Practical approach to programmable analog circuits with memristors
aa r X i v : . [ phy s i c s . i n s - d e t ] J a n Practical approach to programmable analog circuitswith memristors
Yuriy V. Pershin and Massimiliano Di Ventra
Abstract —We suggest an approach to use memristors (resis-tors with memory) in programmable analog circuits. Our ideaconsists in a circuit design in which low voltages are appliedto memristors during their operation as analog circuit elementsand high voltages are used to program the memristor’s states.This way, as it was demonstrated in recent experiments, thestate of memristors does not essentially change during analogmode operation. As an example of our approach, we have builtseveral programmable analog circuits demonstrating memristor-based programming of threshold, gain and frequency. In thesecircuits the role of memristor is played by a memristor emulatordeveloped by us.
Index Terms —Memory, Resistance, Analog circuits, Analogmemories.
I. I
NTRODUCTION T HE recent experimental demonstration of resistivityswitching in TiO thin films [1] and the establishmentof a link between this result and more than thirty-year-old theoretical description of memristors [2] (resistors withmemory) have attracted a lot of attention to this exciting field.Memristive behavior is found in many different systems [1],[2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14],[15], [16], [17], [18], [19], [20]. To interpret the experimentalobservations and predict a circuit behaviour, a number oftheoretical models were developed [2], [3], [21], [22], [1],[23], [24], [25], [26], [27], [28], [29] including SPICE models[27], [28], [29]. Memristors offer a nonvolatile memory stor-age within a simple device structure attractive for potentialapplications in electronics including the field of neuromorphiccircuits as well, namely circuits which mimic the function andoperation of neural cell networks in biological systems [30],[31]. Until now, most of the potential applications that havebeen proposed for these systems have relied on a binary modeof operation (on and off states of a memristor) while theunderstanding that memristors can be used as truly analogmemory elements is only emerging [30], [21], [32].A weaker interest in analog applications of memristors canbe partially justified by the perception that TiO thin filmsbehave as ideal memristors. According to Chua’s definition[2], the internal state of an ideal memristor depends on theintegral of the voltage or current over time. Therefore, the useof ideal memristors as analog elements in, e.g., programmable Yu. V. Pershin is with the Department of Physics and Astronomy and USCNanocenter, University of South Carolina, Columbia, SC, 29208e-mail: [email protected]. Di Ventra is with the Department of Physics, University of California,San Diego, La Jolla, California 92093-0319e-mail: [email protected] received August XX, 2009; revised November YY, 2009. analog circuits seems to be limited since their internal state,once programmed, would change significantly due to a dccomponent in current or applied voltage. Only perfect acsignals would not significantly change the memristor state sothat the use of such devices seems to appear narrow. However,experimentally realizable memristors [10] are not ideal. Infact, these devices belong to the much more general class ofmemristive systems [3] (see the definition below) allowing fora more complex behavior, which is at the basis of our proposal.At this point, a note should be made about the commonlyused terminology and the terminology used in this paper.It appears that in the recent literature the term memristorhas been used for both ideal memristors [2] and memristivedevices and systems [3]. Indeed, we expect all experimentalrealizations of such devices to be not ideal. Therefore, there isno reason for two different names. This convention is used inthe present paper, so that the term memristor will refer to all memristive systems and “ideal memristor” will be understoodonly in the sense of the definition in Ref. [2].In the present paper, we suggest an approach to use re-sistors with memory in analog circuits based on threshold-type behaviour of experimentally studied solid state memris-tors [9], [10]. Our main idea is to use low voltages in theanalog mode of operation and high voltage pulses in orderto program the memristor’s state. In this way, we obtain acircuit element whose mode of operation is close to thatof a digital potentiometer but its realization is much moresimple. Our scheme represents an important application of anew class of emerging systems collectively called memory-circuit elements ( memelements ) [22]. Therefore, it may be ofinterest to scientists from such diverse disciplines as electricalengineering (in particular, from the area of tunable resistanceresearch [33], [34]), physics, materials science, and evenneuroscience.This paper is organized as follows. Our approach to usememristors in programmable analog circuits is introducedin Sec. II. In Sec. III, we present a memristor emulator -an electronic circuit whose response is similar to that of amemristor. We will use this circuit in Sec. IV to demonstrateseveral applications of memristors in analog circuits includ-ing programmable threshold comparator, programmable gainamplifier, programmable switching thresholds Schmitt trigger,and programmable frequency relaxation oscillator. Concludingremarks are given in Sec. V.
II. D
EFINITIONS AND MAIN CONCEPT
A. Circuit elements with memory
Before describing our approach, let us give formal def-initions of memristors and briefly discuss their properties.To this end, we start from the general definition of circuitelements with memory which include also memcapacitors andmeminductors [22]. Let us then introduce a set of n statevariables x that describe the internal state of the system. Letus call u ( t ) and y ( t ) any two input and output variables[22], [35] that denote input and output of the system, suchas the current, charge, voltage, or flux. With g we indicate ageneralized response function. We then define a general classof n th-order u -controlled memory devices as those describedby the relations [22] y ( t ) = g ( x, u, t ) u ( t ) (1) ˙ x = f ( x, u, t ) (2)where f is a continuous n -dimensional vector function, andwe assume on physical grounds that, given an initial state u ( t = t ) at time t , Eq. (2) admits a unique solution.Memcapacitive and meminductive systems are special casesof Eqs. (1) and (2), where the two constitutive variables thatdefine them are charge and voltage for the memcapacitance,and current and flux for the meminductance. The propertiesof these systems can be found in Ref. [22]. In this paperwe will instead be concerned with a third class of memorydevices - memristive systems. Using Eqs. (1) and (2), we candefine an n th-order voltage-controlled memristive system asthat satisfying I ( t ) = R − M ( x, V M , t ) V M ( t ) (3) ˙ x = f ( x, V M , t ) (4)where x is a vector representing n internal state variables, V M ( t ) and I ( t ) denote the voltage and current across the de-vice. The quantity R M is a scalar, called the memristance (formemory resistance) and its inverse R − M is called memductance(for memory conductance).Similarly, an n th-order current-controlled memristive sys-tem is described by V M ( t ) = R ( x, I, t ) I ( t ) (5) ˙ x = f ( x, I, t ) (6)A charge-controlled memristor is a particular case of Eqs. (5)and (6), when R depends only on the charge, namely V M ( t ) = R ( q ( t )) I ( t ) , (7)with the charge related to the current via time derivative: I = dq/dt .Several noteworthy properties can be identified for mem-ristors [3]. For instance, these devices are passive provided R − M ( x, V M , t ) > in Eq. (3), and do not store energy. Drivenby a periodic current input, they also exhibit a “pinched hys-teretic loop” in their current-voltage characteristics. Moreover,a memristor behaves as a linear resistor in the limit of infinitefrequency and as a non-linear resistor in the limit of zerofrequency, assuming Eq. (4) admits a steady-state solution. +V+V pr Q V pp Q QQ V pn to analog -V pr M circuitry M Fig. 1. Memristor-based digital potentiometer consisting of the memristorM and a couple of FETs Q and Q . The external control signals V pp and V pn are used to set (program) the resistance of memristor R M betweentwo limiting values R and R . Here, V pr is the memristor’s programmingvoltage that should exceed the threshold voltage of memristor. The reason for this behavior is due to the system’s ability toadjust to a slow change in bias (for low frequencies) and thereverse: its inability to respond to extremely high-frequencyoscillations. We will explicitly demonstrate these propertieswith our memristor emulator presented in Sec. III.
B. Memristors in programmable analog circuits
Our main idea of using memristors in analog circuits isbased on the following observation of the experimental results:the rate of memristance change depends essentially on themagnitude of applied voltage [9], [10]. At voltages belowa certain threshold, the change of memristance is extremelyslow, whereas at voltages above the threshold, V T , it is fast.Therefore, we suggest to use memristors in analog circuitsin such a way that in the analog mode of operation (whenthe memristor performs a useful function as an analog circuitelement) only voltages of small magnitude (below the thresh-old) are applied to the device, while higher-amplitude voltages(above the threshold) are used only for programming. Theprogramming voltages can be applied in the form of pulses.Each pulse changes the resistance of memristor by a discreteamount.In this way, programmable memristors operate basically asdigital potentiometers. However, there are several potentialadvantages of memristor-based digital potentiometers over thetraditional ones. In particular, the size of a memristor canbe very small, down to 30 × [31], allowing for higherdensity chips/smaller electronic components. The operation ofmemristor-based digital potentiometers requires less transistorssince the information about resistance is written directly intoa memristive medium. Finally, the resistance is rememberedin the analog form potentially allowing for higher resolution.Recent examples of integration of TiO memristors withconventional silicon electronics [36] and of a silicon-basedmemristive systems [11] have demonstrated the practical fea-sibility of integrated memristor-based electronic components.Fig. 1 shows a simple memristor-based digital potentiometerwhich can operate as part of an analog circuit as we demon-strate below. For the sake of simplicity, one of the memristor’sterminals is connected to the ground while another terminal is connected to analog circuitry and a pair of field effecttransistors (FETs) that are used to program the memristor state.We use two external control signals V pp and V pn to open/closethe FETs when needed. When one of these FETs is open, theprogramming voltage ± V pr exceeding the threshold voltage ofmemristor V T is applied and memristor’s resistance changesin the direction determined by the applied voltage sign.For definiteness, let us assume that the application of posi-tive voltage increases R M and application of negative voltagedecreases R M . Then, the following protocol of programmingcan be employed. Since, at t = 0 we start with a possiblyunknown value of R M , the latter can be driven into the R max state (state of maximum resistance) by connection ofmemristor M to + V pr (using the “on” state of Q ) forsufficiently long time (this time should be selected longer,or at least equal to the time required to switch R M from R min - state of minimum resistance - to R max ). After that,a connection of M to − V pr (provided by the “on” state ofQ ) for a specific amount of time will switch R M into adesired state. These manipulations can also be performed byapplication of a certain number of positive and negative fixed-width pulses. In fact, we have recently used pulse control ofmemristors in memristive neural networks [30].The physical properties of TiO memristors were reportedand discussed in several papers [1], [10], [23]. The firsttheoretical model of TiO memristors was suggested alreadyin Ref. [1]. This model does not include any kind of threshold-type behaviour and simply assumes that the resistance isproportional to the charge flown through device. However,subsequent work [10] has clearly shown that the charge-basedmodel fails to describe the experimental results. In particular,Fig. 3b of Ref. [10] shows that a switching of memristorstate occurs at high voltages while at low voltages sweep-down and sweep-up curves coincide. A first activation-typemodel explaining this feature was proposed by us [21] andhas appeared in October 2008 in the cond-mat arxive [37]. Alater publication [23] (in early 2009) explains the activation-type behaviour of TiO memristors by a non-linear dopantdrift in which, at low voltages, the change in memristorstate is exponentially suppressed. These types of memristorbehaviour are precisely those needed for our proposal ofanalog programming.III. M EMRISTOR EMULATOR
As of today, memristors are not yet available on the market.Therefore, in order to study memristor-based programmablecircuits, we have built a memristor emulator [30]. The latteris a simple electronic scheme (see Fig. 2a) which can sim-ulate a wide range of memristive systems. In particular, wehave recently used it to simulate the behavior of synapsesin simple neural networks [30]. The main element of thememristor emulator is a digital potentiometer whose resistanceis continuously updated by a microcontroller and determinedby pre-programmed equations of current-controlled or voltage-controlled memristive systems. A general form of equationsdescribing a voltage-controlled memristive system is given byEqs. (3) and (4). These equations involve a voltage drop on
AAW
ADC V IN+ V B V IN- V M MicrocontrollerMicrocontroller a Initialization b Updating R M Sampling V M Updating R M Waiting for theCalculating R M Waiting for thenext sampling time
Fig. 2. (Color online) a Memristor emulator consisting of the following units:a digital potentiometer, an analog-to-digital converter and a microcontroller.The A (or B) terminal and the Wiper of the digital potentiometer serveas the external connections of the memristor emulator. The resistance ofthe digital potentiometer is determined by a code written into it by themicrocontroller. The code is calculated by the microcontroller according toEqs. (3) and (4). The analog-to-digital converter provides the value of voltageapplied to the memristor emulator needed for the digital potentiometer codecalculation. The applied voltage can be later converted to the current since themicrocontroller knows the value of the digital potentiometer resistance and acurrent-controlled memristive system can be realized. In our implementation,we used a 256 positions 10k Ω digital potentiometer AD5206 from AnalogDevice and microcontroller dsPIC30F2011 from Microchip with internal12bits ADC. b Block scheme of the memristor emulator’s algorithm. the memristor V M measured by the analog-to-digital converter(ADC). A block scheme of the memristor emulator operationalgorithm is shown in Fig. 2b. The algorithm’s steps are self-explanatory.In our experiments, we use an activation-type model ofmemristor [21], [30] inspired by recent experimental results[10]. Within our model, R M = x and Eq. (4) is written as(with the resistance acquiring the limiting values R min and R max ) ˙ x = ( βV M + 0 . α − β ) [ | V M + V T | − | V M − V T | ]) × θ ( x − R min ) θ ( R max − x ) , (8)where α and β are constants defining memristance rate ofchange below and above the threshold voltage V T ; V M is thevoltage on memristor and θ ( · ) is the step function. To testthat our emulator does indeed behave as a memristor, we haveused the circuit shown in the inset of Fig. 3, in which anac voltage is applied to the memristor emulator connected inseries with a resistor which was used to determine the current.The obtained current-voltage (I-V) curves, presented in Fig.3, demonstrate typical features of memristive systems such aspinched hysteresis loops and frequency-dependent hysteresis. -2 -1 0 1 2-2-1012 I ( m A ) V M (V) R + M V ( t ) - Fig. 3. (Color online) I-V curves obtained with a memristor emulator wiredas shown in the inset. The model given by Eq. (8) was used with α = 0 , β = 62 k Ω / V · s, V T = 1 . V, R min = 1 k Ω and R max = 10 k Ω . We used V ( t ) = V sin(2 πft ) as the applied voltage with V = 2 . V and f ’s asindicated on the plot. The curves are noisy because of the small value of R = 100Ω used to find the current and of limited resolution of our dataacquisition system. We found that for the cases shown in this plot the initialvalue of R M (in the present case equal to k Ω ) does not affect the long-timelimit of the I-V curves.Parameter Real memristor Memristor emulatorResistance range Determined by the structure 50 Ω < R < k Ω Discretization of
R R changes continuously 256 stepsFrequency Any . HzResponse Determined by the structure Determined by pre − programmed functionApplied V Less than the breakdown 0,+5V or -2.5,+2.5Vvoltage of the structureSupply V Not needed 0,+5V or -2.5,+2.5VMax. continuous I Determined by the structure ± mATABLE IC OMPARISON TABLE OF A SOLID - STATE MEMRISTOR AND PRESENTVERSION OF MEMRISTOR EMULATOR . Table I shows a summary of main characteristics of a realmemristor and of the present version of memristor emulator.In the case of memristor emulator, its characteristics aremainly limited by the electronic component that we use andcan be significantly varied using different types of electroniccomponents. As it is shown in the Table I, the resistanceof the present memristor emulator can be tuned between50 Ω (this low threshold is determined by unavoidable wiperresistance) and 10k Ω in 256 steps, as determined by thedigital potentiometer used. The ADC sampling frequency of1kHz limits the characteristic frequency of signals applied tomemristor to approximately 50Hz (20 points per period arereasonably enough to simulate a real memristor response).The voltage and current ratings of memristor emulator aredetermined by absolute maximum ratings of AD5206 digitalpotentiometer chip. By using a different hardware, the charac-teristics of memristor emulator can be improved. For example,the resolution can be easily increased to 1024 steps using adifferent digital potentiometer, and characteristic operationalfrequency can be as high as several tens of MHz using, e.g., a modern 2 Giga-sample ultra-high-speed ADC (such as, forexample, ADC10D1000 offered by National Semiconductor).However, for the purpose of demonstration, high resolutionand high frequencies are not required. This allows us to useinexpensive electronic components.Moreover, in order to have a good correspondence betweenthe response of a real memristor and that of a memristoremulator, it is extremely important to have an appropriatedevice model. This model, in terms of equations, is pre-programmed into the microcontroller and the behavior of thememristor emulator would follow closely the given model(within the limits listed in Table I). The model we employ(given by Eq. (8)) is quite simple. However, it contains allimportant physics of solid-state memristive devices whosebehavior derives from activation-type processes.IV. A PPLICATIONS
There is certainly a number of analog circuits where a mem-ristor can operate under the conditions described in Section II.Here, we show few examples of those applications.
A. Programmable threshold comparator
Let us start by demonstrating memristor-based pro-grammable analog circuit operations with arguably the sim-plest case - a programmable threshold comparator as shownin Fig. 4a. The design of this circuit, as well as of all othercircuits discussed below, involves the memristor-based digitalpotentiometer block shown in Fig. 1. In the analog mode ofoperation, both FETs are off. In all our practical examples webuild a scheme in such a way that the maximum voltage dropon memristor is always smaller than the threshold voltage ofmemristor which was selected to be equal to 1.75V. Whenpossible, it is desirable to apply to memristor as low voltagesas possible in order to further reduce the slow change ofmemristor state below its threshold.In the programmable threshold comparator circuit, thecomparator threshold is determined by the voltage on thememristor given by V − = V cc R M / ( R M + R ) , (9)where V cc = 2 . V is the power supply voltage, and R is thevalue of the resistance of the “standard” resistor. If the signalamplitude at the positive input V in exceeds V − , then outputsignal V out is equal to the saturation voltage of the operationalamplifier (which in the present case is close to +2 . V). In theopposite case, V out is close to − . V.Fig. 4b shows the operation of the programmable thresholdcomparator scheme when a sinusoidal voltage with an ampli-tude of 1.3V is applied to its input. At the initial moment oftime, the resistance of memristor is set to R M =10k Ω . Thatmeans that V − = 1 . V and V in exceeds V − only for ashort period of time. As a result, we observe a set of narrowpositive pulses at the output V out . The application of a train ofnegative pulses in the time interval between 4 and 8 secondsre-programs the memristor state by lowering the comparatorthreshold. This can be observed in a smaller V − and wideroutput pulses starting at t = 8 s. +2 5V+2.5V Q V in V pp a + AQ Q V out - -2.5V A Q V pn M R V +2.5V M R V - -101 -2-1012 0 2 4 6 8 10 12-3-2-1012 V i n ( V ) V - ( V ) b V ou t ( V ) Time (s)
Fig. 4. (Color online) a Schematics of a programmable threshold com-parator. Here, M is the memristor, R = 10 k Ω and A is an operationalamplifier. In our experiments, we used operational amplifier model TLV2770(Texas Instruments) that was powered by a dual polarity ± b Programmable threshold comparator response to the input voltage V in = V sin(2 πft ) with V = 1 . V and f = 1 Hz, and several negativeprogramming pulses of 10ms width applied in the time interval between 4and 8 seconds. V in is the input voltage applied to the positive input of theoperational amplifier, V − is the voltage on the negative input of the operationalamplifier, and V out is the signal at the output of the operational amplifier.Each 10ms voltage pulse changes R M by approximately 430 Ω causing agradual decrease of the comparator threshold. B. Programmable gain amplifier
The next circuit we consider is a programmable gain ampli-fier whose general scheme is shown in Fig. 5a. As in the usualnon-inverting amplifier configuration, the input signal V in isapplied to the positive input of the operational amplifier A ,while one of the two resistors, connected to the negative inputis substituted by a memristor M . The gain of such amplifier( R is the value of the resistance of the “standard” resistor) V out /V in = 1 + R /R M (10)is determined by the value of memristance R M which can beselected (programmed) between two limiting values R min and R max using two field-effect transistors (FETs). The desirableregime of operation that minimizes the voltage applied to thememristor is R M ≪ R .Figs. 5b and 5c demonstrate operation of the programmablegain amplifier shown in Fig. 5a. The amplifier’s gain is +2 5V+2.5V Q V in V pp a + AQ Q V out - -2.5V A Q C V pn M R VM R V - V in =0.2V b V out -2 V o lt a g e ( V ) Time (s) V - V in =0.2V c V out -2 V o lt a g e ( V ) Time (s) V - Fig. 5. a . Schematics of a programmable gain amplifier with memristor. Theoperational amplifier A is connected in the standard non-inverting amplifierconfiguration with a memristor M replacing a resistor. Two FETs (Q andQ ) are used to program the resistance of the memristor thus selecting theamplifier gain. The capacitor C = 0 . µ F is used for noise suppression. b .Programming of gain using 10ms width pulses. In these measurements, theinput voltage V in = 0 . V is permanently supplied while positive and negativepulses change the state of memristor and correspondingly circuit’s gain. Asa result, we observe a set of steps in the output signal. c . Coarser control ofgain using 20ms width pulses. controlled using pulses of constant width. The pulse widthis 10ms in Fig. 5b and 20ms in Fig. 5c. In both cases, at theinitial moment of time t = 0 , the memristor is in its highestresistance state R M = R max = 10 k Ω and, according to Eq.(10), the circuit gain is about 2. Correspondingly, the inputsignal V in = 0 . V results in V out = 0 . V at that time, as itcan be seen in Figs. 5b and 5c.Application of pulses at V − changes the value of R M and, correspondingly, the gain. Each negative pulse at V − decreases the value of R M while each positive pulse increases M R V + + +2.5V A V out
V M - A Q V pp Q V in V pn -2.5V V pn a -2-10123 0 1 2 3 4 5 6 7-1012 V o lt a g e ( V ) V in V out b V + ( V ) Time (s)
Fig. 6. (Color online) a Schematics of a programmable switching thresh-olds Schmitt trigger with memristor. Here, R = 10 k Ω . b Programmableswitching thresholds Schmitt trigger response to the input voltage V in = V sin(2 πft ) with V = 1 . V and f = 1 Hz, and several positiveprogramming pulses of 10ms width applied in the time interval between 2and 4 seconds. R M . Steps in the output signal V out (separated by spikesdue to the voltage pulses during programming) correspond todifferent values of circuit’s gain (the input voltage V in is keptconstant during the experiment). For the selected memristorparameters, the circuit gain changes approximately from 2 to11. We demonstrate in Fig. 5c that longer pulses produce largerchanges in R M allowing for coarser control of the gain. C. Programmable switching thresholds Schmitt trigger
Fig. 6a shows schematics of a programmable switchingthresholds Schmitt trigger in the inverted configuration. Thiscircuit behaves as an inverted comparator with the switchingthresholds given by ± ( R M /R ) V sat where, for our circuit, V sat = 2 . V. When we apply programming pulses to M , itsresistance R M changes as well as the switching thresholdsof Schmitt trigger. This type of behaviour is clearly seen inFig. 6b. Here, at the initial moment of time, the memristor isin a low resistance state and, correspondingly, the switchingthresholds are low. Therefore, the switchings (changes of V out from -2.5V to 2.5V and vice-versa) occur close to the momentof time when V in is close to zero as it follows from the +2 5V+2.5V Q V pp R V - - AQ Q C V out a + -2.5V A Q V pn M R V + M R V + -2-1010.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5-2-1012 V - V + b V o lt a g e ( V ) V ou t Time (s)
Fig. 7. (Color online) a Schematics of a programmable frequency relaxationoscillator with memristor. Here, R = R = 10 k Ω and C = 10 µ F. b Oscillating signals in the different points of the programmable frequency re-laxation oscillator. The lower panel demonstrates an increase in the oscillationfrequency as R M is decreased by several negative pulses applied to V + . positions of the intersection points of V in with V out curvesin the upper panel of Fig. 6b.A train of positive pulses applied to the memristor (see V + curve in Fig. 6b) in a time interval between 2 and 4seconds increases R M and, consequently, the Schmitt trigger’sswitching threshold. As a result, the upper panel in Fig. 6bdemonstrates that switching of V out occurs at different valuesof V in when t > s. D. Programmable frequency relaxation oscillator
As a final example we consider a programmable frequencyrelaxation oscillator. This is schematically shown in Fig. 7a.The relaxation oscillator is a well-known circuit which auto-matically oscillates because of the negative feedback added toa Schmitt trigger by an RC circuit. The period of oscillationsis determined by both the RC components and switchingthresholds of the Schmitt trigger. Therefore, in order to con-trol the relaxation oscillator frequency, we use a memristor-based digital potentiometer to vary switching thresholds of theSchmitt trigger, similarly to what is shown in Fig. 6a.Fig. 7b demonstrates that a decrease of the memristorresistance R M results in an increase of the relaxation oscillatorfrequency. As it is demonstrated in the upper panel of Fig. 7b,the decrease of the switching threshold results in a faster ca-pacitor charging time (because now the capacitor has to charge to a smaller voltage). Via this mechanism, the memristor-based digital potentiometer then determines the frequency ofoscillations. V. D ISCUSSION AND C ONCLUSION
Having demonstrated the operation of memristor-based pro-grammable analog circuits, we would like to discuss certainpractical aspects of using the suggested approach with solid-state memristors that are presently investigated. First of all,it should be mentioned that although the rate of memristancechange at low voltages is very small as observed in certainexperiments [10], it can eventually lead to a drift (increase ordecrease depending on particular application scheme) of R M at long times. Currently, it is difficult to estimate this effect, inpart because of the lack of experimental data. However, thisparasitic effect becomes less important at low voltages appliedto the memristor and, in practice, can be corrected by periodicre-setting of R M or/and circuit calibration. We also can notexclude the possibility that the drift would become importantonly after several months or years of device operation.Another important point is the precision of the memristorstate programming. To program a desired value of memris-tance with a given precision we should either have enough in-formation on the memristor operation model, parameters (andhave reproducibly-operating memristors), or provide pulsesof precise amplitude and duration. Alternatively we needto implement, electronically, a suitable calibration procedureallowing for correction of non-precise programming. Technicalsolutions for the tasks mentioned above can certainly be found.Concerning reproducibility of memristive behavior, experi-ments with TiO thin films demonstrate a significant amountof noise in hysteresis curves [10]. Possibly, the resistancechange effect in colossal magnetoresistive thin films [9] ismore suitable for analog-mode memristor applications.In conclusion, we have suggested an approach for a practicalapplication of memristors in programmable analog circuits.To demonstrate experimentally our approach, we have builtseveral memristor-based programmable analog circuits usinga memristor emulator. The latter is a scheme that uses inex-pensive off-the-shelf components and can, therefore, be builtquite easily in any electronic lab. By applying a train of pulses,we have achieved programming of different properties suchas threshold, gain and frequency. We have thus shown that amemristor with a control scheme provides a simple realizationof digital potentiometers and, therefore, can find useful andbroad-range applications in electronics.A CKNOWLEDGMENT
The authors are indebted to B. Mouttet for pointing outRef. [9] to us. This work has been partially funded by theNSF grant No. DMR-0802830.R
EFERENCES[1] D. B. Strukov, G. S. Snider, D. R. Stewart, and R. S. Williams, “Themissing memristor found,”
Nature , vol. 453, no. 7191, pp. 80–83, 2008.[2] L. O. Chua, “Memristor - the missing circuit element,”
IEEE Trans.Circuit Theory , vol. 18, no. 5, pp. 507–519, 1971. [3] L. O. Chua and S. M. Kang, “Memristive devices and systems,”
Proc.IEEE , vol. 64, no. 2, pp. 209–223, 1976.[4] Y. V. Pershin and M. Di Ventra, “Spin memristive systems: Spin memoryeffects in semiconductor spintronics,”
Phys. Rev. B , vol. 78, no. 11, pp.113 309/1–4, 2008.[5] T. Driscoll, H. T. Kim, B. G. Chae, M. Di Ventra, and D. N. Basov,“Phase-transition driven memristive system,”
Appl. Phys. Lett. , vol. 95,no. 4, pp. 043 503/1–3, 2009.[6] T. Driscoll, H.-T. Kim, B.-G. Chae, B.-J. Kim, Y.-W. Lee, N. M.Jokerst, S. Palit, D. R. Smith, M. Di Ventra, and D. N. Basov, “MemoryMetamaterials,”
Science , vol. 325, no. 5947, pp. 1518–1521, SEP 182009.[7] Y. V. Pershin and M. Di Ventra, “Frequency doubling and memoryeffects in the spin Hall effect,”
Phys. Rev. B , vol. 79, no. 15, pp.153 307/1–4, 2009.[8] X. Wang, Y. Chen, H. Xi, H. Li, and D. Dimitrov, “Spintronic memris-tor through spin-torque-induced magnetization motion,”
IEEE ElectronDevice Lett. , vol. 30, no. 3, pp. 294–297, 2009.[9] S. Liu, N. Wu, and A. Ignatiev, “Electric-pulse-induced reversibleresistance change effect in magnetoresistive films,”
Appl. Phys. Lett. ,vol. 76, no. 19, pp. 2749–2751, 2000.[10] J. J. Yang, M. D. Pickett, X. Li, D. A. A. Ohlberg, D. R. Stewart, andR. S. Williams, “Memristive switching mechanism for metal/oxide/metalnanodevices,”
Nat. Nanotechnol. , vol. 3, no. 7, pp. 429–433, 2008.[11] S. H. Jo, K.-H. Kim, and W. Lu, “High-density crossbar arrays based ona si memristive system,”
Nano Lett. , vol. 9, no. 2, pp. 870–874, 2009.[12] D. Stewart, D. Ohlberg, P. Beck, Y. Chen, R. Williams, J. Jeppesen,K. Nielsen, and J. Stoddart, “Molecule-independent electrical switchingin pt/organic monolayer/ti devices,”
Nano Lett. , vol. 4, no. 1, pp. 133–136, 2004.[13] V. V. Erokhin, T. S. Berzina, and M. P. Fontana, “Polymeric elementsfor adaptive networks,”
Crystallogr. Rep. , vol. 52, no. 1, pp. 159–166,2007.[14] H. Choi, H. Jung, J. Lee, J. Yoon, J. Park, D.-J. Seong, W. Lee,M. Hasan, G.-Y. Jung, and H. Hwang, “An electrically modifiablesynapse array of resistive switching memory,”
Nanotechnology , vol. 20,no. 34, pp. 345 201/1–5, 2009.[15] N. Gergel-Hackett, B. Hamadani, B. Dunlap, J. Suehle, C. Richter,C. Hacker, and D. Gundlach, “A flexible solution-processed memristor,”
IEEE Electron Device Lett. , vol. 30, no. 7, pp. 706–708, 2009.[16] X. Chen, G. Wu, P. Jiang, W. Liu, and D. Bao, “Colossal resistanceswitching effect in pt/spinel-mgzno/pt devices for nonvolatile memoryapplications,”
Appl. Phys. Lett. , vol. 94, no. 3, pp. 033 501/1–3, 2009.[17] K. Oka, T. Yanagida, K. Nagashima, H. Tanaka, and T. Kawai, “Non-volatile bipolar resistive memory switching in single crystalline NiOheterostructured nanowires,”
J. Am. Chem. Soc. , vol. 131, no. 10, pp.3434–3435, 2009.[18] L. Chen, Z. Liu, Y. Xia, K. Yin, L. Gao, and J. Yin, “Electricalfield induced precipitation reaction and percolation in Ag30Ge17Se53amorphous electrolyte films,”
Appl. Phys. Lett. , vol. 94, no. 16, pp.162 112/1–3, 2009.[19] B. Standley, W. Bao, H. Zhang, J. Bruck, C. N. Lau, and M. Bockrath,“Graphene-based atomic-scale switches,”
Nano Lett. , vol. 8, no. 10, pp.3345–3349, 2008.[20] G. S. Rose and M. R. Stan, Jr., “A programmable majority logic arrayusing molecular scale electronics,”
IEEE Trans. Circuits Syst. I , vol. 54,no. 11, pp. 2380–2390, NOV 2007.[21] Y. V. Pershin, S. La Fontaine, and M. Di Ventra, “Memristive model ofamoeba’s learning,”
Phys. Rev. E , vol. 80, p. 021926, 2009.[22] M. Di Ventra, Y. V. Pershin, and L. O. Chua, “Circuit elements withmemory: memristors, memcapacitors and meminductors,”
Proc. IEEE ,vol. 97, pp. 1717–1724, 2009.[23] D. B. Strukov and R. S. Williams, “Exponential ionic drift: fast switchingand low volatility of thin-film memristors,”
Appl. Phys. A-Mater. Sci.Process. , vol. 94, no. 3, pp. 515–519, 2009.[24] D. B. Strukov, J. L. Borghetti, and R. S. Williams, “Coupled ionicand electronic transport model of thin-film semiconductor memristivebehavior,”
Small , vol. 5, no. 9, pp. 1058–1063, 2009.[25] C. Cagli, F. Nardi, and D. Ielmini, “Modeling of set/reset operations inNiO-based resistive-switching memory devices,”
IEEE Trans. ElectronDevices , vol. 56, no. 8, pp. 1712–1720, 2009.[26] Y. N. Joglekar and S. J. Wolf, “The elusive memristor: properties of basicelectrical circuits,”
Eur. J. Phys. , vol. 30, no. 4, pp. 661–675, 2009.[27] S. Benderli and T. A. Wey, “On spice macromodelling of TiO2 mem-ristors,”
Electron. Lett. , vol. 45, no. 7, pp. 377–378, 2009. [28] Z. Biolek, D. Biolek, and V. Biolkova, “Spice model of memristor withnonlinear dopant drift,”
Radioengineering , vol. 18, no. 2, Part 2, pp.210–214, 2009.[29] ——, “Spice modeling of memristive, memcapacitative and meminduc-tive systems,”
Proc. of ECCTD ’09, European Conference on CircuitTheory and Design , pp. 249–252, August 23-27, 2009.[30] Y. V. Pershin and M. Di Ventra, “Experimental demonstration ofassociative memory with memristive neural networks,” arXiv:0905.2935 ,2009.[31] G. S. Snider, “Cortical computing with memristive nanodevices,”
Sci-DAC Review , vol. 10, pp. 58–65, 2008.[32] A. Delgado, “Input - output linearization of memristive systems,”
Nan-otechnology Materials and Devices Conference, Traverse City, Michi-gan , June 2-5, 2009.[33] E. Ozalevli and P. E. Hasler, “Tunable highly linear floating-gateCMOS resistor using common-mode linearization technique,”
IEEETrans. Circuits Syst. I , vol. 55, no. 4, pp. 999–1010, MAY 2008.[34] K. H. Wee and R. Sarpeshkar, “An Electronically Tunable Linear orNonlinear MOS Resistor,”
IEEE Trans. Circuits Syst. I , vol. 55, no. 9,pp. 2573–2583, OCT 2008.[35] L. Chua, “Nonlinear circuit foundations for nanodevices, part I: Thefour-element torus,”
Proc. IEEE , vol. 91, no. 11, pp. 1830–1859, NOV2003.[36] J. Borghetti, Z. Li, J. Straznicky, X. Li, D. A. A. Ohlberg, W. Wu, D. R.Stewart, and R. S. Williams, “A hybrid nanomemristor/transistor logiccircuit capable of self-programming,”
Proc. Natl. Acad. Sci. U. S. A. ,vol. 106, no. 6, pp. 1699–1703, 2009.[37] Y. V. Pershin, S. La Fontaine, and M. Di Ventra, “Memristive model ofamoeba’s learning,” arXiv:0810.4179arXiv:0810.4179