One-Way ANOVA modelling for RRAM reset curves alez 1 , Ana M. - - PowerPoint PPT Presentation

One-Way ANOVA modelling for RRAM reset curves

Christian J. Acal Gonz´ alez1, Ana M. Aguilera1,

• M. Carmen Aguilera-Morillo2, Francisco Jim´

enez-Molinos3, Juan B. Rold´ an3

1Departamento de Estad´

ıstica e I.O. Universidad de Granada

2Departamento de Estad´

ıstica. Universidad Carlos III de Madrid

3Departamento de Electr´

• nica 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

1

Introduction

2

Device description and measurement RRAMs operation Experimental Data Used devices and purpose

3

Functional Data Analysis 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

4

Results

5

Future directions

6

References

• C. Acal chracal@ugr.es

2 / 36

Index

1

Introduction

2

Device description and measurement RRAMs operation Experimental Data Used devices and purpose

3

Functional Data Analysis 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

4

Results

5

Future directions

6

References

• 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

1

Introduction

2

Device description and measurement RRAMs operation Experimental Data Used devices and purpose

3

Functional Data Analysis 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

4

Results

5

Future directions

6

References

• C. Acal chracal@ugr.es

7 / 36

Index

1

Introduction

2

Device description and measurement RRAMs operation Experimental Data Used devices and purpose

3

Functional Data Analysis 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

4

Results

5

Future directions

6

References

• 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

1

Introduction

2

Device description and measurement RRAMs operation Experimental Data Used devices and purpose

3

Functional Data Analysis 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

4

Results

5

Future directions

6

References

• 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

• f 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

1

Introduction

2

Device description and measurement RRAMs operation Experimental Data Used devices and purpose

3

Functional Data Analysis 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

4

Results

5

Future directions

6

References

• 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

1

Introduction

2

Device description and measurement RRAMs operation Experimental Data Used devices and purpose

3

Functional Data Analysis 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

4

Results

5

Future directions

6

References

• C. Acal chracal@ugr.es

14 / 36

Index

1

Introduction

2

Device description and measurement RRAMs operation Experimental Data Used devices and purpose

3

Functional Data Analysis 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

4

Results

5

Future directions

6

References

• 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

• r more general functions

Problem

1

Curves are not defined on the same domain

2

We have discrete observations of each reset curve at a finite set of current values until the reset point

Solution

1

Synchronization of curves in the same interval

2

P-spline smoothing from discrete observations

3

FANOVA

• C. Acal chracal@ugr.es

17 / 36

Index

1

Introduction

2

Device description and measurement RRAMs operation Experimental Data Used devices and purpose

3

Functional Data Analysis 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

4

Results

5

Future directions

6

References

• 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 nh reset curves {Ii(v) : i = 1, ..., nh; v ∈ [0, Vi−reset]}, where Vi−reset is the voltage to reset and h denotes the h-th device Discrete observations of each reset curve Ii(v) so that each curve Ii(v) is observed at ki = Vi−reset ∗ 103 discrete equally spaced sampling points vj = j ∗ 10−3 (j = 1, ..., ki)

Curve registration: transforming the domain [0, Vi−reset] of each reset curve in the interval [0,1] by the function v/Vi−reset

Sample of synchronized curves I ∗

i (u) = Ii(u ∗ Vi−reset) u ∈ [0, 1]

Each curve has a new set of arguments in [0,1] uij = vj Vi−reset = j ki (j = 1, ..., ki)

• C. Acal chracal@ugr.es

19 / 36

Index

1

Introduction

2

Device description and measurement RRAMs operation Experimental Data Used devices and purpose

3

Functional Data Analysis 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

4

Results

5

Future directions

6

References

• C. Acal chracal@ugr.es

20 / 36

Functional Data Analysis

Functional reconstruction of reset curve

Smooth reset curves observed with error

I ∗

ij = Ii(uij) + ǫij

j = 1, ..., ki; i = 1, ..., nh

Basis expansion of reset curves

I ∗

i (u) = p

• j=1

aijφj(u), i = 1, ..., nh

B-spline smoothing

Regression splines (non-penalized least squares) Smoothing splines (penalized least squares: continuos penalty) Penalized splines (penalized least squares: discrete penalty)

• C. Acal chracal@ugr.es

21 / 36

Functional Data Analysis

Functional reconstruction of reset curve

Advantages of P-splines

P-splines provide the lowest approximation errors (more accurate approximation of sample curves) Less numerical complexity and computational cost The choice and position of knots is not determinant so that it is sufficient to choose a relatively large number of equally spaced basis knots (Ruppert et al., 2003)

Penalized splines

DPMSEd (ai|I ∗

i ) = (I ∗ i − Φiai)′ (I ∗ i − Φiai) + λa′ iPdai

Pd =

• △d′ △d (d-order diference operator: △d)

ˆ ai = (Φ′

iΦi + λPd)−1Φ′ iI ∗ i

• C. Acal chracal@ugr.es

22 / 36

Index

1

Introduction

2

Device description and measurement RRAMs operation Experimental Data Used devices and purpose

3

Functional Data Analysis 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

4

Results

5

Future directions

6

References

• C. Acal chracal@ugr.es

23 / 36

Functional Data Analysis

Functional analysis of variance of registered reset curvesd

Sample of reset curves independent and identically distributed

{I ∗

hi(u) : i = 1, ..., nh; h = 1, ..., m; u ∈ [0, 1]}

Functional analysis of variance

Aim: H0 : µ1(t) = µ2(t) = ... = µm(t) H1 : some different Linear model for curve I ∗

hi(u):

I ∗

hi(u) = µ(u) + αh(u) + ǫhi(u), h = 1, ..., m, i = 1, ..., nm

µ(u) overall mean function of the m samples αh(u) = µh(u) − µ(u) h-th main-effect function, h = 1, ..., m ǫhi(u) subject-effect functions (error)

• C. Acal chracal@ugr.es

24 / 36

Functional Data Analysis

Functional analysis of variance of registered reset curves

Constraint: m

h=1 nhαh(u) = 0

⇓ ˆ µ(u) = I

∗(u)

ˆ αh(u) = I

∗ h(u) − I ∗(u)

ˆ ǫhi(u) = I ∗

hi(u) − I ∗ h(u)

• C. Acal chracal@ugr.es

25 / 36

Functional Data Analysis

Functional analysis of variance of registered reset curves

Constraint: m

h=1 nhαh(u) = 0

⇓ ˆ µ(u) = I

∗(u)

ˆ αh(u) = I

∗ h(u) − I ∗(u)

ˆ ǫhi(u) = I ∗

hi(u) − I ∗ h(u)

I

∗(u) and I ∗ h(u) are respectively the sample grand mean function and the

sample group mean functions: I

∗(u) = 1

n

m

• h=1

nh

• i=1

I ∗

hi(u)

I

∗ h(u) = 1

nh

nh

• i=1

I ∗

hi(u), h = 1, ..., m

• C. Acal chracal@ugr.es

25 / 36

Functional Data Analysis

Functional analysis of variance of registered reset curves

Considering basis expansion of curves I ∗

hi(u) = a′ hiΦ(u) with

ahi = (ahi1, ..., ahip)′ and Φ(u) = (φ1(u), ..., φp(u))′, I

∗(u) = 1

n

m

• h=1

nh

• i=1

I ∗

hi(u) = 1

n

m

• h=1

nh

• i=1

a′

hiΦ(u)

I

∗ h(u) = 1

nh

nh

• i=1

I ∗

hi(u) = 1

nh

nh

• i=1

a′

hiΦ(u), h = 1, ..., m

• C. Acal chracal@ugr.es

26 / 36

Functional Data Analysis

Functional analysis of variance of registered reset curves

Considering basis expansion of curves I ∗

hi(u) = a′ hiΦ(u) with

ahi = (ahi1, ..., ahip)′ and Φ(u) = (φ1(u), ..., φp(u))′, I

∗(u) = 1

n

m

• h=1

nh

• i=1

I ∗

hi(u) = 1

n

m

• h=1

nh

• i=1

a′

hiΦ(u)

I

∗ h(u) = 1

nh

nh

• i=1

I ∗

hi(u) = 1

nh

nh

• i=1

a′

hiΦ(u), h = 1, ..., m

⇓ ˆ µ(u) = a′Φ(u) ˆ αh(u) = a′

hΦ(u) − a′Φ(u)

ˆ ǫhi(u) = a′

hiΦ(u) − a′ hΦ(u)

⇓

• C. Acal chracal@ugr.es

26 / 36

Functional Data Analysis

Functional analysis of variance of registered reset curves

Considering basis expansion of curves I ∗

hi(u) = a′ hiΦ(u) with

ahi = (ahi1, ..., ahip)′ and Φ(u) = (φ1(u), ..., φp(u))′, I

∗(u) = 1

n

m

• h=1

nh

• i=1

I ∗

hi(u) = 1

n

m

• h=1

nh

• i=1

a′

hiΦ(u)

I

∗ h(u) = 1

nh

nh

• i=1

I ∗

hi(u) = 1

nh

nh

• i=1

a′

hiΦ(u), h = 1, ..., m

⇓ ˆ µ(u) = a′Φ(u) ˆ αh(u) = a′

hΦ(u) − a′Φ(u)

ˆ ǫhi(u) = a′

hiΦ(u) − a′ hΦ(u)

⇓

Result

FANOVA is equivalent to MANOVA of matrix A(m

h=1 nh×p)

• C. Acal chracal@ugr.es

26 / 36

Index

1

Introduction

2

Device description and measurement RRAMs operation Experimental Data Used devices and purpose

3

Functional Data Analysis 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

4

Results

5

Future directions

6

References

• C. Acal chracal@ugr.es

27 / 36

Results

Functional reconstruction of reset curves

For each device, penalized splines Size of the basis equal to 20 functions with 17 equally spaced knots λ = 0.5 matrix of basis coefficients: A4757×20

• C. Acal chracal@ugr.es

28 / 36

Results

Problem: Great dimension

• C. Acal chracal@ugr.es

29 / 36

Results

Problem: Great dimension

Solution: Functional Principal Component Analysis

• 1. j-th principal component score (in general):

ξij = 1 (I ∗

i (u) − I ∗(u))fj(u)du, i = 1, ..., n

• 2. Multivariate PCA of AΨ1/2 matrix (Oca˜

na et al., 2007) A = (aij)i=1,...,n;j=1,...,K matrix of basis coefficients Ψ =

• < φj, φk >L2[0,1]
• j,k=1,...,K matrix of inner products
• 3. Principal components reconstruction

I ∗q

i

(u) = ¯ I ∗(u) +

q

• i=1

ξijfj (u)

• C. Acal chracal@ugr.es

29 / 36

Results

% of variance explained by the first four principal components

PC Percentage of variance 1 99.6393 2 0.284 3 0.0464 4 0.0202 Principal component descomposition of the registered reset curves can be trucated in the first term

• C. Acal chracal@ugr.es

30 / 36

Results

% of variance explained by the first four principal components

PC Percentage of variance 1 99.6393 2 0.284 3 0.0464 4 0.0202 Principal component descomposition of the registered reset curves can be trucated in the first term One-Way ANOVA for the first principal component

• C. Acal chracal@ugr.es

30 / 36

Results

Assumptions of ANOVA

Assumption of independence Assumption of homogeneity of variance Assumption of normality

• C. Acal chracal@ugr.es

31 / 36

Results

Assumptions of ANOVA

Assumption of independence Assumption of homogeneity of variance Assumption of normality

• 1. QQplot
• 2. Levene’s Test for Homogeneity of Variance (p value < 2.2e-16)
• C. Acal chracal@ugr.es

31 / 36

Results

Non-parametric tests

1 Kruskal-Wallis Rank Sum Test (in R ’kruskal.test’)

Pairwise Wilcoxon Rank Sum Tests (in R ’pairwise.wilcox.test’)

2 Mood’s median test (in R ’mood.medtest’; library RVAideMemoire)

Conclusion: There are significant physical differences between considering different materials and thicknesses

• C. Acal chracal@ugr.es

32 / 36

Index

1

Introduction

2

Device description and measurement RRAMs operation Experimental Data Used devices and purpose

3

Functional Data Analysis 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

4

Results

5

Future directions

6

References

• C. Acal chracal@ugr.es

33 / 36

Future directions

Interpretation of results in terms of simulated data by controlling the dimensions of the conductor filament Joint modelling of reset-set cicles by using mixed ARIMA-FPCA models Using rotation varimax for improving the interpretation of the principal components ....

Acknowledgments

The authors thank the support of the Spanish Ministry of Economy and Competitiveness, Spain under project MTM2017-88708-P and also, supported by the FEDER program, Spain.

• C. Acal chracal@ugr.es

34 / 36

Index

1

Introduction

2

Device description and measurement RRAMs operation Experimental Data Used devices and purpose

3

Functional Data Analysis 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

4

Results

5

Future directions

6

References

• C. Acal chracal@ugr.es

35 / 36

References

• M. C. Aguilera-Morillo, A. M. Aguilera, F. Jimenez-Molinos, J. B.

Rold´ an, ’Stochastic modelling of Random Access Memories reset transitions’, Mathematics and Computers in Simulation, in press. https://doi.org/10.1016/j.matcom.2018.11.016

• T. G´
• recki and L. Smaga, ’A comparison of tests for the one-way

ANOVA problem for functional data’, Springer, 30 (4), 987-1010, 2015.

• M. Villena, M. Gonz´

alez, J. Rold´ en, F. Campabadal, F. Jim´ enez-Molinos, F. G´

• mez-Campos, and J. Su˜

n´ e, ’An in-depth study

• f thermal effects in reset transitions in hfo2 based rrams’, Solid-State

Electronics, vol. 111, pp. 47–51, 2015. J.T. Zhang, (2013) ’Analysis of Variances for Functional Data’, London: Chapman and Hall.

• C. Acal chracal@ugr.es

36 / 36