One-Way ANOVA modelling for RRAM reset curves alez 1 , Ana M. Aguilera 1 , Christian J. Acal Gonz´ M. Carmen Aguilera-Morillo 2 , Francisco Jim´ enez-Molinos 3 , an 3 Juan B. Rold´ 1 Departamento de Estad´ ıstica e I.O. Universidad de Granada 2 Departamento de Estad´ ıstica. Universidad Carlos III de Madrid 3 Departamento de Electr´ onica y Tecnolog´ ıa de los Computadores. Universidad de Granada III International Workshop on Advances in Functional Data Analysis Castro Urdiales (Cantabria), Spain, May 23, 2019 C. Acal chracal@ugr.es 1 / 36
Index Introduction 1 Device description and measurement 2 RRAMs operation Experimental Data Used devices and purpose Functional Data Analysis 3 Functional modelling of reset curves Registration of reset curves in the interval [0,1] Functional reconstruction of reset curves Functional analysis of variance of registered reset curves Results 4 Future directions 5 References 6 C. Acal chracal@ugr.es 2 / 36
Index Introduction 1 Device description and measurement 2 RRAMs operation Experimental Data Used devices and purpose Functional Data Analysis 3 Functional modelling of reset curves Registration of reset curves in the interval [0,1] Functional reconstruction of reset curves Functional analysis of variance of registered reset curves Results 4 Future directions 5 References 6 C. Acal chracal@ugr.es 3 / 36
Introduction Motivation C. Acal chracal@ugr.es 4 / 36
Introduction Why this success? Decrease in the size of the cells that make up these memories C. Acal chracal@ugr.es 5 / 36
Introduction Why this success? Decrease in the size of the cells that make up these memories Problems This reduction can not be undefined and the possibilities of using them in the future are very small C. Acal chracal@ugr.es 5 / 36
Introduction Solution: New dispositive One of the strong candidates for future nonvolatile applications are RRAMs C. Acal chracal@ugr.es 6 / 36
Introduction Solution: New dispositive One of the strong candidates for future nonvolatile applications are RRAMs Advantages of RRAMs High-speed reading and writing Low consumption Long endurance They can be reduced CMOS technology compatibility A very simple physical structure C. Acal chracal@ugr.es 6 / 36
Introduction Solution: New dispositive One of the strong candidates for future nonvolatile applications are RRAMs Advantages of RRAMs High-speed reading and writing Low consumption Long endurance They can be reduced CMOS technology compatibility A very simple physical structure Previous steps We must study the statistics behind RRAM variability C. Acal chracal@ugr.es 6 / 36
Index Introduction 1 Device description and measurement 2 RRAMs operation Experimental Data Used devices and purpose Functional Data Analysis 3 Functional modelling of reset curves Registration of reset curves in the interval [0,1] Functional reconstruction of reset curves Functional analysis of variance of registered reset curves Results 4 Future directions 5 References 6 C. Acal chracal@ugr.es 7 / 36
Index Introduction 1 Device description and measurement 2 RRAMs operation Experimental Data Used devices and purpose Functional Data Analysis 3 Functional modelling of reset curves Registration of reset curves in the interval [0,1] Functional reconstruction of reset curves Functional analysis of variance of registered reset curves Results 4 Future directions 5 References 6 C. Acal chracal@ugr.es 8 / 36
Device description and measurement RRAMs operation RRAMs operation is based on the stochastic nature of resistive switching processes The device resistance changes from a High Resistance State (HRS) to a Low Resistance State (LRS) The result is a sample of current-voltage curves corresponding to the reset-set cycles The variability is translated to different voltages and currents related to set and reset processes for each cycle C. Acal chracal@ugr.es 9 / 36
Index Introduction 1 Device description and measurement 2 RRAMs operation Experimental Data Used devices and purpose Functional Data Analysis 3 Functional modelling of reset curves Registration of reset curves in the interval [0,1] Functional reconstruction of reset curves Functional analysis of variance of registered reset curves Results 4 Future directions 5 References 6 C. Acal chracal@ugr.es 10 / 36
Device description and measurement Experimental Data Variability in cycle to cycle change in the I-V curves The current/voltage curves change from cycle to cycle because the process of filament formation is random The reset points are determined by the sudden drop of the current (rupture of the conductive filament) The set points are characterized by the creation of the conductive filament C. Acal chracal@ugr.es 11 / 36
Index Introduction 1 Device description and measurement 2 RRAMs operation Experimental Data Used devices and purpose Functional Data Analysis 3 Functional modelling of reset curves Registration of reset curves in the interval [0,1] Functional reconstruction of reset curves Functional analysis of variance of registered reset curves Results 4 Future directions 5 References 6 C. Acal chracal@ugr.es 12 / 36
Device description and measurement Used devices and purpose Type of devices In the study, we have information about Copper of 20 nanometre (233 reset curves) Nickel of 10 nanometre (1742 reset curves) Nickel of 20 nanometre (2782 reset cuves) Purpose Detecting if there are significant physical differences between RRAM device technologies considering different materials and thicknesses Solution One-way analysis of variance for functional data (FANOVA) C. Acal chracal@ugr.es 13 / 36
Index Introduction 1 Device description and measurement 2 RRAMs operation Experimental Data Used devices and purpose Functional Data Analysis 3 Functional modelling of reset curves Registration of reset curves in the interval [0,1] Functional reconstruction of reset curves Functional analysis of variance of registered reset curves Results 4 Future directions 5 References 6 C. Acal chracal@ugr.es 14 / 36
Index Introduction 1 Device description and measurement 2 RRAMs operation Experimental Data Used devices and purpose Functional Data Analysis 3 Functional modelling of reset curves Registration of reset curves in the interval [0,1] Functional reconstruction of reset curves Functional analysis of variance of registered reset curves Results 4 Future directions 5 References 6 C. Acal chracal@ugr.es 15 / 36
Functional Data Analysis Functional modelling of reset curves Aim of researching Using advanced mathematical techniques (FDA) to model the stochastic nature of the RRAMs devices Current/Voltage curves The intensity of current (amps) is a function of the supplied voltage (volts) C. Acal chracal@ugr.es 16 / 36
Functional Data Analysis Functional modelling of reset curves FDA: set of statistical methods for the analysis of samples of curves or more general functions Problem Curves are not defined on the same domain 1 We have discrete observations of each reset curve at a finite set of 2 current values until the reset point Solution Synchronization of curves in the same interval 1 P-spline smoothing from discrete observations 2 FANOVA 3 C. Acal chracal@ugr.es 17 / 36
Index Introduction 1 Device description and measurement 2 RRAMs operation Experimental Data Used devices and purpose Functional Data Analysis 3 Functional modelling of reset curves Registration of reset curves in the interval [0,1] Functional reconstruction of reset curves Functional analysis of variance of registered reset curves Results 4 Future directions 5 References 6 C. Acal chracal@ugr.es 18 / 36
Functional Data Analysis Registration of reset curves in the interval [0,1] Data For each device (h=1,...,m) Sample of n h reset curves { I i ( v ) : i = 1 , ..., n h ; v ∈ [0 , V i − reset ] } , where V i − reset is the voltage to reset and h denotes the h -th device Discrete observations of each reset curve I i ( v ) so that each curve I i ( v ) is observed at k i = V i − reset ∗ 10 3 discrete equally spaced sampling points v j = j ∗ 10 − 3 ( j = 1 , ..., k i ) Curve registration: transforming the domain [0 , V i − reset ] of each reset curve in the interval [0,1] by the function v / V i − reset Sample of synchronized curves I ∗ i ( u ) = I i ( u ∗ V i − reset ) u ∈ [0 , 1] Each curve has a new set of arguments in [0,1] v j = j u ij = ( j = 1 , ..., k i ) V i − reset k i C. Acal chracal@ugr.es 19 / 36
Index Introduction 1 Device description and measurement 2 RRAMs operation Experimental Data Used devices and purpose Functional Data Analysis 3 Functional modelling of reset curves Registration of reset curves in the interval [0,1] Functional reconstruction of reset curves Functional analysis of variance of registered reset curves Results 4 Future directions 5 References 6 C. Acal chracal@ugr.es 20 / 36
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