On memristor ideality and reciprocity

Panayiotis S. Georgiou a,n, Mauricio Barahona b, Sophia N. Yaliraki c,Emmanuel M. Drakakis d,nn

a Department of Bioengineering, Imperial College London, United Kingdomb Department of Mathematics, Imperial College London, United Kingdomc Department of Chemistry, Imperial College London, United Kingdomd Department of Bioengineering, Imperial College, London, South Kensington Campus, London SW7 2AZ, United Kingdom

a r t i c l e i n f o

Article history:Received 4 March 2014Received in revised form24 June 2014Accepted 11 August 2014Available online 6 September 2014

Keywords:MemristorReciprocity

a b s t r a c t

Starting from the constitutive properties that underpin the ideal memristor as originally dened by LeonChua, we identify the conditions under which two memristors comply with the reciprocity theorem. Inparticular, we explore the minimal set of requirements for an ideal memristor and the physicalimplications of not complying with these criteria. Then, we show that reciprocity is satised whentwo identical ideal memristors with the same initial memristance are connected such that the output ofthe rst one is taken as the input of the second; the output of the second memristor then matchesexactly the initial input of the rst device. We also discuss under which conditions non-ideal memristorscan be reciprocal and how this property may be exploited in applications.& 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/3.0/).

1. Introduction

In 1971 Leon Chua postulated the existence of the memristor,a new fundamental passive two-terminal circuit element to beadded to the classic trio of the resistor, the capacitor and theinductor [1]. As its name indicates, the memristor behavessimilarly to a conventional resistor but it has memory in the sensethat its instantaneous resistance depends on the history of itsinput. A few years later, Chua and Kang extended the conceptualframework to include systems that share common characteristicswith memristors but whose behavior cannot be adequately cap-tured by the stricter original denition of the memristor. To enablethe modeling of such systems, they introduced the term memris-tive systems [2]. For several decades, experimental devices exhibit-ing memristive behavior remained unclassied as such [3,4], withresearch interest mostly conned to theoretical studies [5,6]. Thisstate of affairs changed in 2008, when Hewlett Packard (HP)fabricated a nano-scale device whose operation was describedusing a memristor model, thus reigniting experimental andtheoretical interest in this elusive element [7].

The fabrication of the HP memristor has attracted the atten-tion of the research community, which has been motivated ingreat measure by the many potential applications of memristors.This activity has resulted in the fabrication of a wide range of

experimental memristive devices demonstrating attractive proper-ties such as nano-scale dimensions [4], low-power consumption[8], high-speed resistance switching [9], and intrinsic non-volatilememory [10]. These properties render memristors ideal candidatesfor improving the performance of existing applications (e.g.computer memories, re-congurable digital circuits [11,12]) orenabling new ones (e.g. learning circuits) [13].

The experimental devices presented so far exhibit qualitativecharacteristics that point to a memristor (e.g. nonlinearity, mem-ory, hysteresis, zero-crossing), but they deviate signicantly fromthe strict denition of a memristor as originally articulated byChua [4,1416]. Thus, they can only be satisfactorily modeledunder the broader denition of a memristive system. For thisreason, the terms memristor and memristive system are fre-quently used interchangeably in the literature, although they havedifferent mathematical properties.

In our view, both notions can be useful depending on theproblem being tackled. When it comes to device modeling,utilizing memristive systems is necessary if an accurate model isto be developed. However, theoretical studies aimed at under-standing the fundamental characteristics of memristors as isolatedelements or as part of larger networks, or in combination withother circuit elements, are greatly simplied by the use of thestricter denition of the memristor. Henceforth, we will refer tothe original element as introduced in [1] as the ideal memristor.

Here, we revisit the fundamental mathematical propertiesunderpinning the ideal memristor and highlight the implicationsof omitting any of them. In doing so, one can understand the originof the observable behavior of non-ideal devices, and identify the

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/mejo

Microelectronics Journal

http://dx.doi.org/10.1016/j.mejo.2014.08.0030026-2692/& 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/).

n Corresponding author.nn Principal corresponding author.E-mail addresses: ps.georgiou@imperial.ac.uk (P.S. Georgiou),

e.drakakis@imperial.ac.uk (E.M. Drakakis).

Microelectronics Journal 45 (2014) 13631371

factors responsible for their non-ideal operation. Moreover, manyof the theoretical results developed for the ideal element can beapplied to the non-ideal case, if its operation is appropriatelyrestricted. We will refer to such memristive devices as piecewiseideal. Such piecewise ideal memristors can exhibit propertiesusually associated with strict ideality.

In recent work, we have shown that the dynamics of idealmemristors complies with the Bernoulli differential equation [1719]. This reformulation allowed us to obtain a set of general solutionsand to evaluate analytically the output response of HP's idealmemristor model for various input signals, both as a single elementand as part of series or parallel network congurations. Additionally,using the analytic expressions it was possible to investigate variousproperties of ideal memristors such as hysteresis, harmonic distortionand the effect of parasitic resistance [18,19].

In this paper, we build on these results to investigate anotherinteresting property of ideal memristors, namely under whatconditions do ideal memristors behave as reciprocal elements. Inparticular, we identify the requirements on the input/outputsignals and the device itself which lead to compliance with thereciprocity theorem [20]. When these requirements are met,memristors exhibit a useful behavior that may be exploited inapplications. The nal part of this work discusses the applicabilityof reciprocity on non-ideal devices and its potential applications.

2. The ideal memristor

2.1. Analysis of the ideal memristor

This section introduces and analyzes the ideal memristor [18].The theory presented provides us with the necessary tools toinvestigate the compliance of ideal memristors with the recipro-city theorem (Section 3).

The memristor is a 2-terminal passive circuit element thatrelates the charge q:

qt Z t1

i d q0Z t0i d; 1

with the ux-linkage

t Z t1

v d0Z t0v d; 2

where v is the voltage across and i is the current through thememristor, and q0 and 0 are respectively the initial charge andux-linkage values at t0.

The memristor is characterized by the constitutive relation:

fMq; 0; 3which relates the variables q and . More simply, the memristor isa circuit element whose inputoutput response is dened by achargeux (q) curve [1,21].

An ideal memristor is characterized by a unique and time-invariant q curve complying with the criteria

1.1 nonlinear,1.2 continuously differentiable,1.3 strictly monotonically increasing.

These three criteria for ideality were extracted from the originalpublication introducing the memristor [1]. The only exception isthat we impose strict monotonicity rather than the lighter require-ment imposed by Leon Chua of non-strict monotonicity.

Assuming an ideal memristor, it is always possible to expressthe constitutive relation (3) as an explicit function of both q and :

^q; 4

q q^: 5The memristor is then referred to as charge-controlled, for thecase (4), or ux-controlled, for the case (5). Under the sameassumptions, it follows that

q^1q ^q and ^ 1 q^: 6Using (1) and (2), and taking the time derivative of (4) and (5)

results in the representation of the memristor on the currentvoltage (iv) plane. For the charge-controlled case it is given by

vMqit; 7whereMq d^q=dq is the incremental memristance, measuredin Ohms (), which corresponds to the gradient of the q curveat an operating point (OP) Q qa;a (Fig. 1). Similarly, for the ux-controlled case, the iv representation is given by

iWvt; 8where W dq^=d is the incremental memductance, mea-sured in Siemens (S), which corresponds to the gradient of the qcurve at an OP Q qa;a [1,22].

From (7) and (8), we see that at any OP Q qa;a along the qcurve the following holds [3]:

Mq 1W vtit ; 9

namely the incremental memristance is equal to both the instan-taneous resistance (vt=it) and the reciprocal of the incrementalmemductance. This result is of practical importance as it relatesthe memristance (or memductance) with the driving signal and itsoutput: (9) indicates that the incremental memristance can beobtained from experimental data by sampling the input andoutput of the device and evaluating their ratio.

The denition of ideal memristors allows also the substitutionof (6) in (9):

Mq 1Wq^1q

3 M^1 1W: 10

For an ideal memristor, the q curve is uniquely invertible, andits memristance (memductance) is uniquely dened at any pointalong the q curve irrespective of whether it is driven by avoltage or by a current input [23]. Therefore, the two representa-tions, charge-controlled and ux-controlled, are equivalent to eachother and can be used interchangeably to our convenience. On theother hand, for a non-ideal memristor each value of q or maycorrespond to multiple memristance values. Therefore, (10) doesnot hold in general. In this case, (10) may hold locally only if theq curve is appropriately restricted to be monotonic sub-func-tions, each one complying with the criteria of ideality [24].

Eqs. (7) and (8) are the representation of the memristor on theiv plane and can be viewed as a generalization of Ohm's law witha dynamic and nonlinear resistance that depends on the history ofthe input [22]. These expressions reveal key features of theresponse: the multiplicative output function; the zero-crossing; thenon-volatile memory; and the relation between the incrementalmemristance and the instantaneous resistance [1,22]. More speci-cally, the output of the component at any time is equal to theproduct between the input and a nonlinear function representingthe memristance (multiplicative output function) and it is zerowhenever the input is zero (zero-crossing property). Furthermore,replacing q in (7) with (1) and in (8) with (2) yields

vt MZ t1

i d

it; 11

it WZ t1

v d

vt; 12

P.S. Georgiou et al. / Microelectronics Journal 45 (2014) 136313711364

which clearly reveal the non-volatile memory property of thememristor [18]. The value of the memristance (memductance) isdetermined by the entire past history of the input signal. There-fore, the memristance (memductance) keeps changing as long asan input is applied on the device. Once the signal is removed thememristor will maintain its state indenitely, or until the timeinstant at which an input is applied again [1,5,25].

Fig. 1a shows a q curve satisfying the criteria listed aboveand therefore representing an ideal memristor. Its correspondingiv response is shown in Fig. 1b when the device is driven by asinewave input. The gure also illustrates that in the general

nonlinear case

tqta

d^qtdqt

vtita

dv^itdit : 13

Namely the instantaneous (t=qt) is not equal to the incre-mental (d^qt=dqt) memristance; and the instantaneous resis-tance (vt=it) is not equal to the incremental (dv^it=dit)resistance. These four quantities only become equal when thememristor degenerates to the special case of the linear resistor.However, the incremental memristance is always equal to theinstantaneous resistance and reduces to a constant value in thelinear case. In this work we only refer to the incrementalmemristance and instantaneous resistance, so the terms incre-mental and instantaneous will be generally omitted.

2.2. Exploring the criteria for ideality

We now describe the effect of each of the conditions (1.1)(1.3)on the q curve of an ideal memristor and discuss the con-sequences of not complying with these criteria through examples.

An ideal memristor is characterized by a q curve complyingwith the properties:

Nonlinear: The nonlinearity of the q curve distinguishes thememristor from a linear resistor. From (7) and (8), it wasestablished that the memristance is equal to the slope of theq curve. Therefore, a linear q corresponds to a device witha constant memristance, which is indistinguishable from aconstant linear resistor according to (9).

Continuously differentiable: This property, which ensures thatthe derivative (d^q=dq) of (4) exists and is continuous,implies that the memristance (i.e., the gradient of q) is acontinuous function dened at every point along the q curveand is nite: jMqjo1; 8q. Moreover, from (9), it followsthat Wa0; 8. By applying the same arguments but rever-sing the roles of memristance and memductance, we deducethat the memductance is also nite and the memristance isnon-zero. Putting everything together leads to 0a jMqjo1; 8q and 0a jWjo1; 8. This result can be furtherrestricted to show that the memristance and the memductanceare in fact positive by taking into account the increasing mono-tonicity requirement (1.3): 0oMqo1; 8q, and 0oWo1; 8.

Strictly monotonically increasing: A strictly monotonic qcurve has a unique inverse such that (6) holds with theimplications already explained in the previous section. If theq curve is monotonically increasing, its gradient (i.e., thememristance) is positive and non-zero leading to a strictlypassive device, i.e., a device which does not require an internalpower source to operate. A direct consequence of theserestrictions is that the q curve of an ideal memristor mustbe a one-to-one function. It should be stressed here thatalthough an ideal memristor complying with the three criteriadetailed here is always passive, the converse is not always true(see Table 1).

An ideal memristor characterized by a q curve satisfying theabove criteria has a unique response for any input waveform andinitial state. More specically, assuming the same initial memri-stanceM0 (or memductance W0), a specic amount of charge q(or ux ) owing through the device induces the same changein the memristance (memductance) taking also into account thepolarity of the driving signal. As we will see later throughexamples, this does not necessary hold for non-ideal devices.

Fig. 1. (a) An ideal q curve satisfying the criteria (1.1)(1.3) and (b) itscorresponding iv response when driven by a sinusoidal excitation. In bothgures the same operating point (OP) Q qa;a is shown at t ta . At Q in(a) the instantaneous memristance t=qt and the incremental memristanceMq d^q=dq are shown. For the same OP (b) shows the instantaneousresistance vt=it Mq and the incremental resistance dv^i=di. The incrementalmemristance is equal to the instantaneous resistance when evaluated at the sameOP. However, all four quantities will be equal to each other only when the qcurve becomes linear. In this case the memristor becomes indistinguishable from alinear resistor.

P.S. Georgiou et al. / Microelectronics Journal 45 (2014) 13631371 1365

Table 1 illustrates three examples of non-ideal q curves andtheir respective iv responses when the system is driven by asinusoidal current input i i0 sin t.

Example 1 in Table 1 violates the monotonicity requirement(1.3). In particular, the function q is increasing (Mq40) up toa maximum (Mq 0) and then it decreases (Mqo0) down to alocal minimum (Mq 0) after which it increases (Mq40).When the device is in the region of operation whereMqo0, thevoltage and current have opposite polarities. This results in apartially active memristor with an iv response that is notrestricted to the rst and the third quadrant of the plane. Thetwo local extrema of q correspond to the additional four zero-crossings of the iv function (other than the origin) at which theoutput voltage is zero although the input current is non-zero. Sucha memristor can only exist as an active device.

Example 2 in Table 1 shows a q curve that is not strictlyincreasing due to a point of inection thus violating again (1.3). Atthe point of inection, the memristance is zero. On the iv planethis appears as two points (in addition to the origin) at which thevoltage output is zero although the current input is non-zero. Atthese two points the memristor is instantaneously acting as aresistanceless wire without a voltage drop across it.

The response ensuing from Example 2 is non-unique in thesense that starting from the same initial memristance the sameamount of charge q can cause different changes in the memri-stance. More specically, assume that the q function consistsof two branches: a rst branch on the left of the point of inec-tion and the second on its right. For every point on the leftbranch there is a corresponding point on the right branch thatshares the same memristance. However, driving the same amountof charge q through the device causes the memristance to

decrease, if the OP lies on the left branch, but it causes thememristance to increase, if the OP lies on the right branch.Therefore, the same input can cause two different responsesdepending on which of the two branches the OP lies on. The sameobservations apply in Example 1 for the two increasing sub-functions of the q curve.

Example 3 in Table 1 shows a q curve which is a piecewise(double-valued) function violating (1.2) and (1.3). This couldcorrespond to a device where each of its characteristic sub-functions is chosen depending on the polarity of the input. As aresult, the iv response shown is not symmetric for a periodiczero-mean input. Additionally, when switching from one branch ofq to the other, discontinuities are introduced in the memri-stance as the gradient is not uniquely dened at the switchingpoint. This example can still be classied as passive since its sub-functions are strictly increasing. However, it cannot be classiedas an ideal memristor. This example highlights that passivitydoes not imply ideality. On the other hand, an ideal q curveguarantees passivity.

It is also instructive to note that the response of the memristorin Example 3 is unique although q is double-valued. Because ofthe double-valued q curve, the device may have the same initialmemristance on both branches. Thus applying the same chargeqon each of these initial states will result in two different responses.However, once the polarity of the input is taken into account thisambiguity is immediately resolved.

Example 3 in Table 1 is the most interesting case of the non-ideal memristors. Although its q function is not ideal, each of itsconstituent sub-functions satises all the criteria of ideality. Thesememristors will be referred to as piecewise ideal. This is animportant characteristic since it may lead to properties of ideal

Table 1Examples of non-ideal chargeux curves with their respective currentvoltage responses under a sinusoidal input. The rst example is also compared with the linear caseq q indicated by the black solid line.

Example Chargeux Currentvoltage Comments and q

Ex. 1 The q function is not monotonic hence the iv response is not restricted in the rst and thethird quadrant and the memristor cannot be passive. The extrema of q correspond to zeromemristance and appear as four additional zero-crossings on the iv. The linear q shows that alinear resistor is the degenerative case of a linear memristor

q q34:5q26q

Ex. 2 The q function is not strictly increasing hence the memristance is zero at the point ofinection. In the iv plane this appears as two additional zero-crossings at which the voltage iszero for a non-zero input current

q q33q23q

Ex. 3 The q is a double-valued function hence it is not monotonic. Also, at the transition pointbetween the two branches it is non-continuously differentiable. This causes discontinuities in thememristance and results to a non-symmetric iv response

q q5q=3; itZ0e1:55q1=10; ito0

(

P.S. Georgiou et al. / Microelectronics Journal 45 (2014) 136313711366

memristors to apply separately on each of the sub-functions of theq curve. We expect many practical devices to exhibit such qcurves [16,15].

3. Memristors as reciprocal elements

In the previous section we have analyzed the ideal memristorand its fundamental constituent properties. In particular, we haveshown how these properties restrict the form of the q curve,and thus the behavior of the ideal device. In this section, weidentify the conditions under which ideal memristors comply withthe reciprocity theorem. The conguration that satises this theo-rem leads to an interesting behavior: if the output of a memristoris used to drive another identical memristor, the output of thesecond reproduces exactly the waveform used to drive the rstdevice. We also explore the applicability of reciprocity to non-idealmemristors. Finally, we discuss the idea of exploiting reciprocityfor signal encryption.

3.1. Denition of reciprocity

We rst introduce some notation. As shown in Fig. 2 thereciprocity theorem considers two states of the same network.Consider a circuit network consisting of N branches with j

denoting the jth branch. The single prime (0) is used to denotequantities of the rst state, and the double prime (00) to denotethose of the second state. Each state is a set of N pairs of branchvoltage drops and currents resulting from two distinct excitationsof the same frequency. The two states can also be viewed as twodifferent experiments on the same network, or on two identicalones [26,27]. Hence, the voltage and current of the jth branch aredenoted v0j; i

0j for the rst experiment and v

00j ; i

00j for the second

experiment.Consider two states of the same circuit network resulting from

two distinct input signals having the same frequency. The networkis reciprocal if it satises the condition [27]

N

j 1i0jv00j i00j v0j 0; 14

where the summation is taken over all the branches of thenetwork. Although we will focus here on a one-port network,the denition extends to multi-port networks. Additionally, it canbe shown that a network composed of reciprocal elements is itselfreciprocal. Note that the condition (14) may also be stated in termsof port voltages and currents. However, the two are equivalent andthere is no need to introduce this second representation [27].

3.2. Conditions under which ideal memristors are reciprocal

We now investigate the conditions under which an idealmemristor satises the reciprocity condition stated in (14). Asshown in Fig. 3, we will focus on a network consisting of only asingle memristor, therefore, (14) reduces to the following condi-tion for a single branch:

i01v001 i001v01 0; 15

which by using (7) can be re-expressed in terms of the memris-tances:

M01 M001; 16where M01 v01=i01 and M001 v001=i001. The memristance depends ontime t through its controlling variable, which is either the chargeq qt or the ux t. Depending on the choice of inputsignal in each of the two experiments, the reciprocity condition(16) becomes one of the following four conditions:

M01q M001q; 17a

M01 M001; 17b

M01q M001; 17c

One-portNetwork

SameOne-portNetwork

Experiment 1 Experiment 2

Fig. 2. Two experiments, or states, of the same one-port network consisting of Nbranches. In each experiment ip, vp denote the port current and voltage droprespectively and ij, vj denote respectively the current and voltage drop of the jthbranch, where j 1;2;3;;N. The signals of the rst experiment are indicated by asingle prime (0) and those of the second, by a double prime (00).

Experiment 1

Experiment 2

Fig. 3. Two experiments on the same, or two identical memristors. The two memristors share an identical q curve. The branch voltage, current and the memristance aredenoted by i1, v1 andM1 respectively, while the single (0) and double (00) prime indicate the rst and second experiments respectively.

P.S. Georgiou et al. / Microelectronics Journal 45 (2014) 13631371 1367

M01 M001q: 17dIt is important to emphasize that the appropriate condition musthold for all tZ0.

Consider rst the cases (17a) and (17b), in which both experi-ments have the same controlling quantity. The reciprocity theoremrequires that the two experiments are performed on the samememristor (i.e. characterized by an identical q curve); hencethe two memristances M01 and M001 will also have an identicalform. Consider also equal initial memristances M01q0 M001q0.The equality in (17a) holds if q q01 q001; 8 t40, whereq01t

R t0 i

01 d and q001t

R t0 i

001 d. The new equality

q q01 q001 is only true for all t if i01 i001. Similarly, in the case of(17b) the requirement is satised if v01 v001. Hence, the rst twoconditions are straightforward and do not provide any valuableinsights; they simply verify that, for two identical ideal memris-tors, applying the same input should always have exactly the sameeffect at all instants in time.

Consider now the other two cases (17c) and (17d), in which thecontrolling quantity is different in each experiment. We willdiscuss (17c) but the same arguments hold for (17d) as well. Theinput and output signals satisfying these two conditions reveal aninteresting property of ideal memristors. The rst experiment in(17c) is current-driven with input i01 and q

01t

R t0 i

01 d, while

the second experiment is voltage-driven with input v001 and001t

R t0 v

001 d. The two memristors are identical and hence

share the same q curve. We also assume that the initialmemristance is identical in both cases: M01q0 M0010. It thenfollows that (17c) can be re-expressed as follows:

M1q01t M1001t 8 t40; 18where it is important to note that the rst memristance is viewedas a function of q and the second memristance is a function of .As we have seen in (4), for ideal memristors, the ux is uniquelydened by an explicit function of the charge, thus (18) can be re-expressed as

M1q01t M1^ 001q001t 8 t40: 19Additionally, for ideal memristors each value of the chargecorresponds to a unique memristance. Hence (19) is true only ifq01 q001, or equivalently, if i01 i001; 8 t. Since i01 i001, then it followsfrom (19) that

v01 i01tM1q01 i001tM1^001q001 v001: 20Therefore, the reciprocity property holds for two identical andideal memristors congured such that the output of the rst isequal to the input of the second and provided their initialmemristances,M01q0 andM0010, are the same.

As part of the derivation of (20) we obtained the requirementthat i01 i001, namely the input of the rst memristor is equal to theoutput of the second. If we consider the case (17d), in which therst experiment is ux-controlled and the second charge-con-trolled, it is possible to establish the same result but with the rolesof the voltage and current reversed. This reveals an importantconsequence of reciprocity for ideal memristors connected suchthat the output of the rst is the input of the second memristor:the second memristor cancels the effect of the rst one andreturns as its output the original input of the rst memristor.

In conclusion, the reciprocity condition (15) holds for two idealand identical memristors with the same initial memristance when

output0 input00: 21The effect of this conguration is then

input0 output00; 22where the input and the output can be voltage or current wave-forms. Fig. 4a illustrates the case (17d) and shows how the second

memristor cancels the effect of the rst one and outputs theoriginal input waveform used to drive the rst memristor.

It is important to note that although the discussion above hasfocused on single memristors, the results on reciprocity extend tonetworks consisting of ideal memristors. This follows from the factthat a network of reciprocal elements, is itself reciprocal [27]. In thecase of memristors, we may arrive at the same conclusion follow-ing an alternative path: Chua's Closure and Existence and Unique-ness theorems show that a network of ideal memristors isequivalent to a single unique memristor [1]. Therefore, if theequivalent memristors representing the two networks fulll theconditions summarized above, they will behave as reciprocal.

Fig. 4be demonstrates the operation of two ideal, identical HPmemristors [7] with the same initial memristance arranged suchthat the output of a voltage-driven memristor is fed as an input toa second current-driven, as in Fig. 4a. We show the response tofour different input waveforms and demonstrate how the secondmemristor cancels the effect of the rst one and outputs thewaveform originally fed as an input to the rst memristor.

This canceling property of memristors depends on the set ofrequirements detailed above. If any of these conditions is notsatised, then the reciprocity property is not guaranteed to hold.Fig. 5 illustrates two such scenarios. Fig. 5a shows the case wherethe two memristors are identical but their initial memristance isnot. Clearly, the output of the second memristor is not the same asthe rst input. A similar lack of perfect cancellation is observed inFig. 5b where the two memristors are not identical because theydo not share the same q curves.

3.3. Reciprocity for non-ideal memristors

For a non-ideal memristor it is not always possible to uniquelyexpress the ux as an explicit function of the charge (and viceversa), thus (19) and the steps that follow do not hold in general.However, it is still possible to operate non-ideal memristors, andspecically piecewise ideal memristors, as reciprocal elementslocally, as we now show.

First, consider two identical, non-ideal memristors with thesame initial memristance characterized by the q curve shown inExample 1 of Table 1. Irrespective of whether the controllingquantities of the two experiments are as in (17c) or (17d), if theinput to the rst memristor operates the device only within theincreasing part of the q curve (i.e. the locally ideal sub-function)then the devices will appear as complying with the reciprocitytheorem. On the other hand, if the memristors have their control-ling quantities according to (17c) and the input to the rst device issuch that it is driven beyond the maximum of the q curve, thenbecause the q curve does not have a unique inverse, at least intheory, it is possible for the second memristor to follow multiplepaths. Depending on which path is followed, the congurationmay, or may not comply with reciprocity.

Consider now two identical, non-ideal memristors with thesame initial memristance characterized either by the q curve ofExample 2 or Example 3 of Table 1. In both cases the congurationwill satisfy the reciprocity theorem for any input signal irrespec-tive of the region of operation. This is because for any value of thecharge there is a unique value of the ux and vice versa. We wouldlike to note here that the apparent ambiguity in the third exampleis resolved by taking into account the polarity of the input signal.Both these examples, especially the third one, are of particularvalue since they indicate the potential applicability of reciprocityon real non-ideal devices. An example of the operation of such aconguration of interconnected piecewise ideal memristors isshown in Fig. 6.

It may be possible to utilize the compliance of memristors withthe reciprocity theorem in practical applications. For example, a

P.S. Georgiou et al. / Microelectronics Journal 45 (2014) 136313711368

Fig. 4. Demonstration of the effect of reciprocity. The two memristors are congured as in (a) with the output of the rst one used to drive the second (i01 i001) and assumingthat the two devices are ideal, identical and have the same initial memristance such that (14) is satised. As a result of this conguration the output of the second memristor isidentical to the input of the rst memristor (v01 v001). This effect is demonstrated in (b)(e) for various input waveforms. The rst column of plots shows the input of the rstmemristor (v01), the second column the output of the rst, which is also the input of the second (i

01 i001), and the third column shows the output of the second memristor

superimposed on the input of the rst one to illustrate that the two waveforms match exactly. It is assumed that the memristors are characterized by HP's ideal memristormodel [7] whose output response can be found in Ref. [18]. The input waveforms used are: (b) Sinusoidal, (c) Sawtooth, (d) Sum of two sinusoidals and (e) Product of twosinusoidals.

P.S. Georgiou et al. / Microelectronics Journal 45 (2014) 13631371 1369

proposed application is to use a block of memristors for non-linearly transforming, or encrypting, a signal that needs to besecurely transmitted. According to the reciprocity theorem thesignal can only be decrypted if the receiver has an identical copy ofthe block used to encrypt the signal at the transmitter. In otherwords, at both ends of this encrypted communication system theblock of memristors and its initial state will be acting as theunique key for encrypting/decrypting the signal. If the receiverlacks an exact copy of the transmitter's block of memristors thenthe signal will not be retrieved in its original form. This couldprevent unauthorized interception of the transmitted information.This proposed application will be the object of future research.

4. Discussion

In this work we have investigated and identied the conditionsunder which ideal memristors comply with the reciprocity theo-rem of classical circuit theory. Our investigation began rst byclarifying the properties that give rise to the ideal memristor andhow these properties shape the characteristic q curve of theideal device. Devices that fail to comply with the ideality require-ments may exhibit effects such as zero memristance (or inniteconductance); non-unique responses; or inexistence as passivedevices. We then identied the requirements on the input/outputsignals and the device itself under which ideal memristors behave

Fig. 5. Failure to satisfy the reciprocity theorem because the two memristors are not identical. It is assumed that the memristors are ideal and characterized by HP's model[7] as in Fig. 4. The devices in the two experiments are connected such that the output of the rst one drives the second. In (a) the two memristors differ in their initialmemristance. In (b) the memristors are characterized by non-identical q curves due to different device widths. As a result in both examples the output of the secondmemristor does not match exactly the input of the rst one (v01av

001). The input waveforms used are: (a) Sawtooth and (b) Product of two sinusoidals.

Fig. 6. Reciprocity in non-ideal memristors. The memristors in the two experiments are characterized by the same non-ideal q curve of Example 3 in Table 1 and share thesame initial memristance. As in Fig. 4a the output of the rst devices is fed as an input to the second. The conguration satises reciprocity because each of the monotonicsub-functions of the q is locally ideal (piecewise ideal) and the device has a unique response at each point along the curve. As a result, the nal output matches the initialinput (v001 v01).

P.S. Georgiou et al. / Microelectronics Journal 45 (2014) 136313711370

as reciprocal elements. More specically, it was shown that twoideal and identical memristors with the same initial memristancecomply with the reciprocity theorem in (14) when the output ofthe rst memristor is used as the input of the second device.A direct consequence of this conguration is that the secondmemristor cancels the effect of the rst one and outputs theoriginal input waveform used to drive the rst memristor. In thecase of identical, non-ideal memristors with the same initialmemristance, reciprocity may be satised depending on thespecic q curve characterizing the device and the input signalapplied. More specically, if the device is operated within a locallyideal region of its q curve satisfying (1.1)(1.3) then thereciprocity theorem will still be satised (e.g. Example 3 inTable 1). A proposed application discussed is to exploit reciprocityin an encrypted communication system. In such a system thenetwork of memristors itself and its initial state will be theencryption/decryption key.

The study of reciprocity showcases the use of the idealmemristor as a fundamental tool for investigating the propertiesof these devices and for extending our understanding of non-idealdevices. For example, by viewing non-ideal devices as locally idealmay allow us to apply results developed for the ideal memristor tonon-ideal components.

While signicant research on practical non-ideal devices con-tinues, through our work we would like to demonstrate that thereis also a lot to be learned by studying the ideal element asoriginally dened by Leon Chua [1]. The results presented hereand in our previous works [1719] can be used as examples tomotivate the use of this approach as an alternative means ofcomprehending the behavior of practical memristive devices [20].

Acknowledgments

The authors would like to thank EPSRC for supporting Dr.Georgiou through the Doctoral Prize Fellowship and Prof. RobertSpence (Imperial College) for his insightful comments.

References

[1] L.O. Chua, Memristorthe missing circuit element, IEEE Trans. Circuit Theory18 (5) (1971) 507519.

[2] L.O. Chua, S.M. Kang, Memristive devices and systems, Proc. IEEE 64 (2) (1976)209223.

[3] C. Baatar, W. Porod, T. Roska, Cellular Nanoscale Sensory Wave Computing, 1sted., Springer Publishing Company Incorporated, New York, USA, 2009.

[4] R. Waser, R. Dittmann, G. Staikov, K. Szot, Redox-based resistive switchingmemoriesnanoionic mechanisms, prospects, and challenges, Adv. Mater. 21(2526) (2009) 26322663.

[5] G.F. Oster, D.M. Auslander, The memristor: a new bond graph element, J. Dyn.Syst.Trans. Asme 94 (3) (1972) 249252.

[6] E.M. Drakakis, A.J. Payne, A bernoulli cell-based investigation of the non-lineardynamics in log-domain structures, Analog Integr. Circuits Signal Process. 22(2000) 127146.

[7] D.B. Strukov, G.S. Snider, D.R. Stewart, W.R. Stanley, The missing memristorfound, Nature 453 (7191) (2008) 8083.

[8] C. Schindler, M. Weides, M.N. Kozicki, R. Waser, Low current resistive switch-ing in CuSiO2 cells, Appl. Phys. Lett. 92 (12) (2008) 122910.

[9] A.C. Torrezan, J.P. Strachan, G. Medeiros-Ribeiro, R.S. Williams, Sub-nanosecond switching of a tantalum oxide memristor, Nanotechnology 22(48) (2011) 485203.

[10] S.H. Jo, K.-H. Kim, W. Lu, High-density crossbar arrays based on a Simemristive system, Nano Lett. 9 (2) (2009) 870874.

[11] I. Vourkas, G. Sirakoulis, A novel design and modeling paradigm formemristor-based crossbar circuits, IEEE Trans. Nanotechnol. 11 (6) (2012)11511159.

[12] I. Vourkas, G.C. Sirakoulis, Memristor-based combinational circuits: a designmethodology for encoders/decoders, Microelectr. J. 45 (1) (2014) 5970.

[13] D.B. Strukov, D.R. Stewart, J.L. Borghetti, X. Li, M.D. Pickett, G. Ribeiro, W.Robinett, G.S. Snider, J.P. Strachan, W. Wu, Q. Xia, J.J. Yang, W.R. Stanley, HybridCMOS/memristor circuits, in: IEEE International Symposium on Systems andCircuits.

[14] J.J. Yang, M.D. Pickett, X. Li, D.A.A. Ohlberg, D.R. Stewart, W.R. Stanley,Memristive switching mechanism for metal/oxide/metal nanodevices, Nat.Nanotechnol. 3 (7) (2008) 429433.

[15] F. Alibart, L. Gao, B.D. Hoskins, D.B. Strukov, High precision tuning of state formemristive devices by adaptable variation-tolerant algorithm, Nanotechnol-ogy 23 (7) (2012) 075201.

[16] S.H. Jo, K.-H. Kim, T. Chang, S. Gaba, W. Lu, Si memristive devices applied tomemory and neuromorphic circuits, in: IEEE International Symposium onSystems and Circuits, 2010, pp. 1316.

[17] E.M. Drakakis, S. Yaliraki, M. Barahona, Memristors and bernoulli dynamics, in:IEEE CNNA, 2010, pp. 16.

[18] P.S. Georgiou, S.N. Yaliraki, E.M. Drakakis, M. Barahona, Quantitative measureof hysteresis for memristors through explicit dynamics, Proc. R. Soc. A 468(2144) (2012) 22102229.

[19] P. Georgiou, M. Barahona, S. Yaliraki, E. Drakakis, Device properties of bernoullimemristors, Proc. IEEE 100 (6) (2012) 19381950.

[20] P.S. Georgiou, M. Barahona, S.N. Yaliraki, E.M. Drakakis, Ideal memristors asreciprocal elements, in: ICECS 20th Conference, 2013, pp. 301304.

[21] L.O. Chua, Device modeling via basic nonlinear circuit elements, IEEE Trans.Circuits Syst. 27 (11) (1980) 10141044.

[22] L.O. Chua, Resistance switching memories are memristors, Appl. Phys. A:Mater. Sci. Process. 102 (2011) 765783.

[23] Y.V. Pershin, M. Di Ventra, Memory effects in complex materials and nanoscalesystems, Adv. Phys. 60 (2) (2011) 145227.

[24] T.M. Apostol, Mathematical Analysis, Addison-Wesley Series in Mathematics,1st ed., Addison-Wesley Pub. Co., Reading, Massachusetts, USA, 1957.

[25] G.F. Oster, A note on memristors, IEEE Trans. Circuits Syst. 21 (1) (1974) 152.[26] P. Peneld, R. Spence, S. Duinker, Tellegen's Theorem and Electrical Networks,

Research Monographs, MIT Press, Cambridge, Massachusetts, USA, 1970.[27] R. Spence, Linear Active Networks, Wiley-Interscience, Chichester, UK, 1970.

P.S. Georgiou et al. / Microelectronics Journal 45 (2014) 13631371 1371

On memristor ideality and reciprocityIntroductionThe ideal memristorAnalysis of the ideal memristorExploring the criteria for ideality

Memristors as reciprocal elementsDefinition of reciprocityConditions under which ideal memristors are reciprocalReciprocity for non-ideal memristors

DiscussionAcknowledgmentsReferences