A Demonstration of Implication Logic Based on Volatile (Diffusive) Memristors
11 A Demonstration of Implication Logic Based onVolatile (Diffusive) Memristors
Yuriy V. Pershin,
Senior Member, IEEE
Abstract —Implication logic gates that are based on volatilememristors are demonstrated experimentally with the use ofrelay-based volatile memristor emulators of an original design.The fabricated logic circuit involves two volatile memristors andit is capable of performing four fundamental logic functions(two types of material implication and the negations thereof).Moreover, current-voltage characteristics of individual emulatorsare recorded and self-sustained oscillations in a resistor-volatilememristor circuit are found. The developed emulator offers agreat potential for memristive circuit experiments because of itssimplicity, similarity of response with volatile memristors, andlow cost. Our findings, which are based on emulators, can easilybe reproduced with physical volatile memristors and, thus, openup possibilities for emerging in-memory computing architectures.
Index Terms —memristors, logic gates, threshold voltage, in-memory computing
I. I
NTRODUCTION D URING the past decade, memristor technology has ex-perienced an explosive growth, which has the potentialto revolutionise information processing and storage. The keyadvantage of memristors [1], [2] (as well as memcapacitorsand meminductors [3]) over the traditional electronic com-ponents is the possibility to store and process informationon the same physical location. Memristive behavior has beenobserved in many systems and devices [4]. Up to now,however, most attention has been focused on devices withnon-volatile storage capability [5]. The future applications ofnon-traditional memristors are still not fully understood, andtheir theoretical and circuit-level models are still at an earlystage of development.The present paper explores an in-memory computing appli-cation of volatile memristors, namely, memristors capable ofstoring information only when connected to a power source.Specifically, we will limit ourselves to devices exhibiting twopossible resistance states (ON and OFF states) in a finiterange of voltages and switching to the OFF state when asmaller voltage is applied. Several physical systems satisfythese requirements, including NEMS switches [6], [7], Mottmemristors [8], graphene field emitters [9], and diffusivememristors [10]. The last system has recently attracted atten-tion because of its promising characteristics for the use inartificial neural networks [10], random signal generators [11],and sensing applications [12]. Physically, in such diffusivememristors Ag atoms spread under electrical bias and regroup
Y. V. Pershin is with the Department of Physics and Astron-omy, University of South Carolina, Columbia, SC 29208 USA (e-mail:[email protected]).Manuscript received June ..., 2018; revised .... spontaneously under zero/small bias because of interfacial en-ergy minimization [10], [11]. Moreover, it was shown that twoMott memristors can be used to build a neuristor [13], whichis an electronic analog of the Hodgkin-Huxley axon. In whatfollows we will keep our discussion general, without referringto any particular physical realisation of volatile memristors. Inour electronic circuit experiments, the volatile memristors arerepresented by emulators built out of conventional electronicscomponents (resistors and relays). The volatile memristoremulator is developed as a part of the present work.This paper experimentally demonstrates the implicationlogic [14] gates based on volatile memristors. Previously,in-memory logic gates were realised experimentally by em-ploying bipolar non-volatile memristors [14], and exploredtheoretically based on bipolar non-volatile memristors (see,e.g., [15], [16], [17], [18], [19]), unipolar non-volatile mem-ristors [20], [21], memcapacitors [22], [23], and volatilememristors (graphene field emitters [24]). The advantage ofusing memory devices in logic circuits is that they can servesimultaneously as a gate and latch. Here, we employ volatilememristors to create a polymorphic implication logic circuitand we demonstrate four kinds of fundamental logic gatesusing the same circuit. To the best of our knowledge, this is thefirst experimental realisation of implication logic gates basedon volatile memristors. Moreover, the response of individualemulators is explored. It is found that a simple resistor-volatilememristor circuit can exhibit self-sustained oscillations with apattern involving both regular and random components.The rest of this paper is organised as follows. Section II-Aintroduces the relay-based emulator of volatile memristors andprovides details on its specific realisation and response. Self-sustained oscillations in the resistor-volatile memristor circuitare briefly considered in Section II-B. Section III presents theimplication logic gates based on volatile memristors. One ofour main results is the experimental demonstration of fourfundamental logic functions using the same circuit, which iscontained in Section III. Our concluding remarks are given inSection IV.II. V
OLATILE M EMRISTOR E MULATOR
A. Emulator
Memristor emulators [25] are valuable tools for circuitprototyping when physical memristors are not accessible. Anumber of emulator designs are available in the literaturebased either on analog [26], [27], [28] or digital [29], [30]techniques. With rare exceptions [31], [32], a common featureof memristor emulators is the need for an external power a r X i v : . [ c s . ET ] S e p (a) RELAY M R int R c = (b) -3 -2 -1 0 1 2 3-10.0-5.000.005.0010.0 C u rr e n t ( m A ) Voltage (V)
Emulator 1 Emulator 2 Emulator 3
Fig. 1. (a) Schematics of volatile memristor emulator. An effective two-terminal volatile memristive system is formed by connecting the relay coilin parallel with series-connected resistor and reed switch. (b) Current-voltagecharacteristics of three physically different emulators with R int = 680 Ω . source. Here, we show that the volatile memristors can beemulated in a very simple way and at low cost. The proposedemulator operates without an external power source and itdemonstrates a high similarity to the response of volatilememristors.Fig. 1(a) shows the emulator schematics. The volatile mem-ristor emulator consists of a reed relay and resistor, formingan effective two-terminal memristive system. At lower appliedvoltages, the reed switch is open. In this case, the emulatorresistance equals the coil resistance R OF F = R c . Meanwhile,at higher applied voltages, the switch is closed and the totalresistance is R ON = R c R int / ( R c + R int ) . The intervalbetween the pull-in and drop-out voltages of relay is thebistability (memory) region.Three identical volatile memristor emulators were createdand their current-voltage characteristics were measured. Inthe present experiments, reed relays with the coil resistanceof R c = 600 Ω and nominal operating voltage of 5 V areemployed (part number HI05-1A66, Standex-Meder Electron-ics). Fig. 1(b) shows that the current-voltage characteristics ofdifferent emulators are very close to each other. According toFig. 1(b), at positive voltages, the OFF to ON transition occursat V th ≈ . V, while the ON to OFF transition takes place at V hold ≈ . V. Moreover, the hysteresis region in the negativedomain is slightly shifted to lower voltage amplitudes, whichis likely due to an asymmetry in the reed switch response withrespect to the magnetic field direction.In addition, mention should be made of the inductiveeffects originating from the relay coil. The relay coil can berepresented by a resistor and inductor connected in series anddescribed by the impedance of the form Z = (cid:113) R int + ( ωL ) ,where ω is the angular frequency of input and L is the coilinductance. It follows from this expression that the resistive (a) RELAY M R V ( t ) V out R int (b) V o lt a g e ( V ) Time (arb. units) V V out (c) T i m e ( m s )
7 V6 . 5 V6 V5 . 5 V5 V4 . 5 V
Fig. 2. (a) Resistor-volatile memristor circuit. (b) Applied and output voltagesas functions of time in the resistor-volatile memristor circuit with R int =220 Ω and R = 680 Ω . (c) Digitised V out measured at the output of acomparator at several constant values of the applied voltage V (indicated onthe plot). The curves are shifted for clarity. response is dominant at lower frequencies and transforms intoan inductive response at higher frequencies. The transitionfrequency ν t can be estimated from the condition of equalcontributions of the resistive and inductive components to theimpedance, namely: R int = 2 πν t L . In the present realisationof the emulator, the coil inductance L = 0 . H leads to ν t = 560 Hz. The inductive effects should be consideredwhen designing circuits with relay-based emulators operatingat higher frequencies.
B. Resistor-volatile Memristor Circuit
To better understand the emulator behavior in electroniccircuits, consider a circuit of series-connected resistor andmemristor subjected to an applied voltage V ( t ) (see Fig. 2(a)).An interesting (and potentially useful) feature of this circuitis the possibility of self-sustained memristance (memory re-sistance [2]) oscillations. These oscillations are clearly seenin Fig. 2(b) showing the response of resistor-volatile mem-ristor circuit to the applied voltage of sawtooth wave form.Technically speaking, the oscillations occur at such appliedvoltages when in the R ON state of memristor V M > V th andin the R OF F state V M < V hold . Under these conditions, thememristor will continuously switch back and forth betweenits low- and high-resistance states. In fact, the same oscilla-tion mechanism works in systems with negative differentialresistance.To derive the necessary condition for the oscillations, con-sider the resistor-volatile memristor circuit at the onset ofswitching, namely: assuming that the voltage across M is V M = V th and R M = R OF F . In this case, the applied voltage ˜ V is given by ˜ V = R + R OF F R OF F V th . (1)The switching into R ON drops V M to V (cid:48) M = R ON R + R ON ˜ V . (2)If, after this switching, V (cid:48) M < V hold then the memristor willswitch back into the R OF F state, and so on. In other words, theresistor-volatile memristor circuit will exhibit self-sustainedoscillations. By combining the inequality V (cid:48) M < V hold withEqs. (1), (2) one finds R ON ( R + R OF F ) R OF F ( R + R ON ) V th < V hold , (3)which is the necessary condition for the existence of circuitinstability. Clearly, the circuit is stable in the limit of R ON → R OF F (note that V th > V hold ) and unstable at some smallervalues of R ON .In the measurements, the signal from the resistor-volatilememristor circuit was transformed to the standard 0 V-(+5V) logic levels using a comparator with the threshold voltageset at about . V. Fig. 2(c) presents examples of comparatoroutput for several constant values of the applied voltage V .This plot demonstrates that both the frequency and probabilityof logic ”1” in the output signal depend on V . The outputsignal contains the regular (most clearly seen at V = 5 Vcurve) and random components, as well as a combination offrequencies ( V = 6 V curve). From the physics point of view,the random component can be associated with probabilisticsticking/unsticking of relay reeds and/or their complex dy-namics under Fig. 2(a) circuit conditions.III. I
MPLICATION L OGIC G ATES
The implication logic circuit that is considered in this workis slightly different from the circuit based on non-volatilememristors [14]. Modifications are needed to ensure that the (a) R E L AY R E L AY RM M S V ( t ) V ( t ) V (b) Fig. 3. (a) Implication logic circuit. In the present measurements, the switchS is implemented by a relay. (b) Photograph of experimental setup. Twomemristor emulators are located in the center, while the switch S is to theright. Operational amplifiers are used as buffers. volatile memristors stay in their bistable (hysteresis) regionsbetween the operations. The selected circuit design and itsexperimental realization are presented in Fig. 3. In particular,Fig. 3(a) shows a circuit comprising two volatile memristors,resistor, and switch. Three voltage sources are used to drive thecircuit. It is convenient to split the calculation sequence intothree phases: initialisation, hold, and calculation. The switchis closed in the initialisation and hold phases, and it is openedwithin the calculation phase to induce a gate operation.To store the information, S is kept closed and V = 1 . Vis applied to M and M ( V = V = V ). To initialise thememristor state, 0 V or 5 V is applied to a given memristor.The calculation is performed from the hold phase by changing V and V to desired values, setting V , opening and closingthe switch (the calculation phase). The entire calculationsequence consists of the following steps: initialisation, hold,calculation, hold. Note that the calculation results are storedin the final states of memristors. The final states of thememristors were monitored using small resistors connectedin series with memristors and measuring the voltage dropsacross these resistors. To eliminate the effect of self-sustainedoscillations discussed in Sec. II-B (or at least to reduce it tosome insignificant areas in the parameters space), emulatorswith relatively large R int = 680 Ω were used in combinationwith a smaller common resistor ( R = 220 Ω ).Diagrams of logic operations are obtained following theapproach introduced in Ref. [23]. For each pair of input (a) V I M P C o p y M S e t t o 1
S e t t o 0 V ( V ) (b) S e t t o 0C o p y M I M P S e t t o 1 V V ( V ) Fig. 4. Logic gate type as a function of voltage amplitudes V and V (in the calculation phase). Plot (a) shows the gate type related to the finalstate of M . Plot (b) presents the gate type related to the final state of M .The measurements were performed at V = − . V. Here, IMP denotesM → M , and IMP denotes M → M . combinations ((0,0), (0,1) (1,0), (1,1)), the final states ofmemristors are measured after applying the entire calculationsequence. The type of logic operation is identified usinga code [23], [24] calculated based on the final states ofmemristors. The code is an integer number (from 0 to 15)that encodes the logic operation type and it is calculated asdescribed in Ref. [23]. The code is translated to the logicoperation type with the help of the table from Ref. [23].Overall, our measurements confirm the possibility of logicoperations previously predicted theoretically for the case ofgraphene autoemission memristors [24].Fig. 4 presents measurement results based on Fig. 3 circuittaken at V = − . V. The logic operation type is plotted as afunction of V and V applied in the calculation phase. Fig. 4indicates that two types of material implication, M → M andM → M , can be realised by Fig. 3 circuit at V = − . V.Moreover, note that Fig. 4(a) and Fig. 4(b) can be transformedto each other under the flip across the V = V diagonal and N O T ( I M P )C o p y M S e t t o 1S e t t o 0 V V ( V ) Fig. 5. Logic gate type as a function of voltage amplitudes V and V . Thisplot is generated based on the final result stored in M . The measurementswere performed at V = − . V. Here, NOT(IMP) is the negation ofimplication. interchange of indices 1 and 2 in the operation type (thisproperty stems from the symmetric connections of M andM in the circuit).The circuit functionality is changed when V is shifted to-1.2 V. The map of logic functions for this case is presentedin Fig. 5 for the result stored in M . Fig. 5 shows that thereis a region of voltages in which the negation of implicationNOT(IMP ) is realised. Symmetrically, under appropriate con-ditions, the final state of M stores the result of another typeof the negation of implication (NOT(IMP )).IV. D ISCUSSION AND C ONCLUSION
In conclusion, the possibility of in-memory computingbased on volatile (diffusive) memristors has been demon-strated. Using two volatile memristor emulators, the impli-cation logic circuit was created and four kinds of fundamentallogic gates were shown experimentally. This type of operationis fundamentally different from the case of traditional logicgates having a predetermined functionality. Specifically, inaddition to the trivial operations (set to 1, set to 0 andcopy the initial states), the following fundamental [33] logicgates have been demonstrated: IMP , IMP , NOT(IMP ),and NOT(IMP ). Moreover, self-sustained oscillations weremeasured in the resistor-volatile memristor circuit. It has beenfound that the voltage oscillations involve both regular andrandom components, which shows the potential application ofthe resistor-volatile memristor circuit in the area of randomnumber generation.The present work broadens the opportunities for exploitingvolatile memristors for information processing and storage.Compared to the traditional implication logic circuits that arebased on non-volatile memristors [14], volatile memristivecircuits are slightly more complex because volatile devicesrequire a power source to store information. We note thatseveral non-idealities of real memristors such as the stochastic component in their dynamics [11] and variability of deviceparameters were not captured by our emulators. Such non-idealities (which do also exist in non-volatile memristors) mustbe considered when designing future in-memory computingcircuits/systems.The volatile memristor emulator that was introduced in thiswork offers a low cost and simple design alternative to physicalmemristors for the use in rapid memristive circuit prototyping.Conceptually, its operation principles can help to explain theoperation of volatile memristors, as well as of circuits basedthereof. It is anticipated that the volatile memristor emulatorsmay also find use in undergraduate teaching laboratories toteach memristor technology and provide relevant hands-onexperience. A CKNOWLEDGEMENT
The author would like to thank V. A. Slipko for our usefuldiscussions. R
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