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  1. 1. SEMINAR ON “ Memristors “ Submitted by SHISHIR S BELUR Reg No.1PI09TE089In partial fulfillment of the requirement for the award of degree in Bachelor ofEngineering in Telecommunication for the academic session Aug –Dec 2011 Carried out at PESIT, Bangalore Under the guidance of Ms. Bharati V Kalghatgi Lecturer Dept. of TE PESIT, Bangalore In the academic session Aug–Dec 2011 P E S INSTITUTE OF TECHNOLOGY 100 Feet Ring Road, BSK III Stage, Bangalore - 85 (An Autonomous Institute under VTU, Belgaum) education for the real world I
  2. 2. CERTIFICATEThis is to certify that the seminar entitled “ Memristors ” is a bonafidework carried out by Shishir S Belur Reg. No.: 1PI09TE089 at PESIT, Bangalorein partial fulfillment of the requirement for the award of degree inBachelor of Engineering in Telecommunication of VisveswaraiahTechnological University for the academic session Aug –Dec 2011. Signature of the seminar Guide Signature of the HOD(Name & Designation of the faculty) (Name & Designation of HOD) II
  3. 3. ACKNOWLEDGEMENTI would like to thank the faculty of the Department of Telecommunication,PESIT for having given me this opportunity to present a seminar onMemristors, whose guidance, advice and assistance helped formulate andpresent this seminar. My special thanks to Ms. Bharati.V.Kalghatgi for havingguided me all through my efforts but for which this would not have fructified. III
  5. 5. FIGURE INDEX Page No.Figure 1 about four basic circuit elements 3Figure 2 about the three fundamental circuit elements 4Figure 3 about the symbol of a memristor 5Figure 4 about V-I characteristics of a memristor 8Figure 5 about HP memristor 9Figure 6 about voltage applied to a memristor 15Figure 7 about current through a memristor 16Figure 8 about charge-flux curve of a memristor 16Figure 9 about current-voltage curve for f=1 Hz 16Figure 10 about current-voltage curve for f=1.5 Hz 17Figure 11 about current-voltage curve for f=2 Hz 17Figure 12 about adjusting the memristance 19Figure 13 about various arithmetic operations 21Figure 14: Circuit symbols for memcapacitor and meminductor 23 V
  6. 6. ABSTRACTSince the dawn of electronics, weve had only three types of circuitcomponents-resistors, inductors and capacitors. But in 1971, UC Berkeleyresearcher Leon Chua theorized the possibility of a fourth type of component,one that would be able to measure the flow of electric current in his paperMemristor-The Missing Circuit Element. The three fundamental circuit components- resistors, inductors andcapacitors are used to define four fundamental circuit variables which areelectric current, voltage, charge and magnetic flux. Resistors are used torelate current to voltage, capacitors to relate voltage to charge and inductorsto relate current to magnetic flux. But there was no element which could relatecharge to magnetic flux. This lead to the idea and development of memristors. Memristor is a concatenation of “memory resistors”. The most notableproperty of a memristor is that it can save its electronic state even when thecurrent is turned off, making it a great candidate to replace todays flashmemory. An outstanding feature is its ability to remember a range of electricalstates rather than the simplistic "on" and "off" states that todays digitalprocessors recognize. Memristor-based computers could be capable of farmore complex tasks. HP has already started produced an oxygen depleted titaniummemristor. 1
  7. 7. Memristors 1. INTRODUCTIONIn circuit theory, the three basic two-terminal devices — namely the resistor,the capacitor and the inductor are well understood. These elements are definedin terms of the relation between two of the four fundamental circuit variables,namely, current, voltage, charge and flux. The current is defined as the timederivative of the charge. According to Faraday‗s law, the voltage is defined asthe time derivative of the flux. A resistor is defined by the relationshipbetween voltage and current, the capacitor is defined by the relationshipbetween charge and voltage and the inductor is defined by the relationshipbetween flux and current. Out of the six possible combinations of the fourfundamental circuit variables, five are defined. In 1971, Prof. Leon Chuaproposed that there should be a fourth fundamental circuit element to set upthe relation between charge and magnetic flux and complete the symmetry asshown on the next page in Fig. 1. 2
  8. 8. Memristors Fig.1: Four basic circuit elements Prof. Leon Chua named this the memristor, a short for memory resistor.The memristor has a memristance and provides a functional relation betweencharge and flux. In 2008, Stanley Williams, at Hewlett Packard, announced thefirst fabricated memristor. 3
  9. 9. Memristors 2. MEMRISTOR THEORY2.1 Origin of the MemristorThere are four fundamental circuit variables in circuit theory. They are current,voltage, charge and flux. There are six possible combinations of the fourfundamental circuit variables. We have a good understanding of five of thepossible six combinations. The three basic two-terminal devices of circuittheory namely, the resistor, the capacitor and the inductor are defined in termsof the relation between two of the four fundamental circuit variables. Aresistor is defined by the relationship between voltage and current, thecapacitor is defined by the relationship between charge and voltage and theinductor is defined by the relationship between flux and current. In addition,the current is defined as the time derivative of the charge and according toFaraday‗s law, the voltage is defined as the time derivative of the flux. Theserelations are shown in Fig. 2. 4
  10. 10. Memristors Fig.2: The three circuit elements defined as a relation between four circuit variables2.2 Definition of a MemristorMemristor, the contraction of memory resistor, is a passive device thatprovides a functional relation between charge and flux. It is defined as a two-terminal circuit element in which the flux between the two terminals is afunction of the amount of electric charge that has passed through the device.Memristor is not an energy storage element. Fig. 3 shows the symbol for amemristor. Fig.3: Symbol of the memristor A memristor is said to be charge-controlled if the relation between fluxand charge is expressed as a function of electric charge and it is said to be flux-controlled if the relation between flux and charge is expressed as a function ofthe flux linkage.2.3 What is Memristance?Memristance is a property of the memristor. When charge flows in a directionthrough a circuit, the resistance of the memristor increases. When it flows inthe opposite direction, the resistance of the memristor decreases. If the appliedvoltage is turned off, thus stopping the flow of charge, the memristor 5
  11. 11. Memristorsremembers the last resistance that it had. When the flow of charge is startedagain, the resistance of the circuit will be what it was when it was last active. The memristor is essentially a two-terminal variable resistor, withresistance dependent upon the amount of charge q that has passed between theterminals. To relate the memristor to the resistor, capacitor, and inductor, it ishelpful to isolate the term M(q), which characterizes the device, and write it asa differential equation:where Q is defined by and ϕ is defined by The variable Φ ("magnetic flux linkage") is generalized from the circuitcharacteristic of an inductor. The symbol Φ may simply be regarded as theintegral of voltage over time. Thus, the memristor is formally defined as a two-terminal element inwhich the flux linkage (or integral of voltage) Φ between the terminals is afunction of the amount of electric charge Q that has passed through the device.Each memristor is characterized by its memristance function describing thecharge-dependent rate of change of flux with charge. Substituting that the flux is simply the time integral of the voltage, andcharge is the time integral of current, we may write the more convenient form 6
  12. 12. Memristors It can be inferred from this that memristance is simply charge-dependent resistance. If M(q(t)) is a constant, then we obtain Ohmslaw R(t) = V(t)/ I(t).However, the equation is not equivalentbecause q(t) and M(q(t)) will vary with time.Solving for voltage as a function of time we obtain This equation reveals that memristance defines a linear relationshipbetween current and voltage, as long as M does not vary with charge.Furthermore, the memristor is static if no current is applied. If I(t) = 0, wefind V(t) = 0 and M(t) is constant. This is the essence of the memory effect. The power consumption characteristic recalls that of a resistor, I2R As long as M(q(t)) varies little, such as under alternating current, thememristor will appear as a constant resistor.2.4 Properties of a Memristor2.4.1 Φ-q Curve of a MemristorThe Φ-q curve of a memristor is a monotonically increasing. The memristanceM(q) is the slope of the Φ-q curve. According to the memristor passivitycondition, a memristor is passive if and only if memristance M(q) is non-negative. If M(q) ≥ 0, then the instantaneous power dissipated by the 7
  13. 13. Memristorsmemristor, , is always positive and so the memristor is apassive device. The memristor is purely dissipative, like a resistor.2.4.2 Current–Voltage Curve of a MemristorAn important fingerprint of a memristor is the pinched hysteresis loop currentvoltage characteristic. For a memristor excited by a periodic signal, when thevoltage v(t) is zero, the current i(t) is also zero and vice versa. Thus, bothvoltage v(t) and current i(t) have identical zero-crossing. Another signature ofthe memristor is that the ―pinched hysteresis loop‖ shrinks with the increase inthe excitation frequency. Figure 4 shows the ―pinched hysteresis loop‖ and anexample of the loop shrinking with the increase in frequency. In fact, when theexcitation frequency increases towards infinity, the memristor behaves as anormal resistor. Fig. 4: The pinched hysteresis loop and the loop shrinking with the increase in frequency 8
  14. 14. Memristors 3. MODEL OF THE MEMRISTOR FROM HP LABSIn 2008, thirty-seven years after Chua proposed the memristor, StanleyWilliams and his group at HP Labs realized the memristor in device form. Torealize a memristor, they used a very thin film of titanium dioxide (TiO2). Thethin film is sandwiched between two platinum (Pt) contacts and one side ofTiO2 is doped with oxygen vacancies. The oxygen vacancies are positivelycharged ions. Thus, there is a TiO2 junction where one side is doped and theother side is undoped. The device established by HP is shown in Fig. 5. Fig. 5: Schematic of HP memristor In Fig.5, D is the device length and w is the length of the doped region.Pure TiO2 is a semiconductor and has high resistivity. The doped oxygenvacancies make the TiO2-x material conductive. The working of the memristor 9
  15. 15. Memristorsestablished by HP is as follows. When a positive voltage is applied, thepositively charged oxygen vacancies in the TiO2-x layer are repelled, movingthem towards the undoped TiO2 layer. As a result, the boundary between thetwo materials moves, causing an increase in the percentage of the conductingTiO2-x layer. This increases the conductivity of the whole device. When anegative voltage is applied, the positively charged oxygen vacancies areattracted, pulling them out of TiO2 layer. This increases the amount ofinsulating TiO2, thus increasing the resistivity of the whole device. When thevoltage is turned off, the oxygen vacancies do not move. The boundarybetween the two titanium dioxide layers is frozen. This is how the memristorremembers the voltage last applied. The simple mathematical model of the HP memristor is given bywhere has the dimensions of magnetic flux. is the average driftvelocity and has the units cm2/sV; D is the thickness of titanium-dioxide film; and are on-state and off- state resistances; and q(t) is the totalcharge passing through the memristor device.3.1 Linear Drift ModelLet us assume a uniform electric field across the device. Therefore, there is alinear relationship between drift-diffusion velocity and the net electric field.The state equation can be written as Integrating this gives, 10
  16. 16. Memristorswhere is the initial length of w . The speed of drift under a uniformelectric field across the device is then given by In a uniform field D= . In this case, defines the amount ofcharge required to move the boundary from , where w 0, to distance , where w D. Therefore, . Thus, If then, The amount of charge that is passed through the channel over therequired charge for a conductive channel is given as , then Substituting , we get If we assume that the initial charge , thenand 11
  17. 17. Memristors Where and is the memristive value at . Thus theMemristance at a time t is given by , Where . When >> , .Substituting this in , when we get, )Since , the solution isForIf , then the internal state of the memristor is The current-voltage relationship in this case is 12
  18. 18. Memristors This shows the inverse-square relation between memristance and TiO2thickness, D. Thus, for smaller values of D, the memristance shows improvedcharacteristics. Nowadays, memristance becomes more important forunderstanding as the dimensions of electronic devices are shrinking tonanometre scale. 13
  19. 19. Memristors 4. BENEFITS OF USING MEMRISTORSThe advantages of using memristors are as given below:  It provides greater resiliency and reliability when power is interrupted in data centers.  Memory devices built using memristors have greater data density  Combines the jobs of working memory and hard drives into one tiny device.  Faster and less expensive than present day devices  Uses less energy and produces less heat.  Would allow for a quicker boot up since information is not lost when the device is turned off.  Operating outside of 0s and 1s allows it to imitate brain functions.  Eliminates the need to write computer programs that replicate small parts of the brain.  The information is not lost when the device is turned off.  Has the capacity to remember the charge that flows through it at a given point in time.A very important advantage of memristors is that when used in a device, it canhold any value between 0 and 1. However present day digital devices can holdonly 1 or 0. This makes devices implemented using memristors capable ofhandling more data. 14
  20. 20. Memristors 5. RESULTS AND SIMULATIONS5.1 Simulation Results— Using SPICE modelFor this simulation, the width D of the TiO2 film is considered to be 10 nmand the dopant mobility = . The values assumed are =1KΩ, =100KΩ and the initial resistance required to model theinitial conditions of the capacitor is assumed to be 80KΩ. The simulationresults are shown below in Figs. 5,6,7,8,9 and 10 . Fig. 6 An input voltage applied to the memristor. 15
  21. 21. Memristors Fig. 7: Waveform of the current through the memristor. Fig. 8: Charge-versus-flux curve for memristor. Fig. 9: Current-versus-voltage curve for input frequency of 1 Hz. 16
  22. 22. Memristors Fig. 10: Current-versus-voltage curve for input frequency of 1.5 Hz. Fig. 11: Current-versus-voltage curve for input frequency of 2 Hz.These results are very much consistent with the theoretical graphs which weexpect and this shows that the memristor which we have developed till now isaccurate. 17
  23. 23. Memristors 6. POTENTIAL APPLICATIONS OF MEMERISTOR6.1 Two-state Charge-controlled MemristorThe slope of the Φ–q curve gives the memristance. The two values of thememristance can be considered as two different states which can be used asbinary states. The memristor holds logical values as impedance state and not asvoltages. The resistance can be changed from one state to another by applyingappropriate voltage.6.2 Memristor MemoryMemristors can be used as non-volatile memory, allowing greater data densitythan hard drives. The memristor based crossbar latch memory prototyped byHP can fit 100 gigabits within a square centimetre. HP also claims thatmemristor memory can handle up to 1,000,000 read/write cycles beforedegradation, compared to flash at 100,000 cycles. In addition, memristors alsoconsume less power.In memristor memories, the reading operation is performed by applying avoltage lesser than the threshold value. The memristor will conduct even atthis voltage if it is ―on‖. If it is ―off‖ then it will not conduct. To write one ofthe logic levels (0 or 1) a voltage greater than the threshold value is applied.To write the other logic level, a voltage of opposite polarity whose magnitudeis greater than the threshold voltage is applied. This turns the memristor ―off‖. 18
  24. 24. MemristorsMemristors can ―remember‖ even when the power is turned off. Thus, thecomputers developed using memristors will have no boot up time. Thecomputer can be turned on, like turning on a light switch and it will instantlydisplay all information that was there on it when it was turned off.6.3 Basic arithmetic operationsFor performing any arithmetic operation such as addition, subtraction,multiplication or division, at first, two operands should be represented by someways. In almost all of currently working circuits, signal values are representedby voltage or current. However, as explained in previous section, analogvalues can be represented by the memristance of the memristor as well. Figure11 shows the typical circuit that can be used for adjusting the memristance ofone memristor to the predetermined input value, i.e Vin. Fig. 12: Typical circuit for adjusting the memristance of the memristor with the predetermined value.In this figure, the coefficient is considered to make the dropping voltage acrossthe memristor to be meaningful and reasonable. The absolute value of thevoltage dropped across the memristor at any time will be aM. If aM be lower 19
  25. 25. Memristorsthan aVin, the output of the opamp will be at its lowest value, i.e. 0 volt, whichwill cause the left currentsource to derive the memristor. Passing current from the memristor in thisdirection will increase its memristance. On the other hand, if aM be higherthan aVin, the output of the opamp will be at its highest value, i.e. 5 volt,which will cause the left current source to derive the memristor. Passingcurrent from the memristor in this direction willdecrease its memristance. As a result, final value of the voltage which dropsacross the memristor, i.e aM, will be equal to aVin and therefore by this way,the memristance of the memristor will be set to Vin . Now, this adjustedmemristor can be used as an operand for performing arithmetic operations. Addition Subtraction 20
  26. 26. Memristors Multiplication DivisionFig. 13 shows how the various arithmetic operations can be achieved using memristors. 21
  27. 27. Memristors 7. CONCLUSION AND FUTURE RESEARCH7.1 ConclusionThis report presents a detailed study of the memristor. The properties of thememristor and the model proposed by HP are discussed. This model issimulated by subjecting it to various input voltages and noting the resultsobtained. This report also presents a brief insight into the potential applicationsof the memristor.Nanotechnology is fast emerging, and nanoscale devices automatically bring inmemristive functions. Thus, memristors might revolutionize the 21st centuryas radically as the transistor in the 20th century. Memristor memories havealready been developed and the researchers at HP believe that they can offer aproduct with a storage density of about 20 gigabytes per square centimetre by2013.Leon Chua rightly said ―It‗s time to rewrite all the Electronics Engineeringbooks‖.7.2 Future ResearchRecently, researchers have defined two new memdevices- memcapacitor andmeminductor, thus generalizing the concept of memory devices to capacitorsand inductors. These devices also show ―pinched‖ hysteresis loops in twoconstitutive variables— charge—voltage for the memcapacitor and current—flux for meminductor. Figure 13 shows the symbols for the memcapacitor andthe meminductor. 22
  28. 28. Memristors Fig. 14: Circuit symbols for memcapacitor and meminductorMemristors are not lossless devices. As non-volatile memories, memristors donot consume power when idle but they do dissipate energy when they arebeing read or written. Hence, there is a need to invent lossless non-volatiledevice. Memcapacitors and meminductors are good contenders as they arelossless devices. 23
  29. 29. BIBLIOGRAPHY[1][2] Dmitri B. Strukov, Gregory S. Snider, Duncan R. Stewart & R. StanleyWilliams, ―The missing memristor found‖, Vol 453| 1 May 2008|doi:10.1038/nature06932[3][4] O. Kavehei, A. Iqbal, Y. S. Kim, K. Eshraghian, S. F. Al-Sarawi, D.Abbott, ―The Fourth Element: Characteristics, Modelling, and ElectromagneticTheory of the Memristor‖[5][6][7][8] 24