- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Dissipation and fluctuations of CMOS ring oscillators close to criticality

Ashwin Gopal1, Massimiliano Esposito1, and Jan Meibohm2

  • 1Complex Systems and Statistical Mechanics, Department of Physics and Materials Science, University of Luxembourg, 30 Avenue des Hauts-Fourneaux, L-4362 Esch-sur-Alzette, Luxembourg
  • 2Theoretical Physics Unit, Institute for Physics and Astronomy, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany

PDF

Phys. Rev. B 113, 245308 – Published 24 June, 2026

DOI: https://doi.org/10.1103/xbjq-gj57

Abstract

We analyze a thermodynamically consistent model of Complementary metal-oxide-semiconductor-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 sublinearly with the system-size, in contrast to the behavior far from criticality.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (69)

  1. A. Goldbeter, Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behaviour (Cambridge University Press, Cambridge, 1996).
  2. B. Novák and J. J. Tyson, Design principles of biochemical oscillators, Nat. Rev. Mol. Cell Biol. 9, 981 (2008).
  3. S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Chapman and Hall/CRC, London, 2024).
  4. G. Milburn, The thermodynamics of clocks, Contemp. Phys. 61, 69 (2020).
  5. G. P. Agrawal, Nonlinear fiber optics, in Nonlinear Science at the Dawn of the 21st Century (Springer, New York, 2000), pp. 195–211.
  6. E. Buks and I. Martin, Self-excited oscillation and synchronization of an on-fiber optomechanical cavity, Phys. Rev. E 100, 032202 (2019).
  7. D. Rockwell and E. Naudascher, Self-sustaining oscillations of flow past cavities, J. Fluids Eng. 100, 152 (1978).
  8. M. Golubitsky, I. Stewart, and D. Schaeffer, Singularities and groups in bifurcation theory II, Applied Mathematical Sciences (Springer, New York, 1988), Vol. 69.
  9. G. Nicolis, Introduction to Nonlinear Science (Cambridge University Press, Cambridge, 1995).
  10. J. Meibohm and M. Esposito, Minimum-dissipation principle for synchronized stochastic oscillators far from equilibrium, Phys. Rev. E 110, L042102 (2024).
  11. J. Meibohm and M. Esposito, Small-amplitude synchronization in driven Potts models, Phys. Rev. E 110, 044114 (2024).
  12. P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys. 49, 435 (1977).
  13. I. R. Epstein and K. Showalter, Nonlinear chemical dynamics: Oscillations, patterns, and chaos, J. Phys. Chem. 100, 13132 (1996).
  14. T. Herpich, J. Thingna, and M. Esposito, Collective power: Minimal model for thermodynamics of nonequilibrium phase transitions, Phys. Rev. X 8, 031056 (2018).
  15. B. Nguyen, U. Seifert, and A. C. Barato, Phase transition in thermodynamically consistent biochemical oscillators, J. Chem. Phys. 149, 045101 (2018).
  16. A. Gopal, N. Freitas, and M. Esposito, Information thermodynamics for Markov jump processes coupled to underdamped diffusion: Application to nanoelectromechanics, arXiv:2412.03226.
  17. Y. Taur and T. H. Ning, Fundamentals of Modern VLSI Devices (Cambridge University Press, Cambridge, 2021).
  18. N. Freitas, J.-C. Delvenne, and M. Esposito, Stochastic thermodynamics of nonlinear electronic circuits: A realistic framework for computing around kT, Phys. Rev. X 11, 031064 (2021).
  19. C. Y. Gao and D. T. Limmer, Principles of low dissipation computing from a stochastic circuit model, Phys. Rev. Res. 3, 033169 (2021).
  20. A. Gopal, When nonlinear electronics meets thermal noise at the nanoscale: A stochastic thermodynamics approach, Ph.D. thesis, University of Luxembourg, 2024.
  21. P. Helms, S. W. Chen, and D. T. Limmer, Stochastic thermodynamic bounds on logical circuit operation, Phys. Rev. E 111, 034110 (2025).
  22. G. Schrom and S. Selberherr, Ultralow-power CMOS technologies, in 1996 International Semiconductor Conference. 19th Edition. CAS'96 Proceedings (IEEE, New York, 1996), Vol. 1, pp. 237–246.
  23. K. Kim, Ultra low power CMOS design, Ph.D. thesis, Auburn University, 2011.
  24. C. Mead, Neuromorphic electronic systems, Proc. IEEE 78, 1629 (1990).
  25. V. Milo, G. Malavena, C. Monzio Compagnoni, and D. Ielmini, Memristive and CMOS devices for neuromorphic computing, Materials 13, 166 (2020).
  26. A. Jelinčič, O. Lockwood, A. Garlapati, P. Schillinger, I. Chuang, G. Verdon, and T. McCourt, An efficient probabilistic hardware architecture for diffusion-like models, arXiv:2510.23972.
  27. R. J. Baker, CMOS: Circuit Design, Layout, and Simulation (Wiley, New York, 2019).
  28. B. Razavi, Design of CMOS Phase-Locked Loops: From Circuit Level to Architecture Level (Cambridge University Press, Cambridge, 2020).
  29. M. Mandal and B. C. Sarkar, Ring oscillators: Characteristics and applications, Ind. J. Pure Appl. Phys. 48, 136 (2010).
  30. J. de Cos, A. Suarez, and F. Ramirez, Analysis of oscillation modes in free-running ring oscillators, IEEE Trans. Microwave Theory Tech. 60, 3137 (2012).
  31. A. Hajimiri, S. Limotyrakis, and T. H. Lee, Jitter and phase noise in ring oscillators, IEEE J. Solid-State Circuits 34, 790 (1999).
  32. S. Robson, B. Leung, and G. Gong, Truly random number generator based on a ring oscillator utilizing last passage time, IEEE Trans. Circuits Syst. II 61, 937 (2014).
  33. M. Bucci, L. Germani, R. Luzzi, A. Trifiletti, and M. Varanonuovo, A high-speed oscillator-based truly random number source for cryptographic applications on a smart card IC, IEEE Trans. Comput. 52, 403 (2003).
  34. B. Sunar, W. J. Martin, and D. R. Stinson, A provably secure true random number generator with built-in tolerance to active attacks, IEEE Trans. Comput. 56, 109 (2007).
  35. D. Liu, Z. Liu, L. Li, and X. Zou, A low-cost low-power ring oscillator-based truly random number generator for encryption on smart cards, IEEE Trans. Circuits Syst. II 63, 608 (2016).
  36. Ü. Güler and G. Dündar, Modeling phase noise and jitter in subthreshold region and assessing the randomness performance of CMOS ring oscillators, in 2012 International Conference on Synthesis, Modeling, Analysis and Simulation Methods and Applications to Circuit Design (SMACD) (IEEE, New York, 2012), pp. 257–260.
  37. Ü. Güler and G. Dündar, Modeling CMOS ring oscillator performance as a randomness source, IEEE Trans. Circuits Syst. I 61, 712 (2014).
  38. U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys. 75, 126001 (2012).
  39. C. Van den Broeck and M. Esposito, Ensemble and trajectory thermodynamics: A brief introduction, Physica A 418, 6 (2015).
  40. L. Peliti and S. Pigolotti, Stochastic Thermodynamics: An Introduction (Princeton University Press, Princeton, NJ, 2021).
  41. A. Demir, A. Mehrotra, and J. Roychowdhury, Phase noise and timing jitter in oscillators, in Proceedings of the IEEE 1998 Custom Integrated Circuits Conference (Cat. No. 98CH36143) (IEEE, New York, 1998), pp. 45–48.
  42. W. Vance and J. Ross, Fluctuations near limit cycles in chemical reaction systems, J. Chem. Phys. 105, 479 (1996).
  43. P. Gaspard, The correlation time of mesoscopic chemical clocks, J. Chem. Phys. 117, 8905 (2002).
  44. E. Knobloch and K. Wiesenfeld, Bifurcations in fluctuating systems: The center-manifold approach, J. Stat. Phys. 33, 611 (1983).
  45. P. Coullet, C. Elphick, and E. Tirapegui, Normal form of a Hopf bifurcation with noise, Phys. Lett. A 111, 277 (1985).
  46. A. Fraikin and H. Lemarchand, Stochastic analysis of a Hopf bifurcation: Master equation approach, J. Stat. Phys. 41, 531 (1985).
  47. T. Xiao, J. Ma, Z. Hou, and H. Xin, Effects of internal noise in mesoscopic chemical systems near Hopf bifurcation, New J. Phys. 9, 403 (2007).
  48. Z. Cao, H. Jiang, and Z. Hou, Design principles for biochemical oscillations with limited energy resources, Phys. Rev. Res. 2, 043331 (2020).
  49. A. J. McKane and T. J. Newman, Predator-prey cycles from resonant amplification of demographic stochasticity, Phys. Rev. Lett. 94, 218102 (2005).
  50. R. P. Boland, T. Galla, and A. J. McKane, How limit cycles and quasi-cycles are related in systems with intrinsic noise, J. Stat. Mech. (2008) P09001.
  51. B. Remlein, V. Weissmann, and U. Seifert, Coherence of oscillations in the weak-noise limit, Phys. Rev. E 105, 064101 (2022).
  52. A. Gopal, M. Esposito, and N. Freitas, Large deviations theory for noisy nonlinear electronics: CMOS inverter as a case study, Phys. Rev. B 106, 155303 (2022).
  53. P. J. Davis, Circulant Matrices (Wiley, New York, 1979), Vol. 120.
  54. D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem. 81, 2340 (1977).
  55. J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Appl. Math. Sci. 42 (1983).
  56. E. Ott, Chaos in Dynamical Systems (Cambridge University Press, Cambridge, 2002).
  57. G. Falasco and M. Esposito, Macroscopic stochastic thermodynamics, Rev. Mod. Phys. 97, 015002 (2025).
  58. For finite-size discrete systems, the analog of L is the so-called inflow rate [69], which converges to L in the thermodynamic limit [11].
  59. C. W. Gardiner, Handbook of stochastic methods for physics, chemistry and the natural sciences, Springer Ser. Synergetics (1985).
  60. T. J. Xiao, Z. Hou, and H. Xin, Entropy production and fluctuation theorem along a stochastic limit cycle, J. Chem. Phys. 129, 114506 (2008).
  61. N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (CRC Press, Boca Raton, FL, 2018).
  62. Y. Cao, D. T. Gillespie, and L. R. Petzold, Adaptive explicit-implicit tau-leaping method with automatic tau selection, J. Chem. Phys. 126, 224101 (2007).
  63. A. C. Barato and U. Seifert, Cost and precision of Brownian clocks, Phys. Rev. X 6, 041053 (2016).
  64. D. Santolin and G. Falasco, Dissipation bounds the coherence of stochastic limit cycles, Phys. Rev. Lett. 135, 057101 (2025).
  65. B. D. Sahoo, Ring oscillator based sub-1V leaky integrate-and-fire neuron circuit, in 2017 IEEE International Symposium on Circuits and Systems (ISCAS) (IEEE, New York, 2017), pp. 1–4.
  66. M. Rahaman Nayan and O. Hassan, Frequency tunable CMOS ring oscillator-based Ising machine, Int. J. Circuit Theory Appl. 53, 2958 (2025).
  67. Y. E. Gonul and B. Taskin, Multi-phase coupled CMOS ring oscillator based Potts machine, in Proceedings of the 43rd IEEE/ACM International Conference on Computer-Aided Design (IEEE, New York, 2024), pp. 1–9.
  68. N. A. Aadit, A. Grimaldi, G. Finocchio, and K. Y. Camsari, Physics-inspired Ising computing with ring oscillator activated p-bits, in 2022 IEEE 22nd International Conference on Nanotechnology (NANO) (IEEE, New York, 2022), pp. 393–396.
  69. M. Baiesi and G. Falasco, Inflow rate, a time-symmetric observable obeying fluctuation relations, Phys. Rev. E 92, 042162 (2015).

Outline

Information

Phys. Rev. B 113, 245308– Published 24 June, 2026
  • Received 9 December 2025
  • Revised 29 May 2026
  • Accepted 8 June 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation