N oise on R esistive S witching : a F okker -P lanck A pproach G.A. - PowerPoint PPT Presentation

N oise on R esistive S witching : a F okker -P lanck A pproach G.A. Patterson 1 , D.F. Grosz 2 , 3 , and P .I. Fierens 1 , 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 - 7 th International Conference on Unsolved Problems on Noise Barcelona, Spain - July 16, 2015

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 ◮ ...

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 ◮ ...

Storage and transmission assisted by noise + noise + 16 bits SR w clk ... dx + in out 0.5 @ 1 T B 0.4 @ 15 T B @ 30 T B Error probability 0.3 0.2 0.1 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Noise power [V 2 ] [Pessacg et al . CNSNS, 2015] [Patterson et al . Physica A, 2010]

Resistive Switching 70 R h 60 ◮ Change of resistance under the action 50 Resistance [Ω] of an external field 40 ◮ First reported in 1962 by Hickmott 30 ◮ Binary oxides, transition metal oxides, R l 20 organic materials, etc . 10 −1.00 −0.50 0.00 0.50 1.00 ◮ Potential application of RS in the area Current [A] 0.15 of non-volatile memories R l 0.10 Current [A] 0.05 TE R h Dielectric 0.00 −0.05 BE −0.10 −3 −2 −1 0 1 2 Voltage [V]

Motivation: Hysteretic device Output Output Input Input What is the role of noise in such a system?

Motivation: Hysteretic device Output Output Input Input What is the role of noise in such a system?

Numerical model 8 v ( t ) = R ( x ) i ( t ) 4 Current 0 dx dt = F ( x ) i ( t ) −4 R ( x ) = ( 1 − δ R x ) −8 −2 −1 0 1 2 Voltage 0 x 1 F(x) 1.0 0.8 Resistance 0.6 Ron x Roff (1-x) 0.4 0 1 0.2 −2 −1 0 1 2 Voltage [Strukov et al . Nature, 2008]

Noise in resistive switching Internal noise dx dt = F ( x ) i ( t ) + η ( t ) � η ( t ) η ( t ′ ) � = Γ δ ( t − t ′ )

Fokker-Planck equation Langevin: Stochastic differential equation √ dx = F ( x ) i ( t ) dt + Γ dw ◮ w : Wiener process ◮ F ( x ) i ( t ) : drift coefficient ◮ √ Γ : diffusion coefficient F-P: Partial differential equation ∂ 2 � F ( x ) i ( t ) p ( x , t ) � + Γ ∂ t p ( x , t ) = − ∂ ∂ ∂ x 2 p ( x , t ) ∂ x 2

Results: Influence of internal noise 10 0 2 10 0 3 10 −3 4 10 −2 10 −4 10 −6 PDF 1-x 10 −6 10 −9 Γ = 10 -2 10 −8 Γ = 10 -1 Γ = 0 10 −12 Γ = 10 1 Γ = 2·10 -16 10 −10 Γ = 2·10 -8 Γ = 2·10 0 10 −12 10 −15 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 Time Position +1 � 2 � v ( τ b ) F ( y ) � P s ( x ) ∝ exp R ( y ) dy Γ τ b x -1 As Γ increases the PDF broadens

Results: Influence of internal noise 10 0 2 10 0 3 10 −3 4 10 −2 10 −4 10 −6 PDF 1-x 10 −6 10 −9 Γ = 10 -2 10 −8 Γ = 10 -1 Γ = 0 10 −12 Γ = 10 1 Γ = 2·10 -16 10 −10 Γ = 2·10 -8 Γ = 2·10 0 10 −12 10 −15 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 Time Position +1 � 2 � v ( τ b ) F ( y ) � P s ( x ) ∝ exp R ( y ) dy Γ τ b x -1 As Γ increases the PDF broadens

Results: Influence of internal noise 1.0 10 0 Stationary solution 0.8 10 −2 Positive Negative 10 −4 0.6 x 1-x 10 −6 0.4 Δx 10 −8 0.2 Γ = 0 Γ = 2·10 -16 10 −10 Γ = 2·10 -8 0.0 Γ = 2·10 0 10 −12 0 1 2 3 4 10 −10 10 −8 10 −6 10 −4 10 −2 10 0 10 2 Time Γ +1 ∆ R ∝ ∆ x τ b -1 The stationary solution is not reached for every τ b

Results: Influence of internal noise 1.0 10 0 Stationary solution 0.8 10 −2 Positive Negative 10 −4 0.6 τ b = 1.0 Negative x 1-x 10 −6 0.4 Δx 10 −8 0.2 Γ = 0 Γ = 2·10 -16 10 −10 Γ = 2·10 -8 0.0 Γ = 2·10 0 10 −12 0 1 2 3 4 10 −10 10 −8 10 −6 10 −4 10 −2 10 0 10 2 Time Γ +1 ∆ R ∝ ∆ x τ b -1 The stationary solution is not reached for every τ b

Results: Influence of internal noise 10 0 10 0 10 0 10 −2 10 − 1 10 − 1 10 −4 10 − 2 10 − 2 1 - x max 1-x x min 10 −6 10 − 3 10 − 3 τ b = 1.0 τ b = 1.0 10 − 4 10 − 4 Stat. (F-P) Deterministic 10 −8 SDE Stat. (F-P) Γ = 0 10 − 5 10 − 5 SDE 10 −10 Γ = 2·10 -16 Γ = 2·10 -8 10 − 6 10 − 6 Γ = 2·10 0 10 − 10 10 − 8 10 − 6 10 − 4 10 − 2 10 0 10 2 10 − 10 10 − 8 10 − 6 10 − 4 10 − 2 10 0 10 2 10 −12 Γ Γ 0 1 2 3 4 Time ◮ Low noise amplitude → Deterministic evolution +1 ◮ High noise amplitude → Evolution constrained by τ b -1 noise

Results: EPIR 3.5 3.0 2.5 2.0 EPIR 1.5 τ b = 1.0 Fokker-Planck ◮ Internal noise enhances the EPIR ratio for 1.0 SDE a given initial condition and pulsewidth τ b = 2.0 0.5 Fokker-Planck ◮ Good agreement between SDE & the F-P 0.0 SDE approach −0.5 10 −10 10 −8 10 −6 10 −4 10 −2 10 0 10 2 10 4 Γ EPIR = R h − R l R l

Results: Influence of external noise 0.35 0.30 τ b = 1.0 SDE 0.25 0.20 EPIR 0.15 ◮ External noise only has the effect of degrading 0.10 the EPIR ratio 0.05 0.00 ◮ Same results with the Fokker-Planck approach... −0.05 (see UPON2015 extended abstract) 10 −10 10 −8 10 −6 10 −4 10 −2 10 0 10 2 10 4 Γ dt = F ( x ) dx R ( x ) ( v ( t ) + η ( t ))

But... experimental results 0.8 0.8 noise noise 2 0.6 added added 0.6 0.4 Current [A] Current 0.2 0.4 0.0 −0.2 0.2 −0.4 0.0 −0.6 0.000 0.001 1.998 1.999 2.000 2.001 2.002 Time [s] 28 Resistance [Ω] 24 20 External noise does enhance 16 the resistive contrast! 12 0 400 800 1200 1600 2000 # pulse [Patterson et al . PRE, 2013]

But... experimental results 0.8 0.8 noise noise 2 0.6 added added 0.6 0.4 Current [A] Current 0.2 0.4 0.0 −0.2 0.2 −0.4 0.0 −0.6 0.000 0.001 1.998 1.999 2.000 2.001 2.002 Time [s] 28 Resistance [Ω] 24 20 External noise does enhance 16 the resistive contrast! 12 0 400 800 1200 1600 2000 # pulse [Patterson et al . PRE, 2013]

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?

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