TitleAuthorsAbstractReferencesFigure 1 The four fundamental two-terminal circuit elements: resistor, capacitor, inductor and memristor.Figure 2 The coupled variable-resistor model for a memristor.Figure 3 Simulations of a voltage-driven memristive device.LETTERSThe missing memristor foundDmitri B. Strukov1, Gregory S. Snider1, Duncan R. Stewart1& R. Stanley Williams1Anyone who ever took an electronics laboratory class will be fami-liar with the fundamental passive circuit elements: the resistor, thecapacitor and the inductor. However, in 1971 Leon Chua reasonedfrom symmetry arguments that there should be a fourth fun-damental element, which he called a memristor (short for memoryresistor)1. Although he showed that such an element has manyinteresting and valuable circuit properties, until now no one haspresented either a useful physical model or an example of a mem-ristor. Here we show, using a simple analytical example, that mem-ristance arises naturally in nanoscale systems in which solid-stateelectronic and ionic transport are coupled under an external biasvoltage. These results serve as the foundation for understanding awide range of hysteretic current–voltage behaviour observed inmany nanoscale electronic devices2–19that involve the motion ofcharged atomic or molecular species, in particular certain tita-nium dioxide cross-point switches20–22.More specifically, Chua noted that there are six different math-ematical relations connecting pairs of the four fundamental circuitvariables: electric current i, voltage v, charge q and magnetic flux Q.One of these relations (the charge is the time integral of the current)is determined from the definitions of two of the variables, andanother (the flux is the time integral of the electromotive force, orvoltage) is determined from Faraday’s law of induction. Thus, thereshould be four basic circuit elements described by the remainingrelations between the variables (Fig. 1). The ‘missing’ element—thememristor, with memristance M—provides a functional relationbetween charge and flux, dQ 5 Mdq.In the case of linear elements, in which M is a constant, memri-stance is identical to resistance and, thus, is of no special interest.However, if M is itself a function of q, yielding a nonlinear circuitelement, then the situation is more interesting. The i–v characteristicof such a nonlinear relation between q and Q for a sinusoidal inputis generally a frequency-dependent Lissajous figure1, and no com-bination of nonlinear resistive, capacitive and inductive componentscan duplicate the circuit properties of a nonlinear memristor(although including active circuit elements such as amplifiers cando so)1. Because most valuable circuit functions are attributable tononlinear device characteristics, memristors compatible with inte-grated circuits could provide new circuit functions such as electronicresistance switching at extremely high two-terminal device densities.However, until now there has not been a material realization of amemristor.The most basic mathematical definition of a current-controlledmemristor for circuit analysis is the differential formv~R(w)i ð1Þdwdt~i ð2Þwhere w is the state variable of the device and R is a generalizedresistance that depends upon the internal state of the device. In thiscase the state variable is just the charge, but no one has been able topropose a physical model that satisfies these simple equations. In1976 Chua and Kang generalized the memristor concept to a muchbroader class of nonlinear dynamical systems they called memristivesystems23, described by the equationsv~R(w,i)i ð3Þdwdt~f (w,i) ð4Þwhere w can be a set of state variables and R and f can in general beexplicit functions of time. Here, for simplicity, we restrict the discus-sion to current-controlled, time-invariant, one-port devices. Notethat, unlike in a memristor, the flux in memristive systems is nolonger uniquely defined by the charge. However, equation (3) doesserve to distinguish a memristive system from an arbitrary dynamicaldevice; no current flows through the memristive system when thevoltage drop across it is zero. Chua and Kang showed that the i–vcharacteristics of some devices and systems, notably thermistors,Josephson junctions, neon bulbs and even the Hodgkin–Huxleymodel of the neuron, can be modelled using memristive equations23.Nevertheless, there was no direct connection between the mathe-matics and the physical properties of any practical system, andhence, almost forty years later, the concepts have not been widelyadopted.Here we present a physical model of a two-terminal electricaldevice that behaves like a perfect memristor for a certain restricted1HP Labs, 1501 Page Mill Road, Palo Alto, California 94304, USA.Resistordv = RdiCapacitordq = CdvInductordj = LdiMemristordj = MdqMemristive systemsqvijdj = vdtdq = idtFigure 1|The four fundamental two-terminal circuit elements: resistor,capacitor, inductor and memristor. Resistors and memristors are subsets ofa more general class of dynamical devices, memristive systems. Note that R,C, L and M can be functions of the independent variable in their definingequations, yielding nonlinear elements. For example, a charge-controlledmemristor is defined by a single-valued function M(q).Vol 453|1 May 2008|doi:10.1038/nature0693280Nature PublishingGroup©2008range of the state variable w and as a memristive system for another,wider (but still bounded), range of w. This intuitive model producesrich hysteretic behaviour controlled by the intrinsic nonlinearity ofM and the boundary conditions on the state variable w. The resultsprovide a simplified explanation for reports of current–voltageanomalies, including switching and hysteretic conductance, multipleconductance states and apparent negative differential resistance,especially in thin-film, two-terminal nanoscale devices, that havebeen appearing in the literature for nearly 50 years2–4.Electrical switching in thin-film devices has recently attractedrenewed attention, because such a technology may enable functionalscaling of logic and memory circuits well beyond the limits of com-plementary metal–oxide–semiconductors24,25. The microscopicnature of resistance switching and charge transport in such devicesis still under debate, but one proposal is that the hysteresisrequires some sort of atomic rearrangement that modulates theelectronic current. On the basis of this proposition, we consider athin semiconductor film