I. INTRODUCTION

Recently, there has been growing interest in oscillatory neural networks (ONNs) as a solution to address the bottlenecks of von Neumann architecture in traditional computing ^[1,2,3,4,5]. ONNs rely on the dynamics of coupled oscillators to emulate the interaction observed among oscillating signals in the human brain. By utilizing the phases of oscillators instead of signal amplitudes, ONNs offer the potential for significant improvements in energy efficiency. Building ONNs as hardware requires artificial oscillators capable of continuously generating oscillating signals, as depicted in Fig. 1(a). When these oscillators are coupled, their independently generated output signals interact and synchronize over time according to the strength of their coupling. This synchronization encodes information using the phase difference between the oscillators. While ONNs based on complex complementary metal-oxide-semiconductor (CMOS) circuits have been reported, they should have large hardware area and high power consumption ^[6,7]. Alternatively, various oscillators utilizing compact devices such as metal-insulator transition devices, resistive switching devices, and magnetic tunnel junction devices have been proposed ^{[8,9,10,11,12,13]}.

A bistable resistor is also a promising candidate for device-based oscillators because it can be implemented using well-developed CMOS technology ^[14]. To induce oscillation in a bistable resistor, the single transistor latch (STL) phenomenon is predominantly employed ^{[15,16,17,18]}. However, when STL is activated by impact ionization (I.I.), it becomes sensitive to temperature due to the influence of temperature on the I.I. rate ^[19]. This thermal sensitivity poses a challenge for reliable computing. Similarly, thermal instability is a persistent issue for other oscillator devices mentioned earlier. For instance, metal-insulator transition devices and filamentary resistive switching devices are also temperature-sensitive because their switching mechanisms are closely tied to thermal dynamics ^[20,21]. Therefore, temperature regulation is essential to effectively utilize these devices for ONNs because varying temperatures can lead to disparate outcomes. Specifically, if the properties of the two coupled oscillators fluctuate due to temperature inconsistencies across the chip, it can alter their synchronization status. This challenge exacerbates when deploying ONNs in environments where temperature fluctuations are substantial.

In this work, a reliable ONN utilizing a thermally stable single transistor-based oscillator (1T-oscillator) is demonstrated. By applying a high negative bias to the gate electrode of the MOSFET, temperature invariant band-to-band tunneling (BTBT) can be utilized to enable STL instead of the temperature sensitive I.I. ^[22]. The characteristics of coupled oscillators, where two oscillators are connected with a coupling capacitor (${C}_{\rm c}$), as illustrated in Fig. 1(b), are investigated under varying temperatures based on simulations, reflecting the measured oscillation properties of the 1T-oscillator. Finally, ONN for vertex coloring is constructed to show its reliable operations against thermal variations.

Fig. 1. (a) Schematic of the oscillatory neural network (ONN). (b) The circuit structure of coupled oscillators composes ONN. Single transistor-based oscillators (1T-oscillators) are coupled with a coupling capacitor ($C_{\rm c}$) while applying input current ($I_{\rm in}$) and monitoring output voltage ($V_{\rm out}$) to enable the oscillation behaviors. $C_{\rm par}$ represents the parasitic capacitor.

II. RESULTS AND DISCUSSION

When the input current ($I_{\rm in}$) is applied to the oscillator utilizing STL such as bistable resistors and MOSFETs, the drain voltage ($V_{\rm D}$) increases due to the accumulation of positive charges, causing electrons to be emitted from the source through thermionic emission, thus initiating I.I., as depicted in Fig. 2(a) ^[14,15]. Consequently, electron-hole pairs are created, with the resulting holes stored in the floating body, reducing the potential barrier between the source and the floating body. This leads to further injection of electrons into the floating body and amplifying I.I., which constitutes a form of positive feedback. Therefore, as $V_{\rm D}$ reaches the latch-up voltage ($V_{\rm latch}$), the oscillator undergoes a sudden transition from a high-resistance state (HRS) to a low-resistance state (LRS), which is a common procedure to trigger STL. Consequently, the output voltage ($V_{\rm out}$), corresponding to $V_{\rm D}$, oscillates over time when $I_{\rm in}$ is applied, while the source is grounded. However, the oscillation properties are sensitive to temperature variations because both thermionic emission and the I.I. rate change with temperature ^[19]. Oscillation frequency tends to be higher at elevated temperatures since $V_{\rm latch}$ decreases due to activated thermionic emission, as shown in Fig. 2(b). It is important to note that the measured oscillation characteristics in Fig. 2 are based on a previous paper ^[22]. In the previous paper, a vertical silicon nanowire gate-all-around MOSFET with a diameter of 640 nm and a body length of 300 nm was used as the 1T-oscillator, where the gate dielectric and polycrystalline silicon gate surrounded the silicon pillar to form the floating body.

Fig. 2. (a) The energy band diagram under a gate voltage ($V_{\rm G}$) of $-1.5$ V, which triggers the single transistor latch (STL) via impact ionization (I.I.). (b) Measured and simulated oscillation behaviors at $25^\circ$C and $50^\circ$C with applied $V_G$ of $-1.5$ V. (c) The energy band diagram under $V_{\rm G}$ of $-3$ V, triggering STL via band-to-band tunneling (BTBT). (d) Measured and simulated oscillation behaviors at $25^\circ$C and $50^\circ$C with applied $V_{\rm G}$ of $-3$ V.

Table 1. The parameters for simulation to model the oscillation behaviors.

In the MOSFET, the body potential can be controlled by adjusting the gate voltage ($V_{\rm G}$). Consequently, STL and oscillation characteristics are significantly influenced by $V_{\rm G}$. When a highly negative $V_{\rm G}$ is applied, thermally invariant BTBT can be used to trigger STL, rather than the thermally sensitive I.I., due to a notable energy bending between the floating body and the drain, as illustrated in Fig. 2(c). Electrons emitted from the source to the floating body via thermionic emission are effectively suppressed by the heightened potential barrier between the floating body and the source. Instead, the holes generated by BTBT accumulate in the floating body, prompting STL. Consequently, the oscillation characteristics remain stable irrespective of temperature when applying $V_{\rm G}$ of $-3$ V, as shown in Fig. 2(d).

To validate the effectiveness of the thermally stable 1T-oscillator for ONN applications, the oscillator was modeled as a voltage-controlled threshold switch along with a parallel parasitic capacitor ($C_{\rm par}$) in SPICE simulation. All four measured cases shown in Figs. 2(b) and 2(d) were modeled, with the simulated data closely aligning with the measured results. The simulation parameters, including top oscillating voltage ($V_{\rm top}$), bottom oscillating voltage ($V_{\rm bottom}$), on-state resistance ($R_{\rm on}$), $I_{\rm in}$, and $C_{\rm par}$ for each case were summarized in Table 1. It should be noted that $V_{\rm top}$ and $V_{\rm bottom}$ are the same as $V_{\rm latch}$ and the resting voltage in STL behavior ^[17,18]. Since $I_{\rm in}$ and $C_{\rm par}$ are constant regardless of the temperatures, $V_{\rm top}$ and $V_{\rm bottom}$ are the most dominant parameter that impact on the synchronization of oscillators.

Fig. 3. Coupled oscillator operations when STL was triggered by I.I. (applying $V_{\rm G}$ of $-1.5$ V). $V_{\rm out}$-time when the temperatures of both oscillators 1 and 2 were (a) $25^\circ$C and (b) $50^\circ$C. (c) $V_{\rm out}$-time when the temperatures of oscillator 1 and 2 were $25^\circ$C and $50^\circ$C, respectively. (d) The phase difference between the two oscillators after 0.5 s.

Fig. 3 shows the oscillation characteristics of coupled oscillators triggered by STL using I.I. (applying $V_{\rm G}$ of -1.5 V). As shown in Fig. 3(a), two oscillators (oscillators 1 and 2) were synchronized with a phase difference close to 180$^\circ$. It should be noted that a 50 ms delay was applied between $I_{\rm in}$ of the two oscillators to establish an initial signal difference. When the temperature of both oscillators was elevated to 50 $^\circ$C, they are still synchronized but phases were changed due to increased oscillation frequency attributed to lowered $V_{\rm top}$, as shown in Fig. 3(b). A more significant issue arises when the temperatures of the two oscillators differ. As shown in Fig. 3(c), two oscillators became asynchronized when oscillator 1 operates at $25^\circ$C and oscillator 2 at $50^\circ$C. This discrepancy indicates that temperature non-uniformity within the chip can disrupt synchronization, thereby compromising the reliability of ONN. Fig. 3(d) represents the extracted phase difference between the two oscillators after 0.5 s, showing synchronization when temperatures are equal and asynchronization when temperatures differ.

Fig. 4. Coupled oscillator operations when STL was triggered by BTBT (applying $V_{\rm G}$ of $-3$ V). $V_{\rm out}$-time when the temperatures of both oscillators 1 and 2 were (a) $25^\circ$C and (b) $50^\circ$C. (c) $V_{\rm out}$-time when the temperatures of oscillators 1 and 2 were $25^\circ$C and $50^\circ$C, respectively. (d) The phase difference between the two oscillators after 0.5 s.

Fig. 4 shows the oscillation characteristics of coupled oscillators triggered by STL using BTBT (applying $V_{\rm G}$ of $-3$ V). As shown in Figs. 4(a)-4(c), two oscillators achieved synchronization with a phase difference close to $180^\circ$, irrespective of temperature. This synchronization was due to the stable operation of 1T-oscillators. Notably, synchronization persisted even when one oscillator operated at $25^\circ$C and the other at $50^\circ$C. Hence, reliable operation of ONN is feasible despite temperature variations.

$C_{\rm c}$ is the key parameter that governs the synchronization of coupled oscillators. In practical ONN applications, $C_{\rm c}$ is adjusted based on the connectivity between two nodes. Fig. 5 shows the phase difference between two oscillators as a function of $C_{\rm c}$, with the oscillators at temperatures of $25^\circ$C and $50^\circ$C. The phase difference was extracted at 0.5 s. Synchronization was observed over a wide range of $C_{\rm c}$ values only when BTBT was used (applying $V_{\rm G}$ of $-3$ V). This suggests that stable ONN performance can be maintained across different temperatures and $C_{\rm c}$ values by utilizing BTBT.

Fig. 5. The phase difference between the two oscillators as a function of coupling capacitance ($C_{\rm c}$) when the temperatures of the oscillators are $25^\circ$C and $50^\circ$C.

Fig. 6. (a) Input graph showing vertex coloring when all oscillators were at a temperature of $25^\circ$C. (b) $V_{\rm out}$-time and phase plot when STL was triggered by I.I. (applying $V_{\rm G}$ of $-1.5$ V) for (a). (c) $V_{\rm out}$-time and phase plot when STL was triggered by BTBT (applying $V_{\rm G}$ of $-3$ V) for (a). (d) Input graph illustrating vertex coloring when the temperatures of oscillators 1, 3, and 5 were set to $25^\circ$C while those of oscillators 2 and 4 were set to $50^\circ$C. (e) $V_{\rm out}$-time and phase plot when STL was triggered by I.I. (applying $V_{\rm G}$ of $-1.5$ V) for (d). (f) $V_{\rm out}$-time and phase plot when STL was triggered by BTBT (applying $V_{\rm G}$ of $-3$ V) for (d).

To demonstrate the reliability of ONN in practical applications, vertex coloring was implemented. The vertex coloring problem is a well-known example of nondeterministic polynomial time hard (NP-hard) combinatorial optimization problems ^[23,24,25]. Its objective is to assign different colors to nodes connected by edges, and it finds applications in various fields such as scheduling, routing, and resource allocation. Fig. 6(a) shows an input graph having five nodes interconnected by six edges. Each node in this graph corresponds to an oscillator (labeled 1${\sim}$5), and when they are connected by edges, $C_{\rm c}$ was placed between them. From the oscillation characteristics, the phase of each oscillator was extracted with respect to the phase of $0^\circ$ in the reference oscillator 1, and the phase plots were drawn. Whenever the phase difference between two oscillators exceeded $90^\circ$, different colors were assigned to the corresponding nodes.

As shown in Figs. 6(b) and 6(c), successful coloring was achieved for both cases by applying $V_{\rm G}$ of $-1.5$ V and $-3$ V when all oscillator temperatures were set to $25^\circ$C. Fig. 6(d) shows the same input graph, but with varying oscillator temperatures: oscillators 1, 3, and 5 were at $25^\circ$C, while oscillators 2 and 4 were at $50^\circ$C. When STL was triggered by I.I. (applying $V_{\rm G}$ of $-1.5$ V), coloring failed because the oscillators corresponding to connected nodes were asynchronized due to different temperatures, as shown in Fig. 6(e). Otherwise, when STL was triggered by BTBT (applying $V_{\rm G}$ of $-3$ V), successful coloring was achieved with the correct answer because the oscillation characteristics remained unchanged, and the oscillators corresponding to the connected nodes were synchronized, as shown in Fig. 6(f). Therefore, a reliable ONN was implemented using thermally stable 1T-oscillators. It should be noted that the solution time can be decreased by increasing $I_{\rm in}$ to increase the oscillation frequency ^[15].

III. CONCLUSIONS

In this work, a method to enhance the reliability of ONN was demonstrated using the thermally stable 1T-oscillator. Depending on the gate voltage, there were two mechanisms for supplying holes into the floating body and triggering STL: I.I. with low negative gate voltage and BTBT with high negative gate voltage. When STL was triggered by the thermally invariant BTBT, stable oscillation characteristics and coupled oscillator operations were achieved. Finally, reliable ONN operation was attained even with varying temperatures, as confirmed through vertex coloring tasks. This work will pave the way to establish a design guideline for creating reliable neuromorphic systems.