Computing with Coupled Relaxation Oscillators

N. Shukla, S. Datta, A. Parihar and A. Raychowdhury

Department of Electrical Engineering, The Pennsylvania State University, University Park,, PA, 16802, USA; Department of Electrical Engineering, University of Notre Dame, Notre Dame,, IN, 46556, USA

School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta,, GA, 30332, USA

1 Introduction

Complementary metal-oxide-semiconductor (CMOS) transistors supporting the Boolean computational framework have been the backbone of modern computation, and have fueled the information technology revolution for the past four decades. However, there is a class of computing tasks, such as associative processing, which involves the determination of a degree of match (or association) between two quantities in multidimensional space, for which the Boolean approach consumes a significant amount of computing resources. Mathematically, the quantification of the degree of match between two quantities requires the calculation of a distance (using a suitable distance norm) between two points that represents the two quantities in a high-dimensional space. It turns out that the so-called “Boolean bottleneck” arises from the requirement of a significant number of power-intensive multiply–accumulate (MAC) operations involved in the calculation of the distance norm.

Associative computing-based applications such as pattern recognition, visual saliency, and image segmentation are becoming increasingly pervasive as we aim to design intelligent video analytic systems¹ with the eventual goal of realizing a “visual cortex on a silicon platform,” to mimic human vision on a hardware platform. Consequently, the computational inefficiency of CMOS in associative computing, along with the increasing relevance of associative processing-based applications, motivates the development of an alternate non-Boolean computational approach.

In this chapter, we focus on our recent work in experimentally demonstrating tunable coupled relaxation oscillators, based on the phenomenon of insulator–metal transition in vanadium dioxide (VO₂), as an efficient non-Boolean computational primitive capable of performing the visual saliency task. Our approach is an embodiment of “let the physics do the computing” paradigm and judiciously exploits the phase synchronization dynamics of pairs of coupled oscillators to compute a fractional distance norm that is suitable for associative computing.

2 Vanadium dioxide-based relaxation oscillators

VO₂ is a transition metal oxide that exhibits an abrupt insulator-to-metal transition (IMT) at ∼340 K in its bulk single-crystal form. The insulating state (under no strain) has a monoclinic M1 crystal structure with an optically measured bandgap of 0.6 eV, which abruptly collapses into a metallic phase, accompanied by a structural transformation to the rutile crystal structure (with no bandgap).^{2 3} The abrupt collapse of the bandgap amplifies the free-carrier concentration and manifests itself as a sharp change in resistivity (up to five orders in magnitude in bulk single crystal). Further, the IMT in VO₂ can be induced using temperature, as in Fig. 1(a), strain,⁴ doping,⁵ and electrical stimulus, as in Fig. 1(b). The phase transition can be triggered on ultrafast timescales (∼75 fs) as experimentally demonstrated by time-resolved optical pumping.⁶ However, the exact physics and the origin of the phase transition have been topics of debate for the past several decades; various models have been proposed to explain the IMT in VO₂, attributing it to varying degrees of contribution from a Mott–Hubbard type correlated phase transition as well as from a Peierls-like structural instability.^{7 8}

We can harness the electrically induced large, abrupt, and hysteretic change in resistivity of the VO₂ to implement VO₂-based relaxation oscillators by connecting a resistive load R_S in series, as shown in Fig. 2(a). When a voltage is applied across the circuit so that the electric field across the VO₂ device exceeds the critical field, E > E₂, the IMT is triggered resulting in a negative differential resistance (NDR) characteristic.⁹ The NDR is characterized by a decrease in resistivity and a simultaneous reduction in the electric field across the VO₂ device. If, as a result, the electric field across the VO₂ drops below a critical value, E < E₁, the metallic phase becomes unstable and the VO₂ channel undergoes the metal–insulator transition (MIT) and returns to the insulating state with a corresponding increase in the electric field across the VO₂. Subsequently, the process repeats itself in a self-sustaining manner, resulting in sustained electrical oscillations.

Representing this scenario through an electrical load line, as in Fig. 2(b), a VO₂ device operating on load line 2 will oscillate, since the electric field across the VO₂ channel drops below the critical field for MIT, whereas a device operating on load line 1 will not exhibit oscillatory behavior.⁹ The resulting representative time-domain waveform of the relaxation oscillator is shown in Fig. 2(c).

The properties of the oscillator such as frequency cannot be tuned with a constant R_S, thereby constraining its utility in computational circuits. The tuning capability can be added by replacing the resistor R_S with a metal-oxide semiconductor field-effect transistor (MOSFET) (drain of the MOSFET is connected to the VO₂) and utilizing it as a voltage-controlled variable resistor, thereby creating a hybrid-vanadium dioxide metal-oxide semiconductor field-effect transistor (HVFET) oscillator,¹⁰ illustrated in Fig. 3. Modulating the gate-to-source voltage V_GS of the MOSFET changes the resistance in series with the VO₂ and changes the operating frequency of the oscillator. The voltage-controlled oscillator (VCO) function is shown in Fig. 3(c).

3 Experimental demonstration of pairwise coupled HVFET oscillators

To make such oscillators relevant to associative computing applications, we explore in detail their synchronization dynamics. Figure 4(a) illustrates the schematic of the capacitively coupled HVFET oscillators. The use of capacitive coupling is motivated by its high-pass filtering characteristics that block any DC interaction while simultaneously allowing for the exchange of reactive power among the oscillator pair, allowing the oscillators to synchronize.⁹ The resulting time-domain waveforms of the synchronized oscillators are shown in Fig. 4(b), while the corresponding power spectrum is shown in Fig. 4(c). Further, as shown in Fig. 4(d), the resonant frequency of the coupled oscillators can be tuned by modulating the input gate voltage difference ΔV_GS = V_GS,2 − V_GS,1.¹⁰

4 Computing with pairwise coupled HVFET oscillators

For performing computation, we investigate using simulations, the phase dynamics of the synchronized HVFET oscillators using the equivalent circuit illustrated in Fig. 5(a). Figure 5(b) shows the simulated voltage flow diagrams for the coupled oscillators (at different input gate voltages), as the individual oscillators in the coupled configuration traverse between their respective insulating and the metallic states.^{11 12} The simulations reveal that the shape of the periodic orbit in the phase space (i.e., the relative phase difference between the oscillators) is a strong function of the difference between the input voltages V_GS,1 and V_GS,2. For instance, when the gate voltage inputs to the two oscillators are equal, say V_GS1 = V_GS2 = 0.3 V, the oscillators have a high out-of-phase locking character, with only a small fraction of the periodic orbit lying in the phase space that represents in-phase locking (shown in gray in Fig. 5(b)). On the other hand, as the input voltage difference between the oscillators increases, the periodic orbit evolves to having a higher percentage of in-phase locking character.¹³

This sensitivity of the periodic orbit of the HVFET oscillators can be harnessed for associative computing. By encoding the two quantities that need to be compared in the gate input voltages, the evolving periodic orbit of the coupled oscillators in phase space can be exploited to calculate the degree of match between them. To quantify the periodic orbit, we define a time-averaged xor measure by (i) thresholding the analog output to a binary stream, (ii) applying the xor operation on these binary values at every time instance, and, finally, (iii) averaging the xor output for a finite time duration (over at least three to four complete periodic orbits). This metric quantifies the degree of in-phase locking between the oscillators, and a higher value indicates higher in-phase locking in the periodic orbit. It is discernible from Fig. 5(b) that the time-averaged xor value is minimized when the two inputs are similar and the value subsequently increases with increased separation ΔV_GS between the gate voltage inputs. This is confirmed through both experiments and simulations, compared in Fig. 5(c) and (d), indicating that a quantitative determination of the degree of match can indeed be performed using the pairwise coupled HVFET oscillators.

The pronounced similarity between the 2D contour map of the xor output of the oscillators with the function (x_i^1/2 − y_i^1/2)², illustrated in Fig. 6, indicates that the specific distance norm calculated by the oscillators is the L_0.5 “fractional distance norm.”

5 Associative computing using pairwise coupled oscillators

We investigate the inherent ability of the synchronization dynamics of the coupled oscillators to compute a distance norm for visual saliency determination. Visual attention/saliency is the property of the human brain by which certain aspects of an image are more important aka salient than other aspects of the same image.

Usually, objects in the image that have a high degree of contrast with their neighborhood are salient to the human brain. Figure 7(a) shows a sample image wherein the squirrel and the bark of the tree are more prominent in comparison to the details in the white background. To replicate this functionality with the coupled oscillators, edge detection is performed using an array of pairwise oscillators to approximate the degree of dissimilarity between a given image pixel and its immediate neighbors (distance computation) – see Fig. 7(b). Different edges such as vertical, horizontal, and diagonal are detected based on the selection of neighboring pixels for comparison. As this concept is expanded to include the comparison of pixels within a larger neighborhood (pixels surrounding reference pixel; a 3 × 3 neighborhood is used here), the output approximates the visual saliency (see Fig. 7(c)). For comparison, the same image is also evaluated using a 32 nm CMOS accelerator, as shown in Fig. 7(d) and (e).

The power-performance projection of the HVFET synchronized oscillator-based processing scheme was calculated to be ∼259 µW, as shown in Fig. 8(a). It can be observed that the oscillators performing as the fundamental distance computing functional block consume only 8 µW of power; the remaining power is consumed by the external peripheral circuitry required to read out the xor output of the oscillators. Thus, designing new power-efficient read-out schemes will enable further reduction in power. To evaluate if the oscillator-based computing approach provides a power-performance advantage, the total power requirements of the 32 nm CMOS-based accelerator in performing the same task were also evaluated and the total power consumption is measured to be ∼13.1 mW. Comparing the power requirements for the two computing approaches in Fig. 8(b), we find a substantial advantage (∼50× power reduction) in performance can be attained in associative computing applications using coupled HVFET oscillators.

6 Conclusion

We have experimentally demonstrated coupled relaxation oscillators with input programmable synchronization that harness the insulator–metal transition in VO₂. The phase dynamics of the synchronized oscillators can be used to calculate a fractional distance norm, thereby providing an alternative non-Boolean route to associative computing applications such as visual saliency and pattern matching. Coupled oscillator systems have the potential to achieve a significant advantage in power consumption over their CMOS counterparts in future video analytic applications.

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