Authors: Troy Comi, Aaron Gibson and Joseph Padillaand Joseph PadillaMentor: Jefferson TaftApril 8 2010April 8, 2010Tucson, AZUniversity of ArizonaUniversity of ArizonaCl i l Ci iApril 8, 2010Four Fundamental Circuit Variables:Ch Classical Circuits1.Charge –q2. Voltage –v3. Current –I4. Magnetic Flux –φgφThree Classical Fundamental Circuit Elements:1. Capacitor: dq = C dv2Resistor: dv= R di2.Resistor: dv= R di3. Inductor: dφ = L diTwo Additional Relationships:1. dq = idt2. dφ = v dtModeling of RLC Circuit with Applied VoltageModeling of RLC Circuit with Applied VoltageL Q’’ + R Q’ + Q/C = V(t)April 8, 2010Existence of the Memory Resistorv•An element that is:v•An element that is:• Passive• Dissipativeqi• 2‐terminal• Effectively a resistor with re sistance that depends on ?re sistance that depends on charge that has passed through ‐ memoryStrukov, et. al. Nature Vol 453 (2008) 80‐83April 8, 2010Existence of the Memory Resistorv•Applicationsv•Applications• Higher density circuits• Unique modeling qipossibilities• Generation of special wa veforms?wa veforms• Signal processingStrukov, et. al. Nature Vol 453 (2008) 80‐83April 8, 2010Memristor ‐ The Fourth ElementPd b L Ch i i bd •Proposed by Leon Chua in 1971 using an argument based on symmetry.•Described by the sixth relationship between the four fundamental circuit variables: dφ = M dq•Faraday’s Law of Induction states the induced EMF or voltage in a y gclosed circuit is equal to the time rate of change in magnetic flux. Therefore, the memristor equation can be expressed as the following: M( )iv = M(q)i•Similar equation to resistors described by Ohm’s Law (v = R i). Memristance can reduce to resistance if certain conditions are met.•Memristor combination of “memory” and “resistor”.•Symbol for memristor:Chua, LO. IEEE Transactions on Circuit Theory, Vol. CT‐18, No . 5 (1971)April 8, 2010Properties of Memristors Non‐linear relationship between current and voltage.p g Reduces to resistor for large frequencies as evident in the i‐v characteristic curve. May also reduce to a resistor based on dfi d tt ibldefined state variables. Memory capacities based on different resistances produced by the memristor. Non‐volatile memory possible if the magnetic flux and charge through the memristor have a positive relationship (M > 0). Does not store energy. Similar to classical circuit elements, a system of memristors can also be described as a single memristoralso be described as a single memristor.Chua, LO. IEEE Transactions on Circuit Theory, Vol. CT‐18, No . 5 (1971)April 8, 2010i‐v Characteristic Curve Pinched hysteresis unique to memristors.N bi i f h No combination of other fundamental circuit elements makes elements makes Lissajous figure. Proof by Chua and Kang published in 1976.Strukov, et. al. Nature Vol 453 (2008) 80‐83Chua, LO, Kang SM Proceedings of the IEEE 64 Issue 2 (1976)April 8, 2010Generalization of MemristanceRecall the derived equation for memristance:v = M(q)iThis can be generalized further by considering a set of state variables x = (x1,x2,…,xn). These state variables are dependant on the specific implementation of the memristor. We can use the state variables to make a psubstitution in our memristance equation.v = M(x)i dx/dt = f(x,i)The state variables must be related to the current. This generalization leads to a unique set of equations for different memristors and memristivesystems.Chua, LO, Kang SM Proceedings of the IEEE 64 Issue 2 (1976)April 8, 2010Consider a light bulb In general a light bulb can be thought of as a Example of a Memristive SystemConsider a light bulb. In general, a light bulb can be thought of as a resistor. However, as the filament heats up, the resistance of the bulb increases. This behavior create s a non‐linear resistance which can be described with the following temperature‐dependant equations:V = (R0T)I ≡ M(T)idT/dt= aTi2 ‐b(t4‐t04) ≡f(T,i)dT/dt aTib(tt0) f(T,i)where R0, T0, a, and b are constants. These equations satisfy our conditions for a memristive system.Cunningham, W.J. Journal of Applied Physics (Vol. 23, No . 6) 1952April 8, 2010Difficulties in Finding MemristorsWhy has it taken almost four decades to find a memristor? Memristors are not new. Me mristive properties have been observed by researchers for more than three decades.observed by researchers for more than three decades. Pinched hysteresis curve is a unique property of a memristor. However, researchers did not made the connection with their observationsconnection with their observations. Often described as anomalous inductance/resistance and disregarded in certain practical applications (e.g. Jh Ji)Josephson Junction). Direct link between charge and magnetic flux not necessary.yChua, LO. IEEE Transactions on Circuit Theory, Vol. CT‐18, No . 5 (1971)April 8, 2010Formation and Applications Physical model Electroforming Light emmiting memristor Memristors in logic gates Modeling simple learningMemristorFoundApril 8, 2010MemristorFoundPt PtTiO2‐x1Total ON OFFwt wtRR RDD1ON OFFwt wtvt R R itDDONVdw tRitdt D ONVRwt qtD 21vonoffRMqR qtDDStrukov, et. al. Nature Vol 453 (2008) 80‐83Fif TiOMiApril 8, 2010Formation of TiO2 MemristorV22x2TiO TiO +O22-x2Yang, et. al. Nanotechnology 20 (2009) 21501Fif TiOMiApril 8, 2010Formation of TiO2 MemristorV22x2TiO TiO +O22-x2Yang, et. al. Nanotechnology 20 (2009) 21501April 8, 2010Light Emitting Memristor2+Zakhidov, et. al. Organic Electronics 11 (2010) 150‐153April 8, 2010Memristive Logic GatesBorghetti, et. al. PNAS, Vol. 106, No . 6 (2009) 1699‐1703April 8, 2010Simple “Learning” CircuitPershin, et. al. Physical Review E 80 (2009) 021926April 8, 2010Simple “Learning” CircuitPershin, et. al. Physical Review E 80 (2009) 021926April 8, 2010Simple “Learning” CircuitPershin, et. al. Physical Review E 80 (2009) 021926April 8, 2010HP SimulationsHP Nature PaperOur SimulationHP Nature PaperOur SimulationStrukov, et. al. Nature Vol 453 (2008) 80‐83April 8, 2010HP Simulations Cont.HP Nature PaperOur SimulationHP Nature PaperOur