Revisiting the Memristor Concept within Basic Circuit Theory
Bernardo Tellini, Mauro Bologna, Kristopher J. Chandia, Massimo Macucci
aa r X i v : . [ ee ss . S Y ] F e b Revisiting the Memristor Concept within BasicCircuit Theory
Bernardo Tellini,
Senior Member, IEEE,
Mauro Bologna, Kristopher J. Chand´ıa,
Member, IEEE, andMassimo Macucci
Member, IEEE
Abstract —In this paper we revisit the memristor conceptwithin circuit theory. We start from the definition of the basiccircuit elements, then we introduce the original formulation of thememristor concept and summarize some of the controversies onits nature. We also point out the ambiguities resulting from a nonrigorous usage of the flux linkage concept. After concluding thatthe memristor is not a fourth basic circuit element, promptedby recent claims in the memristor literature, we look intothe application of the memristor concept to electrophysiology,realizing that an approach suitable to explain the observedinductive behavior of the giant squid axon had already beendeveloped in the 1960s, with the introduction of “time-variantresistors.” We also discuss a recent memristor implementationin which the magnetic flux plays a direct role, concluding thatit cannot strictly qualify as a memristor, because its v − i curvecannot exactly pinch at the origin. Finally, we present numericalsimulations of a few memristors and memristive systems, focusingon the behavior in the ϕ − q plane. We show that, contrary towhat happens for the most basic memristor concept, for generalmemristive systems the ϕ − q curve is not single-valued or noteven closed. Index Terms —Circuit theory, Memristive Systems, Memristor,Numerical Simulations.
I. I
NTRODUCTION T HE concept of memristor was first introduced byChua [1] in 1971 as a two-terminal circuit element estab-lishing a relationship between the charge (which is the integralof the current) and the integral of the voltage (which can alsobe defined as “flux linkage”). Already in this pioneering paper,along with the detailed definition of the memristor and ofits properties, a claim appeared that the memristor was to beconsidered as basic as the three classical circuit elements: theresistor, the capacitor, and the inductor.Here we focus on evaluating such a claim and on determin-ing the essential nature of the memristor within circuit theory.The well-established basic elements in circuit theory arethe inductor, the capacitor, and the resistor. Such elementsintroduce independent relationships between pairs of circuitquantities. In their most direct implementations, they can alsobe seen as expressions of different physical laws. Citing theclassical book by Desoer and Kuh [2], we report verbatimthe definition of the inductor: a two-terminal element will be
B. Tellini is with the Dipartimento di Ingegneria dell’Energia, dei Sistemi,del Territorio e delle Costruzioni, University of Pisa, Pisa I-56122, Italy (e-mail: [email protected])M. Bologna and K. Chand´ıa are with the Departamento de Ingenier´ıaEl´ectrica - Electr´onica, Universidad de Tarapac´a, Arica, Chile (e-mail: [email protected], [email protected]).M. Macucci is with the Dipartimento di Ingegneria dell’Informazione,Universit`a di Pisa, Pisa 56122, Italy (e-mail: [email protected]). called an inductor if at any time t its flux ϕ ( t ) and current i ( t ) satisfy a relation defined by a curve in the i, ϕ plane.The essential idea is that there is a relation between theinstantaneous value of the flux ϕ ( t ) and the instantaneousvalue of the current . It is important to point out that from thepoint of view of circuit theory and analysis the flux ϕ ( t ) isthe flux linkage, i.e., the time integral of voltage, which is anactual circuit quantity. In the particular case of the classicalidealized inductor obtained by winding an ideal thin conductor,such an integral happens to coincide with the magnetic fluxlinked with the coil, which in turn corresponds to the actualmagnetic flux piercing it only in the case of a coil with a singleturn. For an inductor, the relationship between the time integralof the voltage and the current can be seen as an expression ofLenz’s law, although this is not the only way to obtain such an“inductive” behavior, which could well be implemented in amore elaborate fashion also in a world without magnetic field(for example with a capacitor and a gyrator).The capacitance establishes a relationship between the volt-age v ( t ) and the integral of the current (i.e., the charge q ( t ) )described by a curve in the q, v plane. In the particular case ofa classical capacitor consisting of two conductors separated bya dielectric, this is an expression of Gauss’ law. Also in thiscase, we report verbatim from [2]: The basic idea is that thereis a relation between the instantaneous value of the charge q ( t ) and the instantaneous value of the voltage v ( t ) . Finally, the resistance establishes a relationship between thevoltage v ( t ) and the current i ( t ) , through a curve on the i, v plane. Linear resistors are implemented through Ohm’s law,i.e., with conductors exhibiting a linear relationship betweencurrent and voltage. In [2], the authors write: The key idea ofa resistor is that there is a relation between the instantaneousvalue of the voltage and the instantaneous value of the current .Also on the basis of the examples provided by Desoer andKuh, this means that such a relation may be independent of thecurrent and constant in time (linear time-independent resistor),dependent on the current but constant in time (non-lineartime-independent resistor), independent of the current but timedependent (linear time-dependent resistor), or dependent onthe current and on time (non-linear time-dependent resistor).Within a strict interpretation, this definition does not in-clude many components of general use, which are commonlyreferred to as resistors because they are characterized by adissipative behavior and the voltage at their terminals dropsto zero whenever the current vanishes (which is equivalent tohaving a “pinched” v − i relationship, an expression commonlyused in the memristor literature). A very simple example is represented by the incandescentlight bulb, which is characterized by a resistance dependingon temperature: since the temperature behavior exhibits adelay with respect to the current behavior (as a result of thethermal inertia of the filament) and depends on the ambienttemperature, this would not fit any of the resistor definitionsby Desoer and Kuh. We will discuss this issue in detail inSec. II.Summarizing, each basic element is associated with asingle-valued curve relating two electrical quantities ( i and v for the resistor, i and ϕ for the inductor, v and q forthe capacitor). Such a curve is in general dependent on time(time-dependent elements), can be time-independent (time-independent elements), or can be just a line (linear elements,and, more in detail, a line with a slope depending on timefor the linear time-dependent elements or just a single line forlinear time-independent elements).To make our argument more rigorous, we provide a workingdefinition for the concept of basic circuit element: a basiccircuit element, in its simplest implementation, establishes arelationship between two circuit quantities that cannot be re-produced by any single other basic circuit element. By simplestimplementation we mean the linear and time-independent case.For example, if two circuit elements are characterized by dif-ferent physical dimensions, they certainly define independentrelationships. Conversely, in the case of a memristor with aconstant value, we obtain the same relationship between fluxlinkage and charge as that implemented by a resistor.In 1976 Chua and Kang [3] extended the concept ofmemristor, indicating the memristor as a special case of aclass of dynamic systems (memristive systems) for which thefollowing relationships hold: ˙ ~x = ~f ( ~x, u, t ) (1) y = g ( ~x, u, t ) u (2)being ~x the vector of the state variables of the system, u a generic input quantity and y a generic output quantity. Inprinciple y and u can be arbitrary physical quantities.In the same 1976 paper, Chua and Kang defined a particularmemristive system in which y is a voltage and u is a currentas a current-controlled memristive one-port. Analogously theydefined a memristive system in which y is a current and u isa voltage as a voltage-controlled memristive one-port.In 2009 Di Ventra etal. [4] analyzed two particular types ofmemristive systems, one with y = q and u = v and the otherwith y = ϕ and u = i , defining them memcapacitive sys-tems and meminductive systems, respectively. A Lagrangian-Hamiltonian formalism for the description of such devices wasthen proposed by Cohen etal. [6] in 2012.More recently, in 2015, Chua [7] provided a detailedoverview, reporting a hierarchy going from a class that isslightly less general than the memristive one-port down to theoriginal 1971 memristor (reclassified as “ideal memristor”).For the purposes of our discussion, we will not needthis more detailed classification (which does not add newcontributions to the basic memristor theory, but it is just a more detailed taxonomy) and will refer to the definitions of the 1971and 1976 papers, limiting ourselves to current-controlled andvoltage-controlled memristive one-ports. It is our impressionthat the 1971 and the 1976 definitions of memristor andmemristive systems, respectively, along with the extension tothe memcapacitor and meminductor, represent a very clearframework, and that the introduction of further distinctionsand/or extensions is not strictly necessary. For example, the“extended memristive device” introduced by Valov etal. [8] isindeed just a circuit in which a memristive system (accordingto the 1976 definition), a nonlinear resistor (according tothe definition by Desoer and Kuh) and a voltage source arepresent. In addition, we believe that it would be convenient touse the word “memristor” only for the device defined in the1971 paper by Chua, avoiding its usage for the more general“memristive systems” (such as, for example, the light bulb).Indeed, as we will discuss in Section III, general memristivesystems are not guaranteed to exhibit a single-valued ϕ − q characteristic, which was one of the fundamental properties ofthe original 1971 memristor.Memristors have received significant attention by the sci-entific community, both for the analysis of circuit behaviorand for possible industrial applications [9],[10],[11],[12]. Suchan interest into the memristor was mainly triggered by the2008 paper by Strukov et al. [13], in which experimental datawere reported showing a behavior of a memristive nature fora nanoscale TiO device.In more detail, a titanium oxide film with a thickness of5 nm was sandwiched between two metal (platinum) elec-trodes and was made up of two layers: an insulating TiO layer and a conducting TiO − x layer (where conduction isdue to oxygen vacancies acting as dopants). Such vacanciesdrift in the presence of an electric field, thereby moving theboundary between the conducting and the insulating layer andthus varying the overall resistance.In [14], the authors criticize the interpretation of such adevice as a memristor on the basis that a “real memristor”should involve magnetic flux. They also state the opinionthat the original hypothesized memristor is still missing andprobably impossible, while recognizing that a real memristormay in principle be discovered.The analysis of the memristor nature that we will presentin the following leads instead to the conclusion that the TiO device can well be considered a memristor (since, as it willbecome apparent from our discussion, the magnetic field is notessential, or even relevant, in the definition of the memristor).In [15] Abraham maintains that the device proposed in[13] is not Chua’s postulate of 1971, while also rejectingthe memristor as a new basic circuit element. He sets outto demonstrate the non basic nature of the memristor withan involved argument based on a periodic table of basicelements and on two rules; one inferred from an analogywith the periodic table of chemical elements and the otherone from the assumption that the definition of basic elementsshould be given outside transient conditions. Overall, Abrahampoints out some of the fundamental problems in the originalmemristor discussion (such as the ambiguity between theactual magnetic flux and the time integral of the voltage) and provides insightful hints into the nature of the memristor,without, however, making a complete case.Our analysis, once we have reached in a more direct waythe same conclusion as Abraham’s that the memristor is nota fourth basic circuit element, focuses on the intimate natureof a current or voltage-controlled memristive one-port. To thispurpose, we provide an extended revisitation and analysis ofsignificant past literature, and discuss the detailed behavior ofa few relevant memristive one-ports, with a special focus onthe associated curves in the ϕ − q plane. Our final conclusionis that the current-controlled or voltage controlled memristiveone-ports, and, specifically, the memristor (“ideal” memristordefined in [1]) can be seen just as generalized resistors.Such a concept was indeed already proposed by AlexanderMauro [16] in 1961, in order to explain the appearance of areactive component in the differential impedance of the giantsquid axon and of the thermistor.The structure of the paper is as follows: Sec. II containsthe main discussion on the basic nature of the memristorand of relevant examples from the literature. It is dividedinto 5 subsections: in Subsection A we summarize the maindefining memristor properties; in Subsection B we provide aclarification about the issue of the flux linkage vs. magneticflux; in Subsection C we present the reasons why we believethat the memristor is not a basic circuit element; in SubsectionD we focus on two enlightening examples in which the appear-ance of a reactive component of the impedance is explainedwith the introduction of a generalized resistor concept; and,finally, in Subsection E we perform an analysis of a recentlypublished memristor implementation, pointing out that its v − i characteristic is not rigorously pinched. In Sec. III, we presenta few numerical examples useful to clarify the properties ofmemristors vs. those of more general memristive systems inthe ϕ − q and v − i planes. In particular, we focus on theconditions needed to obtain a closed or even single-valued (asin the case of the original memristor definition) curve. Finally,we present our conclusions.II. B ASIC D ISCUSSION
A. Memristor properties
Let us start from the analysis of the memristor concept thatwas presented by Chua in his original work [1]: a passiveelement characterized by a relationship of the type f ( q, ϕ ) =0 . In that paper, he also made the distinction between acharge-controlled memristor and a flux-controlled memristorfor which f ( q, ϕ ) can be expressed as a single-valued functionof the charge q or the flux linkage ϕ , respectively. Thus, forexample, for the simplest form of charge-controlled memristor,we have: ϕ = f M ( q ) (3)and the voltage across the memristor is: v ( t ) = M ( q ( t )) i ( t ) . (4) The term M ( q ) is defined memristance and can be expressedas: M ( q ) = df M ( q ) dq . (5)Thus it is actually a differential memristance, because itestablishes a proportionality relationship between the differ-entials of the original quantities ( ϕ and q ) it is supposed torelate. This is not consistent with the definitions for resistors,capacitors and inductors which directly establish a proportion-ality relationship between v and i , q and v , and ϕ and i ,respectively.Thus, in order to be perfectly consistent with the definitionsfor the basic circuit elements, the memristance should insteadhave been defined as the ratio of the flux linkage (the timeintegral of the voltage) to the charge (the time integral of thecurrent), i.e., it should coincide with the M ′ in the relationship ϕ ( t ) = Z t −∞ v ( τ ) dτ = M ′ Z t −∞ i ( τ ) dτ = M ′ q ( t ) (6)As acknowledged by Chua himself [1], if M has no depen-dence on q , the memristor becomes exactly equivalent to anordinary linear and time-independent resistor.Chua also pointed out [1] that the existence of a single-valued relationship between the flux linkage ϕ (time integral ofthe voltage) and the charge q was at the basis of the memristordefinition, in analogy with the relationships mentioned byDesoer and Kuh for the basic circuit elements. He evenpresented a design for a ϕ − q tracer for the experimentalacquisition of such a characteristic.Let us now focus on the more general case of current(voltage) controlled memristive systems, defined by (1) and(2) when u is a current and y is a voltage ( u is a voltage and y is a current).The properties that they are expected to exhibit are:a) In [17] the authors report that the v − i relationshipshows a pinched hysteresis loop for a periodic input signalor equivalently, as stated by Chua in [19], all memristivesystems are characterized by the property that a zero outputcorresponds to a zero input, i.e., that for i = 0 we always have v = 0 ;b) In [18], Ascoli et al. state: The memristor ... is a two-terminal nonlinear dynamic circuit element obeying a Ohmlaw dependent upon the time evolution of its memory state. c) For the particular case of M being only a function of q ,the area of the hysteresis loop decreases with frequency andthe hysteresis loop collapses to a single-valued function in thelimit of f → ∞ , being f the operating frequency.Any of these properties is consistent with the memristivesystem being simply a resistor whose value depends on theinternal state vector ~x , on the instantaneous value of the current(for the current-controlled case) and, possibly, on time.However this does not just correspond to a time-dependentresistor, because, in the commonly accepted interpretationwithin circuit theory, a quantity is time-dependent if its valuehas a direct dependence on time, i.e., its partial derivativewith respect to time is non zero and there is no other indirectdependence on time. In the case of the memristor, instead, the memristancedepends on variables which are in turn functions of time,therefore it has a zero partial derivative with respect to time (inthe case of a “time-independent” memristor), but a non-zerototal derivative with respect to time.In 1961 Mauro [16] introduced the concept of a “time-variant” resistor, an element whose resistance “varies intrin-sically with time by virtue of the fact that the physical stateof the element changes with time,” while a time-dependentresistor is an element whose resistance varies “as an inde-pendent function of time” (in accordance with the definitionby Desoer and Kuh). In Subsections C and D we provide adetailed discussion of this concept.
B. Magnetic flux and flux linkage
As already mentioned, ϕ is just the time integral of thevoltage (i.e., the flux linkage), which is a relevant quantityin circuit analysis (while the magnetic flux is not an actualcircuit quantity, except in the case of ideal inductors, when itcoincides with or is proportional to the flux linkage). This isfully consistent with the definition given by Chua in the firstpage of his original memristor paper of 1971 [1]. However, inSec. 4 of the very same paper, Chua, trying to provide a moregeneral interpretation of the memristor, in terms of a quasi-static solution of Maxwell’s equations, states that the surfaceintegral of the magnetic flux density is proportional to the fluxlinkage. The problem is that such a flux linkage (originallydefined relative to an inductor and equal to the magnetic fluxlinked by each loop of the inductor multiplied by the numberof loops) is not, except for the case of an ideal inductor, theflux linkage used in the first part of the paper for the definitionof the memristor, i.e., the time integral of voltage.This discrepancy is at the origin of the ambiguity (integralof the voltage vs. actual magnetic flux) in the meaning of ϕ that may have contributed to some involved arguments on thenature of the memristor which can be found in the literatureand to the search for a memristor relating the actual magneticflux with the charge, which turns out to be a somewhatill-posed problem: in order to make the integral of voltageproportional to the actual magnetic flux an inductance mustbe present, which will then lead to a ϕ − q curve that cannotbe exactly pinched in the origin.In some papers the meaning of flux linkage used in theoriginal memristor definition appears to have been completelyforgotten, and the memristor is presented as a circuit elementrelating charge and “magnetic flux” [25], [26].More in general, circuit theory is indeed a special case ofMaxwell’s equations only as long as we consider classicalphysics and we limit ourselves to electromagnetic phenomena.If other effects (e.g. chemical, thermal, etc.) are includedor if quantum mechanical effects are taken into account, amultiphysics approach is needed. In the introduction we havealready mentioned the case of the thermistor (which requiresthe inclusion of the heat transport equations) and of theaxon (which requires the inclusion of ion transport equations).Another example can be that of quantum capacitance [27],[28], [29], which arises as a consequence of the quantum (−1,−1) (0,−2)(−1,−2)(−2,−1)(−2,0)(−2,1) (−1,1) (0,1) (1,1)(1,0)(1,−1)(1,−2) α β (−2,−2)−2 11 −1 0−20−1 Fig. 1. Mapping of circuit elements based on the orders of the derivativesof the voltage and of the current between which they establish a relationship;there is a single basic element per diagonal (adapted from Fig. 3 of [33]). confinement energy and the relationship between the chemicalpotential and the charge.If we consider the proper definition of ϕ , the TiO baseddevice presented in [13] can indeed be defined a memristor,as we had anticipated, because it is a resistor whose valueis a function of the charge that has flowed through it, andthus of the time evolution of the current, which makes it aspecific type of time-variant resistor, exhibiting the memristorproperties defined in [1]. C. Matrix representation of circuit elements
In [33] Chua introduces a matrix representation of the circuitelements, which we summarize in Fig. 1. In more detail, eachelement of the matrix can be identified by the pair ( v α , i β ) ,where α and β indicate the column and row of the element,which correspond to the order of the time derivative of thevoltage and of the current, respectively. An element located inthe column α and in the row β relates the time derivative oforder α of the voltage to the time derivative of order β of thecurrent through a single-valued function. Thus, moving alongthe main diagonal, it is possible to identify the resistor and thememristor, which is in the position ( − , − ), while movingalong the diagonal right below the main one we can identifythe capacitor and the memcapacitor ( − , − ), and along thediagonal above the main one we find the inductor and thememinductor ( − , − ) [4].If one were to assume that each element of the matrixcorresponds to an independent basic circuit element, also thememcapacitor and the meminductor would be as basic as theresistor, the capacitor, the inductor, and the memristor.However, it is apparent that it cannot be so, because, firstof all, in the linear case all the elements of the same diagonaldo coincide, which is by itself evidence that independentbasic circuit elements cannot exist along the same diagonal.Furthermore, considering the most direct implementation, eachdiagonal is associated with a specific physical law: Gauss’slaw for the diagonal below the main one, Ohm’s law for themain diagonal, and Lenz’s law for the diagonal above themain one. Indeed also the memristor, as the resistor, is in the end a dissipative element, and is therefore an expressionof Ohm’s law. The dissipative nature of the memristor wasrealized also by Di Ventra and Pershin [5], who point out that,as a result of such dissipative behavior, the memristor violatesthe time-reversal invariance. This intrinsically resistive natureof the memristor has sometimes been overlooked because ofthe mistaken association with the magnetic flux instead of withthe time integral of the voltage.The previously mentioned interpretation of ϕ as the actualmagnetic flux appears to be also at the origin of the choice byWang of the memristor, the memcapacitor and the meminduc-tor as the elements for his “basic element triangle” [35] that heproposed after recognizing the inconsistency of assuming thememristor as a fourth basic circuit element. Such a formulationappears to be reasonable and free from the contradictions ofthe four basic element assumption. It is also consistent withthe position of the memristor, memcapacitor and meminductorin three different diagonals of Fig. 1. Wang reaches thisconclusion mainly because he considers the flux ϕ morefundamental than the voltage, a somewhat understandableargument if ϕ were the actual magnetic flux, which howeveris not, as we have already pointed out.In addition, as previously discussed, the memristance is de-fined exactly as a resistance with more general properties, in-stead of the ratio of the flux linkage to the charge (which wouldstill coincide with a resistor in the linear, time-independentcase). The same is true for the memcapacitor, defined by q = C ( ~x, v, t ) v [5] (instead of R q ( t ) dt = C ′ R v ( t ) dt = C ′ ϕ )and for the meminductor, defined by ϕ = L ( ~x, i, t ) i [5](instead of R ϕ ( t ) dt = L ′ R i ( t ) dt = L ′ q ).Thus it is far simpler to just generalize the definitions of thecircuit elements provided by Desoer and Kuh, by includingtime-variant elements, as proposed by Mauro for the resistor,and to keep the resistor, the capacitor and the inductor asthe basic circuit elements (which in this way would alsoinclude all the properties of the memristor, memcapacitor andmeminductor).Possibly, novel basic elements could be associated to moreexternal diagonals, even though their practical realization isapparently far from being conceived, and a discussion on thisis outside the purpose of the present work.Overall our conclusion is that a fourth basic circuit elementdoes not exist, and that the memristor can be seen simplyas a generalization of the resistor, a time-variant resistor,according to the definition given by Mauro in 1961, whichwe will discuss in detail in the next subsection. A furthergeneralization of the time-variant resistor concept by Mauro,introducing also a direct dependence on time, covers themost general definition of a memristive current controlled (orvoltage controlled) one-port provided by Chua and Kang [3],in which the value of the memristive one-port is a genericfunction of a set of state variables, the current (or the voltagein the case of a voltage controlled memristive one-port), and,possibly, time.As long as we assume a reasonable boundedness conditionfor the memristance (as suggested by Chua in [36] and in [37]to prevent the problems pointed out by Pershin and Di Ventrain [38]), the fact that the curve in the i − v plane is pinched is simply equivalent to stating that there is no energy storagein the device and, thus, that the device is purely dissipative,i.e., a resistor. D. Time-variant resistors
It is now appropriate to go back to the example we men-tioned in the introduction: the filament of the incandescentlight bulb, which is one of the simplest cases that do not fitinto the textbook definition of resistor that we quoted at thebeginning.Already in 1961, Alexander Mauro [16], realized the differ-ence between an incandescent light bulb and a time-dependentresistor, and defined it as a “time-variant” resistor, as men-tioned at the end of Subsection A.In particular, Mauro provided analytical evidence that acircuit containing a thermopositive element (i.e., an elementwhose resistance increases with temperature) such as an incan-descent light bulb exhibits a differential impedance of capac-itive nature (as a consequence of the delay in the temperatureresponse to current variations), while with a thermonegativeelement a differential impedance of inductive nature wouldappear. In the paper by Mauro this was introduced to provide aphysical justification of the so-called “anomalous impedance,”which had been observed in giant squid axons [20]. Indeed,early measurements of the AC impedance of the giant squidaxon had revealed a significant inductive component, whichcould not be explained as a result of magnetic energy storage,due to the absence of a large enough loop. This is the reasonwhy Cole [20] mentioned that this property had been foundshocking, but he accepted it, and correctly attributed it toproperties of the membrane (from the observation that it couldnot be associated with a flow of current parallel to the axon,due to the fact that an asymptotic inductance value was reachedwhen increasing the distance between the electrodes alongthe axon). Mauro, with the case study of the temperature-dependent resistor, was able to present a clear example ofhow a time-variant resistor (a resistor whose value dependson state variables, which in turn depend on time) may exhibita differential inductive behavior. Then in 1965 Cohen andCooley [21] performed a numerical solution of the Nernst-Plank equations for ionic transport through a thin permeablemembrane, finding, for this specific physical system, an induc-tive behavior, as shown in their Figs. 5 and 6, analogous tothat derived by Mauro for a generic thermonegative element.Thus, by 1965 the presence of an inductive component in theimpedance of a membrane, and therefore of an axon, had beenunderstood and explained without requiring the presence ofany magnetic field.It is therefore surprising that in a recent paper [22], thepresence of the axon inductance is reported as a problemthat had remained unsolved for 70 years and that for thesame amount of time had been associated with an “enormousmagnetic field”.In general a magnetic field is not needed to produce aninductance-like behavior, because it can simply arise as aresult of inertia: inertia of carrier motion for the kineticinductance [23], thermal inertia for thermonegative elements, delay associated with ionic diffusion for the axon inductance,etc. It is however important to point out that not all of thesethree examples are of a memristive nature: while in the secondand the last example there is no energy storage, in the firstexample energy is stored in the form of kinetic energy andcan be returned to the circuit. Therefore in the case of kineticinductance we are dealing with an actual inductance which isnot associated with a pinched i − v characteristic, while in theother two cases there is no energy restitution and we have amemristive behavior.The authors of [22] redefine the time-variant resistors ofthe Hodgkin-Huxley [24] paper as time-invariant memristors(time-invariant memristive systems, on the basis of the defini-tions we decided to use in the introduction, since, as we willshow in Sec. III, they are not characterized by a single-valuedcurve in the ϕ − q plane) and then proceed to the calculationof the differential impedance of the overall circuit, finding theexpected inductive component.However, the same calculation could have been performedwithout renaming them, and following, for example, theapproach used by Mauro to treat the temperature-dependentresistor. Furthermore, Mauro provided a clear physical expla-nation of the appearance of an inductance behavior, whilein [22] the inductance is the result of a long and involvedanalytical calculation, without direct physical insight.Thus, both definitions, time-variant resistor and memristivesystem, are acceptable and, if properly handled, lead to thesame results. E. Magnetic core memory device - Φ memristor In relationship to the search for a memristor connectingdirectly the charge with the magnetic flux (instead of the timeintegral of the voltage), Wang et al. [30] have recently pub-lished a paper in which they present a device that they define a“real memristor”. Such a device is based on a conducting wiregoing through a magnetic core: if a rectangular current pulse isapplied to the wire, the resulting magnetic field rotates the coremagnetization, which in turn leads to a variation in time of themagnetic flux, and thus to an induced voltage. This is exactlythe readout process of the old magnetic core memories [32],with the only difference that, while in core memories twoseparate loops were used (one to inject the current and onefor the readout), Wang et al. use a single loop. They find arelationship between the magnetic flux and the charge, and,in particular, a pinched current-voltage relationship (as typicalfor memristors).However they do neglect the voltage induced as a result ofthe geometrical self-inductance of the loop, i.e., the inductiveeffect associated with the magnetic flux variation directlyproduced by the input current. In the case of square currentpulses, the authors state explicitly that they have filteredout (by reducing the oscilloscope bandwidth) the “transientspike,” due to such inductance, which they define as “parasitic”(the spike is indeed visible in the experimental data of theirFig. 10). If instead we consider a sinusoidal excitation, as in(12) of [30], there is another term to be included, again due tothe geometrical loop inductance, which is in quadrature with the one of [30]. Such inductance is indeed not “parasitic,”because it is the source of the very same connection betweenmagnetic flux variation and induced voltage that leads tothe in-phase term. The in-phase term is the result of theflux variation due to the magnetization lining up with thedriving magnetic field created by the injected current, whilethe quadrature term is the result of the flux variation due tothe change of such a driving magnetic field. The alignment ofthe magnetization is the consequence of the dissipative effectconnected with the Gilbert term [31] thus leading to a time-variant resistive component (the memristor-like part). Depend-ing on the waveform and on the magnetic core characteristics,the in-phase term may be much larger than the quadratureterm, but the ratio of their amplitudes must in any case befinite: if the inductive component should vanish, the samewould happen for the dissipative “memristive” component.Thus the ϕ -memristor is not rigorously a memristor, becauseits characteristic v − i curve cannot be exactly pinched.Furthermore, the authors of [30] in their Fig. 9 report curvesfor the voltage and the current as a function of time that arenot consistent with the plot in the i − v plane and with theexperimental results at the top of Fig. 9. Indeed, in the plotreporting the time behavior of the current and the voltage, thecurrent changes sign exactly when the voltage pulse ends. Insuch a case, and in any other case in which the current isreversed after the voltage pulse ends (as in the experimentalresults of Fig. 10), the curve on the i − v plane would notlook like the one reported, but would consist of a collectionof segments without a hysteresis cycle: there would be ahorizontal segment corresponding to the current step, then avertical segment corresponding to the increase and decreaseof the voltage at constant current, and then again a horizontalsegment reaching the negative current value, followed by abehavior symmetric with respect to that during the positivecurrent pulse.Curves with hysteresis such as the one reported by theauthors of [30] for the i − v plane and in their experimentalresults at top of their Fig. 9 are possible only if the current isreversed before the voltage pulse ends (which is also consistentwith the frequencies they report for the experimental result andthe duration of the voltage pulses in Fig. 10).In order to further clarify the properties of different typesof memristive one-ports, in the next Section we present afew numerical examples for relevant cases: the thermistor, theaxon, and a memristor (according to a strict interpretationof the 1971 definition by Chua) made up of resistors andcharge controlled switches. For all the examples, we reportthe associated ϕ − q curve and discuss the conditions underwhich such a curve is single-valued.III. M EMRISTIVE C IRCUIT M ODELS
A memristive functionality can be implemented in many dif-ferent ways, exploiting a variety of different physical systems,even relatively exotic ones [39], since all that is needed is adissipative effect and a mechanism that can vary the resistanceunder the control of a state variable.Here we focus on the analysis of a few simple but mean-ingful examples. − − − − − V o l t ag e ( V ) Fig. 2. Thermistor v − i curve for f = 1 µ Hz, δ = 0 . mW/ ◦ C, R = 8 k Ω , T = 298 K, β = 3460 K, C = 1 mJ/K. We start with a numerical analysis of the thermistor, whichis the first memristive device that was studied by Mauro in1961 to provide an example of how a differential impedancewith a reactive nature can appear in the absence of actualphysical capacitors or inductors and of energy storage. Weconsider the thermistor model of [3], which is based on the fol-lowing equations, including the dependence of the thermistorresistance on temperature (which is the state variable), as wellas the time evolution of temperature as a function of the heatcapacitance, the thermal resistance to the outside environment,and the dissipated electrical power. v = R ( T ) exp (cid:20) β (cid:18) T − T (cid:19)(cid:21) i, (7) p ( t ) = v ( t ) i ( t ) = δ ( T − T ) + C dTdt , (8) dTdt = − δC ( T − T ) + R ( T ) C exp (cid:20)(cid:18) βT − βT (cid:19)(cid:21) i , (9)where δ = 0 . mW/ ◦ C, R = 8 k Ω , T = 298 K, β = 3460 K, C = 1 mJ/K.We first report (Fig. 2) the v − i curve obtained by cycling thecurrent in the low-frequency limit (1 µ Hz), which correspondsto the result in Fig. 2 of [3]: in this case the hysteresisdisappears and a single-valued relationship between voltageand current is established, since dynamical effects due to thedelayed response of the state variable (the temperature) to thetime behavior of the dissipated electrical power are negligibleat such a low frequency, which is much smaller than thereciprocal of the relevant time constant of the system.In Fig. 3, we report the associated curve in the ϕ − q plane, which, in this particular case without hysteresis, issingle-valued. Therefore at vanishingly small frequency thethermistor does have a property (an unambiguous definitionthrough a curve in the ϕ − q plane) that was reported asa fundamental characteristic of a memristor in [1]. Such an F l u x ( k V s ) Fig. 3. Thermistor ϕ − q curve for f = 1 µ Hz, δ = 0 . mW / ◦ C, R = 8 k Ω , T = 298 K, β = 3460 K, C = 1 mJ/K. unambiguous relationship between the two defining quantities(e.g., voltage and current in the case of the resistor) is alsoat the basis of the basic circuit element definitions by Desoerand Kuh [2].We then consider (Fig. 4) the v − i curve of the same deviceobtained cycling the current at a higher frequency (0.01 Hz):in this case a hysteresis appears, as a result of the delay of thethermal response. The red portion of the curve correspondsto the descending half-period of the current, while the blackportion is relative to the ascending half-period. It is to benoted that neither the ascending nor the descending branchesare symmetric around the origin, which implies that a differentamount of flux linkage is transferred in the first and in the thirdquadrant.A complementary asymmetry is present in the other branch,which leads to a hysteresis in the ϕ − q plane. The curvein the ϕ − q plane is not single-valued, neither with respectto ϕ nor with respect to q , although it is closed (Fig. 5).As a consequence, at such a frequency this thermistor modeldoes not exhibit the properties of the memristor defined of1971, although it still belongs to the more general category oftime-variant resistors [and fits in the later definition (1976) ofmemristive systems].If we cycle the current at a relatively high frequency(compared to the reciprocal of the characteristic time constantof the device), the hysteresis disappears and we obtain asubstantially linear v − i curve, with a slope determined bythe value of the reached equilibrium resistance, as shown inFig. 6. Accordingly, also in the ϕ − q plane we obtain a linear,single-valued curve (Fig. 7).We now briefly study the circuit model for the potassiumchannel of the axon formulated by Hodgkin and Huxley [24], − − − − − − V o l t ag e ( V ) Fig. 4. Thermistor v − i curve for f = 10 mHz, δ = 0 . mW / ◦ C, R =8 k Ω , T = 298 K, β = 3460 K, C = 1 mJ/K. F l u x ( V s ) Fig. 5. Thermistor ϕ − q curve for f = 10 mHz, δ = 0 . mW / ◦ C, R =8 k Ω , T = 298 K, β = 3460 K, C = 1 mJ/K. in the form from [22]. We have i K = G K ( x ) v K (10) G K ( x ) = g K x (11) dx dt = γ ( v K + E K )exp (cid:16) v K + E K ˆ V − (cid:17) (1 − x ) − τ (cid:20) exp (cid:18) v K + E K V (cid:19)(cid:21) x (12)We performed numerical simulations with the parameters g K = 36 S, E K = 10 V, γ = 0 . V − s − , ˆ V = 10 V, τ = 8 s. − − − − − V o l t ag e ( V ) Fig. 6. Thermistor v − i curve for f = 10 Hz, δ = 0 . mW / ◦ C, R = 8 k Ω , T = 298 K, β = 3460 K, C = 1 mJ/K. . . . . . . . . . F l u x ( V s ) Fig. 7. Thermistor ϕ − q curve for f = 10 Hz, δ = 0 . mW / ◦ C, R = 8 k Ω , T = 298 K, β = 3460 K, C = 1 mJ/K. In Fig. 8 we report the v − i curve obtained cycling at afrequency of 50 mHz, which is comparable with the reciprocalof the time constant τ . We observe that it pinches in the origin,according to (10), and that it is not symmetric around theorigin, due to the particular nature of the defining equations.As a result of this lack of symmetry, we do not have asingle-valued curve in the ϕ − q plane, and in this case we donot have a complementary asymmetry as the one that exists inthe case of the thermistor (Fig. 5), therefore the curve in the ϕ − q plane is not even closed: we show such a curve for 4 − . − . − . − .
005 0 .
000 0 . − . − . − . − . . . . . . V o l t ag e ( V ) Fig. 8. Axon model v − i curve for the following parameters: f = 50 mHz, g K = 36 S, E K = 10 V, γ = 0 . V − s − , ˆ V = 10 V, τ = 8 s. − . − . − . − . − . − . F l u x ( V s ) Fig. 9. Axon model ϕ − q curve for the following parameters: f = 50 mHz, g K = 36 S, E K = 10 V, γ = 0 . V − s − , ˆ V = 10 V, τ = 8 s. periods in Fig. 9. As a consequence, at each cycle there is a nettransfer of charge, with a value corresponding to the distancebetween two consecutive intersections with the horizontal axis.Let us now consider the circuit of Fig. 10, for which weassume charge dependent switches. As a consequence, we havean overall resistance depending on the charge q that has flowedthrough the device. The current source i s provides the inputsignal, and i s is assumed to vary as i s ( t ) = A sin( ωt ) .The closing/opening sequences of the 10 switches are asfollows: Closing sequence - switch 1 closes once the electricalcharge q that has flowed through R satisfies q ≥ q =2 A/ ω . The second switch closes when q ≥ q , while the i th switch closes when q ≥ iq , thus the the th switchcloses when q ≥ q . At the end of the positive semi-period, Fig. 10. Schematic representation of a memristor obtained by switchingresistors as a function of the charge q . − − V o l t ag e ( V ) Fig. 11. Pinched hysteresis loop for the circuit of Fig. 10 with time-independent R , and A = 1 , f = 0 . Hz, R = R = ... = R = 0 . , q = 2 A/ (11 ω ) . a charge q = 11 q has flowed through R , and all the switchesare closed.Opening sequence - this sequence takes place in the negativesemi-period, therefore the charge q now decreases: the thswitch opens when q ≤ q , the i th switch opens when q ≤ iq , and switch 1 opens when q ≤ q . At the end of thissequence all switches are open and a new cycle can start.As a first case, we assume that all the resistors are time-independent.Simulation results are shown in Fig. 11: we notice that theloop is pinched at the origin and is symmetric around theorigin. Such a symmetry implies the single-valuedness of the ϕ − q curve, which is reported in Fig. 12. Therefore this simplemodel is compliant with the original requirement [1] for amemristor, which includes the unambiguous definition througha curve in the ϕ − q plane.The width of the loop in Fig. 11 depends on the frequencyof the current. This is shown in Fig. 13: proceeding clockwisefrom the top-left subplot we have: f = 0 . Hz, f = 0 . Hz, f = 0 . Hz, and f = 0 . Hz. We clearly observe howthe loop area decreases for higher frequencies, for whichthe charge transfer is smaller, therefore the charge-dependent F l u x ( V s ) Fig. 12. ϕ − q curve for the circuit of Fig. 10 with time-independent R , and A = 1 , f = 0 . Hz, R = R = ... = R = 0 . , q = 2 A/ (11 ω ) .Fig. 13. Pinched hysteresis loops for the circuit of Fig. 10. Proceedingclockwise from the top-left subplot the selected frequencies are: f = 0 . Hz, f = 0 . Hz, f = 0 . Hz, f = 0 . Hz. The other parameters are the sameadopted for Fig. 11. resistor variation decreases, too. With reference to Fig. 10, thismeans that a smaller number of switches is operated at higherfrequencies.The corresponding ϕ − q curves are reported in Fig. 14,where they are arranged in a clockwise fashion as in Fig. 13.In the second case, R is instead a resistor whose value hasa periodic dependence on time, according to: R ( t ) = (cid:26) < t ≤ T /
21 Ω if
T / < t ≤ T (13) Fig. 14. ϕ − q curves for the circuit of Fig. 10. Proceeding clockwise fromthe top-left subplot the selected frequencies are: f = 0 . Hz, f = 0 . Hz, f = 0 . Hz, f = 0 . Hz. The other parameters are the same adopted forFigs. 11,13. − − V o l t ag e ( V ) Fig. 15. Pinched hysteresis loop for the circuit of Fig. 10 with time-dependent R , and A = 1 , f = 0 . Hz, R = R = ... = R = 0 . , q = 2 A/ (11 ω ) . Simulation results are shown in Fig. 15: we can observe thatthe v − i curve is still pinched at the origin, as a result of thepurely resistive nature of the circuit.In particular the average slope in the first quadrant is largerthan that in the third quadrant, as a consequence of the largervalue of the resistance during the positive semi-period. Inthis case the ϕ − q curve is neither single-valued nor closed,because the hysteresis loop is completely asymmetric. We F l u x ( V s ) Fig. 16. ϕ − q curve for the circuit of Fig. 10 with time-dependent R , and A = 1 , f = 0 . Hz, R = R = ... = R = 0 . , q = 2 A/ (11 ω ) . report it in Fig. 16 for two consecutive cycles: after eachcycle the charge returns to zero since the driving quantityis a sinusoidal current, while the flux linkage exhibits a netincrease at each cycle since the integral of the voltage in thepositive semi-period of the current is larger than that in thenegative semi-period.For this circuit, in the high-frequency limit, the v − i hysteresis loop collapses onto two segments, each with a slopeequal to the resistance of R in the corresponding semi-period(see Fig. 17.)We point out that both the memristor circuit with timeindependent resistors and that with a time-dependent resistorsatisfy the passivity criterion adopted by Chua [1], i.e., thatthe instantaneous voltage to current ratio is always positive.This remains true also if the switches are controlled by circuitswith active components (such as transistors), because we canassume that the power needed to operate them is obtainedfrom the current flowing through the device. Indeed, this iswhat happens also in the memristor of [13], where vacancymigration is “powered” by the bias current.In the last simulation we included a capacitor connected inparallel to R , as shown in Fig. 18.Parameter values, in particular C = 0 . F , are chosen insuch a way as to clearly show how the presence of a reactiveelement implies that the v − i curve is not pinched at the originany longer, and thus the resulting device is neither a memristornor a memristive system. This is shown in Fig. 19.IV. C ONCLUSION
We have reconsidered the concept of memristor withincircuit theory, starting from the definitions of basic circuit − . − . . . . V o l t ag e ( V ) Fig. 17. Pinched hysteresis loop for the circuit of Fig. 10 with time-dependent R , and A = 1 , f = 10 Hz, R = R = ... = R = 0 . , q =2 A/ (11 ω ) .Fig. 18. Circuit of Fig. 10 with the inclusion of a capacitor element. elements, in order to understand its actual nature and whetherit is truly a fourth basic circuit element.As a major aspect of our analysis, we have reached theconclusion that the memristor is not a fourth basic circuitelement on the same footing as the traditional basic elements,while it can rather be interpreted as an extension of the resistorconcept.In particular, we have shown that, as long as the input andoutput quantities are a voltage and a current, a memristivesystem can be seen as a generalized time-variant resistor(generalized in the sense that, with respect to the definitionprovided by Mauro in 1961, also an additional direct depen-dence on time is allowed).We have also stressed the relevant point that the meaningfulquantity in the memristor definition is the flux linkage ϕ , tobe intended as the time integral of the voltage and not as theactual magnetic flux. Without this clarification, ambiguitiesmay appear and conclusions that are not fully sound may bedrawn.Motivated by the presence in the recent memristor literatureof references to systems studied in the 50’s and 60’s (suchas the thermistor and the giant squid axon), in which ananomalous reactive behavior had been observed, we have − − − − V o l t ag e ( V ) Fig. 19. Pinched hysteresis loop for a circuit consisting of a resistor dependingon the charge q and of a capacitor. revisited the original papers, realizing that such anomalousbehavior had been understood and explained at the time withconcepts that we can use to provide a general explanationof the nature of the memristor. At the same time we haveanalyzed the evolution of the definition of the memristor andof the memristive systems, starting from the original 1971formulation and its 1976 extension.In particular, we have analyzed the operation of a recentlyproposed “ Φ -memristor,” pointing out that its v − i curvecannot be rigorously pinched in the origin, because couplingwith the magnetic flux involves the unavoidable presence ofan inductance.Finally, we have presented a few numerical examples ofmemristive systems and memristors, showing that, in orderto have a single-valued curve in the ϕ − q plane, the v − i curve must possess particular symmetry properties, which arepresent only in the case of the basic time-invariant memristoras defined by Chua in 1971. Such an essential property is notcharacteristic of generic memristive systems.We believe that we have contributed to a clarification ofthe memristor concept that can be useful for the memristorcommunity, in order to further develop applications and tobetter focus on new and original problems.A CKNOWLEDGMENT
M.B. acknowledges financial support from UTA Mayorproject No. 8765-17. M.M. acknowledges partial support bythe Italian Ministry of Education and Research (MIUR) in theframework of the CrossLab project (Departments of Excel-lence). R
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