Dissipation and fluctuations of CMOS ring oscillators close to criticality

TL;DR

The paper investigates dissipation and fluctuations in CMOS ring oscillators near the onset of coherent oscillations by constructing a thermodynamically consistent ring-oscillator model and performing a Hopf bifurcation analysis at the critical voltage $V^*_{\text{dd}}$. It derives the Hopf normal form for the dominant Fourier mode $k^*=(N-1)/2$, providing explicit expressions for the limit-cycle amplitude $r^*_{k^*}$ and frequency, and shows that entropy production rate $\dot{\sigma}$ decreases in the oscillatory state for $N>3$, with a linear stability-dissipation relation $\Delta\dot{\sigma} \sim [\tfrac{1}{2}-\cos(\tfrac{\pi}{N})]\Delta\mathscr{L}$. By extending to finite-size systems with a stochastic normal form, it demonstrates noise-induced oscillations below the transition and derives a decoherence time $\tau_c$ that scales as $\tau_c \sim \Omega^{1/2}$ near criticality, contrasting with $\tau_c \sim \Omega$ far from criticality. The results connect dynamical criticality with thermodynamic dissipation in mesoscopic CMOS rings and point to potential applications in true random number generation and probabilistic hardware, while highlighting the special case $N=3$ where dissipation increases at onset.

Abstract

We analyze a thermodynamically consistent model of CMOS-based ring oscillators near the onset of coherent voltage oscillations. For driving voltages close to the critical value, we derive the normal form of the Hopf bifurcation that underlies the oscillation transition in the thermodynamic limit. Using this normal form, we determine the phase and amplitude dynamics, and demonstrate that entropy dissipation decreases in the oscillating state for ring oscillators with more than three inverters. These findings culminate in a stability-dissipation relation, which links the observed decrease in dissipation to an increase in the local stability of the oscillating state. Next, we characterize finite-size fluctuations of the amplitude and phase near the critical voltage, using a stochastic version of the normal form. We demonstrate that close to the transition, finite-size fluctuations remain important at arbitrary system size, introducing oscillations even below the critical voltage. We further show that these noise-induced oscillations have an anomalously short decoherence time that scales sub-linearly with the system-size, in contrast to the behavior far from criticality.

Figures (7)

• Figure 1: (a) Logical representation of a ring oscillator as an odd number $N$ of inverters (NOT gates) connected in a loop, i.e $v_N=v_0$. (b) CMOS implementation of the NOT gate (inverter) using pMOS and nMOS transistors, and a bi-Poissonian charge-transport model including the gate-body $C_g$ and drain-source $C_o$ capacitances freitas2021stochastic.
• Figure 2: Dimensionless output voltage vector $\boldsymbol{x} = \boldsymbol{v}/V_\text{T}$ as function of time $t$ below and above the critical voltage $V^*_{\text{dd}}/V_\text{T} \approx 0.7466$ for a $7$-stage ring oscillator with system size $\Omega=10^2$. (a)$\boldsymbol{x}$ as function of $t$ for $\lambda = (V_\text{dd}-V^*_{\text{dd}})/V_\text{T} = -0.1$ as color map. Darker colors correspond to higher voltages (b)$\lambda = 0$. (c)$\lambda = 0.2$. (d) Voltage $v_0$ as function of $t$ for $\lambda=-0.1$ (blue), $\lambda = 0$ (red), and $\lambda=0.2$ (green).
• Figure 3: Steady-state probability density of Fourier modes $z_i$, $i=1,2,3$, for a $N=7$-stage ring oscillator below and above the critical voltage $V^*_{\text{dd}}$, obtained from Gillespie simulations gillespie1977exact with $\Omega=10^2$, averaged over $10^2$ realizations and over a time interval $\Delta t\approx 100\tau_0$. The probability density is shown as a heat map in the complex plane, where the horizontal axes correspond to the real parts, the vertical axes to the imaginary parts of $z_i$. Darker colors indicate higher density, arrows indicate directions of rotation. (a)--(c)$\lambda=-0.1$. (d)--(f)$\lambda=0$. (g)--(i)$\lambda=0.2$.
• Figure 4: Small-amplitude oscillations in the thermodynamic limit as function of $\lambda$ and for different numbers of inverters $N$. (a) Amplitude $r^*_{k^*}$ from Eq. \ref{['eq:rdec']} (dashed lines), from numerical solution of Eqs. \ref{['eq:eom']} (solid lines), and from Gillespie simulations gillespie1977exact with $\Omega=10^4$ (symbols). (b) Angular velocity $|\varphi^*_{k^*}|$ from Eq. \ref{['eq:mod_freq']} (dashed lines), from numerical solution of Eqs. \ref{['eq:eom']} (solid lines), and from Gillespie simulations with $\Omega=10^4$ (symbols).
• Figure 5: (a) Entropy production rate $\dot \sigma$ as function of $\lambda$ for different numbers $N$ of inverters from Eq. \ref{['eq:dsig']} (dashed lines), from numerical solutions of Eq. \ref{['eq:dsigma']}, and from Gillespie simulations with $\Omega=10^3$ (symbols). (b) Stability-dissipation relation \ref{['eq:sdr']} compared with the numerical solution of Eqs. \ref{['eq:eom']} (solid lines) and Gillespie simulations (symbols).