Modelling and Applications of Memristive Devices

[spa] En este trabajo, se ha utilizado un nuevo enfoque en lugar del m´etodo tradicional de tensi´on y corriente para modelar con ´exito ReRAM unipolar y bipolar en el espacio de carga de flujo considerando los dispositivos como memristores. Este enfoque considera un sistema din´amico y permite capturar el complejo comportamiento din´amico de los dispositivos ReRAM mediante el uso de un sistema simple de ecuaci´on diferencial ordinaria que puede implementarse en simulaciones de circuitos. Por lo tanto, es posible simular el comportamiento del dispositivo para cualquier se˜nal de entrada, independientemente de la forma de la se˜nal de entrada. Este enfoque de modelado supera los inconvenientes que aparecen en los modelos complejos tradicionales en el dominio de corriente de voltaje que hacen que la simulaci´on de circuitos grandes sea bastante dif´ıcil o incluso poco pr´actica. Se ha utilizado una relaci´on no lineal entre el flujo y la carga para modelar las transiciones de restablecimiento de los dispositivos unipolares. Esta relaci´on no lineal se extiende a uno por partes para modelar las transiciones de restablecimiento / ajuste de los dispositivos bipolares. La corriente y la tensi´on se pueden obtener derivando con respecto al tiempo las magnitudes, el flujo y la carga principales del modelo. Para los dispositivos memristive ReRM tanto unipolares como bipolares, se han empleado datos simulados y experimentales para desarrollar un modelo que se pueda incluir f´acilmente en los simuladores de circuitos para describir la operaci´on de conmutaci´on resistiva. Se emple´o un modelo simple en el espacio de carga de flujo con un conjunto reducido de par´ametros para ajustar con ´exito los diferentes procesos de reinicio experimental para dispositivos memristive ReRAM unipolar. La variabilidad intr´ınseca en estos dispositivos se ha analizado utilizando el enfoque de carga de flujo. Se emplearon simulaciones f´ısicas de dispositivos con diferentes tama˜nos de filamentos conductores para adaptarse al modelo introducido. Posteriormente, las relaciones entre los par´ametros del modelo y las caracter´ısticas geom´etricas del filamento conductor se caracterizaron en profundidad. Las relaciones obtenidas se utilizaron luego para generar un nuevo conjunto de par´ametros que muestra propiedades estad´ısticas similares a la experimental, lo que demuestra la validez del enfoque. Adem´as, se present´o un modelo para obtener la energ´ıa empleada en el proceso de reinicio. La transici´on de restablecimiento de un dispositivo memristive ReRAM bipolar se ha analizado en funci´on de las consideraciones energ´eticas. Se ha realizado un an´alisis de los resultados experimentales en el espacio de carga de flujo junto al voltaje-corriente habitual. Se ha considerado el efecto de cambiar la pendiente de la se˜nal de entrada en el punto de reinicio, y se ha encontrado un conjunto de ecuaciones para estimar los nuevos par´ametros. Estas ecuaciones, basadas en un an´alisis de energ´ıa cuasiest´atica, permiten la caracterizaci´on de la transici´on de restablecimiento de un dispositivo memristive bipolar ReRAM utilizando solo tres par´ametros adem´as de la pendiente de la se˜nal, una resistencia t´ermica y la temperatura de reinicio del filamento conductor. Se ha desarrollado y probado un modelo por partes para restablecer y establecer transiciones de un dispositivo memristive ReRAM bipolar en el espacio de carga de flujo. El modelo utilizado es muy simple y proporciona resultados de simulaci´on precisos. Tambi´en permite el desarrollo de expresiones simples para la conductancia y el consumo de energ´ıa, as´ı como la caracterizaci´on del dispositivo memristive ReRAM en el dominio de corriente de voltaje mediante el uso de dos puntos para cualquier reinicio o ciclo establecido. Se ha considerado el caso de una se˜nal de entrada en rampa con diferentes pendientes para obtener los par´ametros del modelo, y se han comparado las predicciones del modelo con los resultados experimentales. Finalmente, se ha implementado y probado un modelo cuasi estacionario compacto para el dispositivo memristive bipolar ReRAM para diferentes frecuencias. Finalmente, se ha implementado y probado un modelo cuasi estacionario compacto para el dispositivo memristive bipolar ReRAM para diferentes frecuencias.[cat] En aquest treball, s’ha utilitzat un nou enfocament en lloc del tradicional m`etode de voltatge-corrent per modelar amb `exit el RePram unipolar i bipolar en l’espai de c`arrega de flux considerant els dispositius com memristors. Aquest enfocament considera un sistema din`amic i permet capturar el comportament din`amic complex dels dispositius ReRAM mitjan¸cant l’´us d’un sistema simple d’equaci´o diferencial ordin`aria que es pot implementar en simulacions de circuits. Per tant, ´es possible simular el comportament del dispositiu per a qualsevol senyal d’entrada, independentment de la forma del senyal d’entrada. Aquest enfocament de modelatge supera els inconvenients que apareixen en models complexos tradicionals en el domini de voltatgecorrent que fan que la simulaci´o de circuits grans sigui bastant dif´ıcil o fins i tot pr`actic. S’ha utilitzat una relaci´o no lineal entre flux i c`arrega per modelar les transicions de restauraci´o dels dispositius unipolars. Aquesta relaci´o no lineal s’amplia a una pe¸ca per modelar les transicions de restabliment / establiment de dispositius bipolars. El corrent i la tensi´o es poden obtenir derivant respecte al temps les magnituds model principal, el flux i la c`arrega. Per a dispositius memristius ReRM unipolars i bipolars, s’han utilitzat dades simulades i experimentals per desenvolupar un model que es pugui incloure f`acilment en els simuladors de circuits per descriure l’operaci´o de commutaci´o resistiva. Es va utilitzar un model senzill en l’espai de c`arrega de flux amb un conjunt redu¨ıt de par`ametres per ajustar amb `exit els diferents processos de restabliment experimental per a dispositius memristius unipolars ReRAM. La variabilitat intr´ınseca d’aquests dispositius s’ha analitzat utilitzant l’enfocament de c`arrega de flux. Es van utilitzar simulacions f´ısiques de dispositius amb diferents mides de filaments conductius per adaptar-se al model introdu¨ıt. Posteriorment, es van caracteritzar les relacions entre els par`ametres del model i les caracter´ıstiques geom`etriques del filament conductor. Les relacions obtingudes es van utilitzar llavors per generar un nou conjunt de par`ametres que mostra propietats estad´ıstiques similars a l’experimental, demostrant aix´ı la validesa de l’enfocament. A m´es, es va presentar un model per obtenir l’energia emprada en el proc´es de restabliment. La transici´o de restabliment d’un dispositiu memristiu bipolar ReRAM s’ha analitzat sobre la base de consideracions energ`etiques. S’ha realitzat una an`alisi dels resultats experimentals a l’espai de c`arrega de flux al costat de la tensi´o-corrent habitual. S’ha considerat l’efecte de canviar el pendent del senyal d’entrada al punt de restabliment i s’ha trobat un conjunt d’equacions per estimar els nous par`ametres. Aquestes ecuaciones, basades en una an`alisi d’energia quasiest`atica, permeten la caracteritzaci´o de la transici´o de reset d’un dispositiu memristiu bipolar ReRAM utilitzant nom´es tres par`ametres, a m´es del pendent del senyal, la resist`encia t`ermica i la temperatura de reset del filament conductor. S’ha desenvolupat i provat un model de peces per a la restauraci´o i configuraci´o de transicions d’un dispositiu memristiu bipolar ReRAM a l’espai de c`arrega de flux. El model utilitzat ´es molt senzill i proporciona resultats de simulaci´o precisos. Tamb´e permet el desenvolupament d’expressions simples per a la conducta i el consum d’energia, aix´ı com la caracteritzaci´o del dispositiu memRristard ReRAM en el domini actual de tensi´o utilitzant dos punts per a qualsevol restabliment o cicle establert. S’ha considerat el cas d’una senyal d’entrada de rampa amb diferents vessants per obtenir els par`ametres del model i s’han comparat les prediccions del model amb resultats experimentals. Finalment, s’ha implementat i provat un model compacte gaireb´e estacionari per a dispositius memristius bipolars ReRAM per a diferents freq¨u`encies. Finalment, s’ha implementat i provat un model compacte gaireb´e estacionari per a dispositius memristius bipolars ReRAM per a diferents freq¨u`encies.[eng] In this work, a new approach in place of the traditional Voltage-Current (V − I) method has been used to successfully model unipolar and bipolar ReRAM in Flux-Charge (φ − Q) space by considering the devices as memristors. This approach considers a dynamical system and enables capture of the complex dynamic behavior of ReRAM devices through the use of a simple ordinary differential equation system that can be implemented in circuit simulations. Thus, it is possible to simulate the device behavior for any input signal, regardless the input signal shape. This modeling approach overcomes the drawbacks that appear in traditional complex models in V − I domain which make simulation of large circuits rather difficult or even impractical. A non-linear relation between flux and charge has been used to model the reset transitions of unipolar devices. This non-linear relation is extended to a piecewise one to model the reset/set transitions of bipolar devices. Current and voltage can be obtained by deriving with respect to time the main model magnitudes, flux and charge. For both unipolar and bipolar ReRAM memristive devices, simulated and experimental data have been employed to develop a model that can be easily included in circuit simulators in order to describe resistive switching operation. A simple model in φ − Q space with a reduced set of parameters was employed to fit successfully the different experimental reset processes for unipolar ReRAM memristive devices. The intrinsic variability in these devices has been analysed by using the φ − Q approach. Physical simulations of devices with different conductive filament sizes were employed to fit the model introduced. Afterwards, the relations between the model parameters and the conductive filament geometrical features were characterized in-depth. The obtained relations were then used to generate a new ensemble of parameters that shows similar statistical properties to the experimental one, thus proving the validity of the approach. In addition, a model to obtain the energy employed in the reset process was presented. The reset transition of a bipolar ReRAM memristive device has been analysed based on energy considerations. An analysis of the experimental results has been done in the φ − Q space beside the usual V − I one. The effect of changing the slope of the input signal in the reset point has been considered, and a set of equations to estimate the new parameters has been found. These equations, based on a quasi-static energy analysis, allow the characterization of the reset transition of a bipolar ReRAM memristive device by using only three parameters in addition to the signal slope, a thermal resistance and the reset temperature of the conductive filament. A piecewise model for the reset and set transitions of a bipolar ReRAM memristive device in the φ−Q space has been developed and tested. The model used is very simple and provides accurate simulation results. It also allows the development of simple expressions for the conductance and power consumption, as well as the characterization of the ReRAM memristive device in V − I domain by using two points for any reset or set cycle. The case of a ramp input signal with different slopes has been considered to obtain the model parameters, and the predictions of the model with experimental results have been compared. Finally, a quasi-stationary compact model for bipolar ReRAM memristive device has been implemented and tested for different frequencies

A Simple Quasi-Static Compact Model of Bipolar ReRAM Memristive Devices

[eng] In this brief, we present a quasi-static compact model for bipolar ReRAM memristive devices. This model is based on a piecewise model in flux-charge space for reset and set transitions that has been extended to build a compact model of reset/set transitions for different slopes. In order to show its functionality, we use it to reproduce the behavior of a device fabricated by the CNR-IMM, MDM Laboratory. We discuss the needed parameters extraction procedure for the device. As shown in the results, the implemented model is able to capture the effects of the slope change in the ramp input signal for reset and set by using a set of technological parameters related to the device and information related to the slope of the ramp input voltage signal

Stochastic Computing Emulation of Memristor Cellular Nonlinear Networks

Cellular Nonlinear Networks (CNN) are a concept introduced in 1988 by Leon Chua and Lin Yang as a bio-inspired architecture capable of massively parallel computation. Since then, CNN have been enhanced by incorporating designs that incorporate memristors to profit from their processing and memory capabilities. In addition, Stochastic Computing (SC) can be used to optimize the quantity of required processing elements; thus it provides a lightweight approximate computing framework, quite accurate and effective, however. In this work, we propose utilization of SC in designing and implementing a memristor-based CNN. As a proof of the proposed concept, an example of application is presented. This application combines Matlab and a FPGA in order to create the CNN. The implemented CNN was then used to perform three different real-time applications on a 512 × 512 gray-scale and a 768 × 512 color image: storage of the image, edge detection, and image sharpening. It has to be pointed out that the same CNN was used for the three different tasks, with the sole change of some programmable parameters. Results show an excellent capability with significant accompanying advantages, such as the low number of needed elements further allowing for a low cost FPGA-based system implementation, something confirming the system’s capacity for real time operation

A Switched Capacitor Memristor Emulator Using Stochastic Computing

Due to the increased use of memristors and their many applications, the use of emulators has grown in parallel to avoid some of the difficulties presented by real devices, such as variability and reliability. In this paper, we present a memristive emulator designed using a switched capacitor (SC), that is, an analog component/block and a control part or block implemented using stochastic computing (SCo) and therefore fully digital. Our design is thus a mixed signal circuit. Memristor equations are implemented using stochastic computing to generate the control signals necessary to work with the controllable resistor implemented as a switched capacitor

A Switched Capacitor Memristor Emulator Using Stochastic Computing

Due to the increased use of memristors and their many applications, the use of emulators has grown in parallel to avoid some of the difficulties presented by real devices, such as variability and reliability. In this paper, we present a memristive emulator designed using a switched capacitor (SC), that is, an analog component/block and a control part or block implemented using stochastic computing (SCo) and therefore fully digital. Our design is thus a mixed signal circuit. Memristor equations are implemented using stochastic computing to generate the control signals necessary to work with the controllable resistor implemented as a switched capacitor

Empirical Characterization of ReRAM Devices Using Memory Maps and a Dynamic Route Map

Memristors were proposed in the early 1970s by Leon Chua as a new electrical element linking charge to flux. Since that first introduction, these devices have positioned themselves to be considered as possible fundamental ones for the generations of electronic devices to come. In this paper, we propose a new way to investigate the effects of the electrical variables on the memristance of a device, and we successfully apply this technique to model the behavior of a TiN/Ti/HfO2/W ReRAM structure. To do so, we initially apply the Dynamic Route Map technique in the general case to obtain an approximation to the differential equation that determines the behaviour of the device. This is performed by choosing a variable of interest and observing the evolution of its own temporal derivative versus both its value and the applied voltage. Then, according to this technique, it is possible to obtain an approach to the governing equations with no need to make any assumption about the underlying physical mechanisms, by fitting a function to this. We have used a polynomial function, which allows accurate reproduction of the observed electrical behavior of the measured devices, by integrating the resulting differential equation system

Empirical Characterization of ReRAM Devices Using Memory Maps and a Dynamic Route Map

Memristors were proposed in the early 1970s by Leon Chua as a new electrical element linking charge to flux. Since that first introduction, these devices have positioned themselves to be considered as possible fundamental ones for the generations of electronic devices to come. In this paper, we propose a new way to investigate the effects of the electrical variables on the memristance of a device, and we successfully apply this technique to model the behavior of a TiN/Ti/HfO2/W ReRAM structure. To do so, we initially apply the Dynamic Route Map technique in the general case to obtain an approximation to the differential equation that determines the behaviour of the device. This is performed by choosing a variable of interest and observing the evolution of its own temporal derivative versus both its value and the applied voltage. Then, according to this technique, it is possible to obtain an approach to the governing equations with no need to make any assumption about the underlying physical mechanisms, by fitting a function to this. We have used a polynomial function, which allows accurate reproduction of the observed electrical behavior of the measured devices, by integrating the resulting differential equation system

On the Thermal Models for Resistive Random Access Memory Circuit Simulation

Resistive Random Access Memories (RRAMs) are based on resistive switching (RS) operation and exhibit a set of technological features that make them ideal candidates for applications related to non-volatile memories, neuromorphic computing and hardware cryptography. For the full industrial development of these devices different simulation tools and compact models are needed in order to allow computer-aided design, both at the device and circuit levels. Most of the different RRAM models presented so far in the literature deal with temperature effects since the physical mechanisms behind RS are thermally activated; therefore, an exhaustive description of these effects is essential. As far as we know, no revision papers on thermal models have been published yet; and that is why we deal with this issue here. Using the heat equation as the starting point, we describe the details of its numerical solution for a conventional RRAM structure and, later on, present models of different complexity to integrate thermal effects in complete compact models that account for the kinetics of the chemical reactions behind resistive switching and the current calculation. In particular, we have accounted for different conductive filament geometries, operation regimes, filament lateral heat losses, the use of several temperatures to characterize each conductive filament, among other issues. A 3D numerical solution of the heat equation within a complete RRAM simulator was also taken into account. A general memristor model is also formulated accounting for temperature as one of the state variables to describe electron device operation. In addition, to widen the view from different perspectives, we deal with a thermal model contextualized within the quantum point contact formalism. In this manner, the temperature can be accounted for the description of quantum effects in the RRAM charge transport mechanisms. Finally, the thermometry of conducting filaments and the corresponding models considering different dielectric materials are tackled in depth

On the thermal models for resistive random access memory circuit simulation

Resistive Random Access Memories (RRAMs) are based on resistive switching (RS) operation and exhibit a set of technological features that make them ideal candidates for applications related to non-volatile memories, neuromorphic computing and hardware cryptography. For the full industrial development of these devices different simulation tools and compact models are needed in order to allow computer-aided design, both at the device and circuit levels. Most of the different RRAM models presented so far in the literature deal with temperature effects since the physical mechanisms behind RS are thermally activated; therefore, an exhaustive description of these effects is essential. As far as we know, no revision papers on thermal models have been pub-lished yet; and that is why we deal with this issue here. Using the heat equation as the starting point, we describe the details of its numerical solution for a conventional RRAM structure and, later on, present models of different complexity to integrate thermal effects in complete compact models that account for the kinetics of the chemical reactions behind resistive switching and the current calcu-lation. In particular, we have accounted for different conductive filament geometries, operation re-gimes, filament lateral heat losses, the use of several temperatures to characterize each conductive filament, among other issues. A 3D numerical solution of the heat equation within a complete RRAM simulator was also taken into account. A general memristor model is also formulated ac-counting for temperature as one of the state variables to describe electron device operation. In ad-dition, to widen the view from different perspectives, we deal with a thermal model contextualized within the quantum point contact formalism. In this manner, the temperature can be accounted for the description of quantum effects in the RRAM charge transport mechanisms. Finally, the ther-mometry of conducting filaments and the corresponding models considering different dielectric materials are tackled in depth.Fil: Roldán, Juan B.. Universidad de Granada; EspañaFil: González Cordero, Gerardo. Universidad de Granada; EspañaFil: Picos, Rodrigo. University of Balearic Islands; EspañaFil: Miranda, Enrique. Universitat Autònoma de Barcelona; EspañaFil: Palumbo, Félix Roberto Mario. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Jiménez Molinos, Francisco. Universidad de Granada; EspañaFil: Moreno, Enrique. Centre National de la Recherche Scientifique; FranciaFil: Maldonado, David. Universidad de Granada; EspañaFil: Baldomá, Santiago B.. Universidad Tecnológica Nacional. Facultad Regional Buenos Aires. Unidad de Investigación y Desarrollo de las Ingenierías; ArgentinaFil: Moner Al Chawa, Mohamad. Technische Universität Dresden; AlemaniaFil: de Benito, Carol. University of Balearic Islands; EspañaFil: Stavrinides, Stavros G.. Thermi University Campus; GreciaFil: Suñé, Jordi. Universitat Autònoma de Barcelona; EspañaFil: Chua, Leon O.. University of California; Estados Unido