Mem-elements for Neuromorphic Circuits with Artificial Intelligence Applications

Mem-elements for Neuromorphic Circuits with Artificial Intelligence Applications illustrates recent advances in the field of mem-elements (memristor, memcapacitor, meminductor) and their applications in nonlinear dynamical systems, computer science, analog and digital systems, and in neuromorphic circuits and artificial intelligence. The book is mainly devoted to recent results, critical aspects and perspectives of ongoing research on relevant topics, all involving networks of mem-elements devices in diverse applications. Sections contribute to the discussion of memristive materials and transport mechanisms, presenting various types of physical structures that can be fabricated to realize mem-elements in integrated circuits and device modeling.

As the last decade has seen an increasing interest in recent advances in mem-elements and their applications in neuromorphic circuits and artificial intelligence, this book will attract researchers in various fields.

• Covers a broad range of interdisciplinary topics between mathematics, circuits, realizations, and practical applications related to nonlinear dynamical systems, nanotechnology, analog and digital systems, computer science and artificial intelligence
• Presents recent advances in the field of mem-elements (memristor, memcapacitor, meminductor)
• Includes interesting applications of mem-elements in nonlinear dynamical systems, analog and digital systems, neuromorphic circuits, computer science and artificial intelligence

• Language: English
• Publisher: Academic Press
• Release date: Jun 17, 2021
• ISBN: 9780128232026

Book preview

Part I: Mem-elements and their emulators

Outline

Chapter 1. The fourth circuit element was found: a brief history

Chapter 2. Implementing memristor emulators in hardware

Chapter 3. On the FPGA implementation of chaotic oscillators based on memristive circuits

Chapter 4. Microwave memristive components for smart RF front-end modules

Chapter 5. The modeling of memcapacitor oscillator motion with ANN and its nonlinear control application

Chapter 6. Rich dynamics of memristor based Liénard systems

Chapter 7. Hidden extreme multistability generated from a novel memristive two-scroll chaotic system

Chapter 8. Extreme multistability, hidden chaotic attractors and amplitude controls in an absolute memristor Van der Pol–Duffing circuit: dynamical analysis and electronic implementation

Chapter 9. Memristor-based novel 4D chaotic system without equilibria

Chapter 10. Memristor Helmholtz oscillator: analysis, electronic implementation, synchronization and chaos control using single controller

Chapter 11. Design guidelines for physical implementation of fractional-order integrators and its application in memristive systems

Chapter 12. Control of bursting oscillations in memristor based Wien-bridge oscillator

Chapter 1: The fourth circuit element was found: a brief history

Christos Volosa; Viet-Thanh Phamb,c; Hector Nistazakisd    aLaboratory of Nonlinear Systems, Circuits and Complexity (LaNSCom), Department of Physics, Aristotle University of Thessaloniki, Thessaloniki, Greece

bFaculty of Electrical and Electronic Engineering, Phenikaa Institute for Advanced Study (PIAS), Phenikaa University, Hanoi, Vietnam

cPhenikaa Research and Technology Institute (PRATI), A&A Green Phoenix Group, Hanoi, Vietnam

dFaculty of Physics, Department of Electronics, Computers, Telecommunications and Control, National and Kapodistrian University, Athens, Greece

Abstract

In this chapter, the historic route from the prediction of the existence of the memristor in 1971 to the fabrication of the first physical model in 2008, is presented. In more detail, the first part of this chapter highlights the importance of breakthrough discovery of the memristor by Prof. Leon Chua as the fourth fundamental circuit element. Next, the basic properties of this new element, as well as its characteristic fingerprint, are discussed. The introduction, by Chua and Kang in 1976, of a more generalized class of systems, in regard to the original definition of a memristor, which are called memristive systems, is also presented. The realization of the first physical model of the memristor in the laboratories of Hewlett–Packard and its mathematical description are also discussed. Finally, the importance of this new element due to its possible applications is highlighted.

Keywords

Circuit theory; memristor; memristive systems; memory effect; nonlinear element; pinched hysteresis loop

1.1 Memristor – the first step

The history of electrical circuits' theory has been written based on the three fundamental passive elements, capacitor, resistor, and inductor. The capacitor, was the first circuit element that was discovered in 1745 by the German Ewald Georg von Kleist, who found that charge could be stored by connecting a high-voltage electrostatic generator by a wire to a volume of water in a hand-held glass jar [1]. Next year, the Dutch physicist Pieter van Musschenbroek invented a similar capacitor, which was named the Leyden jar, after the University of Leyden, where he worked [2]. Daniel Gralath was the first to combine several jars in parallel to increase the charge storage capacity [3], while the famous Benjamin Franklin investigated further the Leyden jar and he adopted the term battery [4].

In 1826, Georg Simon Ohm gave a mathematical description of conduction in circuits modeled of Fourier's study of heat conduction. This work continued Ohm's deduction of results from experimental evidence. He was able to propose laws, which went a long way to explaining results of others working on galvanic electricity. This work certainly was a first step in a comprehensive theory, which Ohm was able to give in his famous book, published in the following year, called Die Galvanische Kette, mathematisch bearbeitet [5] which means Galvanic Chain, Mathematically Worked and contained what is known today as the "Ohm laws". In more detail, he investigated that the flow of electric current through a conductor experiences a certain amount of resistance. The magnitude of resistance is dictated by electric properties of the material and by the material geometry. Henceforth, the conductors that exhibit the property of resisting current flow are called resistors.

Inductance was first found by Michael Faraday in 1831, in a simple yet strange way [6]. He wrapped a paper cylinder with wire, attached the ends of the wire to a galvanometer and moved a magnet in and out of the cylinder. The galvanometer reacted to this, revealing the production of a small current. Shortly after this discovery, Reverend Nicholas Calland invented the inductor coil [7]. The earliest version of the inductor consisted of a coil with two terminals at the ends which stored energy inside a magnetic field when a current passed through the coil.

Therefore, from this last invention and for almost 140 years the electrical circuits' theory was spinning around these three basic passive elements. Then in 1971, Leon Chua, a Professor of electrical engineering at the University of California, Berkley, predicted the existence of a fourth fundamental element. Chua became famous for his work on nonlinear circuit theory, as in the decade of 1960 he had established its mathematical foundation [8]. It was an exciting period for electronics engineers, who were working on exotic nonlinear devices, such as the Esaki diode, Josephson junction, varactor and thyristor. For Chua's work on nonlinear circuits and elements, he is acknowledged as the father of nonlinear circuit theory and a large number of awards have been awarded to him, including the IEEE Gustav Robert Kirchhoff Award, a number of other major awards, nine honorary doctorates at major universities around the world and numerous visiting professorships [8].

Unfortunately, during that period circuit theory was concerned only with the prediction of the voltage and current associated with the external terminals of the linear and nonlinear devices interconnected in a circuit and not with the internal physical variables associated with the individual devices in the circuit. Thus, basic nonlinear circuit elements were defined from a black box perspective, independent of their internal composition, material, geometry and architecture.

The basic principles that would predict the fourth fundamental circuit element, the memristor, were first reported by Chua in 1969 in a book [9], and it took him a year to derive and prove mathematically the unique theoretic properties and memory attributed to the memristor. Next year, Chua submitted a seminal paper, which presented the results of his work [10]. As he claimed from the circuit-theoretic point of view, the three fundamental two-terminal circuit elements are defined in terms of a relationship between two of the four fundamental circuit variables, namely the current i, the voltage v, the charge q, and the flux-linkage φ. Out of the six possible combinations of these four variables, five have led to well-known equations. In more detail, two of these relationships are already given by Eqs. (1.1)–(1.2), while three other equations are given, respectively, by the axiomatic definition of the three classical circuit elements, namely, the resistor (R), the capacitor (C) and the inductor (L) (see Eqs. (1.3)–(1.5)). Only one relationship remained undefined, which was the relationship between φ and q (Fig. 1.1). We have

(1.1)

(1.2)

(1.3)

(1.4)

(1.5)

Figure 1.1 The three fundamental two-terminal circuit elements and the fourth unknown element till the discovery by Prof. Leon Chua. Circles represent Eqs. (1.1)– (1.5), which combine the four fundamental circuit variables.

Therefore, Chua concluded that "From the logical as well as axiomatic points of view, it is necessary for the sake of completeness to postulate the existence of a fourth basic two-terminal circuit element which is characterized by a φ–q curve" [10].

This element was named, by Chua, memristor because it behaves somewhat like a nonlinear resistor with memory. The proposed symbol of the memristor is shown in Fig. 1.2.

Figure 1.2 The symbol of the memristor.

1.2 Properties of memristor

Also, Chua in his seminal paper [10] claimed, according to the aforementioned analysis, that by definition a memristor is characterized by a relation of the type:

(1.6)

So, a memristor is called charge-controlled or flux-controlled, if this relation can be expressed as a single-valued function of the charge q or flux φ, respectively. The voltage across a charge-controlled memristor is given by

(1.7)

where

(1.8)

which is called the incremental memristance, since it has the unit of resistance.

In the same way the current through a flux-controlled memristor is given by

(1.9)

where

(1.10)

which is called incremental memductance, since it has the unit of conductance.

From Eqs. (1.8) and (1.10) the reader can observe that the value of the incremental memristance or memductance at any time depends upon the time integral of the memristor's current or voltage, respectively, from to . Therefore, while the memristor behaves like an ordinary resistor at a given instant of time , its resistance or conductance depends on the complete past history of the memristor current or voltage, respectively. This observation motivated Chua to give the name memory resistor, or memristor to this new circuit element. Also, Chua has proved theoretically that a memristor is a nonlinear element because its i–v characteristic is similar to that of a Lissajous pattern. So, as the memristor's voltage or current is specified, the memristor behaves like a linear time-varying resistor. Furthermore, when the memristor's φ–q characteristic curve is a straight line, then , or , and the memristor reduces to a linear time-invariant resistor.

1.2.1 Passivity criterion

In this section the class of memristors, which might be discovered as a physical element without power supply, is investigated through the following passivity criterion according to Chua's analysis in his seminal paper [10].

Theorem

A memristor is characterized by a differentiable charge-controlled φ–q curve is passive if, and only if, its incremental memristance is nonnegative; i.e., .

Proof

The power dissipated by a memristor is given by

(1.11)

By using Eq. (1.7) in Eq. (1.11), the power is

(1.12)

Therefore, if , then and the memristor is passive. For proving the converse the existence of a value in order to suggest that is considered. In this case the differentiability of the φ–q curve implies that there exists an such that , with . By driving the memristor with a current , which is zero for , and such that for , then for sufficiently large t, and hence the memristor is active. □

The passivity criterion shows that only memristors characterized by a monotonically increasing φ–q curve can exist in a device form without internal power supplies.

1.2.2 Closure theorem

Another interesting theorem related with memristor, which has been formulated by Chua in his paper [10], is the closure theorem.

Theorem

A one-port containing only memristors is equivalent to a memristor.

Proof

If we let , , , and denote the current, voltage, charge, and flux of the jth charge-controlled memristor, where , and if we let i and v denote the port current and port voltage of the one-port, then independent Kirchhoff current law equations can be written (assuming the network is connected). So,

(1.13)

where is either 1, −1, or 0 and n is the total number of nodes.

In the same way a system of independent Kirchhoff voltage law equations can be written:

(1.14)

where is either 1, −1, or 0.

By integrating Eqs. (1.13) and (1.14) with respect to time and substituting for in the resulting expressions, we obtain

(1.15)

(1.16)

where and are arbitrary constants of integration. Eqs. (1.15) and (1.16) together constitute a system of independent nonlinear functional equations with unknowns. Hence, solving for φ, we obtain a relation . □

1.2.3 Existence and uniqueness theorem

Finally, Chua has proved the theorem of existence and uniqueness in the case of memristors [10]. According to this, we have the following result.

Theorem

Any network containing only memristors with positive incremental memristances has one, and only one, solution.

Proof

For the proof of this theorem the reader can see the corresponding proof in Ref. [11] for a network containing only nonlinear resistors. □

Therefore, Chua has proved theoretically that a memristor is a nonlinear element because its i–v characteristic is similar to that of a Lissajous pattern. So, a memristor with a non-constant M describes a resistor with a memory, more precisely a resistor which resistance depends on the amount of charge that has passed through the device. Typical responses of a memristor to sinusoidal inputs are depicted in Fig. 1.3. The "pinched hysteresis loop current–voltage characteristic" is an important fingerprint of a memristor. If any device has an i–v hysteresis curve like this, then it is either a memristor or a memristive device. Another signature of the memristor is that the pinched hysteresis loop shrinks with the increase of the excitation frequency. The fundamentality of the memristor can also be deduced from Fig. 1.3, as it is impossible to make a network of capacitors, inductors and resistors with an i–v behavior forming a pinched hysteresis curve.

Figure 1.3 Typical i – v characteristic curves of a memristor driven by a simusoidal voltage input of various frequencies.

In conclusion, some of the more interesting properties of memristor are:

•  Non-linear relationship between current (i) and voltage (v).

•  Similar to classical circuit elements, a system of memristors can also be described as a single memristor.

•  Reduces to resistor for large frequencies as evident in the i–v characteristic curve.

•  Does not store energy.

•  Memory capacities based on different resistances produced by the memristor.

•  Non-volatile memory is possible if the magnetic flux and charge through the memristor have a positive relationship ( ).

1.3 Memristive systems

In 1976 Chua and Kang introduced a more generalized class of systems, in regard to the original definition of a memristor, called memristive systems [12]. These systems have a state variable indicated by w that describes the physical properties of the device at any time. A memristive system is characterized by two equations,

•  the quasi-static conduction equations that relate the voltage across the device to the current through it at any particular time and

•  the dynamical equation, which explicitly asserts that the state variable w is a time varying function f of itself and possibly the current (or voltage) through the device.

An nth-order current-controlled memristive one-port is represented by

(1.17)

where is the n-dimensional state variable of the system.

Also, the nth-order voltage-controlled memristive one-port is defined as

(1.18)

These systems are unconventional in the sense that, while they behave like resistive devices, they can be endowed with a rather exotic variety of dynamic characteristics.

Therefore, similarly to a memristor, a memristive system has the following properties:

•  The memristive system should have a dc curve passing through the origin.

•  For any periodic excitation the i–v characteristic curve should pass through the origin.

•  As the excitation frequency increases toward infinity the memristive system has a linear behavior.

•  The small signal impedance of a memristive system can be resistive, capacitive, or inductive depending on the operating bias point.

1.4 The first physical model of the memristor

Unfortunately, research on memristor was given a lower priority not only by Chua, who had already been awarded the W.R.G. Baker Award by the Institute of Electrical and Electronics Engineers for the potential of the memristor in 1973, but also by the whole research community. This happened due to the fact that the memristor was an exotic new circuit element, as research in circuit theory had been dominated mainly by linear networks. Also, it was not surprising that this element was not even discovered in a device form because it was unnatural to associate charge with flux. As a proof of principle, only three memristor models using operational amplifiers and off-the-shelf electronic components had been presented by Chua in his seminal paper [10]. However, the challenge of fabricating a passive monolithic memristor remained unfulfilled till 2008.

That year, Hewlett–Packard scientists, working at their laboratories in Palo Alto California, announced in Nature [13] that a physical model of memristor had been realized. This research team began reading Chua's papers and trying to understand what was the cause of the pinched hysteresis loops in the devices that they had already made. The big breakthrough and their most significant contribution came in 2006 when they realized that the time derivative of the state variable in Chua's dynamical state equation was comparable to the drift velocity of oxygen vacancies in a titanium dioxide resistive switch.

In more detail, in their scheme, a memory effect is achieved in solid-state thin film two terminal passive device. The memristor, which was realized by HP researchers, is made of a titanium dioxide layer which is located between two platinum electrodes (Fig. 1.4). This layer is of the dimension of several nanometers and if an oxygen dis-bonding occurs, its conductance will rise instantaneously. However, without doping, the layer behaves as an isolator. The area of oxygen dis-bonding is referred to as space-charge region and changes its dimension if an electrical field is applied. This is done by a drift of the charge carriers. The smaller the insulating layer, the higher the conductance of the memristor. Also, the tunnel effect plays a crucial role. Without an external influence the extension of the space-charge region does not change.

Figure 1.4 Cross-section of the first HP memristor consisting of a high conductive (doped) and a low conductive (undoped) part placed in between two platinum electrodes. The boundary between the two parts is dynamic and is moved back and forth by the passing charge carriers.

The internal state x is the extent of the space-charge region, which is restricted in the interval and can be described by the equation

(1.19)

where w is the absolute extent of the space-charge region and D is the absolute extent of the titanium dioxide layer. The memristance can be described by the following equation:

(1.20)

where Ron is the resistance of the maximum conducting state and Roff represents the opposite case. Therefore, when , off, and when , on. The vector containing the internal states of the memristor is one dimensional. For this reason scalar notation is used.

The state equation is

(1.21)

where is the oxygen vacancy mobility and is the current through the device. By using Eq. (1.19) the previous equation can be rewritten as

(1.22)

The dynamics of the memristor can therefore be modeled through the time dependence of the width w of the doped region. Integrating Eq. (1.22) with respect to time,

(1.23)

where is the initial width of the doped region at and q is the amount of charges that have passed through the device. Substituting (1.19), (1.22) into Eq. (1.20) gives

(1.24)

where

(1.25)

and . The term refers to the net resistance at that serves as the device's memory. This term is associated with the memristive state, which is essentially established through a collective contribution, i.e. it depends directly on the amount of all charges that have flown through the device. As a result, we can say that the memristor has the feature to remember whether it was on or off after its power is turned on or off, respectively.

1.5 Memristor's applications

The announcement of the fabrication of the first physical memristor brought a revolution in various scientific fields, as many phenomena in systems, such as in thermistors, of which the internal state depends on temperature [14], spintronic devices, which resistance varies according to their spin polarization [15], and molecules whose resistance changes according to their atomic configuration [16], could be explained now with the use of the memristor. Also, electronic circuits with memory circuit elements could simulate processes typical of biological systems, such as learning and associative memory [17] and the adaptive behavior of unicellular organisms [18].

Therefore, due to the fact that memristor is a two terminal and variable resistance element, it could be used for the following applications.

•  Non-volatile memory applications (NVRAMs) [19]. Memristors can retain memory states, and data, in power-off modes. Non-volatile random access memory is pretty much the first to-market memristor application that will be seen.

•  In remote sensing and low-power applications[20,21]. Coupled with memcapacitors and meminductors, the complementary circuits to the memristor, which allow the storage of charge, memristors can possibly allow for nano-scale low-power memory and distributed state storage, as a further extension of NVRAM capabilities.

•  Crossbar latches as transistor replacements or augmentors[22]. Solid-state memristors can be combined into devices called crossbar latches, which could replace transistors in future computers, taking up a much smaller area.

•  Analog computation and circuit applications[23]. Analog computations embodied a whole area of research which, unfortunately, were not as scalable, reproducible, or dependable as digital solutions. Memristor applications will now allow us to revisit a lot of the analog science that was abandoned in the mid-1960s.

•  Programmable logic and signal processing[24,25]. The memristive applications in these areas will remain relatively the same, because it will only be a change in the underlying physical architecture, allowing their capabilities to expand.

•  Analog filter applications[26]. The memristors can be programmed with low frequency pulses to set its resistance using trick to separate high frequency signal path from the programming pulses, alike to those tricks used with varicap to separate a high frequency signal path from bias voltage. In this way the circuit could be reconfigured while running.

•  Circuits which mimic neuromorphic and biological systems[27,28]. This is a very large area of research, because a large part of the analog science, given in detail above, has to do with advances in cognitive psychology, Artificial Intelligence (AI) modeling, machine learning and recent neurology advances. The ability to map peoples' brain activities under MRI, CAT, and EEG scans is leading to a treasure trove of information about how our brains work. Simple electronic circuits based on an LC network and memristors have been built and used recently to model experiments on adaptive behavior of unicellular organisms. Some types of learning circuits find applications anywhere from pattern recognition to Artificial Neural Networks (ANNs).

1.6 Conclusion

A brief history of the invention in 1971 of the fourth fundamental circuit element, which was called memristor, as well as its fabrication 37 years later, has been presented in this chapter. Also, the basic properties of this new element and especially its characteristic fingerprint, the pinched hysteresis i–v characteristic curve, have been discussed. Furthermore, the more generalized class of systems, in regard to the original definition of a memristor, called memristive systems, were mathematically described. Finally, the applications of this new element in various fields have been highlighted.

References

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Chapter 2: Implementing memristor emulators in hardware

Stavros G. Stavrinidesa; Rodrigo Picosb; Fernando Corintoc; M. Moner Al Chawad; Carola de Benitoe    aSchool of Science and Technology, International Hellenic University, Thessaloniki, Greece

bIndustrial Engineering and Construction Department, University of Balearic Islands, Palma, Spain

cDepartment of Electronics and Telecommunications, Politecnico di Torino, Turin, Italy

dChair of Fundamentals of Electrical Engineering, Technical University of Dresden, Dresden, Germany

ePhysics Department, University of Balearic Islands, Palma, Spain

Abstract

Memristor devices are assumed to be one of the next key technology enablers for many applications, ranging from neural networks and artificial intelligence to the IoT, even sensing elements. However, this technology is not yet mature enough, therefore, these devices are not widely available. Thus, currently, development of new applications can be achieved either by simulation or by utilizing emulators.

In this work we present a general framework on how to implement memristor emulators in hardware. Initially, design and development of passive memristive emulators, using only passive elements, is presented. Then, this framework is further enhanced to include active devices (MOS transistors), as well; and this due to memory retention capabilities improvement, while keeping the passive operation mode of the whole system.

Specific examples are presented, regarding short- and long-term memory, passive memristor emulators. Finally, implementation of flux-controlled, passive, analog memristor emulator, in a standard commercial CMOS technology is described. The proposed memristor emulator has been tested by driving it with both sinusoidal signals and pulses, checking its analog behavior against established memristor signatures and its memory-retain capabilities. Simulation results show the quality of the design, demonstrating all the required fingerprints of an analog long-term memory memristor.

Keywords

Memristors; memristor emulators; design methodology; long-term memory; nonlinear circuits; CMOS

2.1 Introduction

The apparent symmetry between the relations of the four fundamental electrical magnitudes, namely the current i, the voltage v, the charge q and the flux φ, was a pattern that passed unnoticed for many years in circuit theory. It was this idea that led Leon Chua, in the early 1970s, to present the axiomatic introduction and the related description of a fourth (missing at that moment) electrical element, named the memristor [1]. Its name originated from the fact that such an element should behave as a resistor endowed with memory, having these two properties, resistance and the feature of memory, being unified in one element. In fact, memristors had been described many years ago [2], though they had never been in the mainstream of electrical or circuit theory. Besides, Chua's work led to the generalization of a class of devices as well as systems that are inherently nonlinear and governed by a state-dependent, algebraic relation accompanied by a set of differential equations, which are called memristive systems or devices [3].

As a result of the inherent memory feature embodied in memristors, these novel devices are expected to be one of the key technology enablers of a technological breakthrough in integrated circuit (IC) performance growth, beyond and more than Moore [4]. Among others, they are expected to provide us with a solution to the classical problem of the bottleneck in data transmission between memories and processors. The IoT and other edge computing applications are expected to be areas where the introduction of memristors and memristive devices would be beneficial, if not changing radically the related technological landscape. Thus, an increasing number of memristor-based applications has already been proposed; indicatively, new kind of memories (ReRAMs, MRAM, etc.) [5–7], innovative new sensor devices [8,9], or fundamental elements in bio-inspired systems (ANNs)