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 that there should be a fourth fundamental element, which he called a memristor (short for memory resistor). 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 in which 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 devices that involve the motion of charged atomic or molecular species, in particular certain titanium dioxide cross-point switches.

The authors of the International Technology Roadmap for Semiconductors-the industry consensus set of goals established for advancing silicon integrated circuit technology-have challenged the computing research community to find new physical state variables (other than charge or voltage), new devices, and new architectures that offer memory and logic functions beyond those available with standard transistors. Recently, ultra-dense resistive memory arrays built from various two-terminal semiconductor or insulator thin film devices have been demonstrated. Among these, bipolar voltage-actuated switches have been identified as physical realizations of 'memristors' or memristive devices, combining the electrical properties of a memory element and a resistor. Such devices were first hypothesized by Chua in 1971 (ref. 15), and are characterized by one or more state variables that define the resistance of the switch depending upon its voltage history. Here we show that this family of nonlinear dynamical memory devices can also be used for logic operations: we demonstrate that they can execute material implication (IMP), which is a fundamental Boolean logic operation on two variables p and q such that pIMPq is equivalent to (NOTp)ORq. Incorporated within an appropriate circuit, memristive switches can thus perform 'stateful' logic operations for which the same devices serve simultaneously as gates (logic) and latches (memory) that use resistance instead of voltage or charge as the physical state variable.

Memristive devices are promising components for nanoelectronics with applications in nonvolatile memory and storage, defect-tolerant circuitry, and neuromorphic computing. Bipolar resistive switches based on metal oxides such as TiO 2 have been identified as memristive devices primarily based on the "pinched hysteresis loop" that is observed in their current-voltage ͑i-v͒ characteristics. Here we show that the mathematical definition of a memristive device provides the framework for understanding the physical processes involved in bipolar switching and also yields formulas that can be used to compute and predict important electrical and dynamical properties of the device. We applied an electrical characterization and state-evolution procedure in order to capture the switching dynamics of a device and correlate the response with models for the drift diffusion of ionized dopants ͑vacancies͒ in the oxide film. The analysis revealed a notable property of nonlinear memristors: the energy required to switch a metal-oxide device decreases exponentially with increasing applied current.

Teramac is a massively parallel experimental computer built at Hewlett-Packard Laboratories to investigate a wide range of different computational architectures. This machine contains about 220,000 hardware defects, any one of which could prove fatal to a conventional computer, and yet it operated 100 times faster than a high-end single-processor workstation for some of its configurations. The defect-tolerant architecture of Teramac, which incorporates a high communication bandwith that enables it to easi,b route around defects, has significant implications for any future nanometerscale computational paradigm. It may be feasible to chemically synthesize individual electronic components with less than a 100 percent yield, assemble them into systems with appreciable uncertainty in their connectivity, and still create a powerful and reliable data communications network. Future nanoscale computers may consist of extremely large-configuration memories that are programmed for specific tasks by a tutor that locates and tags the defects in the system. T h e last 25 years have witnessed astonishing advances in the fields of microelectron--its and computation. The first integrated circuit mlcro~rocessor, the Intel 4004, was able to perform roughly 5000 binary-coded decimal additions per second with a total power consumption of about 10 W (-500 additions ver Joule) in 1971. whereas mod-. " , ern lnicroprocessors can perform -3 X 106 additions oer Joule. The 1997 National L , Technology Roadlnap for Semiconductors (1) calls for an additional factor of lo3 , . increase in the computational efficiency by the vear 2012. If this eoal is attained. then perfArmance of the silycon-based integrated circuit \+~11 have imoroved bv nearlv seven orders of magnitud; in 40 years, using energy consumed per operation as a metric, with a single manufacturing paradigm. Although co~nplelnentarymetal oxide semiconductor (CMOS) technology is predicted by many researchers to run into significant physical limitations shortly after 2010 (Z), the enerev cost of an addition ooeration -,\+rillstill be nowhere near any fundamental physical limit. A crude estimate of the energy required to add two 10-digit decimal numbers, based on a thermodynamic analysis of nonreversible Boolean logic steps (3, 4) is -100.k.T.ln(2), which implies that 3 x 10'%dditions per Joule can be performed at room temperature without any reversible steps. Thus, there are potentially eight orders of magnitude in colnputational energy efficiency in a nonreversible machine available bevond the limits of CMOS technology. To achieve these f~~rther advances will require a totally different type of computational machinery, but knowing that such a system is in principle possible provides a strong incentive to hunt for it. The requirement for inventing a new technology paradigm has created exciting research opportunities for physical and biological scientists as well as for electrical engineers. Indeed, much of the current interest in interdisciplinary research in areas such as nanofabric...