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If It’s Pinched It’s a Memristor

Abstract

This paper presents an in-depth review of the memristor from a rigorous circuit-theoretic perspective, independent of the material the device is made of. From an experimental perspective, a memristor is best defined as any 2-terminal device that exhibits a pinched hysteresis loop in the voltage-current plane when driven by any periodic voltage or current signal that elicits a periodic response of the same frequency. This definition greatly broadens the scope of memristive devices to encompass even non-semiconductor devices, both organic and inorganic, from many unrelated disciplines, including biology, botany, brain science, etc. For pedagogical reasons, the broad terrain of memristors is partitioned into 3 classes of increasing generality, dubbed Ideal Memristors, Generic Memristors, and Extended Memristors. Each class is distinguished from the others via unique fingerprints and signatures. This paper clarifies many confusing issues, such as non-volatility , DC V-I curves , high-frequency v-i curves, local activity, as well as nonlinear dynamical and bifurcation phenomena that are the hallmarks of memristive devices. Above all, this paper addresses several fundamental issues and questions that many memristor researchers do not comprehend but are afraid to ask.

Keywords

  • Ideal memristors
  • Generic memristors
  • Extended memristors
  • Pinched hysteresis loop
  • Local activity
  • Local passivity
  • Nonlinear dynamical
  • Bifurcation phenomena

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Notes

  1. 1.

    A hysteresis loop is said to be pinched at the origin if it always passes through the origin at all time instants when the input signal waveform is zero. It follows that a pinched hysteresis loop can never intersect the vertical axis except at the origin. For some contrived input signal waveforms, such as Fig. 3 of [2], or for any periodic signal whose frequency tends to infinity (whereupon the area of the pinched hysteresis loop shrinks to zero) the resulting single-valued function in both cases can be interpreted as the limiting Lissajoux figure of a pinched hysteresis loop because for other periodic waveforms, or for moderate frequencies, a pinched hysteresis would emerge.

  2. 2.

    In 1813, Davy was honored and invited to give a lecture and demonstration of his great invention at the London Royal Institute. A picture of Sir Humphry Davy demonstrating his carbon-rod arc discharge lamp is shown in the upper part of Fig. 1c. The huge bank of Volta’s batteries is shown in the lower part of the picture.

  3. 3.

    Inspite of a raging war between England and France in 1807, Bonaparte Napoleon was so impressed by Davy’s numerous fundamental scientific contributions (including the discovery of the potassium and sodium elements, and the invention of the Carbon arc discharge lamp) that he had decided to award France’s most prestigious Prix Napoleon de Institut science prize to Sir Humphry Davy. To bypass the Naval blockade, Napoleon had sent a special courier to sneak into England and deliver to Davy a special diplomatic visa, along with a pouch of gold coins that would cover not only the cost of a luxurious travel between London and Paris for Davy and his wife (lady Jane), but also for a personal servant (who happened to be a young assistant of Davy named Faraday, the inventor of the “inductor”!) to carry their baggages.

  4. 4.

    This simple procedure was proposed by P. Georgiou from the Imperial College London [37].

  5. 5.

    Each lobe area is found by calculating the corresponding Stieltjes integral [41].

  6. 6.

    Diseases involving ion-channel malfunctions are called “Channelopathies” [45, 46]. They include certain forms of Epilepsy, Myotonia, Migraine, Diabetes, etc.

  7. 7.

    The same property can be easily shown to also apply for Extended Memristors.

  8. 8.

    Here we mean an “Ideal” Memristor. For a Generic Memristor, or an Extended Memristor, the preceding section shows that one generally measures a DC V-I curve as the DC voltage is tuned over any voltage interval that would not burnt-out the device.

  9. 9.

    Sgn(●) is the Signum function: \( \begin{array}{*{20}l} {{\text{Sgn}}\,\,\varphi = 1} \hfill & {\varphi > 0} \hfill \\ {\qquad \;\;\; = - 1} \hfill & {\varphi < 0} \hfill \\ \end{array} \)

  10. 10.

    There is a name for this singular circuit element. It is called a Nullator [47].

  11. 11.

    Assuming of course the device behaves like an ideal memristor, or one of its Ideal Generic Memristor siblings.

  12. 12.

    Given any (n + 1)-segment piecewise-linear curve, the simple universal PWL formula in the Appendix can be used to write an exact PWL equation of the curve using only the absolute value function as building blocks [5, 48].

  13. 13.

    We can easily prove that the limiting Lissajoux figure at high frequencies for an Extended Memristor is not a straight line, but a single-valued curve by recalling the proof on page 212 of [52] for Generic Memristors, where the prediction that the state x(t) must tend to the initial state x0 = x(0) remains valild also for Extended Memristors. See [53] for another example.

  14. 14.

    To illustrate that there are many different methods to synthesize and build a locally-active memristor, we opted for a circuit realization which is not derived from the Resistor-to-Memristor mutator presented in Fig. 5b.

  15. 15.

    If the pinched point is located near (but not exactly at) at the origin, we will call it an imperfect memristor.

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Acknowledgements

The author would like to thank Prof. Hyongsuk Kim, and his colleagues Dr. Maheshwar Pd. Sah, and Ram Kaji Budhathoki for their indispensable assistance in the preparation of this paper.

He also wishes to acknowledge financial support from the USA Air force office of Scientific Research under Grant number FA9550-13-1-0136 and from the European Commission Marie Curie Fellowship

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Appendix

Appendix

Universal formula for continuous piecewise-linear functions

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Chua, L. (2019). If It’s Pinched It’s a Memristor. In: Chua, L., Sirakoulis, G., Adamatzky, A. (eds) Handbook of Memristor Networks. Springer, Cham. https://doi.org/10.1007/978-3-319-76375-0_2

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