Memristors and Memelements: Mathematics, Physics and Fiction 3031256247, 9783031256240

Table of contents :
Preface
Acknowledgements
Contents
About the Authors
1 What's in a Name?
1.1 The Notion of Memory
1.2 Response Functions
1.3 Response Functions and Memory Degrees of Freedom
1.4 Memory Elements (Memelements)
1.5 Memory Capacity and Storage
1.6 What Do We Really Measure?
1.6.1 An Example
References
2 Reality Versus Fiction
2.1 Elementary Physics of Fundamental Circuit Elements
2.1.1 Resistors
2.1.2 Capacitors
2.1.3 Inductors
2.2 Experimental Systems with Memory
2.3 Three Types of Memory Circuit Elements
2.4 Ideal Memelements: A Mathematical Fiction?
2.5 Resistors Are Not Capacitors, Are Not Inductors!
2.6 If It's Pinched It's … What?
2.7 Not Even Wrong
References
3 Physical Constraints for Ideal Memelements
3.1 List of Physical Constraints
3.1.1 Strong Sensitivity to Fluctuations
3.1.2 Stochastic Catastrophe
3.1.3 Violation of Landauer Principle
3.1.4 Over-Delayed Switching
3.1.5 Incompatibility with Symmetries of Electrodynamics
3.2 Intrinsic Variability of Memristive Response
References
4 Restoring Sanity: The Memristor Test
4.1 Back to Basic
4.2 The Memristor Test
4.3 Experimental Demonstrations of the Test
4.3.1 ReRAM Devices
4.3.2 The So-Called ``Φ Memristor''
4.4 Impossibility Conjectures
4.5 Ideal Memcapacitors and Meminductors
References
5 The Sociology of the Field and Lesson to Be Learned
5.1 Physical Reality Versus Mathematics
5.2 The Sociology of the Field
5.3 Whispers in the Wind
5.4 Moving Forward
5.5 The Ultimate Lesson
References
Index

Citation preview

SpringerBriefs in Physics Massimiliano Di Ventra · Yuriy V. Pershin

Memristors and Memelements Mathematics, Physics and Fiction

SpringerBriefs in Physics Series Editors Balasubramanian Ananthanarayan, Centre for High Energy Physics, Indian Institute of Science, Bangalore, Karnataka, India Egor Babaev, Department of Physics, Royal Institute of Technology, Stockholm, Sweden Malcolm Bremer, H. H. Wills Physics Laboratory, University of Bristol, Bristol, UK Xavier Calmet, Department of Physics and Astronomy, University of Sussex, Brighton, UK Francesca Di Lodovico, Department of Physics, Queen Mary University of London, London, UK Pablo D. Esquinazi, Institute for Experimental Physics II, University of Leipzig, Leipzig, Germany Maarten Hoogerland, University of Auckland, Auckland, New Zealand Eric Le Ru, School of Chemical and Physical Sciences, Victoria University of Wellington, Kelburn, Wellington, New Zealand Dario Narducci, University of Milano-Bicocca, Milan, Italy James Overduin, Towson University, Towson, MD, USA Vesselin Petkov, Montreal, QC, Canada Stefan Theisen, Max-Planck-Institut für Gravitationsphysik, Golm, Germany Charles H. T. Wang, Department of Physics, University of Aberdeen, Aberdeen, UK James D. Wells, Department of Physics, University of Michigan, Ann Arbor, MI, USA Andrew Whitaker, Department of Physics and Astronomy, Queen’s University Belfast, Belfast, UK

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Massimiliano Di Ventra · Yuriy V. Pershin

Memristors and Memelements Mathematics, Physics and Fiction

Massimiliano Di Ventra Department of Physics University of California, San Diego La Jolla, CA, USA

Yuriy V. Pershin Department of Physics and Astronomy University of South Carolina Columbia, SC, USA

ISSN 2191-5423 ISSN 2191-5431 (electronic) SpringerBriefs in Physics ISBN 978-3-031-25624-0 ISBN 978-3-031-25625-7 (eBook) https://doi.org/10.1007/978-3-031-25625-7 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To our families

Preface

Doubt is the father of invention. Galileo Galilei (1564–1642)

Why this book? Because we have now reached a point in which it is really difficult to separate reality from fiction in the field of memristors and other memory circuit elements (memelements). The word “fiction” (that we put also in the title) may seem too strong, but it is the correct one. Having worked in this field for the past 15 years, we have seen (at best) too many exaggerations, and quite a few plainly wrong and outlandish statements. In fact, many times we agree with Pauli’s famous quote: Some of these statements are “not even wrong.” These include “the memristor is the fourth fundamental circuit element” (seriously doubtful!), “the missing memristor was found” (it was not!), “if it’s pinched it’s a memristor” (not true!), “resistance switching memories are memristors” (demonstrably false!), and many other tall tales and misunderstandings that we will cover in this short book. Important! We want to make absolutely clear that our issue is not with much scientifically solid work (both experimental and theoretical) that has been produced by many researchers in the field. Rather, this book gives us an opportunity to take stock of some of the wrong claims we mention above and help the reader discern between scientifically valid statements and fictional ones. It is our opinion that these statements diminish the importance of, and do not lend credibility to an otherwise flourishing field. That said, it is then clear that the present monograph is not a review of the field. For that, there are some already available. There is indeed a small but growing body of literature that is raising serious objections to the above statements and others we will cover in this monograph. However, these voices are still very feeble and somewhat buried in the mainstream buzz. We hope this book will help amplify such voices.

vii

viii

Preface

Our training as physicists (albeit theoreticians) compels us to see reality via the lenses of experiments, not mathematics. In fact, Mathematical statements do not necessarily correspond to a physical reality!

It is indeed easy to be carried away by the beauty and rigor of mathematics and think that an equation and its consequences represent reality. But this is not always the case. For instance, the concept of infinity is well defined in mathematics, but not in the physical (experimental) sciences, because we cannot measure infinite quantities: Our instruments are finite. We will nonetheless start from mathematical statements since, many times, these are the ones that create confusion. We will then show that some of them do not have a physical counterpart. In fact, we will provide experimental tests to check their veracity. After all, our knowledge of the physical, material world is what we detect with our senses and extensions of our senses (measurement instruments).1 This book then challenges the community to be more grounded in reality rather than fiction. We are aware that this book (as some of our previous publications) may put us at odds with many researchers in this field. To this, we can only answer by paraphrasing Aristotle’s famous quote: Our desire is to befriend everybody, but our biggest desire still is to befriend truth.2

It is only for this reason that we have written this book. May it then serve as a solid anchor for present and future students and researchers in this field, which has still a lot to offer both intellectually and practically. La Jolla, CA, USA Columbia, SC, USA December 2022

1

Massimiliano Di Ventra Yuriy V. Pershin

For a brief and simple introduction to the scientific method see, e.g., the book M. Di Ventra, https://oxford.universitypressscholarship.com/view/10.1093/oso/9780198825623.001. 0001/oso-9780198825623?rskey=4y6M82result=2 The Scientific Method: Reflections from a Practitioner, (Oxford University Press, 2018). 2 And the same philosopher would say “Truth is that which conforms to reality.”

Acknowledgements

We must find time to stop and thank the people who make a difference in our lives. John F. Kennedy (1917–1963)

Several people have worked with us to better understand the fundamentals and applications of memory elements. In particular, we want to acknowledge the fruitful collaborations with Sean Bearden, Dalibor Biolek, Fabrizio Bonani, Dimitri Basov, Alessandro Braggio, Shiva Asapu, Francesco Caravelli, Leonardo Castelano, YuChang Chen, Chih-Chun Chien, Guy Cohen, Timir Datta, Vincent Dowling, Tom Driscoll, Iñigo Egusquiza, Steven La Fontaine, Francesco Giazotto, Claudio Guarcello, Mikhail Katkov, Jinsun Kim, Matt Krems, Lucas Lamata, Alexey Mikhaylov, Haik Manukian, Julian Martinez-Rincon, Franco Nori, Yan Ru Pei, Sebastiano Peotta, Paul Pfeiffer, Daniel Primosch, Mikel Sanz, Forrest Sheldon, Sergey Shevchenko, Valeriy Slipko, Enrique Solano, Alexander Stotland, Carsten Timm, Fabio Traversa, Yuanhang Zhang, and Xiao Zheng. We also want to thank many colleagues (too many to name here) we have discussed with over the years. Their feedback has always been valuable. We particularly thank Leon Chua for our discussions on the topic of this book at an early stage and with whom we proposed the notion of memcapacitive and meminductive elements. Although our research ideas have collided with his at times—due to his insistence on concepts whose validity we do not share—we still remember fondly our initial interactions. We are also grateful to the National Science Foundation, the Department of Energy, and the Defense Advanced Research Projects Agency, for supporting our research in the field. Finally, special thanks go to our families who, through their presence and love, have provided us with invaluable support and continuous encouragement over the years.

ix

Contents

1 What’s in a Name? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 The Notion of Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Response Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Response Functions and Memory Degrees of Freedom . . . . . . . . . . . . 4 1.4 Memory Elements (Memelements) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Memory Capacity and Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.6 What Do We Really Measure? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.6.1 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Reality Versus Fiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Elementary Physics of Fundamental Circuit Elements . . . . . . . . . . . . . 2.1.1 Resistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Inductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Experimental Systems with Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Three Types of Memory Circuit Elements . . . . . . . . . . . . . . . . . . . . . . . 2.4 Ideal Memelements: A Mathematical Fiction? . . . . . . . . . . . . . . . . . . . 2.5 Resistors Are Not Capacitors, Are Not Inductors! . . . . . . . . . . . . . . . . 2.6 If It’s Pinched It’s ... What? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Not Even Wrong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 14 16 16 17 18 19 22 23 25 27

3 Physical Constraints for Ideal Memelements . . . . . . . . . . . . . . . . . . . . . . . 3.1 List of Physical Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Strong Sensitivity to Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Stochastic Catastrophe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Violation of Landauer Principle . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Over-Delayed Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Incompatibility with Symmetries of Electrodynamics . . . . . . 3.2 Intrinsic Variability of Memristive Response . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 30 30 32 34 34 35 35 38 xi

xii

Contents

4 Restoring Sanity: The Memristor Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Back to Basic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Memristor Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Experimental Demonstrations of the Test . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 ReRAM Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 The So-Called “ Memristor” . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Impossibility Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Ideal Memcapacitors and Meminductors . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 43 45 45 46 49 50 51

5 The Sociology of the Field and Lesson to Be Learned . . . . . . . . . . . . . . . 5.1 Physical Reality Versus Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Sociology of the Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Whispers in the Wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Moving Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 The Ultimate Lesson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 53 54 55 55 57 58

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

About the Authors

Massimiliano Di Ventra is Professor of Physics at the University of California, San Diego. His research interests range from condensed matter theory to unconventional computing. He co-edited the textbook Introduction to Nanoscale Science and Technology (Springer, 2004) for undergraduate students; he is single author of the graduate-level textbook Electrical Transport in Nanoscale Systems (Cambridge University Press, 2008), the trade book The Scientific Method: Reflections from a Practitioner (Oxford University Press, 2018), and the monograph MemComputing: Fundamentals and Applications (Oxford University Press, 2022). He has published more than 200 papers in refereed journals and delivered more than 350 invited talks worldwide. He is Fellow of the American Association for the Advancement of Science, the American Physical Society, the Institute of Physics, the IEEE, and Foreign Member of Academia Europaea. In 2018, he was named Highly Cited Researcher by Clarivate Analytics, and he is recipient of the 2020 Feynman Prize for theory in Nanotechnology. Yuriy V. Pershin is Professor of Physics at the University of South Carolina. He received his Ph.D. degree in theoretical physics in 2002 from the University of Konstanz, Germany. His research interests include physics of electronic devices with memory, spintronics, and unconventional computing. He has published about 140 research papers, contributed to several books, and has been awarded one U.S. patent. In 2009, together with Chua and Di Ventra, he introduced the notion of memcapacitors and meminductors. In 2010, Prof. Pershin designed and assembled the first memristive neural network. He is Fellow of the Institute of Physics and Senior Member of the IEEE.

xiii

Chapter 1

What’s in a Name?

A rose by any other name would smell as sweet. William Shakespeare (1564–1616)

Abstract In this chapter we introduce the notion of memory as a general property common to all physical systems. We emphasize that any physical system subject to suitable perturbations showcases some degree of memory, whether this memory is easy or not to detect experimentally. In the Physics literature, the description of memory effects is provided by the theory of response functions, known since at least the 1950s. We then discuss the difference between response functions and observable quantities.

1.1 The Notion of Memory The word “memory” appears in many contexts and, accordingly, can take different meanings. For instance, in (computer) engineering, by memory we typically mean “storage” of information in some physical medium (Hennessy and Patterson 2006). This could be, for instance, in the form of the electric charge on a gate of a floating gate transistor, or the direction of magnetization in a magnetoresistive structure, and so on. However, these are just particular realizations of memory devices. Instead, we are interested in the memory effects of a much broader class of devices known as memory elements or memelements for short (Di Ventra et al. 2009). To proceed, we then need a much more general definition of “memory” that encompasses many different experimental situations. For the purpose of this book we then define memory as follows (Di Ventra 2022):

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Di Ventra and Y. V. Pershin, Memristors and Memelements, SpringerBriefs in Physics, https://doi.org/10.1007/978-3-031-25625-7_1

1

2

1 What’s in a Name?

The non-equilibrium nature of memory

Fig. 1.1 When a physical system is perturbed at some time t0 and its state is affected by such a perturbation at a later time t, we say that the system has memory. This reveals the non-equilibrium (dynamical) nature of memory

Memory of a physical system When the state of a physical system is perturbed by an external signal and the effect of such a perturbation persists in time—namely, it affects the state of the system at a later time—we say that the system possesses memory.

Of course, this definition includes also the meaning of “storage.” The latter is simply “long-term/long-time memory,” typically designed to store information for many years (Freitas and Wilcke 2008). It also encompasses all other definitions of memory that are found in the literature (Di Ventra 2022) as well as effects on other (shorter) time-scales. In fact, this definition explicitly shows the dynamical/nonequilibrium nature of this property (Fig. 1.1).

1.2 Response Functions Having defined the word “memory” we need to ask: Which physical systems showcase it? The short answer is: Physical systems with memory All physical systems showcase some degree of memory.

This is because all physical systems, when subject to external perturbations, cannot respond instantaneously to such perturbations (Kubo 1957). This is true for both classical and quantum systems. In fact, long-time storage is achieved in physical systems with negligible sensitivity to perturbations below a certain threshold. This is the required property for the storage-class memory devices.

1.2 Response Functions

3

In this book, we are only interested in the physical systems of technological importance that are commonly fabricated from the macro- to the nano-scale. All other scales are beyond the scope of this book, irrespective of whether memory effects in the dynamics of physical objects at such scales—such as sub-atomic particles, clusters of galaxies, etc.—can be easily detected experimentally. Now, consider a system subject to an external, input field (like an electric field applied to a conductor) v(t). We want to know how one of its properties or output (e.g., its current density), u(t), varies under v(t). The relation between v(t) and u(t) is known in the literature as response function (Kubo et al. 1985). Let’s call it g (e.g., the conductivity if v is the electric field, and u is the current density). Naïvely we would expect that the response g be always “local” in time, namely, it would depend on the value of the input at the same time t (making it a nonlinear response) or be just a constant (linear response): u(t) = g (v(t)) v(t), time-local r esponse.

(1.1)

This is, however, not true. In fact, Eq. (1.1) serves as a good approximation for certain linear and nonlinear devices, such resistors and diodes (on certain time scales), respectively. However, in general, Eq. (1.1) is just an approximation of the more general relation: t u(t) =

g(t, t  , v(t  ))v(t  )dt  , time non-local/memor y r esponse,

(1.2)

t0

where t0 is the time when the external (input) field is applied, and the response function g(t, t  , v(t  )) depends on two times, not one. In fact, the integral in Eq. (1.2) exemplifies mathematically the notion of memory we have introduced in the previous Sect. 1.1: the property u(t) at time t is the combined reaction to the input v(t) from the initial time t0 until the time t at which we measure u.1 It is only when we make the approximation: approximation

g(t, t  , v(t  ))−−−−−−−→δ(t − t  )g(v(t  )),

(1.3)

where δ(. . .) is the Dirac delta function, that we recover Eq. (1.1) from Eq. (1.2).2 From a physical point of view, this is a very strong approximation precisely because a δ-correlated physical process does not exist in Nature! It is just a mathematical idealization. In fact, the response function depends also on two spatial dimensions, r and r : g(r, r , t, t  , v(t  )) (Kubo 1957). However, for the purpose of this book, we are mainly interested in the response in time, not in space, so we drop the spatial dependence. 2 Assuming that the external field is zero for t  < t and t  > t so that we can extend the integral in 0 Eq. (1.2) from − ∞ to + ∞. 1

4

1 What’s in a Name?

It may be useful to make calculations and for certain applications (e.g., in the case of traditional resistors in electronic circuits), but it is nonetheless an approximation. It is only reasonable when the memory of the response of the system is negligible compared to the time resolution of the instrument we use to detect it.3 Also note that the presence of two times in the response function g(t, t  , v(t  )) means that the latter is non-local in time. This is, in fact, what we often mean by “memory” in Physics. Therefore, in general, we have the following physical equivalence:

Memory

equivalent to

←→

time non-locality.

1.3 Response Functions and Memory Degrees of Freedom Equation (1.2) can be re-written in an equivalent form to take into account memory mechanisms explicitly. This form is more familiar to those working in the field of memelements. To do this, we need some extra physical information. So far, we have just discussed an abstract equation, like (1.2), without regard to the mechanisms that produce memory. In practice, the latter is due, as before, to the inability of certain (microscopic and/or macroscopic) physical properties/degrees of freedom of the system—we typically describe them using internal state variables, x4 —to respond instantaneously to the external field. These physical properties could be the positions of atomic vacancies in the device, or the spin polarization, or some other degree of freedom that provides memory to the system. The interested reader can find in Pershin and Di Ventra (2011) a discussion of several mechanisms that induce memory in a wide variety of complex materials and nanoscale systems. Here, we just assume that some mechanisms are at play, and they follow their own dynamics (Chua and Kang 1976; Di Ventra et al. 2009)5 : dx = f(x, v(t)), dt 3

(1.4)

For instance, if the memory decays in nanoseconds and our instrument can only read data, say, on a time scale of microseconds, then for all practical purposes it will not detect the memory response of the system. 4 Namely variables that describe the system’s state, and we use bold font for vectors (or sets). 5 Note that, in principle, there could also be an explicit dependence on time of the function f, namely the general equation could be dx/dt = f(x, v(t), t) (Pershin and Di Ventra 2011). However, we are not aware of specific devices where this functional dependence is necessary. We therefore neglect it here and in the rest of the book.

1.4 Memory Elements (Memelements)

5

where f is a vector function with the same dimension (number of components) as the vector x. Note that we have also assumed that the dynamics of the memory degrees of freedom depend explicitly on the external field, v(t), and internal state, x. Equation (1.4) describes a continuous-time dynamical system (Perko 2001; Hasselblatt and Katok 2003; Strogatz 2018)—a typical approach employed in the modeling of physical memory devices.

1.4 Memory Elements (Memelements) For practical purposes, it is sufficient to take into account the memory effects originating from the dynamics of internal state variables. This step of the derivation is a bit more involved and we refer the reader to Di Ventra and Pershin (2013) for the explicit derivation specifically done for resistive, capacitive and inductive elements. It results in an equation of the type I (t) = G(x, V )V (t),

(1.5)

where G is again a response function (e.g., the conductance, instead of the conductivity), and I is some property related to u (e.g., the current which is related to the current density via a spatial integration), and V is the external perturbation related to v (e.g., the electric potential whose negative gradient is the electric field, if a negligible magnetic field is present). G(x, V ) contains a time integral of the microscopic response g (e.g., the conductivity). However, in order to avoid overburdening the reader with too much notation, we keep on using the symbol u for I , v for V and g for G. We can then describe a general physical property of a system that showcases memory (time non-locality) with the following coupled equations6 : Memory element or memelement u(t) = g (x, v) v(t) dx = f (x, v) . dt

(1.6) (1.7)

From now on, we will call any physical system or device that satisfies Eqs. (1.6) and (1.7) a memelement (Di Ventra et al. 2009). In fact, we could even generalize beyond Eqs. (1.6) and (1.7) and provide a much more general definition of “memelement” by the following:

6

Assuming no explicit time dependence of either g or f.

6

1 What’s in a Name?

Memelement Any physical system that showcases memory effects in the domain of their operational regime is a memelement.

Here, by “operational regime” we mean the range of physical parameters where the system is used, and where memory (time non-locality) is important for, e.g., technological applications. If the degree of memory is too small to be useful or detectable, we may say that the system is, for all practical purposes, “memory-less” in that range of parameters. Note that the type of memelement is related to a prevailing functionality (or functionalities) of the physical system. In fact, the latter can showcase several response features at once. For instance, a complex material can have memory in both its resistive or capacitive response according to the external applied field (see, e.g., Liu et al. 2006; Salaoru et al. 2014 for examples of these cases). An important remark Although this book is devoted mostly to classical (non-quantum) memelements, the discussion and conclusions pertain to quantum systems as well. See, e.g., Kubo (1957) for the explicit derivation of response functions for quantum systems with memory, and Pershin and Di Ventra (2012) for quantum applications of memelements. Quantum realizations of memelements can be found in, e.g., Pfeiffer et al. (2015), Shevchenko et al. (2016), Kumar et al. (2021), Guo et al. (2022) and Spagnolo et al. (2022).

1.5 Memory Capacity and Storage Having defined a general memelement, let us make the following comment. It would be an overstatement to claim that a single memelement can store the entire history of the applied input. In typical applications, each classical memelement is used to store a one-bit number, or, in some cases, some low-resolution analog information (since it is difficult to precisely set an analog state). As an estimation, the amount of analog information that can be stored in a memelement is less than or equivalent to an 8-bit number (Merced-Grafals et al. 2016). Clearly, such relatively low memory capacity is not sufficient to store a timedependent signal or signals. The memorization of an entire signal would require much larger memory capacity.7 7

Larger memory capacities are available in neural networks (Ganguli et al. 2008).

1.5 Memory Capacity and Storage

7

Fig. 1.2 Example of a potential energy landscape featuring several attractor states located at local energy minima (darker regions) 0.5 0 - 0.5 - 1.0 - 1.5 - 2.0 - 2.5

To understand the storage aspect, we recall that memelements are dynamical systems (Hasselblatt and Katok 2003), Eqs. (1.6) and (1.7).8 When subject to an external signal, their memory states evolve along a trajectory in an N -dimensional space, where N is the number of internal state variables (typical models, such as in Pershin et al. 2009; Strachan et al. 2013; Kvatinsky et al. 2015, are characterized by N = 1 or 2). The non-volatile storage can then only be realized in systems with a set of attractor states corresponding to the local energy minima of a potential energy function, see Fig. 1.2 for an example of such a function with N = 2. For instance, in the resistive random access memory (ReRAM) cells (Valov et al. 2011; Kim et al. 2011; International Roadmap for Devices and Systems 2021), such attractors can be associated with distinct stable configurations of atoms in the device structure. In fact, a memelement only “remembers” the fact that a time-dependent signal was applied sometime in the past to change its state—from possibly an unknown initial or intermediate state—to the present state (assuming that such a switching has occurred). The relationship between the applied signal waveform and corresponding evolution of the internal state is complex, especially in nonlinear models. In such cases, the state dynamics can not be expressed in terms of simple analytic functions of the input. When the system state is modified by the applied input, it is equivalent to say that the system state retains a memory trace of the input (Ganguli et al. 2008). Typically, the memory trace can be tested by a weak non-demolition probe of the response 8

In fact, the dynamics of many memelements involve a significant stochastic component that requires non-deterministic approaches for the analysis of circuits built out of them (Dowling et al. 2021a, b; Slipko and Pershin 2022).

8

1 What’s in a Name?

function, namely a probe that perturbs minimally the response of the system. To keep the state unchanged, the magnitude of the probe signal should be below the switching threshold, and the signal duration should be minimized.

1.6 What Do We Really Measure? At this stage we need to make an important remark. For the sake of clarity let’s refer to Eq. (1.5) and assume that indeed it relates the current I to the voltage V . The response, G, is then simply the conductance. We then ask: How do we measure the conductance (or its inverse, the resistance)? We need to both know (measure) the applied voltage, V , (with a voltmeter) and measure (with an ammeter) the current, I . It is only after these two measurements that we can derive the response G, assuming we know the relation (1.5). In other words the observables9 are the voltage and the current, not the conductance. The latter is a derived quantity, namely it cannot be measured directly, only extracted indirectly. This is true for any response function: they are relations between observable quantities. They are not themselves directly observable. We can then say that Response functions are not observables.

Why are we making such a big deal about it? Because, unlike what has been suggested in some literature (Chua 2014), response functions, in contrast to the observables they relate, may acquire infinite values at certain times. This is due to the fact that, being a relation between two measurable quantities (which cannot take infinite values since we cannot measure infinities, cf. the Preface or see Di Ventra 2018), one of them, say the voltage, V , can become zero at certain times, while the other, the current I , stays finite. At those times, we then see from Eq. (1.5) that the conductance G can become infinite. However, this eventuality (of an infinite value of the response function) does not pose any problem. In fact, if we excluded such cases, we would exclude important physical situations, for instance the transition to a superconducting state, in which the resistance is zero, hence the conductance infinite, or to an insulating state, with zero conductance, and thus infinite resistance.

9

An observable is simply an experimental procedure to determine a physical quantity. For instance, we use a clock to measure time, a ruler to measure position, a voltmeter to measure a voltage, etc.

1.6 What Do We Really Measure?

9

In Sect. 2.6 we will show experimentally that the charge on a ferroelectric capacitor (a memcapacitive system) may be non-zero at zero voltage (an infinite response!). Similarly, it was shown that the capacitance may diverge in multi-layer memcapacitive structures (Martinez-Rincon et al. 2010). Even more, there may be cases when the response function is infinite but the input–output (u − v) curves of Eq. (1.6) pass through the origin.

1.6.1 An Example To see this last point more clearly consider the example reported in Pershin and Di Ventra (2019) (which is, as we shall see in Sect. 2.3, a current-controlled memristive system: the inverse of the relation (1.5) with no explicit time dependence of the response) described by the relations V = R(x, I )I , x˙ = f (x, I ) ,

(1.8) (1.9)

√ with R(x, I ) = g(x)/ |I |.10 Here, V and I are the voltage across and current through the conductor, respectively, while R(x, I ) is its (memory) resistance. We assume a single internal state variable, x, and f (x, I ) is the function describing its evolution, with g(x) ≥ 0 a bounded function of x. Now, let’s assume the current, I , goes to zero so as to satisfy V = lim R(x, I )I = 0. I →0

(1.10)

From the above definition of R(x, I ) we then also have: lim R(x, I ) = ∞,

I →0

(1.11)

for g(x) > 0. However, the model we have just described is a valid memristive element which describes the transition to an insulating state (infinite resistance). Such cases should not then be eliminated as it was previously suggested (Chua 2014), since they represent valid, physical situations.

The choice of the form of R(x, I ) is arbitrary so long as it satisfies Eq. (1.10), namely the resistance increases more slowly than the rate of decrease of the current.

10

10

1 What’s in a Name?

A short summary • A physical system has memory if, when subject to a perturbation, the latter affects the state of the system at a later time. • Any physical system showcases some degree of memory in some of its properties when subject to an external field/perturbation. • Memory elements or memelements are then those systems whose memory can be detected under the appropriate experimental conditions. • The description of the dynamics of memelements is properly done in terms of response functions. • Response functions are derived quantities that relate the external (input) field to the output physical quantity. • Unlike the latter ones, which need to be finite because they are observables (measurable directly), response functions do not need to be finite.

Having discussed the general properties of memelements, and their mathematical description, we can now focus on the sub-class that is mostly studied in electronic circuits: resistors, capacitors, and inductors with memory.

References Chua LO (2014) If it’s pinched it’s a memristor. Semicond Sci Technol 29:104001 Chua LO, Kang SM (1976) Memristive devices and systems. Proc IEEE 64:209–223 Di Ventra M (2018) The scientific method: reflections from a practitioner. Oxford University Press, Oxford Di Ventra M (2022) MemComputing: fundamentals and applications. Oxford University Press, Oxford Di Ventra M, Pershin YV (2013) On the physical properties of memristive, memcapacitive and meminductive systems. Nanotechnology 24:255201 Di Ventra M, Pershin YV, Chua LO (2009) Circuit elements with memory: memristors, memcapacitors, and meminductors. Proc IEEE 97:1717–1724 Dowling V, Slipko V, Pershin Y (2021a) Modeling networks of probabilistic memristors in spice. Radioengineering 30:157–163 Dowling VJ, Slipko VA, Pershin YV (2021b) Probabilistic memristive networks: application of a master equation to networks of binary ReRAM cells. Chaos Solitons Fract 142:110385 Freitas RF, Wilcke WW (2008) Storage-class memory: the next storage system technology. IBM J Res Dev 52(4.5):439–447 Ganguli S, Huh D, Sompolinsky H (2008) Memory traces in dynamical systems. Proc Natl Acad Sci 105(48):18970–18975 Guo YM, Albarrán-Arriagada F, Alaeian H, Solano E, Barrios GA (2022) Quantum memristors with quantum computers. Phys Rev Appl 18:024082 Hasselblatt B, Katok A (2003) A first course in dynamics. Cambridge University Press, Cambridge, UK Hennessy JL, Patterson DA (2006) Computer architecture: a quantitative approach, 4th edn. Morgan Kaufmann Publishers Inc., Burlington, MA

References

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International Roadmap for Devices and Systems (IRDSTM ) (2021) 2021 edn. https://irds.ieee.org/ editions/2021 Kim KM, Jeong DS, Hwang CS (2011) Nanofilamentary resistive switching in binary oxide system; a review on the present status and outlook. Nanotechnology 22(25):254002 Kubo R (1957) Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems. J Phys Soc Jpn 12(6):570–586 Kubo R, Toda M, Hashitsume N (1985) Statistical physics II: nonequilibrium statistical mechanics. Springer-Verlag, Berlin Kumar S, Cárdenas-López FA, Hegade NN, Chen X, Albarrán-Arriagada F, Solano E, Alvarado Barrios G (2021) Entangled quantum memristors. Phys Rev A 104:062605 Kvatinsky S, Ramadan M, Friedman EG, Kolodny A (2015) VTEAM: a general model for voltagecontrolled memristors. IEEE Trans Circuits Syst II Express Briefs 62:786–790 Liu S, Wu N, Ignatiev A, Li J (2006) Electric-pulse-induced capacitance change effect in perovskite oxide thin films. J Appl Phys 100(5):056101 Martinez-Rincon J, Di Ventra M, Pershin YV (2010) Solid-state memcapacitive system with negative and diverging capacitance. Phys Rev B 81:195430 Merced-Grafals EJ, Dávila N, Ge N, Williams RS, Strachan JP (2016) Repeatable, accurate, and high speed multi-level programming of memristor 1T1R arrays for power efficient analog computing applications. Nanotechnology 27(36):365202 Perko L (2001) Differential equations and dynamical systems, vol 7, 3rd edn. Springer Science & Business Media, New York Pershin YV, Di Ventra M (2011) Memory effects in complex materials and nanoscale systems. Adv Phys 60:145–227 Pershin YV, Di Ventra M (2012) Neuromorphic, digital, and quantum computation with memory circuit elements. Proc IEEE 100(6):2071–2080 Pershin YV, Di Ventra M (2019) Comment on “If it’s pinched it’s a memristor” by L. Chua [Semicond. Sci. Technol. 29, 104001 (2014)]. Semicond Sci Technol 34:098001 Pershin YV, La Fontaine S, Di Ventra M (2009) Memristive model of amoeba learning. Phys Rev E 80:021926 Pfeiffer P, Egusquiza IL, Di Ventra M, Sanz M, Solano E (2015) Quantum memristors. Sci Rep 6:29507 Salaoru I, Li Q, Khiat A, Prodromakis T (2014) Coexistence of memory resistance and memory capacitance in TiO2 solid-state devices. Nanoscale Res Lett 9(1):1–7 Shevchenko SN, Pershin YV, Nori F (2016) Qubit-based memcapacitors and meminductors. Phys Rev Appl 6(1):014006 Slipko VA, Pershin YV (2022) Theory of heterogeneous circuits with stochastic memristive devices. IEEE Trans Circuits Syst II Express Briefs 69:214 Spagnolo M, Morris J, Piacentini S, Antesberger M, Massa F, Crespi A, Ceccarelli F, Osellame R, Walther P (2022) Experimental photonic quantum memristor. Nat Photonics 16(4):318–323 Strachan JP, Torrezan AC, Miao F, Pickett MD, Yang JJ, Yi W, Medeiros-Ribeiro G, Williams RS (2013) State dynamics and modeling of tantalum oxide memristors. IEEE Trans Electron Devices 60(7):2194–2202 Strogatz SH (2018) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. CRC Press Valov I, Waser R, Jameson JR, Kozicki MN (2011) Electrochemical metallization memoriesfundamentals, applications, prospects. Nanotechnology 22(25):254003

Chapter 2

Reality Versus Fiction

Truth is stranger than fiction, but it is because fiction is obliged to stick to possibilities; truth isn’t. Mark Twain (1835–1910)

Abstract In this chapter we show that the mathematical definition of memristors and other memory circuit elements (such as memcapacitors and meminductors) is simply a special case of the most general notion of response functions we have introduced in the previous chapter. It will then be clear that (i) memelements are not fundamental circuit elements, and (ii) they have to satisfy specific physical properties. This will allow us to determine which mathematical definitions are consistent with reality, and which are not.

2.1 Elementary Physics of Fundamental Circuit Elements The theory of electronic circuits belongs to the field of classical electrodynamics that studies the time evolution of charges, currents, magnetic fields, etc. (see, e.g., Jackson 1999). The relations between various quantities in electrodynamics, expressed by the Maxwell’s equations, have been formulated based on the empirical observations of Nature, and, in turn, verified by countless experiments.1 Before discussing electrical circuit elements with memory, let us briefly review the electrodynamic relations that define the circuit elements in the absence of memory. Hopefully, this starting point will clarify further the substantial physical difference between these elements, which carries over also when we consider them in the experimental regime where memory is detectable. What we call “memory circuit elements”—memristive, memcapacitive, and meminductive elements; cf. Sect. 2.3—are simply extensions of these three fun1

As in the previous chapter, we are here only concerned with classical physics, and not with quantum phenomena. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Di Ventra and Y. V. Pershin, Memristors and Memelements, SpringerBriefs in Physics, https://doi.org/10.1007/978-3-031-25625-7_2

13

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2 Reality Versus Fiction

L R C Memelements Physical systems Fig. 2.1 Fundamental circuit elements—resistors, capacitors, and inductors (denoted by R, L, and C)—and their extensions to nonlinear and memory response. Note that in this diagram the fundamental circuit elements and their nonlinear versions are “singular” since they do not have memory. This is denoted by dashed curves representing “cuts” in the plane of memelements

damental circuit elements to the physical instances in which their memory response is experimentally measurable under relevant conditions and range of applicability. Figure 2.1 presents the notations used in this book and their relations. In fact, since the basic response mechanism of memory-less components transfers to their generalizations, we can conclude the following: Fundamental circuit elements The only fundamental elements in electronic circuits are linear resistors, capacitors, and inductors.

2.1.1 Resistors When electrons move in a conductor under an applied voltage, V , they experience scattering from ionic vibrations, impurities, and possibly from each other; see Fig. 2.2a. This scattering induces a change of momentum for the electrons, which is the origin of the conductor resistance to the electron flow (current); see, e.g., Di Ventra (2008) for a detailed discussion.

2.1 Elementary Physics of Fundamental Circuit Elements

15

Fig. 2.2 From memory-less (left column) to memory circuit elements (right column). a In a conventional resistor, electrons flow through an unchanging atomic structure (constant resistance). b In ECM cells (resistive devices with memory), the change in the atomic structure driven by the applied field leads to the change in resistance (reprinted from Schindler et al. 2009, with the permission of AIP Publishing). Here, the insets (A)–(D) present four stages of formation and dissolution of a Cu filament. c A conventional capacitor has fixed geometry (constant capacitance). d In the capacitive device based on a buckled membrane, two possible states of the upper membrane lead to two stable capacitance values (© 2011 IEEE. Reprinted, with permission, from Martinez-Rincon and Pershin 2011). e A conventional inductor (a solenoid). f Inductor with a magnetic core based on a semi-hard ferrite. Here, the information is stored in the magnetization of the core, M. Previously, this system was confused with a memristor (for details, see Sect. 4.3.2)

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2 Reality Versus Fiction

In many materials, the voltage is simply proportional to the current, I , with the resistance, R, being the constant of proportionality: R=

V . I

(2.1)

Equation (2.1) is the (macroscopic) Ohm’s law. Resistors are then two-terminal devices that behave according to Ohm’s law and dissipate energy in the form of heat. Importantly, Ohm’s law describes the properties of a material, namely, how the material opposes the flow of current. Although Ohm’s law does not follow from Maxwell’s equations, it can be derived from the theory of response functions that relates the current density, J, to the external electric field, E, via the resistivity, ρ: J = ρE (see, e.g., Di Ventra 2008).

2.1.2 Capacitors Capacitors are components that store electric charge. Their operation is described using the Gauss’s law of electricity (one of the Maxwell’s equations). Mathematically, the capacitance C is the constant of proportionality between voltage, V , across the capacitor and the charge, q, it stores: C=

q . V

(2.2)

The simplest capacitor consists of two parallel plates separated by a vacuum or dielectric medium (Fig. 2.2c). Unlike resistors, capacitors store energy (in the electric field).

2.1.3 Inductors Finally, inductors relate current, I , to magnetic flux, φ. For instance, the inductance of a solenoid (Fig. 2.2e) is equal to the ratio of magnetic flux to current L=

φ . I

(2.3)

Using Faraday’s law of induction, one can show that the electromotive force, E, across the inductor is given by E = −LdI /dt (Jackson 1999). Like capacitors, inductors store energy (in the magnetic field).

2.2 Experimental Systems with Memory

17

2.2 Experimental Systems with Memory Having briefly discussed the “traditional” (memory-less) fundamental circuit elements, let us now consider how the same devices can showcase memory features under specific experimental conditions. There exist many physical mechanisms of memory in electronic systems and devices (Pershin and Di Ventra 2011). To make the following discussion clear, here we introduce specific examples of memory devices that are representatives of the field. Again, our discussion is not meant as a review of the field, rather as a clarification of the fact that the physics of the response of systems with memory is not fundamentally different from that of systems showing no memory effects. Figure 2.2b reports the resistance switching phenomenon in an electrochemical metallization (ECM) cell (also known as Conductive Bridge RAM (CBRAM) as well as Programmable Metallization Cell (PMC)) made of the material sandwich Cu/SiO2 /Pt. ECM cells are emerging as strong candidates for memory applications due to their scalability, speed, and low energy requirements (International Roadmap for Devices and Systems 2021). The current–voltage characteristics of the Cu/SiO2 /Pt cell in Fig. 2.2b indicate the bipolar switching regime. Different resistance states correspond to the creation or annihilation of a conductive filament that is formed in the SiO2 layer (insulator) from atoms of the electrochemically active electrode (Cu in the present case). The low-voltage region in Fig. 2.2b clearly indicates the behavior according to Ohm’s law. Interestingly, while the discovery of the resistance-switching effect in oxide films is commonly attributed to a 1962 paper by Hickmott (1962), there is an earlier (1960) report of a similar effect (Kreinina et al. 1960). Next, consider the capacitor with memory shown in Fig. 2.2d. In this design, a buckled elastic membrane is used as the top plate of a parallel-plate capacitor (Martinez-Rincon and Pershin 2011). The applied stress generates low and high capacitance configurations in the system. It was shown that a voltage pulse of an appropriate amplitude can be used to reliably switch the system into the desired capacitance state (Martinez-Rincon and Pershin 2011). Finally, Fig. 2.2f shows an image of an inductor with memory. Here, the memory state is associated with the direction of magnetization in the magnetic core. In Sect. 4.3.2, we discuss this particular system in more detail. From the point of view of response functions, the devices with memory in Fig. 2.2 are not so different from their conventional counterparts. This is because the physics of the response in these structures remains exactly the same as in the traditional resistors—the Ohm’s law for the resistor with memory (Fig. 2.2b)—and electrodynamics for the capacitor and inductor with memory (Fig. 2.2d, f, respectively). What is different is the presence of memory degrees of freedom whose time scale is accessible under the appropriate experimental conditions.

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2.3 Three Types of Memory Circuit Elements Having clarified the above points, it is now easy to apply the theory of response functions we have introduced in Chap. 1, and embodied in Eqs. (1.6) and (1.7), to the specific quantities we usually discuss in electrical circuits: resistances, capacitances, and inductances.2 It is just a matter of relating the correct input field to the output property we are interested in.3 We summarize these relations here: Memristive element v → I; u → V; g → R

⇒

⎧ ⎨ V (t) = R (x, I ) I (t) . ⎩ dx = f (x, I ) dt

(2.4)

In the above Eq. (2.4), I is the input current and V is the output voltage across the device. This would then be a “current-controlled” system. The memory resistance R (x, I )—nowadays briefly called memristance (Chua and Kang 1976)—takes into account all memory effects induced by the memory degrees of freedom x. Note that a similar equation can be written to describe a “voltage-controlled” device by replacing v → V and u → I in Eqs. (1.6) and (1.7). In that case, the response function, g, is the memory conductance, G (x, V ), or memductance for short. A similar exercise can be done by defining memcapacitive elements (Di Ventra et al. 2009): Memcapacitive element v → V ; u → q; g → C

⇒

⎧ ⎨ q(t) = C (x, V ) V (t) , ⎩ dx = f (x, V ) dt

(2.5)

where now C is the memory capacitance or memcapacitance of the capacitor that holds a charge q and sustains a voltage V . Similarly, we can define meminductive elements (Di Ventra et al. 2009):

2

And any other possible property that pertains to electrical circuits, such as mutual meminductance (Cohen et al. 2012). 3 Recall that a physical system can show several of these memory properties under appropriate experimental conditions (Pershin and Di Ventra 2011).

2.4 Ideal Memelements: A Mathematical Fiction?

19

Meminductive element v → I ; u → φ; g → L

⇒

⎧ ⎨ φ(t) = L (x, I ) I (t) , ⎩ dx = f (x, I ) dt

(2.6)

with L the memory inductance or meminductance when a flux φ is generated by an inductor subject to a current I . These mathematical descriptions can be generalized to memory degrees of freedom that represent fields (like the spin polarization density of spintronic devices) and/or stochastic memory elements where noise is an important component (Stotland and Di Ventra 2012). We refer the reader to the review (Pershin and Di Ventra 2011) for a thorough discussion of all these cases. Here we want to stress again that the quantities R (x, I ), C (x, V ), and L (x, I ) are just response functions (cf. Sect. 1.4). They simply express the fact that, under specific experimental conditions, resistances, capacitances and inductances showcase some degree of memory. Therefore, contrary to many claims found in the literature we can unequivocally conclude that Memory circuit elements are not “fundamental circuit elements.”

They are just extensions of resistors, capacitors, and inductors to the case of memory response. After this discussion, and as we anticipated, it should be clear to the reader that there are only three fundamental circuit elements: linear resistors, capacitors, and inductors. This is not a matter of mathematical definition; it is a matter of physical reality!

2.4 Ideal Memelements: A Mathematical Fiction? Note that the memristive, memcapacitive and meminductive elements we have defined in Sect. 2.3, representing response functions, are experimentally realizable. In other words, the equations we have written do describe the dynamics of actual physical devices. The review (Pershin and Di Ventra 2011) reports a wide range of these systems in various materials and devices, together with their corresponding mathematical description as in Sect. 2.3. However, in the literature, one also finds a specific sub-class of these elements which we could call ideal. The idea of such elements traces back to the work of

20

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Fig. 2.3 Ideal “memristors” have been introduced as the elements coupling charge and flux-linkage (© 2003 IEEE. Reprinted, with permission, from Chua 2003)

Chua (1971). In this work, a “memristor” was postulated as the “fourth fundamental element” of electronic circuits on the basis of “symmetry” arguments, hence as a mathematical axiomatic concept, not a physical one. The reasoning is as follows. As we have discussed in Sect. 2.1, electronic circuits are described using such variables as voltage, current, charge, and flux.4 There are three fundamental elements (resistors, capacitors, and inductors) that are defined using three pairs of circuit variables, as one can see in Fig. 2.3. The charge and current, as well as voltage and flux are coupled with equations of electrodynamics. The supposedly “unused” pair of variables, the charge  t q and the flux-linkage ϕ, which here is simply the integral of the voltage (ϕ = t0 V (t  )dt  ), may thus define another circuit element, which was called a “memristor” (Chua 1971). Therefore, mathematically, the memristor is defined by the relation M (ϕ, q) = 0.

(2.7)

It looks like a nice mathematical expression! But it’s not the end of it. By solving Eq. (2.7) with respect to ϕ and differentiating with respect to time, one arrives at the relation: Ideal memristors V (t) = R(q(t))I (t).

4

(2.8)

Note that electrical circuits are described with the help of Kirchhoff’s laws that involve voltages and currents, and express conservation of charge and energy (Horowitz 2015).

2.4 Ideal Memelements: A Mathematical Fiction?

21

This seemingly innocuous mathematical expression derives from Eq. (2.4) if we assume that the memristance depends only on the charge q(t) that flows through the resistor with, of course, the current I (t) = dq/dt, which serves as the second equation for the memory variable(s) appearing in the definition of memristive elements (cf. t Eq. 2.4). Its integration leads trivially to q(t) = t0 I (t  )dt  .5 An important remark The name “memristor” is sometimes used very liberally to indicate any resistive device with memory, such as Eq. (2.4), which we have properly called “memristive.” However, there is a fundamental difference between Eqs. (2.4) and (2.8): the latter puts severe physical constraints on a device described by it. We will discuss these constraints in Sect. 3.1. These constraints raise serious doubts that such an ideal device can be fabricated. To settle this issue, in Chap. 4 we will introduce a test that can be performed experimentally to unambiguously determine whether a memristive device is indeed an ideal memristor (Pershin and Di Ventra 2019b). So far, no physical device has passed our test.

The question is then: Does Eq. (2.8) represent a real, physical system that can be found in Nature or fabricated in the lab? Or is it simply a mathematical construct? And if so, is it a good approximation in certain cases to the more general memristive expression (2.4) which involves additional degrees of freedom other than the charge? Before answering such questions let’s briefly mention the other two ideal memelements: memcapacitor and meminductor (t0 is always the initial time) (Di Ventra et al. 2009): Ideal (voltage-controlled) memcapacitor ⎡ t ⎤  q(t) = C ⎣ V (t  )dt  ⎦ V (t).

(2.9)

t0

−1 For an ideal charge-controlled memcapacitor, the  relation V (t) = C q(t) cont −1 −1   tains the response function C ≡ C t0 q(t )dt .

5

The initial time t0 can be conveniently taken to be − ∞.

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2 Reality Versus Fiction

Ideal (current-controlled) meminductor ⎡ t ⎤  φ(t) = L ⎣ I (t  )dt  ⎦ I (t),

(2.10)

t0

while for an ideal flux-controlled meminductor I (t) = L −1

 t

  φ(t). φ(t )dt t0

An important remark Similar to the word “memristor” used many times in place of the more correct “memristive element,” also the words “memcapacitor” and “meminductor” are often used in lieu of “memcapacitive” and “meminductive” element, respectively. Since this is not accurate, we will not adopt such nomenclature in this book. This is also in view of the fact that ideal memcapacitors and meminductors, like the ideal memristors, fail to satisfy some strong physical constraints (cf. Sect. 3.1) which makes it doubtful they can be fabricated in the lab. For ideal current-controlled meminductors, a test similar to the one for ideal memristors can be applied since these inductors are defined by the meminductance relation L(q), Eq. (2.10). However, for the other ideal memelements, their response functions need to be measured and plotted as functions of their respective variables.

2.5 Resistors Are Not Capacitors, Are Not Inductors! The title of this section seems obviously trivial, but it has been argued in the literature (Chua 2014) that the memory properties of all memelements can be derived by a simple replacement of “symbols” in their respective equations. For instance, to obtain the memory properties of a memcapacitive system from a memristive one, it was argued (Chua 2014) that we may simply replace in Eq. (2.4) the symbols R with C, I with V , and V with q, so that “magically” we obtain the (physical) memory properties of a memcapacitive system, Eq. (2.5). In other words, it has been argued that memcapacitive and meminductive elements can be defined by a trivial replacement of symbols in the memristor relations, so that there is a correspondence between the hysteresis curves of different types of memory elements (Chua 2014). This cannot be further from the truth!

2.6 If It’s Pinched It’s … What?

23

Although we can always perform such a mathematical substitution as a trivial exercise, by doing so we completely neglect the physics of the systems those equations are supposed to describe. And the physics of resistors is dramatically different from the physics of capacitors or inductors. To make this point clearer let us recall that the memory properties of resistors often derive from the time delay of electronic and/or ionic transport under an electric field (Pershin and Di Ventra 2011). Instead, the memory of memcapacitive systems originates, for instance, from micro-electro-mechanical effects as shown in Fig. 2.2d (Martinez-Rincon and Pershin 2011). These could be due to changes of the relative position of the capacitor plates. Alternatively, memory in capacitors could emerge from the time delay in the relative permittivity of the medium in between the capacitor plates (Martinez-Rincon et al. 2010). Both cases are physically possible and both lead to memory effects in capacitors. But they have nothing to do with how memory arises in resistors! In fact, even from basic circuit theory we know that the properties of the “traditional” (namely without memory) resistors and capacitors, such as the passivity of resistors and the reactance of capacitors, cannot be transformed one into another by a simple “replacement of symbols” (Pershin and Di Ventra 2019a). Following this wrong line of thought can have disastrous consequences. For instance, it has led to the nonsensical results as those in Wang et al. (2019) where a meminductive element has been confused for a memristive element (in fact, for an ideal memristor); see Sect. 4.3.2 and Pershin et al. (2022) for an explicit experimental demonstration that indeed the device discussed in Wang et al. (2019) is not a memristor. All this points again to the danger of misusing mathematics (or just using it mindlessly) in the physical world.

2.6 If It’s Pinched It’s … What? That said, how do we know that we have in hand an ideal memelement, Eqs. (2.8), (2.9), or (2.10), as opposed to the experimentally realizable ones introduced in Sect. 2.2? It was suggested (Chua 2014) that the “clear” evidence for this is a “pinched hysteresis loop.” This means the following. If we drive any memelement (whatever is the form of the equations of motion for the memory degrees of freedom) with a periodic input (whether current, voltage, flux, etc.) we would obtain hysteresis curves in their output variables at appropriate frequencies and amplitudes of the input. Figure 2.4 shows examples of experimentally measured loops for memristive elements. It has been suggested that any two-terminal resistive device that shows such a “pinched” hysteresis loop must be a memristor (Chua 2014) (“If it’s pinched it’s a memristor”). There are several issues with such conclusion.

24

2 Reality Versus Fiction

Fig. 2.4 Examples of experimentally obtained memristive current–voltage loops. a Pinched hysteresis loop of a thermistor. b Twisted hysteresis loop of a ReRAM cell. These graphs show the response of a 1 k (at 25 ◦ C) thermistor (part BC2519-ND, Vishay Intertechnology, Inc.) and b in-house made Ag/SiO2 /W electrochemical metallization cell. Both devices were driven by a triangular voltage waveform

First of all, “pinched” loops like the one in Fig. 2.4a are not the only hysteresis loops one finds experimentally. In fact, while in a pinched loop the ascending and descending curves of the loop are tangent to each other at the pinched point (say, the origin),6 the most common hysteresis loops are actually “twisted” at the crossing point, not pinched, see Fig. 2.4b, Pershin and Di Ventra (2011, 2019a). Second, as we have discussed at length in Sect. 1.4, since the memristance, memcapacitance, or meminductance, are simply response functions, they do not need to be finite, and in the case of memcapacitive and meminductive elements, they do not even need to be positive.7 As a third point, hysteresis curves pinched or twisted at the origin do not need to emerge from non-divergent memristances (or memcapacitances/meminductances). As we have seen in Sect. 1.6.1 we can have both the current and the voltage reaching zero value, Eq. (1.10), while the memristance diverges, Eq. (1.11). There is nothing unphysical about this situation. Same for the other memelements. Finally, even the assumption (Chua 2014) that memcapacitive and meminductive elements should always be pinched at the origin is demonstrably wrong. To show this explicitly consider the ferroelectric capacitor reported in Pershin and Di Ventra (2019a). This device is clearly a memcapacitive system (a capacitor with memory) 6

That is what the word “pinched” means. The passivity criterion of memristive systems instead imposes always a non-negative value on the resistance at any given time (Di Ventra and Pershin 2013). A negative (active) resistance is impossible without power generation and can be emulated using an operational amplifier that uses a supply power (Horowitz 2015).

7

2.7 Not Even Wrong

25 70 60

V(t)

Charge (nC)

50

C

40 30

CM

20 10 0 -10 -20 -6

-4

-2

0

2

4

Voltage (V) Fig. 2.5 Experimentally measured charge-voltage characteristics of a ferroelectric memcapacitor. The circuit used in this measurement is shown in the inset. Used with permission of IOP Publishing, from Pershin and Di Ventra (2019a); permission conveyed through Copyright Clearance Center, Inc.

(Pershin and Di Ventra 2011), and yet its charge versus voltage loop is not pinched or twisted; see Fig. 2.5. This is an experimental result not a theoretical one!

2.7 Not Even Wrong Let’s finally conclude by pointing out an incredibly large number of papers that have appeared in the literature which purport to describe “memristors” with equations that are instead models of either non-linear resistors or certain bi-state systems (namely systems that showcase only two states). In other words all devices considered in these papers are without memory and yet claimed to be memristors or memristive elements! The number of papers that report these senseless results are too many to cite in this short book. We then refer the reader to the paper (Pershin and Di Ventra 2020), where we provide a (partial) list of these papers and explicitly show why they describe devices without memory; see Fig. 2.6. To give the reader just a flavor of how far from the mark these papers are, we report here one of the models which claims to approximate a “memristive element” by an expression of the type ⎧ ⎨ Ron , R M (V˙ M (t)) = Ro f f , ⎩ unchanged,

V˙ M (t) > 0 V˙ M (t) < 0 , V˙ M (t) = 0

(2.11)

26

2 Reality Versus Fiction

Fig. 2.6 Snapshot of the first page of Pershin and Di Ventra (2020) citing a very long (and incomplete) list of papers that claim to discuss “memristors” or “memristive elements” and instead study systems without memory. Reprinted from Pershin and Di Ventra (2020), with permission from Elsevier

where R M is supposed to be the memristance (memory resistance), VM (t) is the voltage across the device, Ron and Ro f f are the low- and high-resistance states of the device, respectively, and the dot denotes the time derivative. To the best of our knowledge, the first use of Eq. (2.11) was proposed in Hu and Wang (2010). Now, if we compare Eq. (2.11) with Eq. (2.4) it is obvious that the device described by this model is not memristive. True memristive elements—Eq. (2.4)—are characterized by a memory of signals applied in the past. Instead, the response of the hypothetical device in Eq. (2.11) is effectively historyindependent. In fact, its instantaneous response is determined only by the sign of the time derivative of the voltage at the same instant of time (no memory of the past). Although the time derivative implies the dependence on the voltage at an infinitesimally close preceding moment of time, this alone is not sufficient for the device to be classified as a memristive element (Pershin and Di Ventra 2020). In another memory-less model, first introduced, to the best of our knowledge, in Wu et al. (2011), the resistance is represented by R (2) M,i j (V j )

=

Rˆ i j , |V j | > Ti , Rˇ i j , |V j | < Ti ,

(2.12)

References

27

where Ti are thresholds, Rˆ i j and Rˇ i j are constants, and V j is the voltage at a node j of the network. It is blatantly obvious that Eq. (2.12) describes just a non-linear resistor. No memory is involved. Unfortunately, these models have flooded the literature and multiple studies based on them have appeared (and still appear!) in leading specialized journals, such as Neural Networks, Neurocomputing, and many others. In fact, it is amazing that despite our warning against these models (Pershin and Di Ventra 2020), they still appear in the literature.8 Following Pauli, we can only say that they are not even wrong! A short summary • By employing the general theory of response functions we have introduced in Chap. 1 we have explicitly shown that memristive, memcapacitive and meminductive elements are just resistors, capacitors, and inductors whose response showcases memory under appropriate experimental conditions. • This means that, unlike what has been claimed in some literature, memelements are not fundamental circuit elements. • This conclusion is valid even for the ideal version of these memelements (assuming they exist). • A “pinched hysteresis loop” under periodic drive is definitely not sufficient for the different memelements to be ideal. • In fact, under periodic drive the response of memelements can be infinite, and in the case of memcapacitive and meminductive elements, even negative. • A non-pinched loop under periodic drive can be easily found in devices. • There are too many models in the literature which purport to describe memelements, but instead are models without memory. We cannot say anything positive about these models other than “they are not even wrong.”

After these considerations, we can now discuss specific physical constraints that are violated by ideal memelements as defined in Sect. 2.4.

References Chua L (2003) Nonlinear circuit foundations for nanodevices. I. The four-element torus. Proc IEEE 91(11):1830–1859 Chua LO (1971) Memristor—the missing circuit element. IEEE Trans Circuit Theory 18:507–519 Chua LO (2014) If it’s pinched it’s a memristor. Semicond Sci Technol 29:104001 Chua LO, Kang SM (1976) Memristive devices and systems. Proc IEEE 64:209–223 Cohen G, Pershin YV, Di Ventra M (2012) Lagrange formalism of memory circuit elements: classical and quantum formulations. Phys Rev B 85:165428 8

This outcome also highlights a massive failure of the refereeing system!

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Di Ventra M (2008) Electrical transport in nanoscale systems. Cambridge University Press, Cambridge, UK Di Ventra M, Pershin YV (2013) On the physical properties of memristive, memcapacitive and meminductive systems. Nanotechnology 24:255201 Di Ventra M, Pershin YV, Chua LO (2009) Circuit elements with memory: memristors, memcapacitors, and meminductors. Proc IEEE 97:1717–1724 Hickmott TW (1962) Low-frequency negative resistance in thin anodic oxide films. J Appl Phys 33(9):2669–2682 Horowitz P (2015) The art of electronics. Cambridge University Press, Cambridge, UK Hu J, Wang J (2010) Global uniform asymptotic stability of memristor-based recurrent neural networks with time delays. In: The 2010 international joint conference on neural networks (IJCNN). IEEE, pp 1–8 International Roadmap for Devices and Systems (IRDSTM ) (2021) 2021 edn. https://irds.ieee.org/ editions/2021 Jackson JD (1999) Classical electrodynamics Kreinina G, Selivanov L, Shumskaya T (1960) Emission and conductivity of the capacitor type cathode. Radio Eng Electron 5(8):1338–1341 Martinez-Rincon J, Pershin YV (2011) Bistable non-volatile elastic membrane memcapacitor exhibiting chaotic behavior. IEEE Trans Electron Dev 58:1809 Martinez-Rincon J, Di Ventra M, Pershin YV (2010) Solid-state memcapacitive system with negative and diverging capacitance. Phys Rev B 81:195430 Pershin YV, Di Ventra M (2011) Memory effects in complex materials and nanoscale systems. Adv Phys 60:145–227 Pershin YV, Di Ventra M (2019a) Comment on “If it’s pinched it’s a memristor” by L. Chua [Semicond. Sci. Technol. 29, 104001 (2014)]. Semicond Sci Technol 34:098001 Pershin YV, Di Ventra M (2019b) A simple test for ideal memristors. J Phys D Appl Phys 52:01LT01 Pershin YV, Di Ventra M (2020) On the validity of memristor modeling in the neural network literature. Neural Netw 121:52–56 Pershin YV, Kim J, Datta T, Di Ventra M (2022) An experimental demonstration of the memristor test. Phys E Low-Dimens Syst Nanostruct 142:115290 Schindler C, Staikov G, Waser R (2009) Electrode kinetics of CuSiO2 -based resistive switching cells: overcoming the voltage-time dilemma of electrochemical metallization memories. Appl Phys Lett 94(7):072109 Stotland A, Di Ventra M (2012) Stochastic memory: memory enhancement due to noise. Phys Rev E 85:011116 Wang FZ, Li L, Shi L, Wu H, Chua LO (2019)  memristor: real memristor found. J Appl Phys 125:054504 Wu A, Zeng Z, Zhu X, Zhang J (2011) Exponential synchronization of memristor-based recurrent neural networks with time delays. Neurocomputing 74(17):3043–3050

Chapter 3

Physical Constraints for Ideal Memelements

Most institutions demand unqualified faith, but the institution of Science makes skepticism a virtue. Robert K. Merton (1910–2003)

Abstract In this chapter we show that ideal memristors (as discussed in the previous Chap. 2) are subject to very strict physical conditions and are unable to protect their memory state against the unavoidable fluctuations, and therefore are susceptible to a stochastic catastrophe. Similar considerations apply to ideal memcapacitors and meminductors. These are strong indications that such devices may be difficult, if not impossible, to realize experimentally.

We read in the literature that “The missing memristor was found in 2008.”1 By this it is meant the ideal memristor defined by the mathematical relation (2.7), namely, a device whose resistance depends only on the charge that flows through it, see Eq. (2.8). Here, we show that this statement is wrong, and what has been observed in Strukov et al. (2008) is simply the response of a resistive memory (a memristive system), Eq. (2.4). In fact, attempts to describe the response of such memristive systems using the ideal memristor’s Eq. (2.8) lead to inadequate results. This is because ideal memristors, like ideal memcapacitors and meminductors, fail to satisfy important physical constraints, which make it doubtful they can be easily realized experimentally. In Chap. 4 we will describe an unambiguous test that can distinguish between ideal memristors and memristive elements (Pershin and Di Ventra 2019), and its application to some resistive memories (Kim et al. 2020).

1

The particular device claimed to be an ideal memristor is a TiO2 film sandwiched by metallic leads (Strukov et al. 2008).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Di Ventra and Y. V. Pershin, Memristors and Memelements, SpringerBriefs in Physics, https://doi.org/10.1007/978-3-031-25625-7_3

29

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3 Physical Constraints for Ideal Memelements

The test we have proposed is much stronger than the signature of a “pinched hysteresis loop” that was suggested in the past (Chua 2014). This is because, as we have seen in Sect. 2.6 such hysteresis loops are also a signature of memristive elements, like thermistors.2

3.1 List of Physical Constraints In what follows we will then focus on ideal memristors, memcapacitors and meminductors, as those described by Eqs. (2.8), (2.9), and (2.10), respectively. The physical constraints we have identified that have to be satisfied by these ideal memelements are the following ones (Di Ventra and Pershin 2013).3

3.1.1 Strong Sensitivity to Fluctuations We have mentioned in Sect. 1.1 that storage is just another word for “long-term/longtime memory.” By “long time” we mean much longer than the time it takes us (via some external device) to read such information. In fact, in computer engineering we truly mean non-volatile information storage (Hennessy and Patterson 2006). However, for this type of storage to occur, there has to be a way for the different states of the memory to be stable.4 From a physical point view this means that there have to be some energetic barriers to prevent these states from disappearing. In other words, these states should be attractors of a dynamical system (Sect. 1.5). We can codify this statement into a general physical principle (Di Ventra and Pershin 2013): Principle of non-volatile information storage Non-volatile information storage without energy barriers that separate distinct memory states is physically impossible.

Of course, the energy barrier needed to store a memory state has to be reasonably high. But how much? This depends on the environment in which the computer memory operates. 2

The closest physical realization of an ideal memristor is the phase-dependent conductance of a Josephson junction, which is, however, just a component of the junction’s total current (Peotta and Di Ventra 2014). To the best of our knowledge, such a device has not been fabricated yet and shown to pass the memristor test we discuss in Chap. 4. 3 This leaves open the possibility that there may be others we have not considered. 4 Again, stable enough to be read by a device or the user, typically at a (very) distant moment of time.

3.1 List of Physical Constraints

31

We can estimate this barrier as follows. Consider the thermal escape time, τ , of a particle trapped in a one-dimensional potential well. It was found that this escape time follows the Arrhenius law (Kramers 1940) τ −1 = at f e

E0 BT

−k

,

(3.1)

where at is a numerical factor of less than one related to damping, f is the oscillation frequency at the bottom of the well (attempt frequency), E 0 is the potential barrier height, k B is the Boltzmann constant, and T is the temperature (Han et al. 1992). Equation (3.1) is also applicable to systems of higher dimensionalities (for details, see, e.g., Han et al. 1992). Consider a minimal model of a ReRAM cell, namely, a solid state system where the position of a specific atom, such as an interstitial one, stores a unit of information. At room temperature, the atoms in solids vibrate at a frequency on the order of 1013 Hz. This frequency can be considered as the attempt frequency f in Eq. (3.1). In the absence of any barrier (which is not true for real materials), the escape time is then on the order of the inverse of the attempt frequency, τ ∼ 10−13 s (fractions of picoseconds)! This time is too short for practical storage. Therefore, for the stored information to be stable over times much longer than any practical reading time, from Eq. (3.1) we see that energy barriers orders of magnitude larger than k B T are required (Landauer 1961), see Fig. 3.1. By construction, these energy barriers do not exist in models of ideal memelements, Eqs. (2.8), (2.9), and (2.10). Therefore, for all these ideal memelements, even very small input signals—applied for a sufficient time—can change the memory state. This implies a high (essentially infinite) sensitivity of ideal memelements to fluctuations in their input variable. In other words, the memory state of ideal memelements 1E15 1E10

(s)

1E5

10 years 1 month 1 hour

1 1E-5 1E-10 1E-15 0.0

0.5

1.0

1.5

2.0

E0 (eV) Fig. 3.1 Escape time as a function of the barrier height at room temperature. This plot was obtained using at = 1, f = 1013 Hz, and k B T = 25.8 meV (room temperature) in Eq. (3.1). The horizontal dashed lines serve as a guide for the eye

32

3 Physical Constraints for Ideal Memelements

Fig. 3.2 Schematic of the diffusive dynamics of the memory state q  (t) of an ideal memristor. Its dynamics are akin to the Brownian (stochastic) motion of a particle in a liquid. Therefore, an ideal memristor, even if it could be fabricated in practice, would be always subject to a stochastic catastrophe, namely it would be unable to protect its state from fluctuations (Di Ventra and Pershin 2013)

is unprotected against fluctuations, and these devices, even if they existed, cannot store information reliably. Note that such a limitation does not hold for memristive, memcapacitive, and meminductive elements whose memory states have physical dynamics (Pershin and Di Ventra 2011). It is precisely the dynamics of the memory degrees of freedom that take into account the energy barriers needed for storage of information.

3.1.2 Stochastic Catastrophe The above conclusion implies another one: the state of ideal memelements diffuses, similar to diffusive dynamics of a Brownian particle (Gardiner 1985), but in the phase space of the system (see schematic in Fig. 3.2). This means that whatever is the state of the ideal memelement at time t, it “degrades” (changes) only because of fluctuations. We have called this phenomenon a stochastic catastrophe (Di Ventra and Pershin 2013). We demonstrate this phenomenon by referring to an ideal current-controlled memristor, Eq. (2.8). Similar considerations hold for the other ideal memelements as well. Suppose there is some additive noise, ξ(t), that affects the memory degree of freedom of the ideal memristor, which is simply the charge q(t) that has flown  t through the device. Let’s call q  (t) the charge in the presence of noise and q(t) = 0 I (t  )dt  the charge in the absence of it.5 The equation of motion for the charge q  (t) is then

5

The initial time is arbitrary and we set it here to t0 = 0.

3.1 List of Physical Constraints

33

dq  = I (t) + ξ(t). dt

(3.2)

Although not necessary, for simplicity let’s assume that ξ(t) is a Gaussian white noise, (3.3) ξ(t) = 0, ξ(t)ξ(t  ) = 2κδ(t − t  ), with · · ·  indicating ensemble average over the realizations of the noise, and all higher cumulants are zero (van Kampen 1992). Here, κ is a positive constant characterizing the noise strength. Let us then integrate both sides of Eq. (3.2). We get 

t



q (t) − q (0) = q(t) +

ξ(t  )dt  .

(3.4)

0

We now move q(t) to the left-hand side of Eq. (3.4), square both sides, and average over the ensemble realizations of the noise. We then find the variance with respect to the deterministic trajectory of the charge 



t t

(q (t) − q (0) − q(t))  = 2

0

ξ(t  )ξ(t  )dt  dt  = 2κt,

(3.5)

0

where we have used the noise property Eq. (3.3) in the last equality. The result expressed by Eq. (3.5) shows that the standard deviation of the internal state (memory) variable (the square root of the variance (3.5)) from the deterministic trajectory q(t) increases with time as the square root of time. This is precisely what one finds for a Brownian particle (Gardiner 1985). Now, in the real world there is no such a thing as “absence of noise.” For instance, if we still consider a conductor, the (intrinsic) thermal agitation of electrons inside any voltage-controlled memristor would be responsible for the well-known Johnson– Nyquist noise (voltage fluctuations) (Johnson 1928; Nyquist 1928). These fluctuations are present irrespective of the strength of the applied voltage. In fact, they are even present in a device that is not connected to any external circuit at all (open circuit) (Di Ventra 2008). Therefore, thermal voltage fluctuations would act as an internal degradation mechanism in such ideal devices. In other words, in the absence of any energy barrier to protect the memory state of the system, an ideal memristor would suffer from a diffusive loss of information (Di Ventra and Pershin 2013). Similar considerations can be made for all other types of noise in electrical circuits as well as for any other ideal memelement as well.

34

3 Physical Constraints for Ideal Memelements

3.1.3 Violation of Landauer Principle Let’s now consider another important physical constraint of ideal memelements that was first realized for memristors by Meuffels and Soni (2012). It is related to the use of ideal memristors (and any other ideal memelement in which heat is generated), in the manipulation of information. It is well known—and goes under the name of Landauer principle (Landauer 1961)—that the logical, hence physical, irreversibility of any computing machine— in the broader sense of a machine that manipulates information—imposes a constraint on the minimal heat generation in any memory device (Landauer 1961). This minimal heat generation is of order k B T ln 2 per machine cycle. This condition is not satisfied by ideal memelements that generate heat, like memristors. To see this, consider the switching of an ideal memristor at constant temperature and pressure. Under these conditions, the thermodynamic properties of the device are properly described by the Gibbs free energy (Kubo et al. 1985). However, as we have also discussed in Sect. 3.1.1, the latter should involve energetic barriers between different information states.6 These energetic barriers are responsible for the heat dissipation which ultimately leads to the minimal condition found by Landauer (1961). However, as we have already mentioned in Sect. 3.1.1, the dynamical equation of memristors, such as Eq. (2.8), does not involve any restrictions on minimal switching energy, thus violating Landauer’s principle. Again, this is just another consequence of not having any energetic barrier to protect memory states in ideal memelements.

3.1.4 Over-Delayed Switching There is yet another issue with ideal memelements: the over-delayed switching effect that is not observed in realistic memelements (Di Ventra and Pershin 2013). By this we mean the following. As an example, let’s consider again an ideal current-controlled memristor described by (2.8). The resistance of this hypothetical device follows the relation R = R(q). Suppose now that the switching of this device occurs in the vicinity of q = 0, and q = 0 is the initial state. We then perturb it so that its state corresponds to a large charge q > 0. In order for us to switch this device back to its original state, we then need a charge − q to flow through it. However, in experiments, it is observed that the switching actually occurs as soon as the applied voltage or current exceeds a threshold value (Waser and Aono 2007; Jo et al. 2009a; Pershin and Di Ventra 2011). Therefore, in real devices, a much smaller amount of charge − q  would be enough to switch the memristor back (q   q). 6

These could be the states of the input/output, not necessarily of the memory degrees of freedom.

3.2 Intrinsic Variability of Memristive Response

35

In other words, experimentally realized “memristors” do not really “track” the charge flowing through them when they are in their limiting states, namely in their high and low resistance states. Indeed, this limitation can be easily checked with the “memristor test” we will discuss in Chap. 4. Similar considerations can be made for ideal memcapacitors and meminductors for their respective variables. In this case, since the response of these elements depends on integrals (see, Eqs. 2.9 and 2.10), the delayed switching takes place as the integrals need time to “unwind.”

3.1.5 Incompatibility with Symmetries of Electrodynamics Finally, there is at least one more problem with ideal memristors which goes to the core of how they have been defined originally (Chua 1971). The symmetry between flux-linkage, ϕ, and charge, q, we have mentioned in Sect. 2.4 is not a symmetry of electrodynamics. The latter is governed by Maxwell’s equations which are invariant under charge conjugation (q → − q), parity (r → − r) and time reversal (t → − t) transformations (Jackson 1999). Note, however, that resistors (as well as memristors, memristive systems, and any dissipative memelements), violate time-reversal invariance since they involve dissipation. Such a dissipation (which converts electric potential energy into heat) can not be reversed by the transformation t → − t. Irrespective, any electronic device and the models used to describe it7 should satisfy the appropriate symmetries of electrodynamics. This is definitely not taken into account in many simple models of ideal memelements (Strukov et al. 2008). For instance, for an ideal memristor, charge conjugation requires R(q) = R(− q), in such a way that Eq. (2.8) remains invariant: the resistance needs to be an even function of the charge. Similar considerations can be made for the other ideal memelements. What about non-ideal, experimentally realizable (mem)elements? In that case, the relevant symmetries of electrodynamics are satisfied by the physical requirements that lead to the equations of motion of the internal state (memory) variables (Pershin and Di Ventra 2011). In other words, if such physical requirements are correctly included, no issue would arise with respect to symmetries.

3.2 Intrinsic Variability of Memristive Response Another critical issue is the experimentally observed strong device-to-device and cycle-to-cycle variability in the response of nearly all memristive devices (Guan et al. 2012; Zhang et al. 2014; Pérez et al. 2019). 7

If the device operation is based solely on microscopic electrodynamics.

36

3 Physical Constraints for Ideal Memelements

Fig. 3.3 a–c Experimentally measured distributions of switching times in Ag-SiO2 cells. d Voltage dependence of the characteristic switching time. Used with permission of RSC Publishing, from Gaba et al. (2013); permission conveyed through Copyright Clearance Center, Inc.

For example, experiments have shown that the switching (wait) times in ECM cells follow the Poisson distribution (Jo et al. 2009b; Gaba et al. 2013, 2014), see Fig. 3.3. A log-normal distribution was observed in the switching TiO2 valence change memories by Medeiros-Ribeiro et al. (2011). Figure 3.4 shows the variability in the response of a Cu/AlOx /W electrochemical metallization cell (Sleiman et al. 2013). One can clearly see in Figs. 3.3 and 3.4 that the same cell behaves differently in different cycles, and the difference between the cycles is significant. It is then obvious that the deterministic models (such as Eqs. 2.4 and 2.8) can only partially reproduce the real device response. This raises concerns regarding the applicability of deterministic models (including the ideal ones) to real devices in general. There are several theoretical approaches to account for the randomness in responses. Previously, we have introduced the model of stochastic memory elements (Pershin and Di Ventra 2011) with dynamics described by a stochastic rather than a deterministic differential equation. Moreover, a probabilistic circuit theory has recently been developed (Dowling et al. 2021a, b) based on the master equation (van Kampen 1992). The description of the circuits combining memristive and reactive components is more complicated, and can be based on the Chapman–Kolmogorov equation (Slipko and Pershin 2021).

3.2 Intrinsic Variability of Memristive Response

37

Fig. 3.4 Cycle-to-cycle variability of the response of a Cu/AlOx /W electrochemical metallization cell. The memory cell shows the OFF-to-ON transition at positive voltage, and ON-to-OFF transition at negative voltage. The structure of this memory device is presented in the inset. Reprinted from Sleiman et al. (2013), with the permission of AIP Publishing

A short summary • Ideal memelements—as those represented by Eqs. (2.8), (2.9), and (2.10)— suffer from some fundamental physical issues. • These are sensitivity to fluctuations which leads to a stochastic catastrophe and violation of the Landauer principle of minimal heat dissipation (for those ideal memelements that dissipate energy). • They also suffer from an over-delayed switching effect and they are incompatible with symmetries of electrodynamics. • All this points to the fact that ideal memelements may be just mathematical concepts and cannot be easily realized experimentally.

In order to provide the reader with some unambiguous way to verify the last point above, we discuss in the next Chap. 4 an easy test to distinguish at least ideal memristors from memristive systems (Pershin and Di Ventra 2019). A similar test can be applied to ideal current-controlled meminductors since they are defined by the meminductance relation L(q), Eq. (2.10).

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3 Physical Constraints for Ideal Memelements

References Chua LO (1971) Memristor—the missing circuit element. IEEE Trans Circuit Theory 18:507–519 Chua LO (2014) If it’s pinched it’s a memristor. Semicond Sci Technol 29:104001 Di Ventra M (2008) Electrical transport in nanoscale systems. Cambridge University Press, Cambridge, UK Di Ventra M, Pershin YV (2013) On the physical properties of memristive, memcapacitive and meminductive systems. Nanotechnology 24:255201 Dowling V, Slipko V, Pershin Y (2021a) Modeling networks of probabilistic memristors in spice. Radioengineering 30:157–163 Dowling VJ, Slipko VA, Pershin YV (2021b) Probabilistic memristive networks: application of a master equation to networks of binary ReRAM cells. Chaos Solitons Fract 142:110385 Gaba S, Sheridan P, Zhou J, Choi S, Lu W (2013) Stochastic memristive devices for computing and neuromorphic applications. Nanoscale 5(13):5872–5878 Gaba S, Knag P, Zhang Z, Lu W (2014) Memristive devices for stochastic computing. In: 2014 IEEE international symposium on circuits and systems (ISCAS). IEEE, pp 2592–2595 Gardiner C (1985) Handbook of stochastic methods: for physics, chemistry and the natural sciences. Springer Guan X, Yu S, Wong HSP (2012) On the switching parameter variation of metal-oxide RRAM—part I: physical modeling and simulation methodology. IEEE Trans Electron Devices 59(4):1172–1182 Han S, Lapointe J, Lukens J (1992) Effect of a two-dimensional potential on the rate of thermally induced escape over the potential barrier. Phys Rev B 46(10):6338 Hennessy JL, Patterson DA (2006) Computer architecture: a quantitative approach, 4th edn. Morgan Kaufmann Publishers Inc., Burlington, MA Jackson JD (1999) Classical electrodynamics Jo SH, Kim KH, Lu W (2009a) High-density crossbar arrays based on a Si memristive system. Nano Lett 9:870–874 Jo SH, Kim KH, Lu W (2009b) Programmable resistance switching in nanoscale two-terminal devices. Nano Lett 9:496–500 Johnson JB (1928) Thermal agitation of electricity in conductors. Phys Rev 32:97 Kim J, Pershin YV, Yin M, Datta T, Di Ventra M (2020) An experimental proof that resistanceswitching memory cells are not memristors. Adv Electron Mater 2000010:1–6 Kramers HA (1940) Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7(4):284–304 Kubo R, Toda M, Hashitsume N (1985) Statistical physics II: nonequilibrium statistical mechanics. Springer-Verlag, Berlin Landauer R (1961) Irreversibility and heat generation in the computing process. IBM J Res Dev 5:261–269 Medeiros-Ribeiro G, Perner F, Carter R, Abdalla H, Pickett MD, Williams RS (2011) Lognormal switching times for titanium dioxide bipolar memristors: origin and resolution. Nanotechnology 22:095702 Meuffels P, Soni R (2012) Fundamental issues and problems in the realization of memristors. arXiv preprint arXiv:1207.7319 Nyquist H (1928) Thermal agitation of electric charge in conductors. Phys Rev 32:110–113 Peotta S, Di Ventra M (2014) Superconducting memristors. Phys Rev Appl 2:034011 Pérez E, Maldonado D, Acal C, Ruiz-Castro J, Alonso F, Aguilera A, Jiménez-Molinos F, Wenger C, Roldán J (2019) Analysis of the statistics of device-to-device and cycle-to-cycle variability in TiN/Ti/Al:HfO2 /TiN RRAMs. Microelectron Eng 214:104–109 Pershin YV, Di Ventra M (2011) Memory effects in complex materials and nanoscale systems. Adv Phys 60:145–227 Pershin YV, Di Ventra M (2019) A simple test for ideal memristors. J Phys D Appl Phys 52:01LT01 Sleiman A, Sayers PW, Mabrook MF (2013) Mechanism of resistive switching in Cu/AlOx /W nonvolatile memory structures. J Appl Phys 113(16):164506

References

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Slipko VA, Pershin YV (2021) Theory of heterogeneous circuits with stochastic memristive devices. IEEE Trans Circuits Syst II Express Briefs 69(1):214–218 Strukov DB, Snider GS, Stewart DR, Williams RS (2008) The missing memristor found. Nature 453(7191):80–83 van Kampen N (1992) Stochastic processes in physics and chemistry. Elsevier, Amsterdam Waser R, Aono M (2007) Nanoionics-based resistive switching memories. Nat Mater 6:833–840 Zhang M, Long S, Wang G, Li Y, Xu X, Liu H, Liu R, Wang M, Li C, Sun P et al (2014) An overview of the switching parameter variation of RRAM. Chin Sci Bull 59(36):5324–5337

Chapter 4

Restoring Sanity: The Memristor Test

Too much sanity may be madness and the maddest of all, to see life as it is and not as it should be. Miguel de Cervantes (1547–1616)

Abstract In this chapter we introduce an unambiguous test to experimentally determine whether a given device is an ideal memristor or not. We demonstrate the test by applying it to resistive random-access memory (ReRAM) cells and the so-called “ memristor.” In both cases, the experimental test clearly shows that, unlike what has been claimed in the literature, these systems are not memristors. Finally, we formulate two conjectures concerning the impossibility of building a model of physical resistance-switching memories based on ideal memristors.

4.1 Back to Basic As we emphasized in Chap. 2, ideal memristors behave as those devices whose resistance, R(q), is a function of only the charge, q, flowing through them; cf. Eq. (2.8).1 This fundamental property follows directly from the memristor definition as a nonlinear element in the charge-flux conjugate variables (Chua 1971). In the previous Chap. 3 we have seen that such a device would suffer from severe physical constraints, which make it doubtful it could be easily realized in practice.2 1

We emphasize that Eq. (2.8) should be considered as a necessary but not sufficient condition for the device to be a memristor, while Eq. (2.7) is the sufficient one. In principle, Eq. (2.8) defines a class of devices behaving like a memristor. 2 We recall from Footnote 2 of Chap. 3 that the phase-dependent conductance of a Josephson junction (a component of the junction’s total current, Peotta and Di Ventra 2014) is the closest physical realization of an ideal memristor. However, we are not aware of experimental realizations of such a device, and most importantly, whether it would pass the memristor test (Pershin and Di Ventra 2019). © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Di Ventra and Y. V. Pershin, Memristors and Memelements, SpringerBriefs in Physics, https://doi.org/10.1007/978-3-031-25625-7_4

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4 Restoring Sanity: The Memristor Test

Now, the claim of “memristor discovery” in Strukov et al. (2008) was based on something different than the direct verification of the fundamental relation R(q). Why is that? And why was it not supported by the reconstruction—from the obtained current–voltage measurements—of the memristor curve in the charge-flux linkage plane? After all, if a device is introduced based on some fundamental property in the relation between its conjugate variables, such a relation needs to be experimentally verified before claiming that such a device is what we think it is. To clarify this point, consider diodes—two-terminal electronic components that are open to current flow in only one direction. From a classification point of view, diodes are essentially non-linear resistors. In fact, in the current–voltage plane, their response is represented by a well-defined non-linear curve; see Fig. 4.1 for an example. Now, say you are given a device and you are asked to prove that it is a diode. To answer such a question, you would certainly test its response to an arbitrary voltage or current waveform (in fact, several voltage and current waveforms) and make sure that it follows a well-defined curve in the current–voltage plane as the one displayed in Fig. 4.1. You would not certainly declare such a device a diode on the basis of some other response(s) or variables. The same procedure should be adopted for ideal memristors. These are also defined as non-linear elements but in a different coordinate system (Chua 1971): charge and flux-linkage plane. Therefore, the response of ideal memristors should be verified using the variables that define them. The same is true for other ideal memelements, such as memcapacitors and meminductors (Di Ventra et al. 2009). After all, as we have discussed at length in Sect. 2.6, showing a “pinched hysteresis loop” is not a valid test to verify if a given device is an ideal memristor (or memcapacitor, or meminductor), since such a feature appears also in non-ideal ones.

5

Current (mA)

Fig. 4.1 Experimentally measured current–voltage characteristics of an 1N4148 silicon diode. This plot shows that diodes can be classified as non-linear resistors

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4.2 The Memristor Test

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4.2 The Memristor Test Since the claim of the “memristor discovery” (Strukov et al. 2008) was not supported by the appropriate measurements of the conjugate variables of a memristor, and a “pinched loop” is not a defining property of this device, we offer a simple experimental test that can be performed to support (or not) such a claim (Pershin and Di Ventra 2019). The test is extremely simple—so simple, in fact, that it can be performed even in undergraduate physics and engineering labs—and its outcome is unambiguous. We then strongly suggest such a test should be done before any claim of “memristor discovery” is put forward in the literature. The test is as follows. Consider an ideal memristor that, at the initial moment of time, is characterized by a charge q0 and corresponding resistance R0 = R M (q0 ). By drawing charges q and − q through the memristor sequentially, say first we add a charge q, and then take out the same charge, we change the state of the memristor to q1 = q0 + q and back to q0 ; see the illustration in Fig. 4.2. Therefore, at the end of this experiment, the resistance of the memristor should return to its initial value, R0 . Since this type of behavior is unique to ideal memristors, it represents a solid foundation for the memristor test. Note that from the experimental point of view, it is more difficult to measure the charge compared to the current (one reason is because the charge measurement equipment is not widely available). To measure the charge directly, we need a device that collects the charge.

RM (q) -q

(q0, R0)

(q1, R1)

q q

Fig. 4.2 An example of memristor evolution. When the charges q and − q are drawn sequentially through an ideal memristor, its state and resistance must return to the initial values. Here, q0 is the initial charge (state), q1 = q0 + q is the intermediate charge, R0 and R1 are the initial and intermediate resistances, respectively

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4 Restoring Sanity: The Memristor Test

V(t)

RM (q)?

DUT

C

q

Fig. 4.3 Schematics of the memristor test circuit. Here, DUT denotes the device under test. The double arrowed line represents the duality between the capacitor charge and the resistance of the DUT if the latter is an ideal memristor. Examples of test voltage waveforms are shown on the right

Fortunately, such devices exist and are known as capacitors (cf. Sect. 2.1.2). The simplest capacitor involves two parallel plates separated by a dielectric medium or vacuum, see Fig. 2.2c. Capacitors store the charge on their plates that hold opposite charges. The memristor test circuit that we propose Pershin and Di Ventra (2019) then combines the device under test (DUT) and a capacitor; see Fig. 4.3. The essential property of the test circuit is the duality between the resistance of the memristor and the capacitor charge. By duality we mean the equivalence (correspondence) between these two quantities. Since the charge flowing though the memristor accumulates on the capacitor, knowledge of the capacitor charge defines completely the resistance of the memristor. Therefore, by charging and discharging the capacitor through the DUT we can directly implement the process depicted in Fig. 4.2. If the DUT is indeed an ideal memristor, its final resistance should always be equal to its initial value provided the capacitor charge returns to its initial value. In principle, this property should be verified for a sufficiently large representative set of initial states of the memristor and applied voltage waveforms. Memristor test: Important points • The return of the memristance to its initial value should be verified for a wide range of initial states of the memristor and applied voltage waveforms. • Particular care should be given to ensure that the capacitor charge returns to its initial value before the measurement of the final memristance. • A single negative result of the test is sufficient to conclude that the tested device is not a memristor.

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Fig. 4.4 Application of the memristor test to a CuSiO2 electrochemical metallization cell. a Measurement of the initial resistance state. b Application of the test waveform. c Measurement of the final resistance state. Reprinted with permission from Kim et al. (2020) (© 2020 WILEY-VCH Verlag GmbH & Co. (KGaA, Weinheim))

4.3 Experimental Demonstrations of the Test Having introduced the memristor test and how it can provide an unambiguous proof of whether a device is truly a memristor or not,3 we provide a few experimental demonstrations of the test for systems that were claimed to be memristors (Chua 2011, 2019; Wang et al. 2019). In all cases, the test fails, thus disproving the claims. In fact, we can say that at the time of the writing of this book: So far, no device has been shown to pass the memristor test.

We stress again that these results are experimental, not mathematical, hence they correspond to physical, objective reality!

4.3.1 ReRAM Devices Let us start with the application of the memristor test to several types of resistive random-access memory (ReRAM) cells (Kim et al. 2020). A bold claim that all these devices are memristors has been made in Chua (2011, 2019). Figure 4.4 presents an example of the test applied to a CuSiO2 electrochemical metallization (ECM) cell (Kim et al. 2020). This experiment was performed using a source measure unit (Keysight B2911A)—a piece of equipment that applies a voltage across a device and measures the current (or vice-versa). 3

In the sense of Eq. (2.8).

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4 Restoring Sanity: The Memristor Test

The left panel in Fig. 4.4a demonstrates the measurement of the initial state of the hypothetical memristor. In order to measure the state, the cell was directly connected to the source-measure unit and a small ramping voltage was applied. By using the slope of the current–voltage curve, the initial resistance was found to be R0 = 53 k. As a second step of the test, the circuit depicted in Fig. 4.3 was assembled with a capacitor of capacitance C = 10 µF. Initially, the capacitor was fully discharged. A triangular voltage pulse (see Fig. 4.4b) was applied, leaving a sufficient interval of time after the pulse to ensure that the capacitor was again fully discharged. After that, the final state of the device was measured (Fig. 4.4c). It was found that the final resistance is 19.5 k, which is significantly smaller than the initial resistance R0 = 53 k, far beyond experimental errors. Since the final resistance is not equal to the initial one—and the condition of the same initial and final capacitor charge were satisfied—this ECM cell has failed the test. To show that such an outcome is not atypical, rather is common to other types of ReRAM cells, the memristor test was also applied to commercially available ECM cells produced by Knowm, Inc. (BS-AF-W and M+SDC Cr devices) (Knowm Inc. 2020). Also these cells failed the test under a wide variety of driving conditions (Kim et al. 2020). Since the response of CuSiO2 and Knowm ECMs is typical of other ReRAM devices, the general conclusion is that ReRAM cells are not memristors. This result unambiguously disproves the previous claim that “resistance switching memories are memristors” (Chua 2011, 2019). It is worth noting that although these results have been published in 2020 (Kim et al. 2020), this erroneous statement is still propagating in the literature. For instance, the paper (Chua 2011) has been cited more than 100 times since the beginning of 2021 alone (from Google Scholar). We shall return to this “sociological,” and worrisome, problem in the next Chap. 5.

4.3.2 The So-Called “ Memristor” Few years ago another claim of “memristor discovery” was reported in the literature (Wang et al. 2019). The device (called by the authors a “ memristor”) was presented as a “real memristor” (Wang et al. 2019).4

4

After we submitted a comment on this paper, the latter has been retracted by the editors of Journal of Applied Physics on technical grounds (Wang et al. 2021), and our comment was never published.

4.3 Experimental Demonstrations of the Test

47

Fig. 4.5 Photograph of a segment of a magnetic-core memory sheet made in USSR in 1980s

500 μm

From a physical point of view, the “ memristor” consists of a current-carrying wire interacting with a magnetic core (Fig. 4.5). This is quite close to the design of a magnetic-core memory. While it should be immediately evident from its components that a wire interacting with a magnetic core—as in Fig. 4.5—is nothing other than an inductor with memory (a meminductive element, cf. Eq. 2.6), the memristor test was nonetheless applied to the “ memristor” (Pershin and Di Ventra 2019), both to clearly invalidate the claim in Wang et al. (2019) and to provide yet another realization of the test to a completely different structure. The main point is that the “ memristor” is not a resistance-switching device! Undeniably, the “ memristor” has an internal state that is physically encoded in the direction of magnetization of the magnetic core. However, unlike the case of ReRAM devices, the direction of magnetization is not directly accessible through low-signal measurements. To measure its state (or rather its flipping), we enhanced the “ memristor” with a pick-up coil, see Fig. 4.6b for an image of this system. The pick-up coil produces a voltage spike each time the direction of magnetization is reversed. The experimental setup used in the measurements was based on the circuit design of Fig. 4.3 and is reprinted in Fig. 4.6a. Figure 4.6b shows the actual experimental device used in our measurements. Figure 4.7 presents the most important results of the measurements. It shows that there is a single magnetization flip per each voltage pulse (negative or positive). Clearly, this type of behavior is not compatible with the memristor model since the capacitor charge is zero right before the beginning of each pulse. If the “ memristor” were indeed a real memristor we should observe two pulses across the pick-up coil per each voltage pulse, which is not what the experimental results of Fig. 4.7 show. Therefore, we can conclude—as we could have trivially done at the outset—that

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4 Restoring Sanity: The Memristor Test

R Pulse generator

Pick-up coil

Amplifier Digital oscilloscope

sw C

(a)

(b) Fig. 4.6 a Electronic circuit employed in the test of the “ memristor”. Reprinted from Pershin et al. (2022), Copyright 2022, with permission from Elsevier. b Photograph of the experimental device in a protective film

6

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4.4 Impossibility Conjectures

49

The hypothetical “ memristor” is not a memristor.

Once again, these experimental demonstrations clearly show the danger of making claims without characterizing the devices under investigation with the appropriate variables, based only on some “mathematical similarities” that bear little or no resemblance at all with physical reality.

4.4 Impossibility Conjectures Having demonstrated experimentally that resistive-switching devices are not ideal memristors (Kim et al. 2020), one could raise the following objection. One may argue that there is no such a thing as an ideal device. In fact, it is true that all physical (experimentally realizable) devices involve some deviations/unwanted behavior that makes their responses non-ideal. For instance, all metal/insulator/metal resistance-switching devices have some capacitive components—simply because of their geometry—that would appear under specific experimental conditions. Same for all capacitors: they would also showcase some parasitic inductive (and resistive) component, and so forth.5 However, a particular device can only be assigned (classified in) to a certain class if the deviation(s) of its response from the ideal one—characteristic of that class model—is (are) small. This is the rationale adduced by Chua to justify the claim that all ReRAM cells are memristors (Chua 2011). His argument is that small corrections to the ideal model are sufficient to explain the behavior of physical ReRAM devices. However, this conjecture has not resulted in any successful model so far. In other words, no model which includes small (perturbative) corrections to the ideal memristor response could satisfactorily account for the experimentally observed features of ReRAM cells (Valov et al. 2011; Pérez et al. 2019; Slesazeck and Mikolajick 2019). Based on experimental data on ReRAM switching and the limitations we have just mentioned, we have then formulated two conjectures about the impossibility of building a viable model of physical resistive-switching memory devices by using only small corrections to an ideal memristor model. Our memristor impossibility conjectures are as follows (Kim et al. 2020)6 :

5

The word “parasitic” is indeed used to indicate an undesired effect of an otherwise ideal element. We recall that according to the Oxford English Dictionary (2022), a “conjecture” is “the formation or offering of an opinion on grounds insufficient to furnish proof; the action or habit of guessing or surmising; conclusion as to what is likely or probable.”

6

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4 Restoring Sanity: The Memristor Test

First memristor impossibility conjecture: “It is impossible to accurately model physical resistance-switching memories by adding small corrections to the ideal memristor model.”

Second memristor impossibility conjecture: “It is impossible to accurately model physical resistance-switching memories by a circuit combining ideal memristors with any kind of non-linear ideal circuit elements.”

Before concluding this section let us make a few remarks regarding these conjectures. The first was formulated based on the fact that the response of physical ReRAM devices is too different from that of ideal elements. In other words, the response of real, experimentally realizable ReRAM devices showcases features which are substantially different from the ones of ideal memristors. In Chap. 3, we have already considered several important deficiencies of ideal models. It is highly unlikely that these deficiencies can be easily overcome by small (perturbative) corrections. The second conjecture has been formulated based on a similar reasoning but at the electronic circuit level. Here, we refer to the ideal elements defined in Chua (2003). It can be also formulated in the strong sense considering only the combinations of ideal memristors with non-linear resistors, capacitors, and inductors.

4.5 Ideal Memcapacitors and Meminductors We expect that similar conclusions can be reached for ideal memcapacitors, Eq. (2.9), and ideal meminductors, Eq. (2.10). However, this direction of study requires further research as the physics of memcapacitors and meminductors is substantially different from that of memristive devices as follows from Sects. 2.1 and 2.2. As we have mentioned in Sect. 2.4, a test similar to the one for ideal memristors can be formulated for ideal current-controlled meminductors. This is because these inductors are defined by the meminductance relation L(q), with q the charge that flows in the device; Eq. (2.10). For the other ideal memelements, however, the test based on the capacitor charge duality can not be used since they depend on variables other than the current. Therefore, other methods of integration (such as numerical) have to be employed to show that their response functions are functions of their respective variables.

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Again, the “pinched hysteresis loop” adduced as “proof” in some literature for the existence of such devices is not a valid test (cf. discussion in Sect. 2.6). We then leave this research direction for future work. A short summary • The mathematical definition of ideal memristors is unambiguous. Its resistance depends only on the charge that flows through it, and nothing else. • A device assigned to a given class should behave according to the class definition. • We have proposed the memristor test to distinguish ideal memristors from all other devices (with or without memory). • The experimental application of such a test to ReRAM cells and the socalled “ memristor” unambiguously demonstrates that such devices are not memristors. • Based on these results, two memristor impossibility conjectures have been formulated. • We expect similar conclusions to apply to ideal memcapacitors and meminductors.

References Chua L (2003) Nonlinear circuit foundations for nanodevices. I. The four-element torus. Proc IEEE 91(11):1830–1859 Chua L (2011) Resistance switching memories are memristors. Appl Phys A 102:765–783 Chua L (2019) Resistance switching memories are memristors. Springer International Publishing, Cham, pp 197–230 Chua LO (1971) Memristor—the missing circuit element. IEEE Trans Circuit Theory 18:507–519 Di Ventra M, Pershin YV, Chua LO (2009) Circuit elements with memory: memristors, memcapacitors, and meminductors. Proc IEEE 97:1717–1724 Kim J, Pershin YV, Yin M, Datta T, Di Ventra M (2020) An experimental proof that resistanceswitching memory cells are not memristors. Adv Electron Mater 2000010:1–6 Knowm Inc. (2020) Self directed channel memristors. https://knowm.org/downloads/Knowm_ Memristors.pdf. Accessed 20 Mar 2020 Oxford English Dictionary (2022) https://www.oed.com Peotta S, Di Ventra M (2014) Superconducting memristors. Phys Rev Appl 2:034011 Pershin YV, Di Ventra M (2019) A simple test for ideal memristors. J Phys D Appl Phys 52:01LT01 Pershin YV, Kim J, Datta T, Di Ventra M (2022) An experimental demonstration of the memristor test. Phys E Low-Dimens Syst Nanostruct 142:115290 Pérez E, Maldonado D, Acal C, Ruiz-Castro J, Alonso F, Aguilera A, Jiménez-Molinos F, Wenger C, Roldán J (2019) Analysis of the statistics of device-to-device and cycle-to-cycle variability in TiN/Ti/Al:HfO2 /TiN RRAMs. Microelectron Eng 214:104–109 Slesazeck S, Mikolajick T (2019) Nanoscale resistive switching memory devices: a review. Nanotechnology 30(35):352003

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Strukov DB, Snider GS, Stewart DR, Williams RS (2008) The missing memristor found. Nature 453(7191):80–83 Valov I, Waser R, Jameson JR, Kozicki MN (2011) Electrochemical metallization memoriesfundamentals, applications, prospects. Nanotechnology 22(25):254003 Wang FZ, Li L, Shi L, Wu H, Chua LO (2019)  memristor: real memristor found. J Appl Phys 125:054504 Wang FZ, Li L, Shi L, Wu H, Chua LO (2021) Retraction: “ memristor: real memristor found” [J. Appl. Phys. 125, 054504 (2019)]. J Appl Phys 129(2):029901

Chapter 5

The Sociology of the Field and Lesson to Be Learned

Whenever you find yourself on the side of the majority, it is time to pause and reflect. Mark Twain (1835–1910)

Abstract In this concluding chapter we summarize our thoughts on the field, its sociology, and where it could go from here. The main lesson we draw is that blind and unquestioned reliance on mathematical definitions to describe physical phenomena can easily lead to wrong conclusions and away from physical reality.

5.1 Physical Reality Versus Mathematics In this short book we have highlighted some of the misleading and wrong statements found in the vast literature of the field of memelements. The most outrageous of all are: • • • •

“The memristor is the fourth fundamental circuit element.” (Seriously doubtful.) “The missing memristor was found in 2008.” (Not at all.) “If it’s pinched it’s a memristor.” (Definitely not.) “All resistance switching memories are memristors.” (Demonstrably false.)

We have shown in this book that none of these statements are true! In fact, we can counter them with these physically correct ones: • The memristor has not been found, definitely not in 2008. • We have serious doubts such an ideal device would ever be found that passes our test (Pershin and Di Ventra 2019). • The word “memristor” may have been coined in 1971 (Chua 1971), but resistive memories, now called memristive elements, have been known long before then (Hickmott 1962; Kreinina et al. 1960).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Di Ventra and Y. V. Pershin, Memristors and Memelements, SpringerBriefs in Physics, https://doi.org/10.1007/978-3-031-25625-7_5

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• Indeed, any physical system showcases some degree of memory (time non-locality: another word for “memory”) under appropriate experimental conditions. This was known at least since the 1950s (Kubo 1957), and possibly even earlier. • It is just a matter of physical reality that any system (whether classical or quantum) subject to an external perturbation would respond by keeping track of their past states. Finally, the reader may ask what is the true relation between memristors and resistance switching memories cells. The answer is1 : ReRAM cells are poor emulators of memristors • ReRAM cells cannot be true memristors as they do not physically couple the flux linkage to the charge. • While some aspects of their response share similarities with those of memristors (e.g., the pinched hysteresis loops), they fail the fundamental response characteristic of memristors—the functional dependence of the resistance on charge.

We reiterate that the correct statements we have put forward in this book are not accurate because of some mathematical definitions or because they have been given to us by some authority (by fiat). They are correct because they correspond to an objective reality, the ultimate judge of any scientific statement.

5.2 The Sociology of the Field And yet, the wrong statements we have previously mentioned and discussed in this book appear unscrupulously in the literature. Why is that? We offer here some personal thoughts that may provide some partial answers. Part of it is certainly due to the clout of some of the researchers in the field who have pushed a particular narrative early on. In that respect, their propaganda has been very successful. Some of it may be related to the uncaring way in which many researchers copy those statements from previous publications without ever questioning their validity. After all, if they have been written in hundreds of publications they must be true, right? Or some of it could simply be due to the choice of the “path of least resistance.” It is certainly an easy way of life to go with the herd’s opinion: a contrarian risks ridicule, ostracism and in the case of modern science, loss of funding or promotions which are always tied to the recommendations of “expert” referees. 1

Emulators are components (which can also be circuits, programs or hardware/systems) that imitate the behavior of other components.

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Why risk it then? Our answer is: because in the end physical reality has always a way of emerging! No amount of mathematics or propaganda (which is typically a byproduct of politics) can change objective reality. Unfortunately, it seems some researchers in this field believe that we should proceed “by consensus” without regard for an objective reality. But that is not Science, it is a bad version of “religion.” That is why we can confidently say (Di Ventra 2018) Science is not a democracy: Nature rules!

It doesn’t matter what the majority claim, if their claims do not correspond to an objective, physical reality. It is the physical, objective world that ultimately decides on the correctness of a scientific statement, not the “consensus” of a group of scientists. If the reverse were true we would not have progress in Science at all!

5.3 Whispers in the Wind It would be unfair to conclude this book by not mentioning that many of the concerns we have raised here are also shared in part by a small but growing number of researchers. We cite here these papers Mouttet (2012), Meuffels and Soni (2012), Georgiou et al. (2014), Vongehr and Meng (2015), Sundqvist et al. (2017), Abraham (2018), Jeltsema and van der Schaft (2020) and Tellini et al. (2021), where some of the same issues have been discussed. However, there may be other publications we may have missed. In a way or another all these papers share the concern that the “ideal memristor” (as discussed in Sect. 2.4) still remains an elusive/idealized concept, and to a lesser extent also the ideal memcapacitors and meminductors. They also question the soundness of adding one or more “fundamental circuit elements” to the three we are all familiar with in circuit theory (Sect. 2.1). At the moment, these papers appear to be just “whispers,” or we should more properly say “whispers in the wind” if we consider the avalanche of papers that are produced in the field which perpetuate the wrong statements we mentioned above. It is our hope that we have given these researchers some exposure and recognition beyond what the community has done so far.

5.4 Moving Forward Finally, we reiterate that our goal for this book was not to diminish the scientifically solid work, both theoretical and experimental, that has been produced by many researchers in this field. Many experimental results and much theory that have been

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generated in the past decades are an incredible testament to the rich science that can be done by studying devices with memory. This fact, coupled with the wide-range technological applications of memelements, makes this field worth pursuing intellectually and investing on. We provide here some suggestions of particularly promising research directions based on our own personal taste and interest. A crucial task in this field is the creation of devices with repeatable (deterministic) response and small device-to-device variation. For application as storage devices, the desirable characteristics can be found in the International Roadmap for Devices and Systems (2021). Progress in devices is likely to require the use of novel materials, materials combinations, and/or physical approaches to the device operation. In fact, high-endurance resistive memories may replace the conventional DRAM2 that is volatile. This would redefine the way computers operate. For instance, it will not be necessary to reload the operating system each time the computer is turned on as the former would always stay in the memory. MemComputing is a word that we coined Di Ventra and Pershin (2013) to describe the plurality of computing approaches with memory (time non-locality). Its formal description has been put forward in Traversa and Di Ventra (2015).3 This type of computation differs substantially from our current model since the storing and processing of information is done on the same physical platform, rather than by spatially separate units. This feature combined with a massively-parallel architecture may eventually result in computing machines showing substantial advantages compared to our traditional computers. Many of the devices we have discussed in this book, although not necessary, could be useful for the hardware implementation of this new computing paradigm. In fact, significant progress has been made in a few related approaches, which are now under active exploration. These include neuromorphic computing (Pershin and Di Ventra 2010; Li et al. 2018), stochastic computing (Knag et al. 2014), reservoir computing (Kulkarni and Teuscher 2012; Zhong et al. 2021), and in-memory boolean computing (Borghetti et al. 2010; Kvatinsky et al. 2013). A particularly interesting related direction of study is the use of digital MemComputing machines (Traversa and Di Ventra 2017) in the solution of combinatorial optimization problems. In fact, the emulation of such machines is already employed in industrial applications (see https://www.memcpu.com/ and Di Ventra 2022 for more details). While most of the above-mentioned studies have relied on some combination of memristive devices with transistors and other memory-less circuit elements, some ideas based on memcapacitive elements have been explored (Pershin and Di Ventra

2

DRAM stands for “dynamic random-access memory” which is used to store programs and data in conventional computers. 3 The “Mem” in MemComputing does not stand for “memristors” or “memelements,” but rather for “memory” which as we have discussed in Sect. 1.1 is a non-equilibrium (dynamical) property of general physical systems. See Di Ventra (2022) for more details.

5.5 The Ultimate Lesson

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2014; Traversa et al. 2014; Demasius et al. 2021). Potentially, computing architectures based on memcapacitive and meminductive elements may consume less power compared to memristive ones.

5.5 The Ultimate Lesson We hope that this book has clarified that certain (very common) statements found in the literature of memristors and memelements have little to do with physical reality, and simply distract, or worse detract, from an otherwise flourishing field which has a strong potential for growth. We do not hold our breath that things will change immediately after the publication of this book. Of course, we expect resistance, criticism and doubt. However, as we have mentioned in the Preface, we feel an obligation to the truth. In fact, our thoughts and concerns go especially to the future generations of scientists and engineers, not just the present ones. We hope our book will at least provide them with the following lesson: The blind and unquestioned reliance on mathematical definitions to describe physical phenomena can easily lead to wrong conclusions and away from physical reality.

Face reality as it is, not as it was or as you wish it to be. Jack Welch (1935–2020)

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References Abraham I (2018) The case for rejecting the memristor as a fundamental circuit element. Sci Rep 8(1):1–9 Borghetti J, Snider GS, Kuekes PJ, Yang JJ, Stewart DR, Williams RS (2010) ‘Memristive’ switches enable ‘stateful’ logic operations via material implication. Nature 464(7290):873–876 Chua LO (1971) Memristor—the missing circuit element. IEEE Trans Circuit Theory 18:507–519 Demasius KU, Kirschen A, Parkin S (2021) Energy-efficient memcapacitor devices for neuromorphic computing. Nat Electron 4(10):748–756 Di Ventra M (2018) The scientific method: reflections from a practitioner. Oxford University Press, Oxford Di Ventra M (2022) MemComputing: fundamentals and applications. Oxford University Press, Oxford Di Ventra M, Pershin YV (2013) The parallel approach. Nat Phys 9:200 Georgiou PS, Barahona M, Yaliraki SN, Drakakis EM (2014) On memristor ideality and reciprocity. Microelectron J 45(11):1363–1371 Hickmott TW (1962) Low-frequency negative resistance in thin anodic oxide films. J Appl Phys 33(9):2669–2682 International Roadmap for Devices and Systems (IRDSTM ) (2021) 2021 edn. https://irds.ieee.org/ editions/2021 Jeltsema D, van der Schaft A (2020) Ideal memcapacitors and meminductors are overunity devices. Sci Rep 10(1):1–8 Knag P, Lu W, Zhang Z (2014) A native stochastic computing architecture enabled by memristors. IEEE Trans Nanotechnol 13(2):283–293 Kreinina G, Selivanov L, Shumskaya T (1960) Emission and conductivity of the capacitor type cathode. Radio Eng Electron 5(8):1338–1341 Kubo R (1957) Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems. J Phys Soc Jpn 12(6):570–586 Kulkarni MS, Teuscher C (2012) Memristor-based reservoir computing. In: 2012 IEEE/ACM international symposium on nanoscale architectures (NANOARCH). IEEE, pp 226–232 Kvatinsky S, Satat G, Wald N, Friedman EG, Kolodny A, Weiser UC (2013) Memristor-based material implication (imply) logic: design principles and methodologies. IEEE Trans Very Large Scale Integr (VLSI) Syst 22(10):2054–2066 Li Y, Wang Z, Midya R, Xia Q, Yang JJ (2018) Review of memristor devices in neuromorphic computing: materials sciences and device challenges. J Phys D Appl Phys 51(50):503002 Meuffels P, Soni R (2012) Fundamental issues and problems in the realization of memristors. arXiv preprint arXiv:1207.7319 Mouttet B (2012) Memresistors and non-memristive zero-crossing hysteresis curves. arXiv preprint arXiv:1201.2626 Pershin YV, Di Ventra M (2010) Experimental demonstration of associative memory with memristive neural networks. Neural Netw 23(7):881–886 Pershin YV, Di Ventra M (2014) Memcapacitive neural networks. Electron Lett 50(3):141–143 Pershin YV, Di Ventra M (2019) A simple test for ideal memristors. J Phys D Appl Phys 52:01LT01 Sundqvist KM, Ferry DK, Kish LB (2017) Memristor equations: incomplete physics and undefined passivity/activity. Fluct Noise Lett 16:1771001 Tellini B, Bologna M, Chandia K, Macucci M (2021) Revisiting the memristor concept within basic circuit theory. Int J Circuit Theory Appl 49:3488–3506 Traversa FL, Di Ventra M (2015) Universal MemComputing machines. IEEE Trans Neural Netw Learn Syst 26(11):2702 Traversa FL, Di Ventra M (2017) Polynomial-time solution of prime factorization and NP-complete problems with digital MemComputing machines. Chaos Interdiscipl J Nonlinear Sci 27:023107 Traversa FL, Bonani F, Pershin YV, Di Ventra M (2014) Dynamic computing random access memory. Nanotechnology 25:285201

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Vongehr S, Meng X (2015) The missing memristor has not been found. Sci Rep 5:11657 Zhong Y, Tang J, Li X, Gao B, Qian H, Wu H (2021) Dynamic memristor-based reservoir computing for high-efficiency temporal signal processing. Nat Commun 12(1):1–9

Index

A Aristotle, viii

B Bi-state system, 25 Brownian particle, 33

C Capacitance, 16 Capacitor, 16, 22 ferroelectric, 24 Circuit element, 13 fundamental, 14, 19 Current density, 16

D Diffusive loss of information, 33 Diode, 42 Duality, 44

E Electric field, 16 Electrodynamics, 13 symmetries, 35 Electromotive force, 16 Energy landscape, 7

F Faraday’s law, 16 Fiction, 13, 19 Fluctuation, 30 Flux, 19

H Heat, 16 minimal generation, 34 Hysteresis loop pinched, 23, 51 twisted, 24

I Inductance, 16 parasitic, 49 Inductor, 16, 22

L Landauer principle, 34

M Mathematics, 53 Maxwell’s equations, 13 Measurement, 8 Memcapacitance, 18 Memcapacitive element, 18 Memcapacitor, 50 charge-controlled, 21 voltage-controlled, 21 MemComputing, 56 Memductance, 18 Memelement, 1, 5 ideal, 19 (see also memory element), 5 stochastic, 36 Meminductance, 19 Meminductive element, 19, 47 Meminductor, 50 current-controlled, 21

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Di Ventra and Y. V. Pershin, Memristors and Memelements, SpringerBriefs in Physics, https://doi.org/10.1007/978-3-031-25625-7

61

62 flux-controlled, 22 Memory, 1, 54 capacity, 6 definition, 1 degrees of freedom, 4 element, 5 long-term, 2 trace, 7 Memristance, 9, 18 Memristive element, 18 response, 35 Memristor, 20, 29, 41 “”, 46 discovery, 42, 43, 46, 53 impossibility conjecture first, 50 second, 50 test, 21, 43

Index Physical reality, 53 Probe non-demolition, 7

R ReRAM, 7, 45, 54 Resistance, 16 parasitic, 49 Resistivity, 16 Resistor, 14, 22 non-linear, 25, 42 Response function, 2, 8, 16 time-local, 3 time non-local, 3

O Observable, 8 Ohm’s law, 16, 17

S Sociology of the field, 54 State, 2 attractor, 7 diffusion, 32 insulating, 9 superconducting, 8 State variables, 4 Stochastic catastrophe, 32 Storage, 30 Storage (see long-term memory) Switching over-delayed, 34

P Passivity criterion, 24 Permittivity, 23

T Thermistor, 30 Time non-locality (see memory)

N Noise Gaussian white, 33 Johnson-Nyquist, 33