SLIDE 1 Noise on Resistive Switching: a Fokker-Planck Approach
G.A. Patterson1, D.F. Grosz2,3, and P .I. Fierens1,2 gpatters@itba.edu.ar
- 1. Instituto Tecnológico de Buenos Aires, Buenos Aires, Argentina
- 2. Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina
- 3. Instituto Balseiro, San Carlos de Bariloche, Argentina
UPON 2015 - 7th International Conference on Unsolved Problems on Noise
Barcelona, Spain - July 16, 2015
SLIDE 2
Motivations
◮ Higher circuit densities lead to smaller
signal-to-noise ratios
◮ There is a prominent role of noise in electronic
circuits
But noise... might not be harmful, after all
◮ Stochastic resonance ◮ Dithering ◮ Synchronization ◮ ...
SLIDE 3
Motivations
◮ Higher circuit densities lead to smaller
signal-to-noise ratios
◮ There is a prominent role of noise in electronic
circuits
But noise... might not be harmful, after all
◮ Stochastic resonance ◮ Dithering ◮ Synchronization ◮ ...
SLIDE 4 Storage and transmission assisted by noise
w
dx
+ +
Error probability
0.0 0.1 0.2 0.3 0.4 0.5
Noise power [V2]
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
@ 1 TB @ 15 TB @ 30 TB
[Patterson et al. Physica A, 2010]
... 16 bits SR
+
in clk noise
[Pessacg et al. CNSNS, 2015]
SLIDE 5 Resistive Switching
◮ Change of resistance under the action
◮ First reported in 1962 by Hickmott ◮ Binary oxides, transition metal oxides,
◮ Potential application of RS in the area
TE Dielectric BE Resistance [Ω] 10 20 30 40 50 60 70 Current [A] −1.00 −0.50 0.00 0.50 1.00
Rh Rl
Current [A] −0.10 −0.05 0.00 0.05 0.10 0.15 Voltage [V] −3 −2 −1 1 2
Rh Rl
SLIDE 6
Motivation: Hysteretic device
Output Input Output Input
What is the role of noise in such a system?
SLIDE 7
Motivation: Hysteretic device
Output Input Output Input
What is the role of noise in such a system?
SLIDE 8 Numerical model
v(t) = R(x)i(t) dx dt = F(x)i(t) R(x) = (1 − δR x)
F(x) 1
x 1 Ron x Roff (1-x)
Current −8 −4 4 8 Voltage −2 −1 1 2 Resistance 0.2 0.4 0.6 0.8 1.0 Voltage −2 −1 1 2
[Strukov et al. Nature, 2008]
SLIDE 9
Noise in resistive switching
Internal noise
dx dt = F(x)i(t) + η(t)
η(t)η(t′) = Γδ(t − t′)
SLIDE 10
Fokker-Planck equation
Langevin: Stochastic differential equation
dx = F(x) i(t) dt +
√ Γdw F-P: Partial differential equation ∂ ∂t p(x, t) = − ∂ ∂x F(x) i(t) p(x, t) + Γ
2
∂2 ∂x2 p(x, t)
◮ w: Wiener process ◮ F(x) i(t): drift coefficient ◮ √
Γ: diffusion coefficient
SLIDE 11 Results: Influence of internal noise
1-x 10−12 10−10 10−8 10−6 10−4 10−2 100 Time 1 2 3 4
Γ = 0 Γ = 2·10-16 Γ = 2·10-8 Γ = 2·100
τb +1
PDF 10−15 10−12 10−9 10−6 10−3 100 Position 0.0 0.2 0.4 0.6 0.8 1.0 2 3 4 Γ = 10-2 Γ = 10-1 Γ = 101
Ps(x) ∝ exp
2 Γ
v(τb) F(y) R(y)dy
- As Γ increases the PDF broadens
SLIDE 12 Results: Influence of internal noise
1-x 10−12 10−10 10−8 10−6 10−4 10−2 100 Time 1 2 3 4
Γ = 0 Γ = 2·10-16 Γ = 2·10-8 Γ = 2·100
τb +1
PDF 10−15 10−12 10−9 10−6 10−3 100 Position 0.0 0.2 0.4 0.6 0.8 1.0 2 3 4 Γ = 10-2 Γ = 10-1 Γ = 101
Ps(x) ∝ exp
2 Γ
v(τb) F(y) R(y)dy
- As Γ increases the PDF broadens
SLIDE 13 Results: Influence of internal noise
1-x 10−12 10−10 10−8 10−6 10−4 10−2 100 Time 1 2 3 4
Γ = 0 Γ = 2·10-16 Γ = 2·10-8 Γ = 2·100
τb +1
x 0.0 0.2 0.4 0.6 0.8 1.0 Γ 10−10 10−8 10−6 10−4 10−2 100 102 Stationary solution Positive Negative Δx
∆R ∝ ∆x
The stationary solution is not reached for every τb
SLIDE 14 Results: Influence of internal noise
1-x 10−12 10−10 10−8 10−6 10−4 10−2 100 Time 1 2 3 4
Γ = 0 Γ = 2·10-16 Γ = 2·10-8 Γ = 2·100
τb +1
x 0.0 0.2 0.4 0.6 0.8 1.0 Γ 10−10 10−8 10−6 10−4 10−2 100 102 Stationary solution Positive Negative τb = 1.0 Negative Δx
∆R ∝ ∆x
The stationary solution is not reached for every τb
SLIDE 15 Results: Influence of internal noise
1-x 10−12 10−10 10−8 10−6 10−4 10−2 100 Time 1 2 3 4
Γ = 0 Γ = 2·10-16 Γ = 2·10-8 Γ = 2·100
τb +1
1 - xmax 10−6 10−5 10−4 10−3 10−2 10−1 100 Γ 10−10 10−8 10−6 10−4 10−2 100 102 τb = 1.0
SDE xmin 10−6 10−5 10−4 10−3 10−2 10−1 100 Γ 10−10 10−8 10−6 10−4 10−2 100 102 τb = 1.0 Deterministic
SDE
◮ Low noise amplitude → Deterministic evolution ◮ High noise amplitude → Evolution constrained by
noise
SLIDE 16
Results: EPIR
EPIR −0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Γ 10−10 10−8 10−6 10−4 10−2 100 102 104 τb = 1.0 Fokker-Planck SDE τb = 2.0 Fokker-Planck SDE
EPIR = Rh − Rl
Rl
◮ Internal noise enhances the EPIR ratio for
a given initial condition and pulsewidth
◮ Good agreement between SDE & the F-P
approach
SLIDE 17 Results: Influence of external noise
EPIR −0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Γ 10−10 10−8 10−6 10−4 10−2 100 102 104 τb = 1.0 SDE
dx dt = F(x) R(x) (v(t) + η(t))
◮ External noise only has the effect of degrading
the EPIR ratio
◮ Same results with the Fokker-Planck approach...
(see UPON2015 extended abstract)
SLIDE 18 But... experimental results
Resistance [Ω] 12 16 20 24 28 # pulse 400 800 1200 1600 2000 Current [A] −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 noise added noise added
[Patterson et al. PRE, 2013]
Current 0.0 0.2 0.4 0.6 0.8 Time [s] 0.000 0.001 1.998 1.999 2.000 2.001 2.002 2
External noise does enhance the resistive contrast!
SLIDE 19 But... experimental results
Resistance [Ω] 12 16 20 24 28 # pulse 400 800 1200 1600 2000 Current [A] −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 noise added noise added
[Patterson et al. PRE, 2013]
Current 0.0 0.2 0.4 0.6 0.8 Time [s] 0.000 0.001 1.998 1.999 2.000 2.001 2.002 2
External noise does enhance the resistive contrast!
SLIDE 20
Conclusions & open questions
Conclusions
◮ Internal noise enhances the contrast between resistive states in a non-harmonic signal ◮ We introduced a Fokker-Planck approach to study the effect of internal noise ◮ We provide an alternative explanation by means of this approach ◮ We found that external noise has only the effect of degrading the resistive contrast
UPON question: What is the role of external noise in RS?
Does it
◮ enhance ion migration? ◮ promote conductive filaments creation?
SLIDE 21
Thank you for your kind attention!
