How to Teach Memristors in EE Undergraduate Courses Jordi Albo-Canals

Giovanni E. Pazienza

LIFAELS La Salle - Ramon Llull University

MTA-SZTAKI & Pazmany University

Barcelona, Spain Email: [email protected]

Budapest, Hungary Email: [email protected]

Abstract— Sin Since ce

a couple couple of years years,, the memr memrist istor or has been in the the me medi dia a spot spotli ligh ghtt and and it is expe expect cted ed to have have a majo majorr impactt on the future techno impac technology logy.. Many scienti scientists sts think that it should be taught in EE undergraduate courses, but there is no general agreement on how to do so. This paper presents several approaches to memristor and a thorough discussion about them, and it is the result of numerous discussions with experts in this area.

I. I NTRODUCTION In his his semi semina nall pape paperr pu publ blis ishe hed d in 1971 1971 [1], [1], Ch Chua ua co conn jectured the existence of a fourth fundamental passiv passivee twotermin terminal al circui circuitt elemen elementt beside besidess the can canoni onical cal res resist istor or,, capacitor paci tor,, and induc inductor tor.. Such element element was called called “memr “memristor istor”” because it has the characteristics of a non-linear resistor with memory, in the sense that its resistance is a function of the charge cha rge that that has ﬂowed ﬂowed thr throug ough h the devic device. e. The mem memris ristor tor has rarely appeared in undergraduate EE courses, sometimes taug taught ht by rese resear arch cher erss who who ha had d ha had d a dire direct ct co cont ntac actt with with Chuaa (se Chu (seee commen comments ts by Kang Kang and Mazie Maziersk rskaa in [2]). [2]). Thi Thiss was a natural situation, since nobody had claimed of having manufa man ufactu ctured red memris memristor torss (in fac fact, t, it is now now known known tha thatt in the last few decades there have been hundreds of papers and patents pate nts on memristiv memristivee devices, devices, even though the authors authors did not identify this characteristic). The scenario changed drastically in 2008 when HP made public its work on memristors [3], which attracted the attention

why memristor should be taught and compared the methods discussed before; in Sec. V, we draw the conclusions of our work. II. B RIEF NOTES ABOUT

MEMRISTORS

Usua Usuall lly y, a me memr mris isto torr is deﬁn deﬁned ed as a ci circ rcui uitt el elem emen entt in which whi ch the relati relations onship hip bet betwee ween n volta voltage ge and cha charge rge has the form: v = M (q )i. Since the memristance M depends on the charge, the element retains a memory of the past events of the input current. In fact, this deﬁnition is not general because a memristor could be deﬁned as an element in which “there is a nonlinear relationship between the integrals of the current and voltage” as already pointed out in [3]. This last deﬁnition has the advantage [5] of not not introducing the magnetic ﬂux, which in fac factt does does not reall really y pla play y a rol rolee in the physica physicall wor workin king g principles of the memristor. It is possible to prove [6] that all devices are characterized by the fact of displaying a pinched hysteresis loop in the current-voltage characteristic, as shown in Fig. 1, are memristors, and vice versa.

of the scientiﬁc community. Since then, many scientists have afﬁrmed afﬁ rmed that “textbook “textbookss should should be rewr rewritten itten”” [4] and some schools have started introducing memristors in their curricula. Ne Neve verth rthele eless, ss, lit little tle has been been wri writte tten n (e. (e.g., g., [5] [5])) on why and how how the mem memris ristor tor should should be introd introduce uced d in und underg ergrad raduat uatee Fig. 1. A pinched hysteresis hysteresis loop in the curren current-vol t-voltage tage characteris characteristic tic is the EE courses. Nowadays, to the best of our knowledge, there hallmark of memristors. is no genera generall stu study dy answer answering ing these these que questi stions ons;; this this pap paper er wants to address them, also illustrating several approaches to When a memristor works with alternating current, the device memristor. The goal of this work is to distill into a few pages switches reversibly between a low resistance state and a high the main considera considerations tions that professors professors teaching memristors memristors resistance state; for this reason, it is considered to be a possible should shou ld make make,, provide provide argumentation argumentation in fav favor or or against against the breakthrough in the technology for non-volatile memories [3]. approaches proposed, and present an essential, yet thorough, III. P OSSIBLE APPROACHES TO MEMRISTORS reference list. approach ch The paper is structured as follows: in Sec. II, we give a brief A. Axiomatic approa Chua introduced the memristor through an ‘axiomatic aptheoretica theor eticall overvie overview w of memristors memristors;; in Sec. III, we desc describe ribe fourr met fou method hodss to teach teach mem memris ristor tors; s; in Sec Sec.. IV, we explai explain n proach’ [1] which deﬁnes four fundamental circuit variables

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– voltage v , current i, charge q , and ﬂux φ – and describes the two-terminal circuit elements as relationships between two of the four variables. There are six independent permutations of two objects in a bank of four, and hence there must be six ways to link v, i, q , φ: two of them correspond to the deﬁnitions deﬁni tions of curre current nt and voltage (Eqs. 1 and 2, respe respectiv ctively), ely), while other three corresp correspond ond to the canonica canonicall circu circuit it eleme elements nts (Eqs. 3, 4, and 5). For the sake of completeness, Eq. 6 has to corres correspon pond d to a fourth fourth fun fundam dament ental al two two-te -termi rminal nal cir circui cuitt

and a resistor can be seen as a spring, a mass, and a dissipative effect (friction), respectively. Now, what about the memristor? The answer to this question was given short after the publication of the original article about memristors (see Fig. 3 in [7], described in more detail in [8]). We report the “mechanical memristor” proposed by those authors in Fig. 2, for ease of reference.

element: the memristor. Deﬁnition of current: dq = = idt

(1)

Deﬁnition of voltage: dφ = vdt v dt

(2)

Resistor: dv = Rdi

(3)

Capacitor: dq = = C dv

(4)

Inductor: dφ = Ldi

(5)

Memristor: dφ = M dq

Fig. 2.

Mechanical Mechanical equiv equivalent alent of a memristo memristor. r.

(6)

From this equation, it is clear that when the memristance M is not a function of q , the memristor behaves like a resistor. B. Memristor as time-varying nonlinear resistor resistor

The memristor is a nonlinear element and consequently the methods based on the Kirchhoff’s laws lead to a system of

This device is composed of a dashpot cylinder with a tapered fri fricti ction on rod att attach ached ed at a certai certain n dis distan tance ce fro from m the dashpo dashpott enclos enc losure ure and a pis piston ton to whi which ch is att attach ached ed a thi thick ck rub rubber ber disc. The diame diameter ter of ﬂexi ﬂexible ble rubber sleeve varie variess with the penetr pen etrati ation on of the pis piston ton d and and hen hence ce the force need needed ed to push pus h the pisto piston n fur furthe therr is a fun functi ction on of d. In oth other er wor words, ds, the incremental resistance depends on the instantaneous piston disp displa lace ceme ment nt.. This This si situ tuat atio ion n is an anal alog ogue ue to th thee one one of a res resist istor or who whose se res resist istanc ancee dep depend endss on the cha charg rgee ﬂown: ﬂown: it is then then rea reason sonabl ablee to mod model el the dashp dashpotot-pis piston ton system system as a memristor [7].

nonlin non linear ear undergraduate dif differ ferent ential ial equati equ ations ons,, which whi ch may be not compoeas easy y to solve for students. However, nonlinear nents are common – e.g., diodes, thermistors, varistors – for which we can study them by using the conventional methods that do not require advanced knowledge of nonlinear analysis, such as: the load linear equation, the assumption veriﬁcation method , and the recursion. We will spend a few words on each D. Teaching memristor through examples of them since the literature on these topics is easily accessible. In EE undergraduate courses, it is usual to introduce funda1) Load linea linearr equation equation:: this is one of the most common mental circuit elements through examples that do not require methods to ﬁnd the working point of the nonlinear compo- more than basic concepts of calculus; memristor should not be nents, nen ts, and it simply simply combin combines es gra graphi phical cally ly the cur curren rentt and an exception to this practice. A good perspective on this topic voltage characteristics with the values deriving from the rest of was offered in [5] which starts from a simple deﬁnition of the circuit. Of course, this technique can only be used when memristance in terms of the combination of a two resistances the manufactur manufacturer er supp supplies lies the working working characteri characteristics stics of the – one with a high value RH and the other with a low value component, as in general happens in the case of memristors. RL – of the form 2) Assu Assumptio mption n and veriﬁcatio veriﬁcation: n: in this case the component is modeled state by state obtaining a non-continuous function; then, the circuit is analyzed assuming that the value modeled for the memristor is correct. 3) Recu Recursio rsion: n: this can be use used d to sol solve ve numer numerica ically lly the nonlinear differential equation resulting from a circuit with a memristor memri stor.. An initial initial val value ue of memristanc memristancee is imposed imposed and then it is recursively corrected until convergence is achieved.

M (x) = x (t) · RL + (1 − x(t)) · RH

(7)

where x(t) is a internal state restricted to the interval [0,1]. In the simplest model, x(t) is proportional to the charge q . Throug Thr ough h a str straig aightf htforw orward ard pro proced cedure ure,, it is pos possib sible le to ﬁnd that there is a nonlinear relationship between the integrals of current and voltage, as required by the deﬁnition of memristor. We also sugges suggestt a simpl simplee exa example mple using of the exponen exponential tial C. Mechanic Mechanical al analogy function, which is in general well-known by the students. If Ba Basi sicc co conc ncep epts ts of EE are are taug taught ht in a larg largee vari variet ety y of the memristance M has the form dis discip ciplin lines es and for some some studen students, ts, pos possib sibly ly com coming ing fro from m M (q ) = R 0 eq/Q different ﬁelds, EE notions can be considered ‘abstract’, especially when lessons are not accompanied by adequate lab where R and Q are reference values for the resistance and 0 0 practices. For this reason, often the working principles of the the charge, respectively. we obtain from Eq. (6) that fundamenta funda mentall circuit circuit elements elements are illustrated illustrated through anal analogy ogy with mechanical devices. For instance, a capacitor, an inductor, dφ = R 0 eq/Q dq 0

0

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Anotherr fre Anothe freque quent nt claim claim is tha thatt the memristo memristorr is not the fourth fundamental fundamental element element since many other relations relationships, hips, involvin inv olving g highe higher-or r-order der derivat derivative ives, s, amon among g the funda fundamenta mentall variables can be deﬁned [13]. It is out of the scope of our paper to confute this assertion, but the axiomatic approach, discussed in Sec. III-A, makes it clear that memristors can be put on the same level as the other canonical circuit elements. Moreover, it is true that other elements with memory can be deﬁned [14] but this result was already predicted in [3] where Fig. 4, which

Fig. 3. Te Temporal mporal evolut evolution ion of the volta voltage ge and the current in the memrist memristor or presented in Sec. III-B.

By integrating, we ﬁnd

φ(q ) = Q 0 R0 eq/Q + K 0

where K is a constant. Supposing that there is a sinusoidal alternating current, the charge has the expression q (t) = Q 0 · sin(t) and, in order to simplify the analysis, we can assume that R0 = 1 and Q0 = 1. Obviously, i(t) = dq dt , so i (t) = cos t. Hence, the equation above can be rewritten as

shows that the broad class of ‘memristive systems’ includes not not only only th thee me memr mris isto torr but but al also so th thee (m (mem em)c )cap apac acit itor or an and d the (mem)inductor, was published. Similar conclusions were drawn in [6] where the existence of the so-called memristive systems – which includes, but it is not limited to, memristors – was already discussed in both from a theoretical and a practical viewpoint. Also, in [15] it was displayed a so-called periodic table of circuit elements from which it is clear that, in general, countless new circuit elements, with and without memory, can be deﬁned.

φ(t) = e sin t + K but v(t) = dφ , and hence dt

v(t) = cos t · esin t The The temp tempor oral al ev evol olut utio ion n of the the volt voltag agee an and d the the cu curr rren entt is presented in Fig. III-D. The voltage-current graph, shown in Fig. 1, shows the two linear regions, one with high resistance (RH = 2, 5Ω) and one with low resistance (RL = 0, 45Ω). IV IV.. D ISCUSSION Before starting the discussion about the different approaches we have presented so far, it is important to pose a fundamental question: Is it really necessary to teach memristors? We answer this question in the ﬁrst part of this section whereas the second part is devoted to a general discussion about the approaches presented above. A. Why should we teach memristors?

Fig. Fig. 4. The The fo four ur funda fundame ment ntal al two-te two-term rmin inal al pass passiv ivee ci circ rcui uitt el elem emen ents ts (from [3]).

Last but not least, the importance of memristors goes beyond circuit theory and EE in general, because memristive behaviors can be easily observed in numerous phenomena. For instance, in a light bulb the resistance of the ﬁlament is not a constant but changes with temperature, exactly as memristance changes with charge. Indeed, Indeed, the memristi memristive ve effect of the light bulb was alread already y cha charac racter terize ized d in [16 [16], ], lon long g bef before ore the for formal mal introduction of memristors. Memristor is not one of the many controve contr oversial rsial techn technologi ologies es often prese presented nted as promis promising ing but rather a realistic effective possibility for new devices in the near near futu future re,, whic which h de dese serv rves es to be wi wide dely ly ta taug ught ht as al also so emphasized by the two IEEE Spectrum cover stories related to memristor published in the last two years [17] and [18].

Some critics argue that the technology required to manufacture memristors makes it impossible to most of the academic institutions to have them; as a consequence, memristors should have have no pla place ce int into o the curri curricul culaa bec becaus ausee it wil willl ne neve verr have have a plac placee in into to th thee la labs bs.. This This thes thesis is has has two two ﬂa ﬂaws ws:: ﬁrst ﬁrst of all, lab practices make extensive use of software simulations, and SPICE models models of memristors memristors are alrea already dy availab available le [9], [1 [10] 0];; se seco cond nd,, th thee te tech chno nolo logy gy prop propos osed ed by HP is not not the the B. How should we teach memristors? Of course, course, there is no best way to teach memristors and only onl y possib possible, le, and other other labs labs have have pro propos posed ed techni technique quess to necess essary ary to ada adapt pt the metho method d to the back backgro ground und and manufacture memristors without nanoscale facilities [11] or on it is nec different supports [12]. Therefore, even though the memristor the expectations of the audience. Nevertheless, it is possible may neve neverr bec become ome an of off-t f-thehe-she shelf lf com compon ponent ent,, it wil willl be to make some general considerations about the four methods possible for a large number of institutions to manufacture, or described above, trying to understand in which context they can be successfully used. at least simulate, memristors.

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In the axiomatic approach (Sec. III-A) the introduction of memristor is straightforward and it was used in articles [19] aimed aim ed at a gen genera erall audien audience. ce. Such Such a sim simple ple met method hodolo ology gy,, though tho ugh,, is dif difﬁcu ﬁcult lt to apply apply when when EE course coursess are preced preceded ed in the curricula by basic physics courses where students get familiar ﬁrst with the physics of the devices. In this context, the resistance is introduced as a measure of the opposition of the atomic structure of a material to the passage of electrons and the capacitance as the quantity of charge hold in a volume.

undergraduate curricula, it will be difﬁcult for memristor to do so. Also, we believe that a mix of the four methods illustrated in III III-B -B is possib possibly ly the more app approp ropria riate te wa way y to introd introduce uce memris mem ristor torss to stu studen dents. ts. The axioma axiomatic tic approa approach ch mus mustt be used used si sinc ncee it is th thee mo most st natu natura rall way way to in intr trod oduc ucee it it,, but but practical examples as well as simulations, which we collected in [21], are in general very popular among students, especially nowadays.

The basic physical mechanisms of the memristor may not be simple to explain and understand, and for this reason freshmen would not get in touch with the memristor at the same time as the other fundamental elements. Also, the axiomatic approach would not work when circuit elements are taught directly in the lab, and their character characteristic isticss are derive derived d direc directly tly by the students stud ents through practical practical examples examples.. The lack of a physical physical memristor would turn this element into a ‘mysterious object’ and hence no attention would be given to it. Teach eachin ing g th thee me memr mris isto torr as sugg sugges este ted d in III III-B -B ha hass the the evident advantage of not requiring any additional knowledge of nonlinear analysis: the memristor can be modeled in the same way as other nonlinear electronic components. Also, SW tools as Matlab or SPICE can help to simulate the behavior of memristors and hence save the students from stepping into the “nonlinear world”. Though, it is clear that we should question why the basic concepts of nonlinear analysis do not ﬁnd place in undergraduate courses. cours es. As Chua himself afﬁrmed in sev several eral intervie interviews ws [2] “most “mo st pro profes fessor sorss are educat educated ed in lin linear ear the theori ories, es, and are illiterate on nonlinear circuits” and this fact, we add, can be a big obstacle to the introduction of memristor in undergraduate course cou rses. s. Then, Then, the met method hodss propos proposed ed can be a tem tempor porary ary solution to this problem but they should not be become the standard approach. Finall Fin ally y, wha whatt dis discus cussed sed in Secs. Secs. III-C III-C and III-D can be deﬁnitely used before or after the detailed theoretical introduction of memristors but they cannot substitute it. The approach should sho uld be sim simila ilarr to the one for the oth other er thr three ee canon canonica icall elements for which theoretical statements can be accompanied by practical analogies. It is important to emphasize that the

ACKNOWLEDGMENT

stu studen dents ts sho should uld not only only hear about memri memristors stors but learn about them, since they can be crucial in future circuit design, as already noticed in recent papers [20]. V. C ONCLUSION Techno echnologic logical al and syst systemati ematicc aspe aspects cts of memristors memristors hav havee been widely discussed in the last few years, but the didactic aspects have not received the attention they deserve. In this paper, we illustrated our viewpoint on them which has matured through intensive discussions with experts in this ﬁeld. One of the conclusi conclusions ons of our discou discourse rse is that that non nonlin linear ear analys ana lysis is should should be tau taught ght wid widely ely and the introd introduct uction ion of memris mem ristor tor is a good good ex excus cusee for it. In a quo quote te attrib attribute uted d to several scientists, and possibly belonging to none of them, “the study of non-linear physics is like the study of non-elephant biology”. Until nonlinear analysis does not ﬁnd place in EE

The authors would like to thank Professor Leon Chua for his constant support and his valuable advice. R EFERENCES [1] L. Chua, “Memristo “Memristor-the r-the missing missing circuit element,” element,” Circuit Theory, IEEE Transactions on, vol. 18, no. 5, pp. 507 – 519, sep. 1971. [2] S. Adee. How do you teach the memri rist stor or?? [O [Onl nlin ine] e].. Avail vailab able le:: ht http tp:/ ://s /spe pect ctrum rum.i .iee eee. e.or org/ g/te tech ch-talk/semiconductors/nanotechnology/do-you-teach-the-memristor [3] D. B. Str Struko ukov v, G. S. Sni Snider der,, D. R. Stewar Stewart, t, and R. S. Wi Willi lliams ams,, “The “The missing memristor found,” Nature, no. 453, pp. 80–83, March 2008. [4] R. C. Jo Johns hnson. on. ’M ’Mis issi sing ng link’ link’ memr memris isto torr cr crea eate ted: d: Rewr Rewrit itee th thee text textbooks? books? [Onlin [Online]. e]. Avail vailable: able: http://www http://www.eeti .eetimes.c mes.com/ele om/electroni ctronicscsnews/4076910/-Missing-link-memristor-created-Rewrite-the-textbooks[5] F. Y. Wang ang,, “Me “Memri mristo storr for int introdu roducto ctory ry physics physics,,” 2008. 2008. [Onlin [Online]. e]. Available: Available: http://arxiv.org/abs/0808.0286 http://arxiv.org/abs/0808.0286 [6] L. Chua and S. M. Kang, “Memristiv “Memristivee devices and systems, systems,”” Proceedings of the IEEE , vol. 64, no. 2, pp. 209 – 223, feb. 1976. [7] G. F. Oste Osterr and and D. M. Ausl Auslan ande der, r, “T “The he memr memris isto tor: r: A new new bond bond Journall of Dynami Dynamicc Sys System tems, s, Measur Measureme ement, nt, and graph element, element,”” Journa Control, vol. vol. 94, 94, no. no. 3, pp. pp. 249–2 249–252 52,, 1972. 1972. [O [Onl nlin ine] e].. Avail vailab able le:: http://link.aip.org/link/?JDS/94/249/1 [8] G. Oster Oster,, “A note on memristors,” memristors,” Circuits and Systems, IEEE Transactions on, vol. 21, no. 1, pp. 152 – 152, jan. 1974. [9] M. Mahvash and A. Parker, Parker, ““A A memristor spice model for designing memristor circuits,” aug. 2010, pp. 989 –992. [10] A. Rak and G. Cse Cserey rey,, “Macro “Macromode modelin ling g of the memristor memristor in spi spice, ce,”” Computer-Aided Design of Integrated Circuits and Systems, IEEE Transactions actio ns on, vol. 29, no. 4, pp. 632 –636, apr. 2010. [11] T. Prodro Prodromakis, makis, K. Michelakis, Michelakis, and C. Toumazou oumazou,, “Switching “Switching mechanisms in microscale memristors,” Electronics Letters, vol. 46, no. 1, pp. 63 –65, jan. 2010. [12] V. Erokhin Erokhin,, “Organic “Organic memristo memristors rs : Basic principles, principles,”” may. may. 2010, pp. 5 –8. [13] B. Mouttet, Mouttet, “The mythology of the memristor, memristor,”” in Circuits and Systems (ISCAS), Proceedings of 2010 IEEE International Symposium on , May 2010. [14] M. Di Ventra, Ventra, Y. Y. Pershin, and L. Chua, “Circuit “Circuit elements with memory: memory: Proceedi oceedings ngs of the Memristors Memri stors,, memcapacitor memcapacitors, s, and memind meminductors uctors,” ,” Pr IEEE , vol. 97, no. 10, pp. 1717 –1724, oct. 2009. [15] L. Chua, “Nonlinear “Nonlinear circuit foundations foundations for nanodevices nanodevices.. i. the fourelement torus,” Proceedings of the IEEE , vol. 91, no. 11, pp. 1830 – 1859, nov. 2003. [16] W. J. Cunningham, Cunningham, “Incandescent “Incandescent lamp bulbs in voltage voltage stabilizers stabilizers,” ,” Journal of Applied Physics, vol. 23, no. 6, pp. 658 –662, jun. 1952. [17] R. Williams, Williams, “How we found the missing memrist memristor, or,”” IEEE spectrum, vol. 45, no. 12, pp. 28–35, 2008. [18] M. Versac Versacee and B. Chandler, Chandler, “The brain of a new machine, machine,”” Spectrum, IEEE , vol. 47, no. 12, pp. 30–37, 2010. [19] J. M. Tour and T. He, “Electronics: “Electronics: The fourth element,” element,” Nature, vol. 453, no. 42-43, 2008. [20] C. K. Volo olos, s, I. M. Kypri Kypriani anidis dis,, S. G. Stavri Stavrinid nides, es, I. N. Stoubo Stouboulos ulos,, and A. N. Anagnostopoulos, Anagnostopoulos, “Memristors: “Memristors: a new approach in nonlinear nonlinear circuits design,” in ICCOM’10: Proceedings of the 14th WSEAS international conference on Communications. Ste Stevens vens Point, Point, Wisconsin Wisconsin,, USA: World Scientiﬁc and Engineering Engineering Academy and Socie Society ty (WSEAS), (WSEAS), 2010, pp. 25–30. [21] J. Albo. [Online]. Available: http://www.salle.url.edu/ jalbo/iscas2011.htm

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