Abstract
Neuromorphic hardware implementation of Boltzmann Machine using a network of stochastic neurons can allow nondeterministic polynomialtime (NP) hard combinatorial optimization problems to be efficiently solved. Efficient implementation of such Boltzmann Machine with simulated annealing desires the statistical parameters of the stochastic neurons to be dynamically tunable, however, there has been limited research on stochastic semiconductor devices with controllable statistical distributions. Here, we demonstrate a reconfigurable tin oxide (SnO_{x})/molybdenum disulfide (MoS_{2}) heterogeneous memristive device that can realize tunable stochastic dynamics in its output sampling characteristics. The device can sample exponentialclass sigmoidal distributions analogous to the FermiDirac distribution of physical systems with quantitatively defined tunable “temperature” effect. A BM composed of these tunable stochastic neuron devices, which can enable simulated annealing with designed “cooling” strategies, is conducted to solve the MAXSAT, a representative in NPhard combinatorial optimization problems. Quantitative insights into the effect of different “cooling” strategies on improving the BM optimization process efficiency are also provided.
Introduction
Stochastic neuron devices are essential for the neural network implementation of key emerging nonvonNeumann computing concepts such as the Boltzmann machines, which are recurrent artificial neural networks with stochastic features analogous to the thermodynamics of realworld physical systems. BM can be used to solve a broad range of combinatorial optimization problems^{1,2} with applications in classification^{3}, pattern recognition^{4}, feature learning, and other emerging computing systems. Deriving its name from the Boltzmann distribution of statistical mechanics, BM possesses an artificial notion of “temperature”, and the controlled evolution of this “temperature” parameter during the optimization process^{5,6}, i.e., the “cooling” strategy, can impact the convergence efficiency of the BM and its chance of reaching a better costenergy minimization (or maximization depending on problem definition). To realize the hardware implementation of the BM that can also allow the “temperature” control and hence the precise execution of desired “cooling” strategy, it is essential to have electronic devices that can generate exponentialclass stochastic sampling with dynamically tunable distribution parameters.
The property of memristor in its deterministic form has been commonly used in applications such as multiplyandaccumulate matrix calculation^{7} and resistorlogic demultiplexers^{8,9,10}. Its stochastic property is often intentionally suppressed^{11,12,13} in such applications for the purpose of achieving accurate and reproducible computational results^{14,15}. On the other hand, rich stochastic property of memristors, which relies on ensembles of random movements of atoms and ions, offers opportunities in energyefficient computing applications^{16,17,18,19,20}. With the stochastic property, one can generate random number^{21} to encrypt information, implement physical unclonable functions^{22}, and realize artificial neurons^{23} with integrateandfire activations. Furthermore, emerging computing schemes can use stochastic memristive device as a building block to emulate biological neural network^{24,25}, whose functions—such as decisionmaking—can leverage the stochastic dynamics of neurons and synapses. However, a common challenge with previous stochastic memristors is the lack of means to precisely control and modulate the probability distribution that is associated with its randomness. Realizing such devices has been difficult because many devicegenerated random features in stochastic memristors or oscillators lack stable probability distribution, which limits the chance of controlling it experimentally^{19,26,27}. Additionally, with only two terminals in a common memristor, where the probability distribution can only be influenced through the twoterminal bias, the probability distribution of the device output cannot be tuned flexibly and precisely.
In this work, we overcome such challenge with a threeterminal stochastic heteromemristor based on tin oxide/MoS_{2} heterostructure, which demonstrates tunable statistical distributions enabled by the gate modulation. The inherent exponentialclass stochastic characteristics of the device arising from the intrinsic randomness and energy distribution in its ionic motions are explored to realize sampling of exponentialclass sigmoidal distributions that resembles the Fermi–Dirac distribution in physical systems. The device incorporates gate modulation that allows the efficient control of the stochastic features in the device output characteristics. The device enables the realization of reconfigurable stochastic neuron and the implementation of Boltzmann machine in which the reconfigurable statistic of the device allows different “cooling” strategies to be implemented during the optimization process. The effect of different “cooling” strategies on improving the optimization process efficiency of the BM is demonstrated experimentally.
Results
Figure 1a shows the schematic of this reconfigurable heteromemristor, where tin oxide serves as filamentswitching layer and is sandwiched between a MoS_{2} layer and Cr/Au top electrodes (TE). The Si substrate serves as a modulating gate bias (V_{g}) that can influence the filamentformation dynamics in the tin oxide layer. The highresolution scanning transmission electron microscopy (HRSTEM) image in Fig. 1b shows the cross section of the fabricated device and reveals that the tin oxide layer is amorphous. An energydispersive Xray spectroscopy (EDX) scan in Fig. 1c indicates the elemental composition. Figure 1d plots the Raman spectra for the SnSe sample before and after oxidation, which leads to the formation of the SnO_{x} layer. All signature modes of SnSe, including the shear mode A_{g}^{1}, the intheplane modes A_{g}^{2} and B_{3g}, and the outofplane mode A_{g}^{3} that are observed before oxidation, and are not detected after oxidation, indicating the full oxidation and amorphization of the SnSe sample^{28}. The tin oxide film can also be synthesized using atomiclayer deposition (ALD)^{29,30,31}, which produces films of similar quality as the direct oxidation method.
Unipolar electrical switching characteristics of the device at V_{g} = 0 V are shown in Fig. 1e. It sets and resets at around 3.2 V and 2.8 V respectively in the positive bias, and at −3.4 V and −3 V, respectively, in the negative bias^{32}. Both the Joule heating and the electricfield driven effect can be playing roles in the device operation. The filamentformation operation can be due to a breakdownlike process with random creation of voltagestressinduced vacancy or defect sites, which is electricfield driven. The Joule heating can be the main effect in filament rupturing. The insertion of the MoS_{2} layer in the device made it possible to adjust the electron energy level in MoS_{2} by externally modulating the gate bias V_{g}, which can modulate both the contactenergy barrier between the MoS_{2} and SnO_{x}, and the conductivity of the MoS_{2} sheet itself (see supplementary information section 4). Hence, as shown in Fig. 1f, as the gate bias decreases from 30 V to −20 V, the electrostatic doping in MoS_{2} and the associated energy level decreases, leading to the reduction in the series conductivity and hence the gradual increase in the set voltage.
The filamentformation process is stochastic due to the inherent random motion of oxygen ions. To extract this stochastic property quantitatively, a statistical study is carried out on the set process. As shown in Fig. 2, the device is initially reset to the highresistance state and a bias V_{TE} is applied to the device for up to 2 s. During each set process, it takes a certain amount of time t (t ≤ 2 s) after the bias voltage is applied for the device to be set. This required bias time until set is stochastic in each trial. Furthermore, there is certain chance that the device may still remain in the highresistance state after 2 s. Figure 2a plots the device current characteristics as a function of time when this reset and set process was repeated for 30 times at V_{TE} = 6 V, 5 V, 4 V, and 3 V, respectively, with V_{g} fixed at 0 V. At V_{TE} = 6 V, the device is successfully set within the first 2 s for all the 30 trials. At V_{TE} = 5 V, 4 V, and 3 V, the device failed to set within the first 2 s in certain cases. Figure 2b shows the histogram probability distribution extracted from 30 trials of the time required, until the device becomes set. If we consider t as a random variable, the probability that the set will occur within an infinitesimal interval \(\triangle t\) at time t can be described by an exponentialclass distribution^{33} function \(P=\frac{\triangle t}{\tau }\cdot {e}^{\frac{t}{\tau }}\) with the wait time t following a Poisson distribution (see supplementary information section 6) and it fits the experimental data well (red lines, Fig. 2b). This experimental observation resembling Poisson random wait time underlying the filamentformation process in the tin oxide memristive device is indicative of its exponentialclass stochastic nature.
Moreover, Fig. 2c plots P_{ss,t<2s} as a function of V_{TE}−V_{TE0} under different gate voltages, which shows exponentialclass sigmoidal distribution function. Here, P_{ss,t<2s} is the probability that the device will successfully set within 2 s and V_{TE0} is the 50% probability biasvoltage point, i.e., P_{ss,t<2s} (V_{TE} = V_{TE0}) = 0.5. With the gate voltage fixed, the chance of the device being set within t < 2 s becomes higher with increasing V_{TE}, following a sigmoidal distribution. It shows that V_{TE} can tune the stochastic property of the set event in the device when V_{g} is fixed. Microscopically, the V_{TE} tunes the filamentformation process by modulating the vacancyhopping barrier height and thus the ionhopping rate. Thus, the device is understandably easier to set at high V_{TE} than low V_{TE}. Under different gate voltages, P_{ss,t<2s} shows a sharper 0to1 transition when V_{g} is 30 V and a wider spread in its 0to1 transition when the V_{g} decreases. Here V_{g} tunes the Fermi level and charge density in the MoS_{2} layer, which modulates the potential distribution between MoS_{2} and tin oxide layer under V_{TE} bias. V_{TE} is more effective in modulating the device when V_{g} is higher, i.e. the MoS_{2} layer has a higher electron carrier density and higher conductivity, and thus leads to a sharper 0to1 transition in the sigmoidal distribution curve.
The set process is achieved by the filament formation through stochastic vacancy generation and hoppingtransport processes. Applying a voltage can reduce the generation and hoppingbarrier height and exponentially enhance the generation and hopping rates. Analytically, the set probability, P_{ss,t<2s}, can be derived as P_{ss,t<2s}\(\; = 1{e}^{{\beta e}^{\alpha ({V}_{{{{{{\rm{TE}}}}}}}{V}_{{{{{{\rm{TE}}}}}}0})}}\), where \(\alpha\) and \(\beta\) are parameters related to the material and device structure (see supplementary information section 7). After further approximation, P_{ss,t<2s} can be simplified to a distribution function that resembles the Fermi–Dirac distribution (see supplementary information section 8):
where T_{eff} is an effective “temperature” term that can be tuned by the gate bias. This expression fits very well with the experimental data in Fig. 2c. The above analytical description is also in agreement with kinetic Monte Carlo simulations, which describes microscopic stochastic process of vacancy generation, hopping, and recombination in filament formation^{34,35}. T_{eff} corresponding to various gate voltages is extracted from the fitting and Fig. 2d plots T_{eff} versus gate voltage V_{g}. A behavioral model is developed to understand the dependence of the T_{eff} on the gatebias voltage. The device is modeled as a memristor in serial combination with a MoS_{2} layer whose resistance (both the sheet resistance and its contact property with the memristive filament) can be modulated by the gate electric field. As a result, T_{eff} can be expressed as \({T}_{{{{{{\rm{eff}}}}}}}\left({V}_{{{{{{\rm{g}}}}}}}\right)={T}_{{{{{{\rm{V}}}}}}0}\left[1+\frac{Z}{\left({V}_{{{{{{\rm{g}}}}}}}{V}_{{{{{{\rm{T}}}}}}}\right)}\right]\), where \({T}_{{{{{{\rm{V}}}}}}0}\) and Z are constants, V_{T} is the threshold voltage (see supplementary information section 9). As shown in Fig. 2d, this model fits well with the experimental data and describes the modulation effect of T_{eff} by V_{g}. We would like to note that the value of T_{eff} has the unit of volt. However, to avoid confusion with the actual electrical bias voltages applied on the device, the unit of T_{eff} will be omitted in the subsequent discussions. The above discussed stochastic process of the filament formation together with the gate voltagedependent “temperature” effect can be used to construct exponentialclass distribution sampling that has broad applications in statistical modeling and computing, with the Boltzmann machine as a typical example.
To demonstrate the unique advantages of these tunable exponentialclass stochastic heteromemristors in computing application, a version of Boltzmann machine that contains a network of stochastic neurons is implemented. The stochastic neurons may fire in response to the input signals and thus drive the searching dynamics of the BM. The BM iterates all possible solutions to search for the best solution by minimizing the systemenergy function. Hardware implementations^{36,37} of such BM are challenging with conventional transistors and would require a large number of devices and complex circuitry. Here we build a BM where each of the stochastic neuron is based on a single tin oxide/MoS_{2} heteromemristor as stochastic switching and simple peripheral circuitry (more details in Methods: BM construction). This implemented BM is used to solve a maximum satisfiability problem (MAXSAT), which is an NPhard combinatorial optimization problem underlying a wide range of key applications, including MaxClique^{38}, correlation clustering^{39}, treewidth computation^{40}, Bayesian network structure learning^{41}, and argumentation dynamics^{42}.
Given a set of Boolean clauses, where each clause is a disjunction of Boolean variables and their negations, the MAXSAT problem^{43} aims to maximize the number of clauses that can be true when truth values are assigned to the Boolean variables. Without the loss of generality, the set of Boolean clauses to be solved in this work are selected to be \(\left\{{{{{{\rm{Ci}}}}}}{{{{{\rm{i}}}}}}={{{{\mathrm{1,2}}}}},\ldots ,5\right\}\), where the clause C1 is \(\left(x\vee y\vee z\right)\); C2 is \(\left({x}^{{\prime} }\vee y\vee z\right)\); C3 is \(\left({x}^{{\prime} }\vee {y}^{{\prime} }\vee z\right)\); C4 is \(\left(x\vee {y}^{{\prime} }\vee {z}^{{\prime} }\right)\) and C5 is \(\left({x}^{{\prime} }\vee y\vee {z}^{{\prime} }\right)\) (shown in Fig. 3a, the Boolean variable \({x}^{{\prime} }\) is the negation of the Boolean variable \(x\)). The optimization task here is to find a state vector \({{{{{\bf{X}}}}}}=\left({x}_{1},\cdots ,{x}_{6}\right)=(x,y,z,{x}^{{\prime} },{y}^{{\prime} },z^{\prime} )\) that can maximize the number of clauses to be true. A MAXSAT can be converted equivalently to a problem that is solvable for the BM^{44,45}. Six stochastic units are used in the BM to realize the activation for each Boolean variable in the state vector \({{{{{\bf{X}}}}}}=\left({x}_{1},\cdots ,{x}_{6}\right)\). Then we build a weight matrix W. The weight \({w}_{{{{{{\rm{ij}}}}}}}\) that is between every two Boolean variables is assigned based on the MAXSAT problem. Solving the MAXSAT is equivalent to minimizing the total energy \(E={{{{{{\bf{X}}}}}}}^{{{{{{\rm{T}}}}}}}{{{{{\bf{WX}}}}}}\) of the BM, where \({{{{{{\bf{X}}}}}}}^{{{{{{\rm{T}}}}}}}\) is the transverse of \({{{{{\bf{X}}}}}}\).
The constructed BM utilizing the tin oxide/MoS_{2} heteromemristors is shown in Fig. 3b and the schematic of the circuit blocks with six stochastic neurons is shown in Fig. 3c. In each iteration step, if the heteromemristor sets, the Boolean value of \({x}_{{{{{{\rm{i}}}}}}}\) would be flipped. If the heteromemristor does not set, the stochastic neuron would not fire and \({x}_{{{{{{\rm{i}}}}}}}\) remains the same. The stochastic neurons are sequentially updated until the BM reaches the optimal solution. In Fig. 3d, we experimentally demonstrated the evolution of the state vector and total energy when the BM started from three different initial states and found the same optimal solution, which is \({{{{{\bf{X}}}}}}=(x,{y},z,{x}^{{\prime} },{y}^{{\prime} },{z}^{{\prime} })=({{{{\mathrm{0,1,1,1,0,0}}}}})\).
As previously shown in Fig. 2d, V_{g} can tune the tin oxide/MoS_{2} heteromemristor to have different T_{eff} during the BM optimization process. T_{eff} of the BM describes the average behaviors of all the stochastic units, in close analogy to the temperature parameter in the Boltzmann distribution that describes the average behavior of particles under different thermal equilibrium states in physical systems. Thus, by controlling T_{eff} in the optimization process that can be achieved via tuning the V_{g}, it is possible to avoid premature convergence issues and facilitate the convergence efficiency associated with the BM. Figure 3e shows the effect of different V_{g} bias on the BM optimization process. During these three different runs of the BM, all the tin oxide/MoS_{2} stochastic heteromemristors are biased at V_{g} = −20 V, 0 V, and 20 V, respectively. The energy evolved differently during these runs each time. The BM is at T_{eff} = 7 when V_{g} = 20 V and converges easily for this particular problem. On the other hand, the BM is at T_{eff} = 50 when V_{g} = −20 V and is less efficient in reaching convergence. For V_{g} = 0 V, the BM is at T_{eff} = 10 and converges at an intermediate rate among the three cases. By counting how many times the BM can reach the global optimal solution out of 50 trial runs, the success rate as a function of V_{g} and T_{eff} is statistically obtained as shown in Fig. 3f. It indicates that the V_{g} and hence the T_{eff} can substantially affect the performance of the BM.
Simulated annealing^{46,47} can be implemented with our BM where the T_{eff} can gradually change during the optimization process to emulate different “cooling” strategy. It is an important approach for efficiently reaching better optimization solutions and for avoiding the premature convergence. Using the gatetunable tin oxide/MoS_{2} device, such “cooling” procedures can be quantitatively implemented during the simulated annealing by translating the designated sequential evolution of T_{eff} into the corresponding series of gate voltage bias conditions following the relation in Fig. 2d. To study the effect of different “cooling” strategies on the efficiency of the BM, four different T_{eff} variation strategies were experimentally applied on the BM. Strategy 1: high T_{eff} in the first three iteration steps followed by low T_{eff} for the remaining iterations in one optimization process (HT to LT), Strategy 2: low T_{eff} in the first three iterations followed by high T_{eff} for the remaining iterations (LT to HT), Strategy 3: maintaining a low T_{eff} in the entire optimization process (LT), and Strategy 4: maintaining a high T_{eff} in the entire optimization process (HT). Figure 4a shows the qualitative schematic about how system energy (color dots) would evolve in the process of searching optimal solutions among multiple possible energy minimums (gray line). To analyze the effect of these “cooling” strategies, typical evolutions of the energy (cost function) during the BM optimization process for the four different strategies were experimentally obtained. As shown in Fig. 4b, using the HT strategy (T_{eff} = 50), the BM is highly active but loses the selectivity for reaching proper convergence. Using the LT strategy (T_{eff} = 5), the BM is significantly less active but possesses higher selectivity that facilitates its convergence to a premature state. Finally, simulated annealing using a “cooling” strategy (HT to LT) enables active initial searches at HT (T_{eff} = 50) and then steady convergence to the minimum energy state at LT (T_{eff} = 5) as shown in the experimental results. Furthermore, Figs. 4c and 4d show the experimentally obtained statistics of success rate in finding the global optimal solution when the different “cooling” strategies are used. Different initial values for the state vectors are used in Figs. 4c and 4d to show the effect from the different initial conditions. Both figures indicate that the HT to LT strategy has the highest success rate for reaching the global optimal solution for this particular problem, while the HT strategy has the lowest success rate. The results are consistent with the simulated performance of the BM (see supplementary information section 10).
To quantitatively understand why T_{eff} can make such a significant difference in the BM optimization process, we analyze the Russel–Rao (RR) similarity^{48} between all the clauses for this particular MAXSAT problem. It is because, as illustrated in Fig. 5a, all the five clauses C1–C5 bear inherent similarity to each other due to the following two constraints: the variable constraint and the clause constraint. On the variable side, a Boolean variable and its negation (two variables connected by red lines) are always logically opposite. For example, \(x\) and \({x}^{{\prime} }\) will always have opposite values. On the clause side, the chance of two clauses both being true is lower if they contain more complementary Boolean variables in each clause. By assigning true values to the variables \(x\), \({y}^{{\prime} }\)and \({z}^{{\prime} }\)(yellow circle), the number of complementary variables (blue circle) between clauses could be easily observed. Counting the number of complementary variables can directly reflect the inner connection and constraint of the clauses. In Fig. 5a, for example, if the clause C4: \(\left(x\vee y^{\prime} \vee z^{\prime} \right)\) is true, then the probability that the clause C2: \(\left({x}^{{\prime} }\vee y\vee z\right)\) also being true is much smaller than the other three clauses since C4 and C2 contain three pairs of complementary variables.
With the BM set to different T_{eff}, the RR similarity matrix among the five clauses based on the experimental data is constructed in Figs. 5b, 5c and 5d. The color and number in each cell quantify the similarity between each pair of clauses indexed by the row and column. It represents the probability when both clauses are true among all cases. For example, a RR similarity of 0.84 between C1 and C2 in Fig. 5b means that by repeatedly running the BM 50 times at T_{eff} = 50, we had C1 and C2, both being true by the end of 42 (out of 50) runs.
The effect of T_{eff} can be explained as follows. We view the RR similarity as the distance measurement of the statistical relationship between each of the two clauses (distance = 1 − RR coefficient) in solution space^{49}. In other words, clauses with RR similarity close to 1 are seen as closely clustered, while the clauses with RR similarity close to 0 are furthermost separated. When T_{eff} is tuned to 50 (Fig. 5b), all the clauses have similar distances in the solution space, since they show close RR similarity between all pairs. As a consequence, BM tends to search widely in the solution space with a high robustness, high stochasticity, and low selectivity, since choosing any solution would look the same to the BM. When T_{eff} is 20 (Fig. 5c), clauses with small distances are closely clustered, giving high RR similarity close to unity for pairs of clauses that can be easily satisfied simultaneously, such as C1 and C2, and a low RR similarity for pairs of clauses that can hardly be satisfied at the same time, such as C1 and C4. At this T_{eff} = 20, the BM gains more selectivity in solution space. When the T_{eff} is 5 (Fig. 5d), all the clauses are either strongly clustered or separated in distance, with distinct either 1 or 0 RR similarity. BM behaves more like a deterministic “machine”. This tends to cause premature convergence as the BM is significantly less active.
Next, a simulated annealing process in the BM with linear cooling is simulated in Fig. 5e. The evolution of the RR similarity matrix indicates that the BM would evolve through all the cases that are discussed above from being fully stochastic toward nearly deterministic as T_{eff} decreases linearly. Thus, the simulated annealing process of a BM could be understood as such: at high T_{eff}, the BM searches solution space globally with high robustness and low selectivity, for the sake of large gradient descent; as the BM cools down, it gains selectivity toward some solutions and can possibly jump out of local minima since T_{eff} still provides enough perturbation; as the BM cools down to the limit, the BM exhibits a stronger selectivity than robustness, preventing itself from jumping out of the optimal zone. Hence, more efficient performance in the BM can be achieved with an appropriate “cooling” strategy.
In summary, tunable stochastic behavior is demonstrated in the tin oxide/MoS_{2} heteromemristor, showing inherent exponentialclass statistical characteristics. The device can sample exponentialclass sigmoidal distributions resembling the Fermi–Dirac distribution in physical systems with tunable distribution parameters to emulate the “temperature” effects. Simulated annealing with control of the “cooling” strategies is demonstrated in the implemented Boltzmann machine for solving combinatorial optimization with respect to a MAXSAT problem. These stochastic neurons based on tin oxide/MoS_{2} heteromemristors with reconfigurable statistical behavior pave the way for implementing selected “cooling” strategies in BM to reach optimal convergence efficiency and can find broad applications in energyefficient computing for learning, clustering, and classification.
Methods
Device fabrication
A thin MoS_{2} layer is first deposited on a Si wafer with a 285nm thermally grown SiO_{2} layer on top. The sample is then treated in an Ar/H_{2}mixed gas environment at 350 °C to clean the MoS_{2} surface. Subsequently, a thin tin oxide layer oxidized from SnSe is deposited on MoS_{2} and serves as filamentswitching layer. Electron beam lithography is then used to transfer the patterns followed by the evaporation of a 10nm/40nm Cr/Au metal stack, which forms the top electrode.
STEM and EDX
A FEI Titan Themis G2 system was used to prepare the HRSTEM images with four detectors and spherical aberration. To observe the crosssection image, the sample was pretreated by depositing chromium and carboncapping layers, then thinned by a focusedion beam (FIB, FEI Helios 450 S) with an acceleration voltage of 30 kV. The HRSTEM image was acquired with an acceleration voltage of 200 kV. EDX signals were collected to identify the elemental component in the cross section, which was integrated within the STEM system.
Raman spectroscopy
A Renishaw inVia Qontor system was used to measure the Raman spectra, which was installed with a ×100 objective lens, a grating (1800 grooves mm^{−1}), and a chargecoupled device camera. The wavelength of the excitation laser was 532 nm (from a solid laser). The Raman spectra resolution is 1.2 cm^{−1} per pixel.
BM construction
The implemented BM prototype contains 24 5bit digitaltoanalog converters (DAC). The digital pattern generation interface (DPGI) and training data acquisition interface (TDAI) are controlled by a Xilinx ML605 FPGA board that carries out information storage and computations. It formed a feedback loop to adjust both input and output patterns at each BM iteration. Depending on different input signals, the BM system adjusts the corresponding output training data accordingly. The BM prototype has six stochastic units, with each unit containing a tin oxide/MoS_{2} heteromemristor that has approximately sigmoidal switching probability upon applied voltages and peripheral circuitry. The peripheral circuitry is consisting of 4 DACs (digitaltoanalog converter) to read digital voltage values and apply to heteromemristor, a dynamic comparator for generating discretestate readout and outputlevel shifters.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
This work is supported in part by the Army Research Office (grant no. W911NF2120128) and National Science Foundation (grant no. CMMI2036359). T.W. and J.G. acknowledge support by National Science Foundation (grant no. 1809770 and 1904580). W.W. acknowledges the support from Air Force Research Laboratory (grant no. FA87501910503).
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X.Y., J.M., and H.W. conceived the project idea. X.Y., J.M. and J.W. fabricated the devices, characterized their electrical performance, and constructed and measured the BM circuit. A.Z., X.Y., M.S.W.C., and Z.Z. contributed to the design of the BM circuit. M.C and M.D. contributed to the device fabrication. W.W. contributed to the understanding of the device operation. T.W, X.Y., J.M., and J.G led the simulation and modeling of the device and BM circuit. H.W. coordinated and supervised the overall research activities. All coauthors contributed to the discussion of the data. X.Y., J.M., T.W., J.G., and H.W. cowrote the paper with inputs from all coauthors.
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The authors declare the following competing interests: H.W. currently also leads the lowdimensional materials research at Taiwan Semiconductor Manufacturing Company (TSMC) Corporate Research. All other authors declare no competing interests.
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Yan, X., Ma, J., Wu, T. et al. Reconfigurable Stochastic neurons based on tin oxide/MoS_{2} heteromemristors for simulated annealing and the Boltzmann machine. Nat Commun 12, 5710 (2021). https://doi.org/10.1038/s41467021260125
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DOI: https://doi.org/10.1038/s41467021260125
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