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.

AFM image of 17 nanodevices with a zoom‐in cartoon schematically shows an individual crosspoint device consisting of two Pt metal electrodes separated by a TiO2 bi‐layer memristive material. By applying an electric field across the memristive material, oxygen vacancies can drift up and down, leading to four current‐transport end‐states. The switching between these end‐states results in a family of nanodevices.

The memristor, the fourth passive circuit element, was predicted theoretically nearly 40 years ago, but we just recently demonstrated both an intentional material system and an analytical model that exhibited the properties of such a device. Here we provide a more physical model based on numerical solutions of coupled drift-diffusion equations for electrons and ions with appropriate boundary conditions. We simulate the dynamics of a two-terminal memristive device based on a semiconductor thin film with mobile dopants that are partially compensated by a small amount of immobile acceptors. We examine the mobile ion distributions, zero-bias potentials, and current-voltage characteristics of the model for both steady-state bias conditions and for dynamical switching to obtain physical insight into the transport processes responsible for memristive behavior in semiconductor films.

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Memristor crossbars were fabricated at 40 nm half-pitch, using nanoimprint lithography on the same substrate with Si metaloxide-semiconductor field effect transistor (MOS FET) arrays to form fully integrated hybrid memory resistor (memristor)/transistor circuits. The digitally configured memristor crossbars were used to perform logic functions, to serve as a routing fabric for interconnecting the FETs and as the target for storing information. As an illustrative demonstration, the compound Boolean logic operation (A AND B) OR (C AND D) was performed with kilohertz frequency inputs, using resistor-based logic in a memristor crossbar with FET inverter/amplifier outputs. By routing the output signal of a logic operation back onto a target memristor inside the array, the crossbar was conditionally configured by setting the state of a nonvolatile switch. Such conditional programming illuminates the way for a variety of self-programmed logic arrays, and for electronic synaptic computing.crossbar ͉ integrated circuit ͉ memristor ͉ nanoimprint lithography

Memristive devices are nonlinear dynamical systems [ 1 ] that exhibit continuous, reversible and nonvolatile resistance changes that depend on the polarity, magnitude and duration of an applied electric fi eld. The memristive properties of metal/ metal oxide/metal (MOM) materials systems were discovered [ 2 , 3 ] in the 1960s and studied extensively for decades without reaching a consensus [4][5][6] on the physical switching mechanism. Recent research revealed that memristive switching is caused by electric fi eld-driven motion of charged dopants that defi ne the interface position between conducting and semiconducting regions of the metal oxide fi lm. [7][8][9][10][11][12] There have also been multiple reports of current-controlled negative differential resistance (CC-NDR) in electroformed MOM devices since the early 1960s (e.g. oxides of V, [13][14][15][16][17] Nb, [ 18 , 19 ] Ta, [ 20 ] Ti, [20][21][22][23] and Fe [ 24 ] ), and there have been a variety of proposals for the physical mechanism. Current work presents persuasive evidence that CC-NDR in these materials is due to a Joule-heating induced metal-insulator transition (MIT). [ 25 , 26 ] When the device is locally self-heated past the critical MIT temperature the resistivity drops abruptly, which has an unstable positive feedback effect on the current and results in the formation of a metallic phase conductive fi lament, [ 15 , 16 ] a necessary condition [ 27 ] for bulk CC-NDR. Independent researchers have recently shown [ 28 , 29 ] that the Magnéli phase Ti 4 O 7 [ 30 , 31 ] can be present in electroformed fi lms of memristive TiO 2 and may act as the source and sink for oxygen vacancies during memristive switching. Ti 4 O 7 is also known to exhibit a metal insulator transition at 155 K, [32][33][34] which opens the possibility to study nanoscale devices that simultaneously exhibit both CC-NDR and memristance.As shown in Figure 1 a, we have observed that electroformed titanium dioxide MOM devices can simultaneously exhibit both memristance and CC-NDR when immersed in liquid He. Here we derive an analytical model for the coexistence of both phenomena, which can be described by two independent mechanisms: (a) at all temperatures the memristance is due to the fi eld-driven motion of oxygen vacancies, and (b) at low temperatures the CC-NDR is caused by an insulator-to-metal phase transition triggered by Joule heating [ 13 , 14 , 25 , 26 ] of an electroformed conduction channel, probably the Magnéli phase Ti 4 O 7 . [ 28 , 29 ] Additionally, we analyze the electrical oscillations that arise from the CC-NDR effect in order to characterize the dynamics of the MIT. Finally, we demonstrate a notable application of such a device: a tunable voltage-controlled oscillator with conversion effi ciency greater than 1% that is capable of injecting AC energy into nanoscale oxide-based circuits.Schematic diagrams and an equivalent circuit of the model are presented in Figure 1 . We have previously reported a model for memristance in TiO 2[ 35 ] that accounts for t...

Memristors are memory resistors promising a rapid integration into future memory technologies. However, progress is still critically limited by a lack of understanding of the physical processes occurring at the nanoscale. Here we correlate device electrical characteristics with local atomic structure, chemistry and temperature. We resolved a single conducting channel that is made up of a reduced phase of the as-deposited titanium oxide. Moreover, we observed sufficient Joule heating to induce a crystallization of the oxide surrounding the channel, with a peculiar pattern that finite element simulations correlated with the existence of a hot spot close to the bottom electrode, thus identifying the switching location. This work reports direct observations in all three dimensions of the internal structure of titanium oxide memristors.