, Rep. of Korea. 4 Division of Nano & Information Technology, Korea University of Science and Technology, Daejeon 34316, Rep. of Korea. Supplementary Information The online version contains supplementary material available at ... References J. Hartmanis, in Current Trends in Theoretical Computer Science , pp. 445 T. Kanao et al., Phys. Rev. Appl. 12 , 024052 (2019) G. Tanaka et al., Neural Netw. 115 , 100 (2019) G. Csaba, W. Porod, Appl. Phys. reviews. 7 , 011302 (2020) N. Mohseni, P.L. McMahon, T. Byrnes, Nat. Reviews Phys. 4 , 363 (2022) A. Marandi et al., Nat. Photonics. 8 , 937 (2014) P.L. McMahon et al., Science. 354 , 614 (2016) I. 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Hady et al., Proceedings of the IEEE 105, 1822 (2017) Supplementary Files 240208OTSNOIMYWLeeSIENrev0.pdf Fig. S1. Synchronization of OTSNO Fig. S2. Phase difference of two coupled oscillators, coupling-dependent output waveforms and phase portraits. Fig. S3. Solving a 5-node MaxCut problem by using coupled OTSNOs. Fig. S4. Power dissipation in the coupling capacitor. Fig. S5. Dependence of the oscillating behavior on the pore size ( d ) of the OTS device. Fig. S6. Temporal evolution of the in-phase component (Amp in ) of the output waveforms of oscillators for Mobius ladder geometries with varying the number of connections (simulation). Table S1. Comparison between electronic oscillator devices. Cite Share Download PDF Status: Published Journal Publication published 22 May, 2024 Read the published version in Nano Convergence → Version 1 posted Reviewers agreed at journal 14 Mar, 2024 Reviewers invited by journal 07 Mar, 2024 Editor assigned by journal 08 Feb, 2024 First submitted to journal 08 Feb, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3943189","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":277210775,"identity":"4847387d-0e07-451a-8b91-bb2e1e7be4dc","order_by":0,"name":"Young Woong Lee","email":"","orcid":"","institution":"Korea Institute of Science and Technology","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Young","middleName":"Woong","lastName":"Lee","suffix":""},{"id":277210776,"identity":"d561faf8-ab4f-4ac5-adc8-61a8fa4744c6","order_by":1,"name":"Seon Jeong Kim","email":"","orcid":"","institution":"Korea Institute of Science and Technology","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Seon","middleName":"Jeong","lastName":"Kim","suffix":""},{"id":277210777,"identity":"3c55171d-b518-4813-b6d0-c8f83365f05c","order_by":2,"name":"Jaewook Kim","email":"","orcid":"","institution":"Korea Institute of Science and Technology","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Jaewook","middleName":"","lastName":"Kim","suffix":""},{"id":277210778,"identity":"b867e832-a1d5-4cf9-aabb-a220c387180c","order_by":3,"name":"Sangheon Kim","email":"","orcid":"","institution":"Korea Institute of Science and Technology","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Sangheon","middleName":"","lastName":"Kim","suffix":""},{"id":277210779,"identity":"a068f789-006c-4039-b633-1d2ebcb4706c","order_by":4,"name":"Jongkil Park","email":"","orcid":"","institution":"Korea Institute of Science and Technology","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Jongkil","middleName":"","lastName":"Park","suffix":""},{"id":277210780,"identity":"68638c49-cae7-499e-99e2-f8c88a0b48bd","order_by":5,"name":"YeonJoo Jeong","email":"","orcid":"","institution":"Korea Institute of Science and Technology","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"YeonJoo","middleName":"","lastName":"Jeong","suffix":""},{"id":277210781,"identity":"68abdc1a-b159-4add-8788-22081a5cb43d","order_by":6,"name":"Jaewook Kim","email":"","orcid":"","institution":"Korea Institute of Science and Technology","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Jaewook","middleName":"","lastName":"Kim","suffix":""},{"id":277210782,"identity":"26af9e8c-882d-4355-96cd-97b329b0b519","order_by":7,"name":"Gyu Weon Hwang","email":"","orcid":"","institution":"Korea Institute of Science and Technology","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Gyu","middleName":"Weon","lastName":"Hwang","suffix":""},{"id":277210783,"identity":"c5799b8e-8146-402a-b677-bacd3da79420","order_by":8,"name":"Seongsik Park","email":"","orcid":"","institution":"Korea Institute of Science and Technology","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Seongsik","middleName":"","lastName":"Park","suffix":""},{"id":277210784,"identity":"7fa008d5-5ec5-4935-8279-c1034fa85fd0","order_by":9,"name":"Bae Ho Park","email":"","orcid":"","institution":"Konkuk University - Seoul Campus: Konkuk University","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Bae","middleName":"Ho","lastName":"Park","suffix":""},{"id":277210785,"identity":"ab5e2adb-ead6-4860-aa54-60fedb9fa017","order_by":10,"name":"Suyoun Lee","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0klEQVRIiWNgGAWjYDACdjBpwcAPohIKiNHCDCYlGCQbQFoMSNFicABEE6NFvpn54IMPFRL2xudXJ354YMAgzy92AL8Wg8NsyYYzzkgkbrvxdrME0GGGM2cnENDCzGMmzdsmkWB24+wGkJYEg9sEtMg383///fcf0GEzzm7+QZQWhsM8bMyMDRKMG/h7txFnC9AvxpI9xyQSZ9zg3WaRYCBB2C/y7c0PP/yosbHn7z+7+eaPCht5fmlCDoMDCbBKCWKVgwD/AVJUj4JRMApGwUgCAE9aQEnhKykTAAAAAElFTkSuQmCC","orcid":"https://orcid.org/0000-0002-5147-6821","institution":"Korea Institute of Science and Technology","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Suyoun","middleName":"","lastName":"Lee","suffix":""}],"badges":[],"createdAt":"2024-02-09 14:04:10","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3943189/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3943189/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1186/s40580-024-00429-2","type":"published","date":"2024-05-23T00:40:02+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":52524767,"identity":"48646571-8950-4a64-9205-f022c7f3b7da","added_by":"auto","created_at":"2024-03-12 15:37:58","extension":"jpeg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":501960,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eFrequency-tunable nano-oscillator based on OTS device.\u003c/strong\u003e (a) 1OTS+1FET structure as an OTS-based nano-oscillator (OTSNO). The inset shows a schematic \u003cem\u003eI\u003c/em\u003e-\u003cem\u003eV\u003c/em\u003e curve of an OTS device. (b) (top) An optical microscope (scalebar=50 mm) and (bottom) a cross-sectional TEM image of an OTS device, where SGS means Sb\u003csub\u003e\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003e(GeSe)\u003csub\u003e1-\u003c/sub\u003e\u003csub\u003e\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003e (\u003cem\u003ex\u003c/em\u003e=0~0.1, scalebar=50 nm). (c) Characteristic \u003cem\u003eI\u003c/em\u003e-\u003cem\u003eV\u003c/em\u003e curve of an OTS device (ten repetitions). (d) Output waveforms of the OTSNO for various \u003cem\u003eV\u003c/em\u003e\u003csub\u003eG\u003c/sub\u003es. (e) Fast Fourier transform (FFT)-amplitude of the output waveforms. (f) Dependence of the natural frequency (\u003cem\u003ef\u003c/em\u003e\u003csub\u003enat\u003c/sub\u003e) on \u003cem\u003eV\u003c/em\u003e\u003csub\u003eG\u003c/sub\u003e, where \u003cem\u003ef\u003c/em\u003e\u003csub\u003enat\u003c/sub\u003e is defined as the position of the primary peak in FFT.\u003c/p\u003e","description":"","filename":"floatimage1.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-3943189/v1/ca210c326fc9d583d6bc054c.jpeg"},{"id":52524765,"identity":"27309631-fb36-4d61-bd9d-217feff1edab","added_by":"auto","created_at":"2024-03-12 15:37:58","extension":"jpeg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":897223,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eSynchronization of OTSNO. \u003c/strong\u003e(a) Circuit diagram of two capacitively-coupled OTSNOs. (b) Output waveforms of two oscillators, where \u003cem\u003eR\u003c/em\u003e\u003csub\u003eL\u003c/sub\u003e=2.5 kW and \u003cem\u003eC\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e=100 pF, respectively. The waveforms are vertically shifted for clarity. (c) Phase portrait of the two coupled oscillators, which clearly shows an attractor curve resembling a butterfly indicating the anti-phase relationship between those two oscillators. (d), (e) Phase difference (\u003cem\u003eDf\u003c/em\u003e) between two coupled oscillators as a function of the load resistance (\u003cem\u003eR\u003c/em\u003e\u003csub\u003eL\u003c/sub\u003e) and the coupling strength ((d) \u003cem\u003eC\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e and (e) \u003cem\u003eR\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e, respectively). In the dark grey region, the oscillators do not show oscillating behavior. (f), (g) \u003cem\u003eDf\u003c/em\u003e as a function of \u003cem\u003eDV\u003c/em\u003e\u003csub\u003eth\u003c/sub\u003e ( = \u003cem\u003eV\u003c/em\u003e\u003csub\u003eth2\u003c/sub\u003e -\u003cem\u003eV\u003c/em\u003e\u003csub\u003eth1\u003c/sub\u003e) and \u003cem\u003eDV\u003c/em\u003e\u003csub\u003eH\u003c/sub\u003e ( = \u003cem\u003eV\u003c/em\u003e\u003csub\u003eH2\u003c/sub\u003e -\u003cem\u003eV\u003c/em\u003e\u003csub\u003eH1\u003c/sub\u003e) for the capacitive coupling (d) and the resistive coupling (e) cases, respectively.\u003c/p\u003e","description":"","filename":"floatimage2.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-3943189/v1/0d2dcce1d0547634c314b5bd.jpeg"},{"id":52524764,"identity":"fd8bea0f-0c30-4f54-a062-4bf2bcc774cc","added_by":"auto","created_at":"2024-03-12 15:37:58","extension":"jpeg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":910646,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eSolving a 14-node MaxCut problem by using coupled OTSNOs. \u003c/strong\u003e(a) A picture of the measurement setup (insets: PCB for configuring connections (orange box) and an array of OTS devices (red box)). (b) 14-node MaxCut problem with a Mobius ladder geometry with cubic connections. (c) The output waveforms of the 14 oscillators, which are divided into two groups (even-numbered and odd-numbered ones). The waveforms are vertically shifted by intention for clarity. The right panel shows an expansion of the region from 40 to 43 ms, clearly showing the AP relationship between the two groups. (d) Temporal evolution of the in-phase component (Amp\u003csub\u003ein\u003c/sub\u003e) of the output waveforms (see the main text).\u003c/p\u003e","description":"","filename":"floatimage3.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-3943189/v1/4b81b606f0247dd744516264.jpeg"},{"id":52524768,"identity":"412f6f79-1474-4ea2-8968-e5943e8e5577","added_by":"auto","created_at":"2024-03-12 15:37:58","extension":"jpeg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":317867,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eEnergy efficiency of the OTSNO-based Ising machine and its scalability. \u003c/strong\u003e(a) Consumed power obtained by \u003cem\u003eP\u003c/em\u003e(\u003cem\u003et\u003c/em\u003e) = \u003cem\u003eI\u003c/em\u003e(\u003cem\u003et\u003c/em\u003e) ´ \u003cem\u003eV\u003c/em\u003e(\u003cem\u003et\u003c/em\u003e) of all oscillators (see text) as a function of time, resulting in an estimation of the consumed energy (~2.3 mJ) for solving the problem. (b) Calculation of the energy consumption per cycle (\u003cem\u003eE\u003c/em\u003e\u003csub\u003ecycle\u003c/sub\u003e) for an OTSNO using an OTS with the pore size (\u003cem\u003ed\u003c/em\u003e) of 500 nm. (c) Dependence of the \u003cem\u003eE\u003c/em\u003e\u003csub\u003ecycle\u003c/sub\u003e of the OTSNO on \u003cem\u003ed\u003c/em\u003e in comparison with PTNO [16, 36].\u003c/p\u003e","description":"","filename":"floatimage4.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-3943189/v1/5e53e5fabdd812db88c1f77b.jpeg"},{"id":52524769,"identity":"ace4002e-37f1-4a52-b46c-52635c84a7f3","added_by":"auto","created_at":"2024-03-12 15:37:58","extension":"jpeg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":674112,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eScalability of the OTSNO-based IM (simulation). \u003c/strong\u003e(a) A Mobius ladder geometry composed of variable numbers of oscillators and connections. (b)~(d) Amp\u003csub\u003ein\u003c/sub\u003e as a function of time with \u003cem\u003eN\u003c/em\u003e=22 (b), 50 (c), 102 (d) with \u003cem\u003eN\u003c/em\u003e\u003csub\u003ec\u003c/sub\u003e being fixed at three. (e), (f) \u003cem\u003eT\u003c/em\u003e\u003csub\u003esol\u003c/sub\u003e (line and symbol, left axis) and \u003cem\u003eP\u003c/em\u003e\u003csub\u003esuccess\u003c/sub\u003e (bar plot, right axis) as a function of the number of nodes (\u003cem\u003eN\u003c/em\u003e) and the number of connections (\u003cem\u003eN\u003c/em\u003e\u003csub\u003ec\u003c/sub\u003e) in the Mobius ladder geometry, respectively. In (e) and (f), \u003cem\u003eN\u003c/em\u003e\u003csub\u003ec\u003c/sub\u003e and \u003cem\u003eN\u003c/em\u003e are fixed at 3 and 22, respectively. For each geometry, the simulation was performed 20 times with random initial phases of oscillators, from which the average and the standard deviation are plotted for \u003cem\u003eT\u003c/em\u003e\u003csub\u003esol\u003c/sub\u003e.\u003c/p\u003e","description":"","filename":"floatimage5.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-3943189/v1/6ade67be7b7fce569d1be151.jpeg"},{"id":57045514,"identity":"92ebc8a9-2eed-4bb9-8acc-882494cbc2ae","added_by":"auto","created_at":"2024-05-24 00:40:09","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3943329,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3943189/v1/7e83a605-60ad-4ca3-a25b-ad48d42f160e.pdf"},{"id":52525342,"identity":"d130c583-e729-4f7a-a587-e04cc0e2f8a6","added_by":"auto","created_at":"2024-03-12 15:45:58","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":1316631,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eFig. S1.\u003c/strong\u003e Synchronization of OTSNO\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFig. S2.\u003c/strong\u003e Phase difference of two coupled oscillators, coupling-dependent output waveforms and phase portraits.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFig. S3.\u003c/strong\u003e Solving a 5-node MaxCut problem by using coupled OTSNOs.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFig. S4.\u003c/strong\u003e Power dissipation in the coupling capacitor.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFig. S5.\u003c/strong\u003e Dependence of the oscillating behavior on the pore size (\u003cem\u003ed\u003c/em\u003e) of the OTS device.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFig. S6.\u003c/strong\u003e Temporal evolution of the in-phase component (Amp\u003csub\u003ein\u003c/sub\u003e) of the output waveforms of oscillators for Mobius ladder geometries with varying the number of connections (simulation).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable S1.\u003c/strong\u003e Comparison between electronic oscillator devices.\u003c/p\u003e","description":"","filename":"240208OTSNOIMYWLeeSIENrev0.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3943189/v1/c0e25269376b5f33c0a5f92f.pdf"}],"financialInterests":"","formattedTitle":"Demonstration of an energy-efficient Ising solver composed of Ovonic Threshold Switch (OTS)-based nano-oscillators (OTSNOs)","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eCombinatorial optimization problems (COPs) are commonly found in everyday life, for example, schedule-planning, travel-planning, resource allocation, etc, and are solved in brute-force forms or rather sophisticated algorithmic forms. As COPs are expanding their application to logistics, routing in IC (integrated circuits) design, drug discovery, etc., there is a growing demand for an energy- and time-efficient solver for COPs since conventional computers with the von Neumann architecture are not good at solving these kinds of problems [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]. A promising alternative computing method is to use the so-called \u0026ldquo;compute-by-physics\u0026rdquo; strategy, where a COP is translated into a complex physical system which, ruled by the physics same as in the problem [\u003cspan additionalcitationids=\"CR3\" citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e], evolves into the ground state leading to a solution to the problem. One example is the Ising machine (IM), a system composed of spins coupled by the exchange interaction, whose ground state naturally gives a solution to the MaxCut problem [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]. In practical implementations of IM, the Ising spins can be replaced with coupled oscillators with phase being quantized at a few values by the same physics governing the Ising spins. Prototypical large-scale IMs were demonstrated by using optical parametric oscillators (OPO) [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e, \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e], showing excellent performance of Ising solvers compared to other competitors. Nevertheless, such OPO-based IM has a drawback in their scalability because they used the long fiber ring cavity requiring a lot of space. To resolve this scalability problem, scalable IMs have been demonstrated by using electronic oscillators such as ring oscillators (ROSCs) [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e], the spin-torque nano-oscillator (STNO), and spin-Hall nano-oscillator (SHNO) based on the magnetic tunnel junction (MTJ) [\u003cspan additionalcitationids=\"CR11\" citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e], and the phase-transition nano-oscillator (PTNO) based on Mott insulators [\u003cspan additionalcitationids=\"CR14 CR15\" citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]. However, due to scalability and/or energy efficiency issues with those oscillator devices, there is a need to find a more scalable and energy-efficient alternative for implementing a practical IM.\u003c/p\u003e \u003cp\u003eIn this work, we propose a novel frequency-tunable nano-oscillator based on the Ovonic threshold switch (OTS) as a solution to this problem. This OTS nano-oscillator (OTSNO) has a structure similar to that of the PTNO with the phase-change switching device replaced by an OTS device (see Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e1\u003c/span\u003ea). Consisting of an amorphous chalcogenide sandwiched between the conducting electrodes, an OTS device shows reversible electrical switching [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e], which is believed to be closely associated with the charging and discharging of the trap states inside the amorphous chalcogenide [\u003cspan additionalcitationids=\"CR19 CR20 CR21 CR22 CR23 CR24 CR25\" citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e]. Such a switching mechanism endows the OTSNO with superior energy efficiency compared to the PTNO which requires Joule heating for the switching. In Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e1\u003c/span\u003ea, note that an OTS device is identified by two characteristic voltage levels, the threshold voltage (\u003cem\u003eV\u003c/em\u003e\u003csub\u003eth\u003c/sub\u003e) and the holding voltage (\u003cem\u003eV\u003c/em\u003e\u003csub\u003eH\u003c/sub\u003e), at which the resistance of the OTS turns low (\u003cem\u003eR\u003c/em\u003e\u003csub\u003eon\u003c/sub\u003e) and high (\u003cem\u003eR\u003c/em\u003e\u003csub\u003eoff\u003c/sub\u003e), respectively, with a very high \u003cem\u003eR\u003c/em\u003e\u003csub\u003eoff\u003c/sub\u003e/\u003cem\u003eR\u003c/em\u003e\u003csub\u003eon\u003c/sub\u003e ratio (~\u0026thinsp;10\u003csup\u003e6\u003c/sup\u003e) [\u003cspan additionalcitationids=\"CR28 CR29\" citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e]. For the 1OTS\u0026thinsp;+\u0026thinsp;1FET structure to be an oscillator, the resistance of the FET (\u003cem\u003eR\u003c/em\u003e\u003csub\u003eFET\u003c/sub\u003e) should have \u003cem\u003eR\u003c/em\u003e\u003csub\u003eon\u003c/sub\u003e\u0026thinsp;\u0026lt;\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eR\u003c/em\u003e\u003csub\u003eFET\u003c/sub\u003e\u0026thinsp;\u0026lt;\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eR\u003c/em\u003e\u003csub\u003eoff\u003c/sub\u003e. Under this condition, the oscillating behavior of the 1OTS\u0026thinsp;+\u0026thinsp;1FET structure is easily understood considering the voltage-dividing relation between the OTS and the FET. When a voltage bias (\u003cem\u003eV\u003c/em\u003e\u003csub\u003ebias\u003c/sub\u003e \u0026gt; \u003cem\u003eV\u003c/em\u003e\u003csub\u003eth\u003c/sub\u003e) is applied to the 1OTS\u0026thinsp;+\u0026thinsp;1FET structure with the OTS in its OFF state, most of \u003cem\u003eV\u003c/em\u003e\u003csub\u003ebias\u003c/sub\u003e is dropped across the OTS because \u003cem\u003eR\u003c/em\u003e\u003csub\u003eFET\u003c/sub\u003e\u0026thinsp;\u0026lt;\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eR\u003c/em\u003e\u003csub\u003eoff\u003c/sub\u003e, turning the OTS into ON state. On the contrary, with the OTS in its ON state, most of \u003cem\u003eV\u003c/em\u003e\u003csub\u003ebias\u003c/sub\u003e is dropped across the FET because \u003cem\u003eR\u003c/em\u003e\u003csub\u003eon\u003c/sub\u003e\u0026thinsp;\u0026lt;\u0026thinsp;\u0026lt;\u0026thinsp;\u003cem\u003eR\u003c/em\u003e\u003csub\u003eFET\u003c/sub\u003e, turning the OTS into OFF state and completing one cycle of oscillation. Since the aforementioned discharging period depends on \u003cem\u003eR\u003c/em\u003e\u003csub\u003eFET\u003c/sub\u003e, the oscillation frequency can be controlled by the gate voltage (\u003cem\u003eV\u003c/em\u003e\u003csub\u003eG\u003c/sub\u003e) of the FET.\u003c/p\u003e \u003cp\u003eFor applying the OTSNO to the IM, the coupling element, mostly a capacitor and/or a resistor, should be carefully selected such that the coupled OTSNOs have an appropriate phase relationship, in-phase (IP) or anti-phase (AP). We have performed a systematic circuit simulation study to examine the phase relationship between the coupled OTSNOs depending on the strength of the coupling element and the tolerance to the variations of the oscillator devices composing the IM. Finally, by using these findings, we have constructed an IM to successfully demonstrate the solution of a MaxCut problem with 14 nodes. Finally, we present a comparison of the OTSNO with other existing nano-oscillators, showing that the OTSNO is a promising candidate for developing large-scale IMs.\u003c/p\u003e"},{"header":"2. Results and Discussion","content":"\u003cp\u003eFigure\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e1\u003c/span\u003eb shows an optical microscope plane-view image (top) and a cross-sectional transmission electron microscope (TEM) image of an OTS device, which has a pore-type structure with a pore size (\u003cem\u003ed\u003c/em\u003e) of 300 nm. The fabrication process of the OTS device is described in detail in the Experimental Section. Figure\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e1\u003c/span\u003ec shows the characteristic current-voltage (\u003cem\u003eI\u003c/em\u003e-\u003cem\u003eV\u003c/em\u003e) curves of the device, which is nominally the same as Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e1\u003c/span\u003eb. In ten repeated measurements, the OTS device shows the variation in \u003cem\u003eV\u003c/em\u003e\u003csub\u003eth\u003c/sub\u003e within 2.5\u0026thinsp;~\u0026thinsp;3.0 V while \u003cem\u003eV\u003c/em\u003e\u003csub\u003eH\u003c/sub\u003e is around 1 V with a negligible variation. In addition, \u003cem\u003eR\u003c/em\u003e\u003csub\u003eoff\u003c/sub\u003e and \u003cem\u003eR\u003c/em\u003e\u003csub\u003eon\u003c/sub\u003e of OTS devices have been read around ~\u0026thinsp;10\u003csup\u003e8\u003c/sup\u003e and ~\u0026thinsp;10\u003csup\u003e2\u003c/sup\u003e Ω, respectively. Figure\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e1\u003c/span\u003ed shows \u003cem\u003eV\u003c/em\u003e\u003csub\u003eout\u003c/sub\u003e waveforms of a typical OTSNO with varying \u003cem\u003eV\u003c/em\u003e\u003csub\u003eG\u003c/sub\u003e, where the OTSNO consists of an OTS device and a FET (n-MOSFET, LND150N3-G, Microchip Inc.). It is clearly shown that the natural frequency (\u003cem\u003ef\u003c/em\u003e\u003csub\u003enat\u003c/sub\u003e) of the OTSNO increases with \u003cem\u003eV\u003c/em\u003e\u003csub\u003eG\u003c/sub\u003e as expected. To quantify \u003cem\u003ef\u003c/em\u003e\u003csub\u003enat\u003c/sub\u003e at each \u003cem\u003eV\u003c/em\u003e\u003csub\u003eG\u003c/sub\u003e, the fast Fourier transform (FFT) is performed as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e1\u003c/span\u003ee and \u003cem\u003ef\u003c/em\u003e\u003csub\u003enat\u003c/sub\u003e, defined as the frequency at the primary peak, is plotted as a function of \u003cem\u003eV\u003c/em\u003e\u003csub\u003eG\u003c/sub\u003e in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e1\u003c/span\u003ef. It is observed that \u003cem\u003ef\u003c/em\u003e\u003csub\u003enat\u003c/sub\u003e is well defined and linearly proportional to \u003cem\u003eV\u003c/em\u003e\u003csub\u003eG\u003c/sub\u003e being adjustable in the range of 0.5\u0026thinsp;~\u0026thinsp;2 MHz. These results clearly show that the oscillation in the OTSNO is composed of fundamental-frequency components and their harmonics while keeping other components negligible.\u003c/p\u003e \u003cp\u003eNext, we investigate the synchronization behavior of coupled OTSNOs (see Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e2\u003c/span\u003ea). Figure\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e2\u003c/span\u003eb shows the output waveforms of two OTSNOs coupled by a capacitor. It is observed that the output waveforms of two oscillators look nearly the same with a certain degree of phase difference, demonstrating that two oscillators are synchronized to each other (for a detailed study of the synchronization behavior, see section S1 in Supplementary Information). In Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e2\u003c/span\u003ec, \u003cem\u003eV\u003c/em\u003e\u003csub\u003eout2\u003c/sub\u003e is plotted as a function of \u003cem\u003eV\u003c/em\u003e\u003csub\u003eout1\u003c/sub\u003e, which presents the so-called \u0026ldquo;phase portrait\u0026rdquo; of a coupled-oscillator system. It shows a butterfly-shaped attractor curve, implying the AP relationship between the two oscillators. The attractor curve appears as a band implying the chaotic nature of the OTSNO [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e, \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e, \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e, \u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e], which is desirable for the application to the IM because it helps the system to find the solution of the network configuration with the global minimum energy [\u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eTo examine the phase stability of two coupled oscillators, we have performed a simulation study where the circuit parameters of the system are systematically varied. Especially, we have investigated two types of coupling - capacitive and resistive coupling - with varying their values, \u003cem\u003eC\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e and \u003cem\u003eR\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e, respectively. In the SPICE (Simulation Program with Integrated Circuit Emphasis) simulation, the OTS device is modeled as a voltage-controlled switch characterized by four parameters (\u003cem\u003eV\u003c/em\u003e\u003csub\u003eth\u003c/sub\u003e, \u003cem\u003eV\u003c/em\u003e\u003csub\u003eH\u003c/sub\u003e, \u003cem\u003eR\u003c/em\u003e\u003csub\u003eon\u003c/sub\u003e, \u003cem\u003eR\u003c/em\u003e\u003csub\u003eoff\u003c/sub\u003e) in parallel connection with a parasitic capacitor. Figure\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e2\u003c/span\u003ed and \u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e2\u003c/span\u003ee show the phase difference (\u003cem\u003eΔφ\u003c/em\u003e) between oscillators #1 and #2 as a function of (\u003cem\u003eC\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e, \u003cem\u003eR\u003c/em\u003e\u003csub\u003eL\u003c/sub\u003e) and (\u003cem\u003eR\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e, \u003cem\u003eR\u003c/em\u003e\u003csub\u003eL\u003c/sub\u003e), respectively. Here, the parameters of both OTS devices are set to (\u003cem\u003eV\u003c/em\u003e\u003csub\u003eth\u003c/sub\u003e, \u003cem\u003eV\u003c/em\u003e\u003csub\u003eH\u003c/sub\u003e, \u003cem\u003eR\u003c/em\u003e\u003csub\u003eon\u003c/sub\u003e, \u003cem\u003eR\u003c/em\u003e\u003csub\u003eoff\u003c/sub\u003e)=(3.3 V, 0.7 V, 150 Ω, 10 MΩ). \u003cem\u003eΔφ\u003c/em\u003e is calculated by 2*π*(\u003cem\u003eT\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e-\u003cem\u003eT\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e)*\u003cem\u003ef\u003c/em\u003e\u003csub\u003esync\u003c/sub\u003e after ~\u0026thinsp;200 cycles, where \u003cem\u003eT\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e-\u003cem\u003eT\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e and \u003cem\u003ef\u003c/em\u003e\u003csub\u003esync\u003c/sub\u003e are the time difference between adjacent peaks of oscillator #1 and #2 and the oscillation frequency in the synchronized state, respectively. It is found that the AP relationship between two oscillators is more stable in the capacitive coupling with \u003cem\u003eC\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e for AP relationship spanning almost two orders (2\u0026thinsp;~\u0026thinsp;200 pF) independent of \u003cem\u003eR\u003c/em\u003e\u003csub\u003eL\u003c/sub\u003e while the range of \u003cem\u003eR\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e for AP relationship is relatively narrow with a strong dependence on \u003cem\u003eR\u003c/em\u003e\u003csub\u003eL\u003c/sub\u003e.\u003c/p\u003e \u003cp\u003eFor the practical application, an important aspect is the tolerance of the coupled oscillator system to the inevitable variation of the OTS device. Figure\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e2\u003c/span\u003ef and \u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e2\u003c/span\u003eg show \u003cem\u003eΔφ\u003c/em\u003e as a function of \u003cem\u003eΔV\u003c/em\u003e\u003csub\u003eth\u003c/sub\u003e (=\u0026thinsp;\u003cem\u003eV\u003c/em\u003e\u003csub\u003eth2\u003c/sub\u003e-\u003cem\u003eV\u003c/em\u003e\u003csub\u003eth1\u003c/sub\u003e) and \u003cem\u003eΔV\u003c/em\u003e\u003csub\u003eH\u003c/sub\u003e (=\u0026thinsp;\u003cem\u003eV\u003c/em\u003e\u003csub\u003eH2\u003c/sub\u003e-\u003cem\u003eV\u003c/em\u003e\u003csub\u003eH1\u003c/sub\u003e), where we have varied \u003cem\u003eV\u003c/em\u003e\u003csub\u003eth\u003c/sub\u003e and \u003cem\u003eV\u003c/em\u003e\u003csub\u003eH\u003c/sub\u003e of OTS #2 systematically while keeping those of OTS #1 fixed. It is shown that, in the case of the capacitive coupling, the AP relationship is robust against the \u003cem\u003eV\u003c/em\u003e\u003csub\u003eth\u003c/sub\u003e-variation of up to \u0026plusmn; 20%, which is significantly higher compared to resistive coupling where the limit is only \u0026plusmn; 5%. The superior stability of the AP relationship in the \u003cem\u003eC\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e-coupled oscillator system can be explained qualitatively by the ability of the coupling capacitor to store energy. In detail, if a voltage difference is generated between two oscillators, then it can be stored in the coupling capacitor. The stored energy is returned to the oscillators making them repel each other, stabilizing the AP relationship. In contrast, since the resistor can\u0026rsquo;t store energy in the \u003cem\u003eR\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e-coupled oscillator system, the voltage difference between oscillators generates heat dissipation in \u003cem\u003eR\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e. This makes the oscillators lose their energy destabilizing the AP relationship in the \u003cem\u003eR\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e-coupled oscillator system.\u003c/p\u003e \u003cp\u003eAnother intriguing finding is that the phase relationship of the coupled OTSNOs in both cases is hardly sensitive to the variation of \u003cem\u003eV\u003c/em\u003e\u003csub\u003eH\u003c/sub\u003e compared to the variation of \u003cem\u003eV\u003c/em\u003e\u003csub\u003eth\u003c/sub\u003e. It is related to the asymmetry of the output waveform (see Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e1\u003c/span\u003ed), which shows a much longer falling period compared to the rising period. Recalling the mechanism of the oscillating behavior of the OTSNO, it is easily found that the peak of the oscillation aligns with the ON-to-OFF transition of the OTS device whereas the valley point corresponds to the OFF-to-ON transition. Consequently, when the OTS reaches its peak, it transitions into a highly resistive state with \u003cem\u003eR\u003c/em\u003e\u003csub\u003eoff\u003c/sub\u003e ~ 10 MΩ. As mentioned above, since the OTS can be described by a parallel connection of a voltage-controlled switch and a parasitic capacitor, such a high \u003cem\u003eR\u003c/em\u003e\u003csub\u003eoff\u003c/sub\u003e results in a large RC delay, and consequently, the elongation of the falling period compared to the rising period. During the falling period, the parasitic capacitor is charged and the voltage across it runs into \u003cem\u003eV\u003c/em\u003e\u003csub\u003eth\u003c/sub\u003e starting from \u003cem\u003eV\u003c/em\u003e\u003csub\u003eH\u003c/sub\u003e. Since the charging time is mainly determined by \u003cem\u003eV\u003c/em\u003e\u003csub\u003eth\u003c/sub\u003e, the variation in \u003cem\u003eV\u003c/em\u003e\u003csub\u003eH\u003c/sub\u003e has little effect on the phase relationship of the coupled OTSNOs. The output waveforms and the phase portraits at points in various regions in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e2\u003c/span\u003ed and \u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e2\u003c/span\u003ee are presented in the Supplementary Information (Fig. S2).\u003c/p\u003e \u003cp\u003eBased on these results, we have built an IM using OTSNOs and coupling capacitors (\u003cem\u003eC\u003c/em\u003e\u003csub\u003ec\u003c/sub\u003e=100 pF) as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e3\u003c/span\u003ea and, as a benchmark test, have tried to solve a MaxCut problem with a Mobius ladder geometry composed of 14 nodes and cubic connections (see Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e3\u003c/span\u003eb). We have also tried another geometry, which is presented in Fig. S3 in the Supplementary Information. Tlhe details of the measurement is described in the Experimental Section. A representative output waveforms of the 14 OTSNOs are presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e3\u003c/span\u003ec. We have employed both the second-harmonic injection locking (SHIL) and simulated annealing (SA) techniques [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e, \u003cspan additionalcitationids=\"CR34\" citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e] to improve the reliability of the solution by keeping the system from being stuck to a local minimum. In this technique, all the oscillators are driven by a DC voltage (\u003cem\u003eV\u003c/em\u003e\u003csub\u003edc\u003c/sub\u003e) and a modulational AC voltage (\u003cem\u003eV\u003c/em\u003e\u003csub\u003eSH\u003c/sub\u003esin(\u003cem\u003eω\u003c/em\u003e\u003csub\u003eSH\u003c/sub\u003e\u003cem\u003et\u003c/em\u003e)), which locks the oscillation frequency to \u003cem\u003eω\u003c/em\u003e\u003csub\u003eSH\u003c/sub\u003e (=\u0026thinsp;2\u003cem\u003eω\u003c/em\u003e\u003csub\u003esync\u003c/sub\u003e\u0026thinsp;~\u0026thinsp;2.75 MHz, where \u003cem\u003eω\u003c/em\u003e\u003csub\u003esync\u003c/sub\u003e is the oscillation frequency in the synchronized state). In the right panel, which is the expansion of the waveforms in the range of 40\u0026thinsp;~\u0026thinsp;43 \u0026micro;s, it is shown that the odd-numbered OTSNOs oscillate with the nearly same phase while the even-numbered ones also show the synchronized behavior with a clearly distinguished phase.\u003c/p\u003e \u003cp\u003eTo quantify the phase difference between oscillators, we have calculated the amplitude of the in-phase component of each \u003cem\u003eV\u003c/em\u003e\u003csub\u003eout\u003c/sub\u003e(\u003cem\u003et\u003c/em\u003e) by applying the low-pass filter (the cut-off frequency of 0.1 MHz) to the product of a reference wave (cos(\u003cem\u003eω\u003c/em\u003e\u003csub\u003eSH\u003c/sub\u003e\u003cem\u003et\u003c/em\u003e)) and \u003cem\u003eV\u003c/em\u003e\u003csub\u003eout\u003c/sub\u003e(\u003cem\u003et\u003c/em\u003e). Such obtained in-phase amplitude (Amp\u003csub\u003ein\u003c/sub\u003e) of the oscillators are plotted as a function of time in Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e3\u003c/span\u003ed. It is clearly observed that the oscillators are separated into two groups with the opposite signs of Amp\u003csub\u003ein\u003c/sub\u003e implying the AP relationship between those two groups of oscillators. In addition, it shows that those two groups are clearly separated 35 \u0026micro;s after applying the bias. Considering that \u003cem\u003eV\u003c/em\u003e\u003csub\u003eSH\u003c/sub\u003e gradually increases over 35 \u0026micro;s in the simulated annealing scheme, it indicates that the time-to-solution (\u003cem\u003eT\u003c/em\u003e\u003csub\u003esol\u003c/sub\u003e) is expected to be at most 35 \u0026micro;s. From repeated implementations of the same experiment, we have observed \u003cem\u003eT\u003c/em\u003e\u003csub\u003esol\u003c/sub\u003e is in the range of 35\u0026thinsp;~\u0026thinsp;40 \u0026micro;s. It is much shorter than \u003cem\u003eT\u003c/em\u003e\u003csub\u003esol\u003c/sub\u003e ~ 4 ms reported in a previous work on the hardware implementation of a PTNO-based Ising solver [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e] for solving an 8-node MaxCut problem although, in a simulation study, \u003cem\u003eT\u003c/em\u003e\u003csub\u003esol\u003c/sub\u003e=30 \u0026micro;s was presented for solving a 100-node MaxCut problem with random cubic connections.\u003c/p\u003e \u003cp\u003eThe energy-to-solution (\u003cem\u003eE\u003c/em\u003e\u003csub\u003esol\u003c/sub\u003e), as another performance metric of the IM, is calculated by integrating the instantaneous power consumption (\u003cem\u003eP\u003c/em\u003e(\u003cem\u003et\u003c/em\u003e)\u0026thinsp;=\u0026thinsp;\u003cem\u003eV\u003c/em\u003e(\u003cem\u003et\u003c/em\u003e)\u0026times;\u003cem\u003eI\u003c/em\u003e(\u003cem\u003et\u003c/em\u003e), where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(V(t)={V_{dc}}+{V_{SH}}\\sin (\\omega t)\\)\u003c/span\u003e\u003c/span\u003e applied to the system (refer to Fig. S3a) and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(I(t)=\\sum\\limits_{{i=1}}^{{14}} {{I_{i,out}}(t)}\\)\u003c/span\u003e\u003c/span\u003e, the sum of the output currents (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({I_{i,out}}(t)\\)\u003c/span\u003e\u003c/span\u003e) at each node) over \u003cem\u003eT\u003c/em\u003e\u003csub\u003esol\u003c/sub\u003e. \u003cem\u003eP\u003c/em\u003e(\u003cem\u003et\u003c/em\u003e) is plotted as a function of time in Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e4\u003c/span\u003ea, leading to an estimation of \u003cem\u003eE\u003c/em\u003e\u003csub\u003esol\u003c/sub\u003e around 2.3 \u0026micro;J. It can not be compared with that of the PTNO-based IM as it was not known in the previous work. Instead, we have tried to compare the energy consumption in a single OTSNO and PTNO because of the negligible energy consumption in the coupling capacitor (see Fig. S4 in the Supplementary Information) as expected in the pure capacitive circuit. The energy consumption per cycle (\u003cem\u003eE\u003c/em\u003e\u003csub\u003ecycle\u003c/sub\u003e) is estimated to be ~\u0026thinsp;1.25 nJ/cycle (see Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e5\u003c/span\u003eb) for a 500 nm (\u003cem\u003ed\u003c/em\u003e: diameter of the pore)-sized OTSNO and ~\u0026thinsp;0.9 nJ/cycle for a 200 nm (the length of VO\u003csub\u003e2\u003c/sub\u003e channel)-sized PTNO, respectively. In the OTS device, \u003cem\u003eE\u003c/em\u003e\u003csub\u003ecycle\u003c/sub\u003e is expected to be scaled with the dimension of the device because of the shrinkage of the switching volume. Therefore, we have investigated the dependence of \u003cem\u003eE\u003c/em\u003e\u003csub\u003ecycle\u003c/sub\u003e of the OTSNO on the diameter of the pore as presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e4\u003c/span\u003ec (for the waveforms of \u003cem\u003eV\u003c/em\u003e(\u003cem\u003et\u003c/em\u003e), \u003cem\u003eI\u003c/em\u003e(\u003cem\u003et\u003c/em\u003e), and \u003cem\u003eP\u003c/em\u003e(\u003cem\u003et\u003c/em\u003e) corresponding to OTSNOs with various pore sizes, see Fig. S5 in Supplementary Information). For comparison, \u003cem\u003eE\u003c/em\u003e\u003csub\u003ecycle\u003c/sub\u003e of the PTNO [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e, \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e] is also located in the same graph, clearly showing that the OTSNO consumes less energy by about an order than the PTNO with similar device dimensions. The difference is attributed to the difference in the switching mechanisms of the PTNO and the OTSNO. In detail, the PTNO is based on the phase transition of the Mott insulator, which requires heating of a part in the channel material up to the transition temperature. In contrast, the OTSNO is believed to be based on the filling and evacuation of the trap states in the OTS, the electronic process in nature, although there is still controversy about its switching mechanism [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e, \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e, \u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e, \u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e]. In addition, Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e4\u003c/span\u003ec shows that \u003cem\u003eE\u003c/em\u003e\u003csub\u003ecycle\u003c/sub\u003e of the OTSNO scales as \u003cem\u003eE\u003c/em\u003e\u003csub\u003ecycle\u003c/sub\u003e ~ \u003cem\u003ed\u003c/em\u003e\u003csup\u003e1.81\u003c/sup\u003e with the diameter of the pore, being close to the expected relation of \u003cem\u003eE\u003c/em\u003e\u003csub\u003ecycle\u003c/sub\u003e ~ \u003cem\u003ed\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e for the case of the uniform current density in the switching material. Therefore, it seems to imply that the reduction in \u003cem\u003eE\u003c/em\u003e\u003csub\u003ecycle\u003c/sub\u003e is attributed to the shrinkage of the switching volume inside the cylinder. As a result, \u003cem\u003eE\u003c/em\u003e\u003csub\u003ecycle\u003c/sub\u003e of the OTSNO extrapolates to ~\u0026thinsp;1 pJ/cycle at \u003cem\u003ed\u003c/em\u003e\u0026thinsp;=\u0026thinsp;10 nm, which enables the development of a highly efficient large-scale IM.\u003c/p\u003e \u003cp\u003eFinally, we have performed a simulation study to investigate the scalability of the OTSNO-based IM. As a typical benchmark task, we have studied the Mobius ladder geometry composed of variable numbers of OTSNOs and connections (see Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e5\u003c/span\u003ea). Figures\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e5\u003c/span\u003eb\u0026thinsp;~\u0026thinsp;5d show the representative examples of Amp\u003csub\u003ein\u003c/sub\u003e of oscillators with varying the number of nodes (\u003cem\u003eN\u003c/em\u003e) from 22 to 102. The correctness of the obtained solution has been verified by comparing the energy of the solution with that of the ground state. From 20 repetitions of the simulation with varying the initial phases of each oscillator randomly, we have obtained the performance metrics, the success probability (\u003cem\u003eP\u003c/em\u003e\u003csub\u003esuccess\u003c/sub\u003e) and \u003cem\u003eT\u003c/em\u003e\u003csub\u003esol\u003c/sub\u003e, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e5\u003c/span\u003ee. Note that, as \u003cem\u003eN\u003c/em\u003e increases, \u003cem\u003eT\u003c/em\u003e\u003csub\u003esol\u003c/sub\u003e is linearly proportional to log(\u003cem\u003eN\u003c/em\u003e) with a slight decrease in \u003cem\u003eP\u003c/em\u003e\u003csub\u003esuccess\u003c/sub\u003e. We have repeated similar simulations with varying the number of connections (\u003cem\u003eN\u003c/em\u003e\u003csub\u003ec\u003c/sub\u003e) in the way of adding connections to the next nearest neighbors in order with a diagonal connection being fixed. Representative examples of Amp\u003csub\u003ein\u003c/sub\u003e of oscillators for various geometries are presented in the Supplementary Information (see Fig. S6). In Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e5\u003c/span\u003ef, it is observed that \u003cem\u003eT\u003c/em\u003e\u003csub\u003esol\u003c/sub\u003e shows a sublinear dependence on \u003cem\u003eN\u003c/em\u003e\u003csub\u003ec\u003c/sub\u003e while \u003cem\u003eP\u003c/em\u003e\u003csub\u003esuccess\u003c/sub\u003e decreases slightly with increasing \u003cem\u003eN\u003c/em\u003e\u003csub\u003ec\u003c/sub\u003e. It implies that, as \u003cem\u003eN\u003c/em\u003e\u003csub\u003ec\u003c/sub\u003e increases, local minima are likely to be formed in the energy landscape and the oscillators are trapped in those local minima. Due to the intrinsic chaotic nature of the OTSNO device as mentioned in the discussion of Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e2\u003c/span\u003ec, we expect that the problem of such local minima might be alleviated in the hardware implementation of the IM using OTSNOs.\u003c/p\u003e"},{"header":"3. Conclusion","content":"\u003cp\u003eIn summary, we have investigated the OTSNO as a candidate nano-oscillator with improved scalability for applications in Ising machines. We have demonstrated that the OTSNO shows robust oscillating behavior with a well-defined frequency, which can be controlled by the gate voltage applied to the FET. We have also demonstrated the synchronization between two coupled OTSNOs with their phase relation (in-phase or anti-phase) controlled by the strength of the coupling elements (electrical resistors or capacitors), which is systematically investigated leading to a conclusion that capacitive coupling provides the larger operation windows with respect to both the value of the coupling capacitance and the device variations of the OTS. Finally, we have implemented an Ising machine composed of capacitively-coupled OTSNOs and demonstrated that the solution to a 14-node MaxCut problem can be obtained in 35 \u0026micro;s while consuming no more than 2.3 \u0026micro;J of energy. Moreover, it is shown that the OTSNO has superior energy scalability with consuming less energy by about an order compared with the PTNO based on the Mott insulator such as VO\u003csub\u003e2\u003c/sub\u003e. We have compared the characteristics of various nano-oscillators in the Supplementary Information (see \u0026ldquo;2. Supplementary Note: Comparison with other oscillators\u0026rdquo;). Based on this comparison and considering that the OTSNO provides an oscillator with a minimum size of 6F\u003csup\u003e2\u003c/sup\u003e with F down to ~\u0026thinsp;14 nm [\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e, \u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e38\u003c/span\u003e], we believe that the OTSNO is highly promising for application to large-scale Ising machines.\u003c/p\u003e"},{"header":"4. Experimental Section","content":"\u003cp\u003e \u003cstrong\u003eFabrication of the OTS\u003c/strong\u003e \u003cp\u003eOTS devices are fabricated with a pore-type structure, where the pore size (\u003cem\u003ed\u003c/em\u003e) is defined by the electron beam lithography in the range of 100\u0026thinsp;~\u0026thinsp;500 nm. 60 nm-thick Sb\u003csub\u003ex\u003c/sub\u003e(GeSe)\u003csub\u003e1\u0026minus;x\u003c/sub\u003e (SGS) layer is used as a switching material because it is found to have a lower threshold voltage (\u003cem\u003eV\u003c/em\u003e\u003csub\u003eth\u003c/sub\u003e) compared to Ge\u003csub\u003e50\u003c/sub\u003eSe\u003csub\u003e50\u003c/sub\u003e. It is deposited by a co-sputtering technique (magnetron RF sputtering) using Ge, Sb, and GeSe\u003csub\u003e2\u003c/sub\u003e targets. Pt and TiN are used as the bottom and top electrodes, respectively.\u003c/p\u003e \u003c/p\u003e \u003cp\u003e \u003cstrong\u003eCharacterization of the OTSNO\u003c/strong\u003e \u003cp\u003eTo configure a frequency-adjustable OTSNO, an OTS device is connected in series to an n-MOSFET device (LND150N3-G, Microchip Inc.). The characteristics of OTS devices and OTSNOs have been investigated using an arbitrary function generator (AFG-3102, Tektronix), an oscilloscope (MSO-58, Tektronix), and a source-measure unit (2635B, Keithley). We have tested several OTSNO devices using OTS devices of various pore sizes and found similar behavior. The results presented in this paper are mainly obtained by using OTS devices with \u003cem\u003ed\u003c/em\u003e\u0026thinsp;=\u0026thinsp;500 nm.\u003c/p\u003e \u003c/p\u003e \u003cp\u003e \u003cstrong\u003eImplementation of the Ising solver\u003c/strong\u003e \u003cp\u003eThe implementation of the Ising machine composed of OTSNOs including the measurement setup is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e3\u003c/span\u003ea and Fig. S3. The OTS devices fabricated on the SiO\u003csub\u003e2\u003c/sub\u003e substrate are loaded onto a custom-made breakout board. Terminals of the OTS devices are connected to the power supply (\u003cem\u003eV\u003c/em\u003e\u003csub\u003edd\u003c/sub\u003e), \u003cem\u003eR\u003c/em\u003e\u003csub\u003eL\u003c/sub\u003e, and \u003cem\u003eC\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e on a specially-designed printed circuit board (PCB). We have used the second-harmonic injection locking (SHIL) technique to lock phases of OTSNO with an oscillation frequency of 2.75 MHz for the injection signal, twice the natural frequency of the single OTSNO. In addition, we have used the simulated annealing technique for the injection signal, whose amplitude gradually increases from 0 V to 2 V over 45 \u0026micro;s. \u003cem\u003eV\u003c/em\u003e\u003csub\u003edc\u003c/sub\u003e, \u003cem\u003eR\u003c/em\u003e\u003csub\u003eL\u003c/sub\u003e, and \u003cem\u003eC\u003c/em\u003e\u003csub\u003eC\u003c/sub\u003e are set to 4 V, 16 kΩ, and 100 pF, respectively. An arbitrary function generator (AFG-3102, Tektronix) has been used to bias the driving dc voltage and the SHIL ac voltage. To read the output waveforms of 14 OTSNOs by using an 8-channel oscilloscope (MSO-58, Tektronix), we have repeated the measurement runs two times. In the first and second runs, 14 OTSNOs have been read half-and-half relative to oscillator #1.\u003c/p\u003e \u003c/p\u003e \u003cp\u003e \u003cstrong\u003eSPICE simulation\u003c/strong\u003e \u003cp\u003eTo perform SPICE simulations, we have used the LTSpice (Analog Device Inc.) software. The OTS device is modeled as a parallel connection between a voltage-controlled switch (VCS) and a parasitic capacitor (100 pF), the former of which is modeled to have hysteretic switching behavior mimicking the OTS with \u003cem\u003eV\u003c/em\u003e\u003csub\u003eth\u003c/sub\u003e = 3.3 V, \u003cem\u003eV\u003c/em\u003e\u003csub\u003eH\u003c/sub\u003e = 1 V, \u003cem\u003eR\u003c/em\u003e\u003csub\u003eon\u003c/sub\u003e = 150 Ω, and \u003cem\u003eR\u003c/em\u003e\u003csub\u003eoff\u003c/sub\u003e = 10\u003csup\u003e6\u003c/sup\u003e Ω.\u003c/p\u003e \u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAcknowledgments\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNot applicable.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eS.L. designed and conceived the experiments with help from Y.W.L. and S.J.K., and S.K. Y.W.L., S.J.K., and S.K. fabricated OTS devices. Y.W.L., S.J.K., and S.K. performed the characterization of devices. S.L. and B.H.P. participated in the analysis of the synchronization. J.K., J.P., Y.J., G.W.H., and S.P. performed a circuit analysis of the behavior of the oscillator device and a simulation of a coupled oscillator network. All authors discussed the data and participated in the writing of the manuscript.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis work was supported by the Korea Institute of Science and Technology (KIST) through 2E32960 and by National Research Foundation program through NRF- 2021M3F3A2A03017782, 2021M3F3A2A01037738, 2021M3F3A2A01037814, 2022R1A2C3004135, and NRF-2021R1A2C3011450. B.H.P. was supported by Korea Basic Science Institute (National Research Facilities and Equipment Center) grant funded by the Ministry of Education. (Grant No. 2022R1A6C101A754).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAvailability of data and materials\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNot applicable.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting interests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare no competing financial interest.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor details\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003csup\u003e1\u003c/sup\u003eCenter for Neuromorphic engineering, Korea Institute of Science and Technology, Seoul 02792, Rep. of Korea. \u003csup\u003e2\u003c/sup\u003eDepartment of Physics, Konkuk University, Seoul 05029, Rep. of Korea. \u003csup\u003e3\u003c/sup\u003eCore Facility Center for Quantum Characterization/Analysis of Two-Dimensional Materials \u0026amp; Heterostructures, Konkuk University, Seoul 05029, Rep. of Korea. \u003csup\u003e4\u003c/sup\u003eDivision of Nano \u0026amp; Information Technology, Korea University of Science and Technology, Daejeon 34316, Rep. of Korea.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eSupplementary Information\u003c/strong\u003e\u003cem\u003e\u0026nbsp;\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eThe online version contains supplementary material available at ...\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eJ. 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Hady et al., Proceedings of the IEEE 105, 1822 (2017)\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":true,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"nano-convergence","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"ncon","sideBox":"Learn more about [Nano Convergence](https://www.springer.com/journal/40580)","snPcode":"40580","submissionUrl":"https://www.editorialmanager.com/ncon/default2.aspx","title":"Nano Convergence","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Open","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Ising solver, Ovonic Threshold Switch (OTS), oscillator-based computing","lastPublishedDoi":"10.21203/rs.3.rs-3943189/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3943189/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"As there is an increasing need for an efficient solver of combinatorial optimization problems, much interest is paid to the Ising machine, which is a novel physics-driven computing system composed of coupled oscillators mimicking the dynamics of the system of coupled electronic spins. In this work, we propose an energy-efficient nano-oscillator, called OTSNO, which is composed of an Ovonic Threshold Switch (OTS) and an electrical resistor. We demonstrate that the OTSNO shows the synchronization behavior, an essential property for the realization of an Ising machine. Furthermore, we have discovered that the capacitive coupling is advantageous over the resistive coupling for the hardware implementation of an Ising solver by providing a larger margin of the variations of components. Finally, we implement an Ising machine composed of capacitively-coupled OTSNOs to demonstrate that the solution to a 14-node MaxCut problem can be obtained in 40 ms while consuming no more than 2.3 mJ of energy. Compared to a previous hardware implementation of the phase-transition nano-oscillator (PTNO)-based Ising machine, the OTSNO-based Ising machine in this work shows the performance of the increased speed by more than one order while consuming less energy by about an order.","manuscriptTitle":"Demonstration of an energy-efficient Ising solver composed of Ovonic Threshold Switch (OTS)-based nano-oscillators (OTSNOs)","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-03-12 15:37:53","doi":"10.21203/rs.3.rs-3943189/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"reviewerAgreed","content":"","date":"2024-03-14T14:29:34+00:00","index":0,"fulltext":""},{"type":"reviewersInvited","content":"","date":"2024-03-08T04:24:27+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2024-02-08T17:18:11+00:00","index":"","fulltext":""},{"type":"submitted","content":"Nano Convergence","date":"2024-02-08T05:15:05+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"nano-convergence","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"ncon","sideBox":"Learn more about [Nano Convergence](https://www.springer.com/journal/40580)","snPcode":"40580","submissionUrl":"https://www.editorialmanager.com/ncon/default2.aspx","title":"Nano Convergence","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Open","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"53501f1b-069a-4e08-b360-0ac1be5c9f12","owner":[],"postedDate":"March 12th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2024-05-24T00:40:02+00:00","versionOfRecord":{"articleIdentity":"rs-3943189","link":"https://doi.org/10.1186/s40580-024-00429-2","journal":{"identity":"nano-convergence","isVorOnly":false,"title":"Nano Convergence"},"publishedOn":"2024-05-23 00:40:02","publishedOnDateReadable":"May 23rd, 2024"},"versionCreatedAt":"2024-03-12 15:37:53","video":"","vorDoi":"10.1186/s40580-024-00429-2","vorDoiUrl":"https://doi.org/10.1186/s40580-024-00429-2","workflowStages":[]},"version":"v1","identity":"rs-3943189","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3943189","identity":"rs-3943189","version":["v1"]},"buildId":"cBFmMYwuxLRRLfASyISRj","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}