The missing memristor found
Dmitri B. Strukov1, Gregory S. Snider1, Duncan R. Stewart1 & R. Stanley Williams1
Anyone who ever took an electronics laboratory class will be familiar with the fundamental passive circuit elements: the resistor, the capacitor and the inductor. However, in 1971 Leon Chua reasoned from symmetry arguments thatthere should be a fourth fundamental element, which he called a memristor (short for memory resistor)1. Although he showed that such an element has many interesting and valuable circuit properties, until now no one has presented either a useful physical model or an example of a memristor. Here we show, using a simple analytical example, that memristance arises naturally in nanoscale systems inwhich solid-state electronic and ionic transport are coupled under an external bias voltage. These results serve as the foundation for understanding a wide range of hysteretic current–voltage behaviour observed in many nanoscale electronic devices2–19 that involve the motion of charged atomic or molecular species, in particular certain titanium dioxide cross-point switches20–22. More specifically,Chua noted that there are six different mathematical relations connecting pairs of the four fundamental circuit variables: 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, and another (the flux is the time integral of the electromotive force, or voltage) isdetermined from Faraday’s law of induction. Thus, there should be four basic circuit elements described by the remaining relations between the variables (Fig. 1). The ‘missing’ element—the memristor, with memristance M—provides a functional relation between charge and flux, dQ 5 Mdq. In the case of linear elements, in which M is a constant, memristance is identical to resistance and, thus, is ofno special interest. However, if M is itself a function of q, yielding a nonlinear circuit element, then the situation is more interesting. The i–v characteristic of such a nonlinear relation between q and Q for a sinusoidal input is generally a frequency-dependent Lissajous figure1, and no combination of nonlinear resistive, capacitive and inductive components can duplicate the circuit propertiesof a nonlinear memristor (although including active circuit elements such as amplifiers can do so)1. Because most valuable circuit functions are attributable to nonlinear device characteristics, memristors compatible with integrated circuits could provide new circuit functions such as electronic resistance switching at extremely high two-terminal device densities. However, until now there has notbeen a material realization of a memristor. The most basic mathematical definition of a current-controlled memristor for circuit analysis is the differential form v~R(w)i ð1Þ propose a physical model that satisfies these simple equations. In 1976 Chua and Kang generalized the memristor concept to a much broader class of nonlinear dynamical systems they called memristive systems23, described by theequations v~R(w,i)i ð3Þ
dw ~f (w,i) ð4Þ dt where w can be a set of state variables and R and f can in general be explicit functions of time. Here, for simplicity, we restrict the discussion to current-controlled, time-invariant, one-port devices. Note that, unlike in a memristor, the flux in memristive systems is no longer uniquely defined by the charge. However, equation (3) does serve todistinguish a memristive system from an arbitrary dynamical device; no current flows through the memristive system when the voltage drop across it is zero. Chua and Kang showed that the i–v characteristics of some devices and systems, notably thermistors, Josephson junctions, neon bulbs and even the Hodgkin–Huxley model of the neuron, can be modelled using memristive equations23. Nevertheless,...