Modelling and Applications of Memristive Devices

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(1)DOCTORAL THESIS 2018. Modeling and Applications of Memristive Devices. Mohamad Moner Al Chawa.

(2) DOCTORAL THESIS 2018 Doctoral Programme of Electronic Engineering. Modeling and Applications of Memristive Devices Mohamad Moner Al Chawa. Thesis Supervisor: Rodrigo Picos.

(3) iii. WE, THE UNDERSIGNED, DECLARE:. That the Thesis titled Modeling and Applications of Memristive Devices, presented by Mohamad Moner Al Chawa to obtain a doctoral degree, has been completed under the supervision of Dr. Rodrigo Picos and meets the requirements to opt for an International Doctorate.. For all intents and purposes, we hereby sign this document:. MSc Eng. Mohamad Moner Al Chawa. Dr. Rodrigo Picos. Palma de Mallorca,.

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(5) Acknowledgement. v.

(6) vi.

(7) Resumen En este trabajo, se ha utilizado un nuevo enfoque en lugar del método tradicional de tensión y corriente para modelar con éxito ReRAM unipolar y bipolar en el espacio de carga de flujo considerando los dispositivos como memristores. Este enfoque considera un sistema dinámico y permite capturar el complejo comportamiento dinámico de los dispositivos ReRAM mediante el uso de un sistema simple de ecuación diferencial ordinaria que puede implementarse en simulaciones de circuitos. Por lo tanto, es posible simular el comportamiento del dispositivo para cualquier señal de entrada, independientemente de la forma de la señal 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 simulación de circuitos grandes sea bastante difı́cil o incluso poco práctica. Se ha utilizado una relación no lineal entre el flujo y la carga para modelar las transiciones de restablecimiento de los dispositivos unipolares. Esta relación no lineal se extiende a uno por partes para modelar las transiciones de restablecimiento / ajuste de los dispositivos bipolares. La corriente y la tensión 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 fácilmente en los simuladores de circuitos para describir la operación de conmutación resistiva. Se empleó un modelo simple en el espacio de carga de flujo con un conjunto reducido de parámetros para ajustar con éxito los diferentes procesos de reinicio experimental para dispositivos memristive ReRAM unipolar. La variabilidad intrı́nseca en estos dispositivos se ha analizado utilizando el enfoque de carga de flujo. Se emplearon simulaciones fı́sicas de dispositivos con diferentes tamaños de filamentos conductores para adaptarse al modelo introducido. Posteriormente, las relaciones entre los parámetros del modelo y las caracterı́sticas geométricas del filamento conductor se caracterizaron vii.

(8) viii en profundidad. Las relaciones obtenidas se utilizaron luego para generar un nuevo conjunto de parámetros que muestra propiedades estadı́sticas similares a la experimental, lo que demuestra la validez del enfoque. Además, se presentó un modelo para obtener la energı́a empleada en el proceso de reinicio. La transición de restablecimiento de un dispositivo memristive ReRAM bipolar se ha analizado en función de las consideraciones energéticas. Se ha realizado un análisis 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 señal de entrada en el punto de reinicio, y se ha encontrado un conjunto de ecuaciones para estimar los nuevos parámetros. Estas ecuaciones, basadas en un análisis de energı́a cuasiestática, permiten la caracterización de la transición de restablecimiento de un dispositivo memristive bipolar ReRAM utilizando solo tres parámetros además de la pendiente de la señal, una resistencia térmica 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 simulación precisos. También permite el desarrollo de expresiones simples para la conductancia y el consumo de energı́a, ası́ como la caracterización 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 señal de entrada en rampa con diferentes pendientes para obtener los parámetros 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..

(9) Resum En aquest treball, s’ha utilitzat un nou enfocament en lloc del tradicional mètode de voltatge-corrent per modelar amb èxit el RePram unipolar i bipolar en l’espai de càrrega de flux considerant els dispositius com memristors. Aquest enfocament considera un sistema dinàmic i permet capturar el comportament dinàmic complex dels dispositius ReRAM mitjançant l’ús d’un sistema simple d’equació diferencial ordinària que es pot implementar en simulacions de circuits. Per tant, és 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 simulació de circuits grans sigui bastant difı́cil o fins i tot pràctic. S’ha utilitzat una relació no lineal entre flux i càrrega per modelar les transicions de restauració dels dispositius unipolars. Aquesta relació no lineal s’amplia a una peça per modelar les transicions de restabliment / establiment de dispositius bipolars. El corrent i la tensió es poden obtenir derivant respecte al temps les magnituds model principal, el flux i la càrrega. Per a dispositius memristius ReRM unipolars i bipolars, s’han utilitzat dades simulades i experimentals per desenvolupar un model que es pugui incloure fàcilment en els simuladors de circuits per descriure l’operació de commutació resistiva. Es va utilitzar un model senzill en l’espai de càrrega de flux amb un conjunt reduı̈t de paràmetres per ajustar amb èxit els diferents processos de restabliment experimental per a dispositius memristius unipolars ReRAM. La variabilitat intrı́nseca d’aquests dispositius s’ha analitzat utilitzant l’enfocament de càrrega de flux. Es van utilitzar simulacions fı́siques de dispositius amb diferents mides de filaments conductius per adaptar-se al model introduı̈t. Posteriorment, es van caracteritzar les relacions entre els paràmetres del model i les caracterı́stiques geomètriques del filament conductor. Les relaix.

(10) x cions obtingudes es van utilitzar llavors per generar un nou conjunt de paràmetres que mostra propietats estadı́stiques similars a l’experimental, demostrant aixı́ la validesa de l’enfocament. A més, es va presentar un model per obtenir l’energia emprada en el procés de restabliment. La transició de restabliment d’un dispositiu memristiu bipolar ReRAM s’ha analitzat sobre la base de consideracions energètiques. S’ha realitzat una anàlisi dels resultats experimentals a l’espai de càrrega de flux al costat de la tensió-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 paràmetres. Aquestes ecuaciones, basades en una anàlisi d’energia quasiestàtica, permeten la caracterització de la transició de reset d’un dispositiu memristiu bipolar ReRAM utilitzant només tres paràmetres, a més del pendent del senyal, la resistència tèrmica i la temperatura de reset del filament conductor. S’ha desenvolupat i provat un model de peces per a la restauració i configuració de transicions d’un dispositiu memristiu bipolar ReRAM a l’espai de càrrega de flux. El model utilitzat és molt senzill i proporciona resultats de simulació precisos. També permet el desenvolupament d’expressions simples per a la conducta i el consum d’energia, aixı́ com la caracterització del dispositiu memRristard ReRAM en el domini actual de tensió 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 paràmetres del model i s’han comparat les prediccions del model amb resultats experimentals. Finalment, s’ha implementat i provat un model compacte gairebé estacionari per a dispositius memristius bipolars ReRAM per a diferents freqüències. Finalment, s’ha implementat i provat un model compacte gairebé estacionari per a dispositius memristius bipolars ReRAM per a diferents freqüències..

(11) Abstract 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. xi.

(12) xii. 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..

(13) List of Publications The work described in this manuscript has been partly published in the following journals and conferences:. • Journals 1. R. Picos, J. B. Roldan, M. M. Al Chawa, P. Garcia-Fernandez, F. Jimenez-Molinos, and E. Garcia-Moreno, Semiempirical modeling of reset transitions in unipolar resistive-switching based memristors, RADIOENGINEERING, vol. 24, no. 2, p. 421, 2015. [DOI : 10.13164/re.2015.0420] (cited by 18) 2. M. M. Al Chawa, R. Picos, J. B. Roldan, F. Jimenez-Molinos, M. A. Villena, and C. de Benito, Exploring resistive switching-based memristors in the charge-flux domain: a modeling approach, International Journal of Circuit Theory and Applications, vol. 46, no. 1, pp. 29–38, 2018, cta.2397. [DOI : 10.1002/cta.2397] (cited by 3) 3. M. M. Al Chawa, C. de Benito, and R. Picos, A simple piecewise model of reset/set transitions in bipolar ReRam memristive devices, IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 65, no. 10, pp. 3469-3480, 2018. [DOI : 10.1109/T CSI.2018.2830412] 4. M. M. Al Chawa, C. de Benito, and R. Picos, Quasi-static energy based analysis of the reset transition in bipolar ReRam memristive devices, IEEE Transactions on Very Large Scale Integration (VLSI) Systems, Submitted 5. M. M. Al Chawa, and R. Picos, Quasi-static compact model of reset/set transitions in bipolar ReRam memristive devices, in preparation. xiii.

(14) xiv. • Conferences 1. R. Picos, M. M. Al Chawa, M. Roca, and E. Garcia-Moreno, A charge-dependent mobility memristor model, in Proceedings of the 10th Spanish Conference on Electron Devices, CDE’2015. IEEE, 2015. 2. E. Garcia-Moreno, R. Picos, and M. M. Al Chawa, Spice model for unipolar RRAM based on a flux-controlled memristor, in Power, Electronics and Computing (ROPEC), 2015 IEEE International Autumn Meeting on . IEEE, 2015, pp. 1–4. (cited by 4) 3. R. Picos, J. Roldan, M. M. Al Chawa, F. Jimenez-Molinos, M. Villena, and E. Garcia-Moreno, Exploring ReRAM-based memristors in the charge-flux domain, a modeling approach, in Memristive Systems (MEMRISYS) 2015 International Conference on. IEEE, 2015, pp. 1–2 (cited by 9) 4. M. M. Al Chawa, R. Picos, E. Garcia-Moreno, S. Stavrinides, J. Roldan, and F. Jimenez-Molinos, An analytical energy model for the reset transition in unipolar resistive-switching RAMs, in Electrotechnical Conference (MELECON), 2016 18th Mediterranean . IEEE, 2016, pp. 1–4. (cited by 4) 5. R. Picos, J. Roldan, M. M. Al Chawa, F. Jimenez-Molinos, and E. Garcia-Moreno, A physically based circuit model to account for variability in memristors with resistive switching operation, in Design of Circuits and Integrated Systems (DCIS), 2016 Conference on . IEEE, 2016, pp. 1–6. (cited by 3) 6. M. M. Al Chawa, R. Picos, E. Covi, S. Brivio, E. Garcia-Moreno, and S. Spiga, φ − Q characterizing of reset transition in bipolar resistive-switching memristive devices, in 11th Spanish Conference on Electron Devices, 2017. IEEE, 2017 (cited by 1) 7. M. M. Al Chawa, A. Rodriguez-Fernandez, M. Bargallo, F. Campabadal, C. de Benito, S. Stavrinides, E. Garcia-Moreno, and R. Picos, Waveform and frequency effects on reset transition in bipolar ReRam in φ − Q space, in International Conference on Memristive Materials, Devices and Systems (MEMRISYS 2017) , 2017 (cited by 2) 8. C. de Benito, M. M. Al Chawa, R. Picos, and E. Garcia-Moreno, A procedure to calculate a delay model for memristive switches, in MDAC workshop at HiPEAC’2017 , 2017 (cited by 3).

(15) xv 9. C. de Benito, M. M. Al Chawa, J. L. Rossello, M. Roca, R. Picos, I. Messaris, and S. Nikolaidis, An analytical delay model for ReRAM memory devices, in Power and Timing Modeling, Optimization and Simulation (PATMOS), 2017 27th International Symposium on . IEEE, 2017, pp. 1–6 10. R. Picos, E. Garcia-Moreno, M. M. Al Chawa, and L. O. Chua, Using memristor formalism in semiconductor device modeling, in Meeting Abstracts , no. 45. The Electrochemical Society, 2017, pp. 2048–2048 (cited by 2) 11. A. Rodriguez-Fernandez, J. Su n e, E. Miranda, M. B. Gonzalez, F. Campabadal, M. M. Al Chawa, and R. Picos, Spice model for the ramp rate effect in the reset characteristic of memristive devices, in Design of Circuits and Integrated Systems (DCIS), 2017 32nd Conference on. IEEE, 2017, pp. 1–4 (cited by 2) 12. E. Garcia-Moreno, R. Picos, M. Roca, and M. M. Al Chawa, Quasi stationary equivalent circuit for unipolar ReRAM memristive devices, in Design of Circuits and Integrated Systems (DCIS), 2017 32nd Conference on. IEEE, 2017. 13. M. M. Al Chawa, C. de Benito, M. Roca, R. Picos, and S. Stavrinides, Design and implementation of passive memristor emulators using a charge-flux approach, in Circuits and Systems (ISCAS), 2018 IEEE International Symposium on . IEEE, 2018, pp. 1–5. (cited by 1) 14. O. Camps, R. Picos, C. de Benito, M. M. Al Chawa, and S. G. Stavrinides, Emulating memristors in a digital environment using stochastic logic, in Modern Circuits and Systems Technologies (MOCAST), 2018 7th International Conference on. IEEE, 2018, pp. 1–4. 15. O. Camps, R. Picos, C. de Benito, M. M. Al Chawa, and S. G. Stavrinides, Effective accuracy estimation and representation error reduction for stochastic logic operations, in Modern Circuits and Systems Technologies (MOCAST), 2018 7th International Conference on. IEEE, 2018, pp. 1–4..

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(17) Contents List of publications. xiii. 1 Introduction 1.1 Memristor History . . . . . . . . . . . . . . . 1.2 Memristor Fingerprints . . . . . . . . . . . . . 1.3 Memristive Devices . . . . . . . . . . . . . . . 1.4 Memristive Mechanisms . . . . . . . . . . . . 1.4.1 Thermochemical Memory . . . . . . . 1.4.2 Valence Change Memory . . . . . . . . 1.4.3 Electrochemical Metallization Memory 1.5 Memristor Models . . . . . . . . . . . . . . . 1.5.1 Physical-Based ReRAM Models . . . . 1.5.2 Memristive Modeling Approach . . . . 1.6 Objectives . . . . . . . . . . . . . . . . . . . . 1.7 Methodology . . . . . . . . . . . . . . . . . . 1.8 Structure of the Thesis . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. 2 Modeling Unipolar ReRAM Memristive Devices 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Device Characterization . . . . . . . . . . . . . . . . . . . . 2.2.1 Devices Under Study . . . . . . . . . . . . . . . . . . 2.2.2 Measurements Setup . . . . . . . . . . . . . . . . . . 2.2.3 Measurements in Flux-Charge Space . . . . . . . . . 2.3 Model Description . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Model Reset/Set Transitions . . . . . . . . . . . . . . 2.3.2 Connection between the Model and the CF Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Variability Modeling Scheme . . . . . . . . . . . . . . 2.3.4 Explicit Monte Carlo Scheme for the Model . . . . . 2.3.5 Energy Dissipation Model . . . . . . . . . . . . . . . 2.4 Discussion and Results . . . . . . . . . . . . . . . . . . . . . xvii. . . . . . . . . . . . . .. 1 1 3 5 6 6 7 7 7 7 9 12 13 14. . . . . . . .. 17 17 17 18 18 19 22 22. . . . . .. 32 34 37 39 41.

(18) xviii 3 Modeling Bipolar ReRAM Memristive Devices 3.1 Introduction . . . . . . . . . . . . . . . . . . . . 3.2 Device Characterization . . . . . . . . . . . . . 3.2.1 Devices Under Study . . . . . . . . . . . 3.2.2 Measurements Setup . . . . . . . . . . . 3.2.3 Measurements in Flux-Charge Space . . 3.2.4 Error Estimation . . . . . . . . . . . . . 3.3 Model Description . . . . . . . . . . . . . . . . 3.3.1 Model Reset Transition . . . . . . . . . . 3.3.2 Energy Based Analysis . . . . . . . . . . 3.3.3 Model Set Transition . . . . . . . . . . . 3.4 Discussion and Results . . . . . . . . . . . . . .. CONTENTS. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. 43 43 44 44 44 46 46 50 50 64 73 81. 4 Implementing Compact Model of Bipolar ReRAM Memristive Devices 85 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.2 Model in Flux-Charge . . . . . . . . . . . . . . . . . . . . . . 85 4.3 Parameters Extraction . . . . . . . . . . . . . . . . . . . . . . 89 4.3.1 Technological Parameters . . . . . . . . . . . . . . . . 89 4.3.2 Model Parameters . . . . . . . . . . . . . . . . . . . . 90 4.4 Model Implementation . . . . . . . . . . . . . . . . . . . . . . 95 4.5 Discussion and Results . . . . . . . . . . . . . . . . . . . . . . 103 5 Conclusions and Future Works. 105.

(19) Chapter 1 Introduction 1.1. Memristor History. In 1960, the first concept of memory resistor device was proposed by Widrow, [1] who christened this device as a ”memistor”. This device had three terminals, and a variable resistance that could be controlled and sensed using the control (DC current) and sensing (AC current) terminals. The significant difference between this device and a transistor was that the memistor resistance was controlled by the instantaneous time integral of the control current, which is the accumulated charge passing through the memistor. Widrow devised the memistor as an electrolytic memory element to form a basic structure for a neural circuit architecture referred to as ADAptive LInear NEuron (ADALINE). After this work, the existence of a fourth passive element with memory properties was stated mathematically in 1971 by L.O. Chua, and was mainly based on theoretical arguments [2]. The original reasoning was based on a missing element which related the electrical charge and the magnetic flux, and would complete the symmetry of the passive electrical elements, as seen in Fig. 1.1. The resulting element would have the properties of a resistance with memory, and thus the word ”memristor” (from ”memory” and ”resistor”) was coined. He defined mathematically two types of memristors; charge-controlled and flux-controlled based on their memristance relation, when it is function of the charge or the flux respectively. This definition was extended later in 1976 [4] by Chua and Kang to include other elements to describe the memristive devices and systems. After more than three decades, Hewlett-Packard (HP) announced the first 1.

(20) 2. CHAPTER 1. INTRODUCTION. Figure 1.1: Resistors and memristors are subsets of a more general class of dynamical devices, memristive systems. Note that R, C, L and M can be functions of the independent variable in their defining equations, yielding nonlinear elements. For example, a charge-controlled memristor is defined by a single-valued function M (q) [3].. actual implementation of memristors from a Resistive Switching Memory device, as seen in Fig. 1.2. Thus, in 2008 a new interest sprang up in this field for researchers under the cover of this new term ”Memristor” [3]. Moreover, it has been observed that if the conductance, inductance or capacitance of a device depends on a state variable, we may get memristive phenomena such as: memristance, meminductance, and memcapacitance, and memristive devices such as: memristor, memcapacitor, and meminductor [5]. In fact, the term ”Memristor or Memristive” refers to a phenomenon more than to an element and we mean with this term ”Memristor” a Resistor with a state variable or more [6], [7]. It is worth to mention that the Resistive Switching Memory device, due to Choi et al. [8], predated the HP work. Actually, in the 1960s, Simmons [9] published the very first report on metal-insulator-metal (MIM) V − I curve,.

(21) 1.2. MEMRISTOR FINGERPRINTS. 3. Figure 1.2: Picture of the original memristor build at HP [3]. It shows an array of 17 oxygen-depleted titanium dioxide devices, imaged by an atomic force microscope. The wires are around 50nm wide. illustrating the hysteresis effect associated with a MIM structure, which characterized the tunneling current behavior.. 1.2. Memristor Fingerprints. As has been defined by Chua in [10], a device can be considered to be a memristor if it presents the fingerprints of a memristor, independently of the particular materials or mechanisms that it is based on. These fingerprints are defined [5,10] by a pinched hysteresis loop restricted to the first and the third quadrants of the V − I plane, whose contour shape in general changes with both the amplitude and frequency of any periodic input voltage or current source, as seen in Fig. 1.3. By increasing the frequency of the source, this pinched hysteresis loop shrinks and tends to a straight line. According to the previous definition, many examples of voltage verse current pinched hysteresis loops have been found in many unrelated fields, such as biology, chemistry, or physics, and observed in many unrelated phenom-.

(22) 4. CHAPTER 1. INTRODUCTION. Figure 1.3: Pinched hysteresis loop.. ena, such as gas discharge arcs, mercury lamps, power conversion devices, and earthquake conductance variations [10], [11], [12]. In this work, we focus on devices showing non-volatile behavior, since they are generally assumed to be the next generation of memory. [13–15]. Any electronic device with only two electrical terminals is usually referred to in the semiconductor industry as a non-volatile memory device (NVM) if the device can exhibit a memory effect over a sufficiently long time period without consuming any power. Resistive switching RAMs (ReRAMs) are one special kind of NVM, that store the memory as a change in its resistance, from a low-resistance state (LRS) to a high-resistance state (HRS), and vice versa. This change can be occurred when the device in set (the process where the device changes from a high resistivity to a low resistivity state) or in reset (the process where the device changes from a low resistivity to a high resistivity state). LRS (or set) corresponds to ON and HRS (or Reset) corresponds to OFF. As has been stated in [10], all 2-terminal nonvolatile memory devices based on resistance switching are memristors. Notice that this classification holds valid regardless of the device material and the physical operating mechanisms, since, as commented before, the key point it that the devices show the corresponding fingerprints. Thus, we can consider bipolar and unipolar resistance switching devices to be a class of memristors..

(23) 1.3. MEMRISTIVE DEVICES. 5. Figure 1.4: Operation modes of memristive devices.. 1.3. Memristive Devices. The two main operation modes of memristive devices are: the unipolar mode and the bipolar mode, which can be seen in Fig. 1.5. The unipolar mode is independent of the polarity of the applied switching signal for set and reset, which can be positive as well as negative. This mode needs higher voltages in the set than the reset process [16]. The bipolar mode needs to apply two switching signals with two different polarities for full operation. A current compliance is needed in all cases where irreversible device breakdown during the set have to be avoided and possible multilevel device switching has to be controlled [17]. The major performance parameters of memristive device are [16], [18]: • Resistance values LRS and HRS and the resistance ratio HRS/LRS. • Write speed: the shortest electrical pulse able to change the resistive state. • Retention time: the time that a resistive state is maintained without applying an external signal to the device. • Endurance: the number of switching cycles can be done before the resistance ratio being an unacceptable value. • Operation energy per bit: the energy needed to to change the resistive state of the device. • Scalability: the possibility to change the size of the memristor device, keeping the memristive effects..

(24) 6. CHAPTER 1. INTRODUCTION. Figure 1.5: Classification of the memristive phenomena [16] • Stackability: the possibility to stack several layers of device on top of one another by fabrication technology. • Multilevel storage: the possibility to store more than one bit of information in one device.. 1.4. Memristive Mechanisms. Resistive switching memristive pheromone can be observed in a wide range of materials due to several physical mechanisms. A classification based on memristive mechanisms was proposed by Waser et al. [16] and can be seen in Fig. 1.5. However, only three mechanisms associated to the filamentary devices which involve chemical effects which relate to redox processes in the MIM device either triggered by temperature or electrical field or both. These atomic switching mechanisms are described in [16], [19], [18]:. 1.4.1. Thermochemical Memory. The main distinctive feature of Thermochemical Memory (TCM) is the unipolar type of switching, the formation and dissolution of the filament occurs at same voltage polarity, whereas VCM and ECM memories are bipolar, the the formation and dissolution of the filament appears at different voltage polarity. In this memory, the switching is dominated by thermally controlled diffusion and redox reaction. The formation in set process is governed by the.

(25) 1.5.. MEMRISTOR MODELS. 7. diffusion of metal ions. However, the dissolution in reset process is considered to be governed by thermal diffusion instead of ion dissolution. Notice that this kind of memory consumes high energy, because it needs a high current level in order to break the filament in the reset process.. 1.4.2. Valence Change Memory. The bipolar switching mechanism of Valence Change Memory (VCM) based on the field induced and temperature accelerated movement of donor type point defects which goes along with a valence change of the metal ion.. 1.4.3. Electrochemical Metallization Memory. Electrochemical Metallization Memory (ECM) is also called Conductive Bridge RAM (CBRAM), Programmable Metallization devices (PCM), and gapless type atomic switch. The bipolar switching mechanism is based on electrode redox reactions nanoionic transport. These memories allow for the highest LRS/HRS ratio up to 108 .. 1.5. Memristor Models. As has been mentioned, resistive switching phenomena had been reported as early as the 1960s [9]. However, ReRAM devices experienced a substantial advance in 2008 by Strukov et al. when they were considered to be memristive systems [3], [2], or memristors, [10]. This modeling approach enables the capture of the complex dynamic behavior of ReRAM devices by using a simple ordinary differential equation system that can be implemented in circuit simulators [20]. In this section, we will discuss the physical-based models for ReRAM and the memristive modeling approach.. 1.5.1. Physical-Based ReRAM Models. As the requirement for precision increases, the needed spatial resolution becomes smaller, the mathematical complexity, physical detail, and computational cost required to perform accurate simulations increase [21]. Thus, electrical models for ReRAM can be classified into four classes according to the simulation scope, expanding the list proposed by Ielmini in [22]:.

(26) 8. CHAPTER 1. INTRODUCTION. Atomistic Scale Density functional theory(DFT) model (few nm3 ) gives the materials comprehending that is significant to simulate device operation through device scale models in order to get a deep knowledge of the structure of materials, ion/atom diffusion and migration mechanisms, and the impact of oxide composition on those aspects. Solution of physical equations based on DFT calculate physical quantities, such as energy barriers for defect generation and migration, band structure, and phonon structure.. Device Scale At the device scale(few tens of nm3 ), physically based device simulations by finite element method (FEM) models and Kinetic Monte Carlo (KMC) models let to understand deeply the switching mechanisms [23]. These simulation models have the profit of giving a V − I characteristics, or response to applied pulses. FEM models solve differential equations for transport of charge carriers (electrons and holes), heat, and ionized defects (oxygen vacancies and cations) while counting on a continuous characterization of the physical entities, such as electric field, temperature and defect concentration. On the other hand, KMC models deal with similar equations with discrete individual defects locally enhancing the conduction through trap-assisted tunneling for instance. As a result, KMC are stochastic, as the position of defects is dictated by Monte Carlo models for generation, recombination and migration, therefore the average switching characteristics can be obtained only from several simulation runs. On the other hand, FEM provides the average switching characteristics while the variation characteristics can be simulated by energy scenes of microscopic parameters, such as the energy barrier for migration.. Circuit Scale The numerical simulations that have been described previously allow the visualization of the local dynamics of defect concentration that lead to set/reset processes, that allow to build of analytical models to simulate ReRAM-based circuits which consist of a more simplified set of equations for macroscopic parameters, such as the average temperature, the geometry of the conduction filament, or the depleted gap. As we will discuss later, these analytical models in fact are memristor models that use the macroscopic parameters as states variables in V − I domain..

(27) 1.5.. MEMRISTOR MODELS. 9. System Scale The circuit level models are useful up to a given level of complexity. Once the number of elements goes up, simulation gets more and more complex [24]. Moreover, the parameters of interest are not voltage and current anymore, but other parameters related to higher level descriptions, like write delay or switching energy. Thus, other kind of models are needed to further simplify the simulations. For instance, models for calculating the energy dissipation are provided in [25], or models for the write delay time in [26, 27].. 1.5.2. Memristive Modeling Approach. Strukov et al. presented a mathematical model based on two series resistors are the doped Ron and the undoped Rof f region resistances. It is assumed that the physical device is of width, D, and the doped region of width, w which is the state variable that changes depending on the charge [3]. The resistance of the device can be written as: M (t) = Ron ·. W (t) W (t) + Rof f · (1 − ) D D. (1.1). and the rate change of the state variable can be expressed as: µV .Ron dW = · i(t) dt D. (1.2). which yields the following formula for w(t): W (t) =. µV .Ron · q(t) D. (1.3). where µV is the average ion mobility and i(t) is the passing current through the memristor and the charge can be written as: q(t) =. di(t) dt. (1.4). However, the first practical model was described by Strukov et al. in [3], has a symmetric behavior that does not describe the real nonlinearities of the memristor. Also, the model is valid only for certain choices of input and initial state. This weakness is related to the adopted window function in this model [28], [29]. Based on that model [3], a whole class of models using different kinds of state variable boundaries, i.e., window functions, was derived [30], [31],.

(28) 10. CHAPTER 1. INTRODUCTION. [32], [33], [34], [35] later in order to have better simulation results. A more complex modeling approach by Pickett et al. and implemented in SPICE by Abdalla et al. [36], [37] was used as a reference for another class of models. Some memristive models have been derived [38], [39], [40] based on a physicsoriented modeling approach [41] that explains the mechanisms at the origin of the complex dynamics observed in the T iO2 -based memristor by means of the Simmons tunnel barrier model [38]. A comprehensive review of SPICE implementations of the above mentioned models can be found in [42]. In fact, the design of new circuits with these devices requires extensive simulations, often involving hundreds of thousands of them, mainly in the case of memories, or bio-inspired circuits. In order to speed up simulations, or to build some actual circuits, some researchers have also opted to implement FPGAs or ASICs memristor emulators [43–47]. This approach is sound, but it is also bulky and requires complex implementations, since modeling of analog behavior using digital circuitry is ungainly, especially in the case of designs containing a large number of elements. Many models have already been proposed in the literature [48–55]. Additionally, variability from cycle to cycle in ReRAMs has also been addressed [56, 57]. However, many models appear to demonstrate drawbacks that make simulation of large circuits rather difficult or even impractical [24, 58]. Also, simulation robustness can be an issue for highly complex models [36]. Moreover, using a general set of equations that do not directly correspond to the actual device physics decreases the predictability [38]. If the basic equations of the model do not well implement the device physics, with or without window functions, as in case of the memristor models in V − I space, this will lead to obtain a low accurate model [20]. However, if only a certain operation state is to be modeled, such as piecewise models, then very accurate models can be applied. Most of the previous works for modeling the memristor behavior used a V − I domain, but in our case we adopted a φ − Q one. This modeling approach considers a dynamical system. Thus, it should be possible to simulate the device behavior for a wide range of input signals, regardless the input signal shape. Another criterion is related to the switching kinetics. A strong nonlinear relationship between pulse height and its width has been observed experimentally. Hence, in fullling this criterion, is essential to simulate, either the memory or the logic applications, using ReRAM memristive devices that are conducted by pulses.. The framework for our approach in this thesis was introduced in 2015 by.

(29) 1.5.. MEMRISTOR MODELS. 11. Corinto et al. in [59]. Corinto et al. introduced voltage first momentum and current first momentum and employed them in the theoretical framework for the memristive systems. Using these momenta as the electrical variables may have some advantages in terms of modeling as has been proposed in the literature in [10, 59, 60] and others [25, 33, 51, 53, 61]. One of the advantages, for instance, is simplification of the explanation of how the device can reset or set without reaching the needed voltage when a series of pulses is applied. Furthermore, taking into account the time factor by using φ − Q space simplifies the differential equations in V − I space. Thus, instead of using a V − I domain, in this work we will use a domain described by the flux (φ) and charge (Q) magnitudes. By taking into our consideration a two-terminal device described by a current I(t) and a voltage V (t) at its terminals, we can define the voltage first momentum φ(t) (also called, for simplicity, flux) and the current first momentum Q(t) (or charge) as follows [59]: Z t φ(t) = V (τ )dτ (1.5) Z Q(t) =. t. I(τ )dτ. (1.6). A detailed discussion on the use of the terms, flux and charge to refer to these integrals can be found in [59]. The ideal memristor can be described in terms of the flux φ(t) and the charge Q(t) by the following nonlinear equation: Q = f (φ, V ). (1.7). However, this definition in (1.7) is limited, and was later extended to include more complex behaviors. This behavior corresponds to a more general class of memristors called extended memristors. They are described by introducing an internal state vector X with k variables. X = (x1 , x2 , .., xk )T. (1.8). We can define the rate of change for these state variables as: dX = g(φ, V, X) dt. (1.9). Additionally, instead of using current and voltage, the extended memristor can be described in terms of the flux φ(t) and the charge Q(t) by the following non-linear equation: Q = f (φ, V, X). (1.10).

(30) 12. CHAPTER 1. INTRODUCTION. Thus, the whole behaviour of the memristor would be described by the set of equations (1.9) and (1.10), while the function f is the static characteristic equation, and g governs the behavior of the state variables when driven by an external voltage V , where for all state variables X: k. X ∂f dxi ∂f dV · + · =0 ∂V dt ∂x dt i i=1. (1.11). It is important to call the reader’s attention to the fact that, this approach fits the memristor fingerprints which have been mentioned in Sec 1.2. Generic memristors are extended memristors with no parasitic effects. That is, f is only dependent on φ and X. Finally, ideal memristors are those generic memristors where no state variable is present. The Venn diagram in Fig. 1.6 shows the memristor universe and the relationship among the classes of memristors [62]. Focusing on NVM, it is worth noticing that (1.9) determines the memory capability of the system. When there is no excitation (that is, V = 0 and, thus, φ = constant), (1.9) is often referred to as the power-off plot (POP) equation, so if the POP equation is zero, the system presents long term memory, while when the POP equation is not zero, the system can present only short term memory. In the case of interest for this work, where we are considering ReRAMs to be memristors [10], the state variables are usually thought to be the radius of the conducting filament (CF), the gap between the rupture CF tip and the electrode, and the temperature of the filament [53, 56, 63–66].. 1.6. Objectives. There are many published models, but there still some pending problems. For instance, modelling the cycle to cycle variability of the characteristics has not been thoroughly addressed in the literature, or the variability when the forcing waveform is changed. Thus, the overall objective of the thesis is to provide some new insights on the modelling of the ReRAM Memristive Devices. Specifically, we focus on the following different aspects of this problem: 1. Studying the different kinds of ReRAM Memristive Devices and their different governing mechanisms. 2. Adopting a new approach with a φ − Q space instead of the V − I one..

(31) 1.7. METHODOLOGY. 13. Figure 1.6: Venn diagram showing the relationship among the classes of memristors [62]. 3. Compact modelling of a given kind of memristor devices (ReRAMs), including the development of the mathematical models.. 1.7. Methodology. The method to fulfil all these objectives is based on these main steps: 1. Study and analysis of devices measurements: These measurements have been obtained either by direct measuring or through other research groups. The unipolar ReRAM memristive devices based on N i/Hf O2 /Si − n+ structures had been fabricated and measured at the IMB-CNM (CSIC) in Barcelona, Spain, where, the bipolar ReRAM memristive devices based on P t/Hf O2 /T iN structures had been fabricated and measured during short scientific mission at the at the CNR-IMM, MDM Laboratory in Milan, Italy. 2. Mathematical modelling of the physical behavior of the devices. The physics governing the device has to be modelled and translated into equations that can be run in a simulator..

(32) 14. CHAPTER 1. INTRODUCTION. 3. Development of a procedure to extract the parameters of the proposed model based on the analysis of experimental data. The specific methods followed to treat the different topics of this thesis are detailed in each chapter.. 1.8. Structure of the Thesis. The manuscript is organized in the following manner: In Chapter 2, a model in φ − Q space for Unipolar ReRAM Memristive Devices based on N i/Hf O2 /Si − n+ structures will be introduced as a first step in this work. We use simulated and experimental data to develop a model for unipolar ReRAM meristive device that can be easily included in circuit simulators. The adopted approach allows to model the conductance with simple expression. Physical simulations of devices with different conductive filament sizes are employed to fit the 3-parameter model for reset transition introduced. Later on the relations between the model parameters and the conductive filament geometrical features are characterized in-depth. In addition, we use the model to estimate the experimental conductive filament radius distribution using a set of 3000 reset cycles. The measured data are reproduced and used to describe the implementation of an explicit Monte Carlo method to include variability in the model for reset transition and to estimate cycle to cycle variations. Finally, a model to obtain the energy employed in the reset process is presented. In Chapter 3, we will employ the proposed model to describe and implement a piecewise model for the reset and set transitions of a bipolar ReRAM memristive devices based on P t/Hf O2 /T iN structures in the φ − Q space, instead of the usual V − I one. 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. We consider the case of a ramp input signal with different slopes to obtain the model parameters, and we compare the predictions of our model with experimental results. In addition, we analyse, from energy considerations, the reset transition of a bipolar ReRAM memristive device. An analysis of the experimental results is done in the φ − Q space beside the usual V − I one. We consider the effect of changing the slope of the input signal in the reset point, and we find a set of equations.

(33) 1.8. STRUCTURE OF THE THESIS. 15. to estimate the new parameters. These equations, based on a quasi-static energy analysis, allow characterizing the reset transition of a bipolar ReRAM memristive device using only three parameters in addition to the signal slope, a thermal resistance and the reset temperature of the conductive filament. In Chapter 4, we will implement in MATLAB a quasi-static compact model for bipolar ReRAM memristive devices. We will use the same device that had been fabricated in CNR-IMM, MDM Laboratory, and the piecewise model equations in φ − Q for reset and set transitions in Chapter 3 to build a compact model of reset/set transitions for different slopes. We will do the parameters extraction for the device based on the results in the previous chapter. The implemented model will be able to capture the slope change in the ramp input signal for reset and set by using a set of parameters related to the device and another one related to the slope of the ramp input voltage signal. Finally, in Chapter 5, we will highlight the main conclusions and future works..

(34) 16. CHAPTER 1. INTRODUCTION.

(35) Chapter 2 Modeling Unipolar ReRAM Memristive Devices 2.1. Introduction. The unipolar ReRAM meristive device is independent of the polarity of the applied switching signal for set and reset, which can be positive as well as negative. This device needs higher voltages in the set than the reset process [16]. In this chapter we use simulated and experimental data to develop a model for unipolar ReRAM meristive device that can be easily included in circuit simulators. The adopted approach allows to model the conductance with simple expression. Physical simulations of devices with different conductive filament sizes are employed to fit the 3-parameter model for reset transition introduced. Later on the relations between the model parameters and the conductive filament geometrical features are characterized in-depth. In addition, we use the model to estimate the experimental conductive filament radius distribution using a set of 3000 reset cycles. The measured data are reproduced and used to describe the implementation of an explicit Monte Carlo method to include variability in the model for reset transition and to estimate cycle to cycle variations. Finally, a model to obtain the energy employed in the reset process is presented. Results using this model show a very good agreement with experimental results.. 2.2. Device Characterization. In this section, a description of the ReRAM memristive devices under study in this work has been introduced. The characterization in φ − Q space was performed by measuring in the V − I space, considering a ramp input signal, 17.

(36) 18CHAPTER 2. MODELING UNIPOLAR RERAM MEMRISTIVE DEVICES. Figure 2.1: Devices under study: fabricated at IMB-CNM (CSIC) on (100) n-type CZ silicon wafers with resistivity (0.007-0.013)Ω/cm. and then integrating numerically the obtained values.. 2.2.1. Devices Under Study. Devices under study had been fabricated and measured at the IMB-CNM (CSIC) in Barcelona, Spain. The devices were based on N i/Hf O2 /Si − n+ structure with an oxide layer 20nm thick, fabricated on (100) n-type CZ silicon wafers with resistivity between 0.007 Ω/cm and 0.013 Ω/cm following a field isolated MIS process (Fig. 2.1 ) . The 20 nm-thick Hf O2 layer was deposited by atomic layer deposition using TDMAH and H2 O as precursors. The 200 nm-thick N i electrode was deposited by magnetron sputtering. The resulting device structures are square devices of 5 x 5 µm2 [67].. 2.2.2. Measurements Setup. Measurements were performed using a HP-4155B semiconductor parameter analyser at temperatures from 40°C to 175°C. The voltage was applied to the top Ni electrode, while the Si substrate was grounded. In order to evaluate the cycle-to-cycle variability, numerous cycles and measurements need to be assessed. For this purpose, a software tool has been developed and implemented in Matlab to control the instrumentation via GPIB (General Purpose Instrumentation Bus) and to smartly detect the set and reset currents. Experimental measurements has been measured at average ramp speed of the input voltage, S = 1V /s, and the source current compliance was set to Ic = 0.1mA, during the set process. Fig. 2.2 shows the applied voltage signal vs. time and measured current vs. time during set/reset cycle. In this chapter, we depend on measurements of three thousand (3000) reset cycles of a single unipolar resistive switching (RS) memristive device.

(37) 2.2. DEVICE CHARACTERIZATION. 19. Figure 2.2: Applied voltage signal vs. time and measured current vs. time during set/reset cycle. after a forming process. Some of the measured curves corresponding to several RS cycles are shown in Fig. 2.3.. 2.2.3. Measurements in Flux-Charge Space. In order to get the device characterization in φ − Q space, some of the reset curves in V − I space have been plotted in Fig. 2.4 for a single device. There is a great dispersion in the shapes of the curves and in the reset voltages and their corresponding reset currents [68], where a reset voltage by definition is the voltage level at which the conductance filament (CF) starts its dissolution process or the voltage level at which the CF is destroyed [69]. It is shown that not only the CF size but also its shape is important in order to determine the reset voltage [23]. As has been reported in the literature, ( [70], [23], [71], [72], [73], [74]) the characteristic conductive filaments responsible for the RS operation in conductive bridge devices are broken when.

(38) 20CHAPTER 2. MODELING UNIPOLAR RERAM MEMRISTIVE DEVICES. Figure 2.3: Experimental current vs. applied voltage for several set/reset transitions in a long RS series for a single device based on a N i/Hf O2 /Si−n+ structure. the reset process takes place. We take profit from this fact to extract the reset voltage (Vrst ) and the reset current (Irst ). In order to extract the reset voltages and reset currents, we will move from the usual representation in the V − I space to a representation in the φ − Q domain, then we can extract in that new domain. The flux φ and the charge Q are defined in (1.5) and (1.6) respectively [59], and have been calculated numerically from the applied input voltage signal and measured current respectively. Thus, Fig. 2.5 plots again the data of Fig. 2.4, but in the φ−Q domain. It is worth noticing that, as expected, after the reset voltage is achieved, the charge remains practically constant since the device current is reduced in several orders of magnitude and there is no contribution to the time integral of the charge. In fact, we first extract the reset flux (φrst ) and the corresponding reset charge (Qrst ); afterwards we recover the corresponding reset voltage and current values. In order to extract Qrst and φrst values, we fit two lines in the φ − Q plot Fig. 2.6 and Fig. 2.7: one to the region where the charge remains constant and the other to the upper part of the monotonously increasing.

(39) 2.2. DEVICE CHARACTERIZATION. 21. Figure 2.4: V − I characteristics of reset transitions corresponding to 100 RS consecutive experimental cycles on a single device (using absolute values for Voltage). curve prior to the plateau. The desired reset point is given by the intersect point of the crossing lines, as depicted in Fig. 2.6 and Fig. 2.7. Fig.2.6 shows the application of the method to a curve with a single current step, while Fig. 2.7 shows the application of this algorithm to a curve corresponding to a cycle with several current steps [23]. From the values of φrst and Qrst , respectively, it is simple to calculate Vrst and Irst , since the flux and charge are monotonically increasing functions. The mean values and standard deviations for all these calculated parameters are provided in Tab. 2.1. So, we can define the reset point in φ − Q domain as the point that after increasing the flux the relating charge still constant. The corresponding charge and flux for this point are: the reset values for the flux and charge which will use to obtain the reset voltage and current. Fig. 2.8 plots Qrst vs. φrst and Irst vs. Vrst (in the inset). The result can be obtained the correlation between them is much higher in the φ − Q variables than in the V − I pairs, which is coherent with the fact that, the.

(40) 22CHAPTER 2. MODELING UNIPOLAR RERAM MEMRISTIVE DEVICES φ − Q domain is the natural space for memristor modeling [10].. 2.3. Model Description. In order to model the behavior of the memristor device, we consider two resistance states. LRS in case of having a CF, and HRS when the CF disappears. Thus, the physical mechanism during the reset transiton can be described as a condcutive path that can be modeled in φ − Q space as a CF, while the mechanism during the set transition can be assigned to a thermionic or hopping conduction that can be modeled as a diode [51–53, 55].. 2.3.1. Model Reset/Set Transitions. Once the φrst , Qrst , Vrst , and Irst are calculated, we can normalize the measurements, scaling the curves in the φ−Q domain in order to obtain Qrst = 1 and φrst = 1. Fig. 2.9 shows the results of the normalized charge vs. the. Figure 2.5: φ − Q characteristics for the same reset cycles shown in Fig 2.4 (using absolute values for Flux)..

(41) 2.3. MODEL DESCRIPTION. 23. Figure 2.6: Q vs. φ for a single plain transition. The fitting lines are shown in red, while the blue curves are the transformed measured data. The reset point is assumed to be located at the crossing point between the red lines. normalized flux. It is apparent that the behavior of the whole set of curves is very similar. We have seen that, after normalizing all the RS reset cycles with the values of Qrst and φrst , the curves can be modeled with a single explicit analytical expression in the φ − Q domain. In this respect, taking into consideration the work of Chua [10] and Shin [33] outlining a work plan for modeling in the φ − Q domain, we propose a non-linear relation between the charge and the flux in the following way for the low resistance state of the device [51, 53]: φ n ) ) (2.1) φrst where φrst , Qrst and n are the fitting parameters. The flux φ can be calculated as the time integral of the voltage while the charge Q is obtained Q = Qrst · f (1, (.

(42) 24CHAPTER 2. MODELING UNIPOLAR RERAM MEMRISTIVE DEVICES as the current time integral from (1.5) and (1.6) respectively [10]. Notice that we follow the notation in [59], where a detailed discussion on the use of the names f lux and charge to refer to these integrals is performed. Notice that f (x, y) is a function that provides the minimum between x and y in a smooth way. For a complete discussion of smoothing functions in device modeling, see [75]. In our case, we have chosen the following: p (2.2) f (x, y) = 0.5(x + y − (x − y 2 ) − 4δ 2 ) The constant δ is a smoothing value,which as been fixed at δ = 10−5 , or we can use in this form directly: n φ ) (2.3) Q = Qrst · min(1, φrst. Figure 2.7: Q vs. φ for a multiple transition. The fitting lines are shown in red, while the blue curves are the transformed measured data. The reset point is assumed to be located at the crossing point between the red lines..

(43) 2.3. MODEL DESCRIPTION. 25. Table 2.1: Mean values and standard deviations of the obtained reset values for the flux, charge, voltage, and current. φrst. Qrst. Vrst. Irst. (V.s) (mC). (V ). (mA). Average. 3.28. 0.56. 1.98. 0.167. Standard deviation. 0.76. 0.25. 0.26. 0.088. Notice that we have not taken into account any residual charge after the reset transition. This simple and explicit model fits the experimental data in a reasonable manner with only three parameters (Qrst , φrst , n). The model fits experimental values fairly well, mostly in the φ − Q domain. A notably accurate fit of experimental results was achieved making use of this. Figure 2.8: Qrst vs. φrst for each reset transition. The inset shows Irst vs. Vrst . The correlation is clearly much better in the first case (using absolute values)..

(44) 26CHAPTER 2. MODELING UNIPOLAR RERAM MEMRISTIVE DEVICES. Figure 2.9: Normalized Q vs normalized φ for each reset transition. The normalization is performed by scaling each curve in order to fit the reset values to one. The discontinuous line corresponds to the model proposed in (2.1). In the case of V − I curves with several CFs, there can be seen Q values above one; in these cases the normalizing value was connected with the first reset event. model [51]. As an example, see Fig. 2.10 and Fig. 2.11, where a reset transition is modeled in the φ − Q space and in the V − I one respectively. In Fig. 2.12 we have represented the extracted model parameters for each RS cycle considered. As can be seen from this figure, it can be concluded that they are strongly correlated through some underlying process, which we assume to be connections between the values of these parameters and the geometry, number of conductive filaments and their radii that control RS processes. This simple model can be complemented with other effects such as contact effects, as already done in other devices (see [76], for instance). It can be also used to simulate the statistical behavior of resistive switching unipolar memristors. It must be noted also that it does not depend on the shape of the input waveform, since the model is, as recommended in [10], in.

(45) 2.3. MODEL DESCRIPTION. 27. Figure 2.10: Q vs. φ for a single reset transition. Experimental data are shown as red marks, while the blue line corresponds with the fitted model. Table 2.2: Extracted mean values and Standard deviations for the parameters of the model described in Tab. 2.1.. Average Standard deviation. φrst (V.s) 3.28 0.76. Qrst n (µC) 562 1.5 255 0.0999. the φ − Q domain. The mean values of the fitting parameters are provided in Tab. 2.2. The values of φrst and Qrst are obviously the same than those given in Tab. 2.1, but are repeated here for completeness. Expressing the model (2.3) in terms of voltage and current is straightforward: i(t) =. dQ dt. =. n d Qrst ·min 1, φ φ rst. dt. (2.4) =. R v(t)dt n d Qrst ·min 1, φ rst. dt. It is also worth noticing that the modeling approach based on charge and.

(46) 28CHAPTER 2. MODELING UNIPOLAR RERAM MEMRISTIVE DEVICES. Figure 2.11: I vs. V for a single reset transition. Experimental data are shown as red marks, while the blue line corresponds with the fitted model. flux is quite similar to some classical approaches to digital test ( [77, 78]). In these procedures the relevant parameter is connected with the charge consumption, also understood as the time integral of the current circulating through the circuit during a time period. Notice also that, in the case of a pulse signal, the flux is simply the pulse height multiplied by the pulse duration. According to the formalism presented by Corinto, Civalleri, and Chua [59], this kind of device can be classified as a flux-controlled extended memristor [10], [4], with an V − I characteristic dependent on a state variable (the minimum radius rmin of the filament, for instance) plus an equation governing the behavior of this variable: i = G(r, v, φ) · v. (2.5). dr = f (r, v, φ) (2.6) dt being G the device conductance and v the applied voltage. Assuming that the radius is the governing variable that depends on the voltage is a reasonable assumption, if we consider the conduction to take place inside a conductive filament [73,79]. As has been shown in [79], the shape of the device conductive filament remains almost unchanged just until the reset process.

(47) 2.3. MODEL DESCRIPTION. 29. is triggered once the self-accelerated process that disrupts the conductive filament starts. Thus, the consideration of the CF radius as a state variable could be reasonable. We do not provide the details of the equation governing the radius (f ), since they can be found in the literature [73]. Comprehensive descriptions of the conductive mechanisms can be also found in, for instance, [70, 80, 81]. It is also important to call the reader’s attention that in the devices under study, the non-linear behavior observed in the i-v curve is mainly caused by a barrier that leads to quantum effects and can be described according with the Quantum Point Contact (QPC) model [70,82]. In particular, at low voltages (far from the reset region) the conductivity of the channel is therefore determined by the conduction across this barrier, whose transparency (and thus G) depends on the voltage applied to this barrier v 0 , which is lower than the total device applied voltage, v, since the CF ohmic part and the series resistance (due to metal paths and setup wires) are in series with this tunneling barrier. In this operation region, the relationship between i and. Figure 2.12: Extracted parameters φrst , Qrst and n represented in 3D plot. Dark dots are the points in the 3D space, while the clear points correspond to projections into the corresponding axis planes.

(48) 30CHAPTER 2. MODELING UNIPOLAR RERAM MEMRISTIVE DEVICES. Figure 2.13: Device conductance (G) vs. flux (φ) calculated using experimental data (as i/v, black square symbols) and the model (via (2.8), red circular symbols) the voltage, v 0 , is given by the following equation: 2eN i= h. 0 1 1 + eα0 [γ−ζev ] 0 ev + ln , α0 1 + eα0 [γ+(1−ζ)ev0 ]. (2.7). being γ and α0 the parameters that define the quantum barrier, whereas ζ is the fraction of v 0 that drops at one of the sides of the barrier [82]. This nonlinear expression does not allow the extraction of a simple algebraic equation for the relation between charge (Q) and flux (φ) by means of integration in (2.7). Therefore, for the sake of simplicity we have assumed a semiempirical expression in line with the development proposed by Chua [10]. This expression was given in (2.3) and it was previously introduced and validated in [51]. In order to illustrate that memristor modeling could be easier in the φ−Q domain, we have analytically calculated the device conductance (G(r, v)) by deriving expression (2.1):.

(49) 2.3. MODEL DESCRIPTION. 31. Figure 2.14: Modeled current vs. voltage for set/reset cycle (using absolute values)..  n−1 φ ·n dQ  Qφrst · , ( φ < φrst ) φrst rst = G= 0 dφ , ( φ > φrst ). (2.8). Fig. 2.13 shows a comparison between the values of G calculated by using the experimental data (as i/v) and by means of expression (2.8). As can be seen, the model in the φ−Q domain provides a reasonably good and compact description of G. It is important to highlight that this approach could be simpler than others that could have been obtained by calculating i/v though (2.7). In this case, we believe that evaluating our approach against the trade off of accuracy and simplicity that is always present in compact modeling, it does pay off. When the device is in the HRS, the current seems to be controlled by a thermionic mechanism. Therefore, the following expression for the current during the HRS has been adopted [52, 55]: v. I = IA · (e va − 1) where IA and vA are fitting parameters.. (2.9).

(50) 32CHAPTER 2. MODELING UNIPOLAR RERAM MEMRISTIVE DEVICES Fig. 2.14 allows comparing the model with experimental measurements; it shows experimental and modeled V − I characteristics for a set/reset transition in semilogarithmic scale when a single CF is involved. Starting with the ReRAM in the HRS, the applied voltage is increased progressively from zero volts until it reaches the LRS (set process), then the voltage is reset to zero. Next a second voltage ramp is applied, starting from zero and going beyond the reset point to examine the reset process. It has been measured at ramp speed of the input voltage, S = 1V /s, and the source current compliance was set to IC = 0.1mA, only during the set process. It can be observed that the (2.9) fits the experimental data in the HRS almost perfectly. Notice that the thermionic current always is existed when the device in HRS or in LRS, however, this current is negligible in case of having a conduction current as a CF [52, 55].. 2.3.2. Connection between the Model and the CF Characteristics. In order to gain some insight into the mechanisms behind our model, we have studied the connections between the model parameters depicted in (2.1) and the features of the simulated structures employed to reproduce experimental measurements. The simulations described memristors with a single CF of different geometrical characteristics. The CFs were characterized by a maximum radius (rmax ) and a minimum radius (rmin ) for a truncated-cone shape, considering that the devices operate in the low resistance state. The CFs were also characterized as well by different quantum barriers (different α values were assumed according to (2.7) following the results presented in [79]). A voltage ramp was employed in the calculations with an equivalent rate of S = 0.1V /s, as in the case of the experimental measurements in [68]. The conduction is considered to take place inside conductive filaments (CFs) [68]. The conductive filaments have been modeled as truncated-coneshaped CFs in this work. These truncated cones, as shown in Fig. 2.15, are characterized by the maximum radius (rmax ), the minimum radius (rmin , this one chosen as a fraction, of the maximum radius), and the CF length, which in our case is taken as (tox ) the thickness of the oxide layer. For the simulator employed, an arbitrary number of these conductive filaments electrically coupled with different initial radii and shapes are considered including the series and Maxwell resistances for each CF [23]. Quantum effects were taken into consideration in addition to the self-consistent temporal solution of the chemical redox equations controlling the CF radius evolution along with the diffusion of metallic species, the heat equation and the current equations in.

(51) 2.3. MODEL DESCRIPTION. 33. the structures under study [70] .. Figure 2.15: Conductive filament geometry assumed in the simulations, the variables shown are explained in [70]. This shape has been checked by means of different characterization techniques [83]. The temporal evolution of the CFs is determined by calculating the diffusion of the metal atoms and the electron transfer reactions (electrochemical processes) that describe the aggregation of metal atoms to the CF. Quantum effects linked to the narrow CF constrictions located close to the Si − n+ electrode contact region are accounted for by using the Quantum Point Contact model [70,79]. The reset voltage in the V − I curve is determined as the point where the current falls abruptly, see more details on this issue in [69]. The model fitting parameters vs. the minimum CF radius are plotted in Figures 2.16, 2.17 and 2.18 after tuning the simulated data. There is a simple correlation between parameters Qrst , and φrst with rmin and α; while n shows a more complex relation with rmin , rmax , and the barrier height, characterized by the α parameter. We can approximately fit these correlations with simple semiempirical expressions as follows: 2 Qrst = Qr · rmin. (2.10). 2 φrst = φr · rmin. (2.11). (φrst )n = rmin (a + b · rmin ) c + d · α1/2. . (2.12).

(52) 34CHAPTER 2. MODELING UNIPOLAR RERAM MEMRISTIVE DEVICES. Figure 2.16: Simulated device charge at the reset point Qrst vs. initial CF rmin . Different α values were employed in the simulated data employed in this plot. where Qr , φr and a, b, c, d are fitting parameters, and rmin represents the CF minimum radius. The fitting parameters employed to reproduce simulated data for the specific curves we have considered are the following: Qr =5.3E-6 C/nm2 , a=45.7E-3 (V ·s)n /nm, b=2.2E-3 (V ·s)n /nm2 , c=1.3718 and, d=0.5842 eV 1/2 . Notice that parameter n can be then calculated as: n=. ln(rmin (a + b · rmin )(c + d · α0.5 )) 2 ln(φr · rmin ). (2.13). It can be seen that for a determined memristor with a minimum radius rmin , the direct use of this value in previous equations would allow the calculation of the parameters needed to use in (2.1).. 2.3.3. Variability Modeling Scheme. The model described in (2.1) was employed to describe a set of more than 3000 reset V −I curves in N i/Hf O2 /Si−n+ devices [68]. The measured data were correctly reproduced, an example is plotted in Fig. 2.10 in the φ − Q domain. We repeated the fitting procedure for the set of curves and obtained groups of (Qrst , φrst , n) values. In Fig. 2.19 we show the distribution of Qrst.

(53) 2.3. MODEL DESCRIPTION. 35. Figure 2.17: Simulated device flux at the reset point (φrst )n vs. initial CF rmin . Different α values were employed in the simulated data employed in this plot. vs. φnrst . It is clear that the fitting parameters are correlated. There are, at least, three different domains (marked with white lines in the figure) in the data set plotted, this fact suggests that a multi valued logic scheme could be feasible for these devices. Using the extracted Qrst parameter and (2.10), we have estimated the initial CF minimum radius in each experimental reset curve, making use of the fact that its dependencies seem to be concentrated on this radius. The histogram of the distribution of minimum radii was plotted in Fig. 2.20. The envelope shape of the histogram is consistent with a W eibull distribution with λ = 12.42 nm and k = 1.48, as shown in a green solid line in Fig. 2.20. As is well known, this distribution is used to describe reliability processes, therefore this result is consistent considering the connection between the CF formation and the typical reliability processes in RS-based memristors. Once the radii have been extracted, (2.12) can be used to estimate the parameter employed to characterize the quantum barrier height, α: α=. c (φrst )n − d · rmin (a + b · rmin ) d. 2 (2.14). where a, b, c and, d are the constants mentioned above, and rmin is.

(54) 36CHAPTER 2. MODELING UNIPOLAR RERAM MEMRISTIVE DEVICES. Figure 2.18: Fitting parameter, n, vs. rmin for different values of rmax . rmax and rmin are the radii of the truncated cone-shaped conductive filaments assumed in the RS memristor analysis.. Figure 2.19: Qrst vs. φnrst . The distribution has been obtained with the fitting parameters extracted from the experimental reset curves. Different domains can be seen, they are marked with white lines..

(55) 2.3. MODEL DESCRIPTION. 37. Figure 2.20: CF minimum radii distribution corresponding to the group of experimentally measured reset cycles considered. The envelope shape is coherent with the W eibull distribution shown in a blue line. the estimated initial CF minimum radius. Using this equation, we obtained a distribution for the values of α, as depicted in Fig. 2.21. We call the reader’s attention to the fact that this distribution shows favored (somehow quantized) values of this α parameter, this result being coherent with the different domains seen in Fig. 2.19. See that in this latter figure the variability makes the distribution of both parameters Qrst and φrst cluster in different domains reflecting the stochastic nature of the processes behind the physics of the device operation. That is the main difference between experimental and simulated results.. 2.3.4. Explicit Monte Carlo Scheme for the Model. In this section we use the previous results reported to describe the implementation of an explicit Monte Carlo method to include variability in our model. This approach allows simulations to estimate cycle to cycle variations. For the sake of clarity we sort out the results obtained so far: we have used an easy approach to fit our proposed model with three fitting parameters (Qrst , φrst , and n) to both simulated and experimental V − I reset curves; we have shown that these fitting parameters are connected with physical CF features that characterize the studied memristors (quantum barrier fea-.