the effect of electrode size on memristor properties: an experimental and theoretical study

Ella Gale, Ben de Lacy Costello and Andrew

Adamatzky

The Effect of Electrode Size on Memristor Properties: An

Experimental and Theoretical Study

A. Which Model of Memristance Works Best

B. What Effect Electrode Size has on Memristor Properties

We Want To Know…

THEORIES OF MEMRISTANCE

M = memristanceq = chargeφ = magnetic flux

CHUA’S PHENOMENALOGICAL

DEFINITION

1. Strukov et al’s Phenomenalogical Model

2. Georgiou et al’s Bernoulli Equations

3. Mem-Con Model

There Are Three Theoretical Memristor Models

1. Strukov et al2. Georgiou et al3. Gale,

1. Phenomenological Model

𝑀 (𝑞 (𝑡 ) )=𝑅off−𝜇𝑣𝐷2 𝑅off 𝑅on𝑞 (𝑡)

Strukov et al, The Missing Memristor Found, Nature, 2008

= ionic mobility of the O+ vacancies

Roff = resistance of TiO2

Ron = resistance of TiO(2-x)This is a 1-D model

• Rewrote Strukov et al’s model as Bernoulli Equations

• Gained Some Analytical Solutions• Predicts the Size of the Hysteresis, ,

in Memristor I-V curves

2. GEORGIOU ET AL’S MODEL

= ‘Dimensionless Lumped Parameter’Contains: • ‘all’ physical dimensions of

device• all parameters of experiment

is related to is related to

The Model Predictions

2. GEORGIOU ET AL’S MODEL

~𝛽=2 𝛽=2𝑉𝑚𝑎𝑥𝜔0𝑅0

2 𝜇𝑣(𝑅𝑜𝑛𝐷 )2

(𝑅𝑜𝑓𝑓𝑅𝑜𝑛−1)

This is a 1-D model

• Universal constants:

• , Experimental constants: product of surface area () and electric field (),

• , Material variable, =, where

3. Memristance, as Derived from Ion Flow

Gale, The Missing Magnetic Flux in the HP Memristor Found, 2011

This is a 3-D model

𝑀 𝑒=𝐶𝑀 ∙𝑀+𝐶2 𝑅𝐶𝑜𝑛=(𝐷−𝑤 (𝑡 ) ) 𝜌𝑜𝑓𝑓

𝐸𝐹

Memory Function Conservation Function

MEM-CON MODEL

𝑅 (𝑡 )=𝑀𝑒+𝑅𝐶𝑜𝑛

Goal: To Investigate Which Theoretical Model Works BestMethod:

A. Spatial Dimension Effects (Strukov and Mem-Con)B. Test Hysteresis Predicitons (Georgiou)

OUR PREMISE

• Strukov et al’s suggests no effect of size of E or F• Georgiou et al suggest no effect of E or F• Mem-Con model suggests that changing E or F will affect

memristance

• Test whether there is an effect of altering E or F

SIZE PREDICTIONS

Our Memristors

• Crossed Aluminium electrodes

• Thin-film (40nm) TiO2 sol-gel layer

• E = 4mm• F = 1, 2, 3, 4 or

5mm1. Gergel-Hackett et al, A Flexible Solution Processed Memristor, IEEE Elec. Dev. Lett., 20092. Gale et al, Aluminium Electrodes Effect the Operation of Titanium Dioxide Sol-Gel Memristors, Submitted 2012

Pictures

Curved (BPS-like) Memristors

Triangular (UPS-like) Memristors

Two Different Types of Memristor Behaviour Seen in Our Lab

The Effect of Varying Electrode Size

TEST 1

CURVED SWITCHING MEMRISTORS

As , ,

Fit Memory Function to as a function of

Fit Conservation Function to as a function of F

Memory function Describes ’s variation with F

Only 1 fitting parameter needed: (, ~ )

CONSERVATION FUNCTION DESCRIBES ’S VARIATION WITH F

One Fitting Parameter, , = Ωm-1 (Bulk value: 1012 Ωm-1)

• Measured and vary with electrode size• This relationship is well described by the Mem-Con

theory• Hysteresis is effected by Electrode Size

• The Mem-Con Theory Correctly Predicts that Memristance Should be a Function of the Three Spatial

Dimensions• The Strukov Theory Incorrectly Asserts that it is Only a

Function of 1 Spatial Dimenion

SO,

Is the Hysteresis Related to the ‘dimensionless lumped parameter’, ?

TEST 2

THE EXAMPLE GIVEN IN GEORGIOU ET AL’S PAPER

Ref…

Simulated ResultVoltage Source Waveforms:• Green:

Bipolar Piece-Wise Linear (analytically calculable)

• Red: Sinusoidal (not analytically calculable)

• Blue: Triangular (analytically calculable)

MEASURED HYSTERESIS VERSUS EXPERIMENTAL VALUES OF

DOES GEORGIOU ET AL’S PREDICTED RELATE TO MEASURED ?

HYSTERESIS SIZE DEPENDS ON F AND RON

• Georgiou et al’s Bernoulli Equation Formulation does not work at predicting hysteresis*

• Electrode Size can be changed to control hysteresis size*• The Mem-Con Model can be used to predict which electrode sizes will give a certain max or min resistance

value (at the same omega)*• All three spatial dimensions of the memristor are

important in describing memristance• The Mem-Con Model is a good model for real world

memrstors

* For Curved Type Devices (see next talk for an explanation)

SUMMARY

Ella Gale, Ben de Lacy Costello and Andrew

Adamatzky

FILAMENTARY EXTENSION OF THE MEM-CON THEORY OF

MEMRISTANCE AND ITS APPLICATION TO TITANIUM DIOXIDE

SOL-GEL MEMRISTORS

Pictures




Memristor Structure and Function

SHAPE OF THE FILAMENT

Extend the Mem-Con Model to Describe Filamentary (Triangular) Memristors

We Want To…

THE MEM-CON THEORY

φ

q

V

I

Definition of the Memristor

Resistor

Capacitor

Inductor Memristor

What the Flux?

𝑑𝜑=𝑀 (𝑞 (𝑡 ) )𝑑𝑞𝑀 (𝑞 (𝑡 ) )=𝑅𝑜𝑓𝑓−𝜇𝑣𝐷2 𝑅𝑜𝑓𝑓 𝑅𝑜𝑛𝑞(𝑡)

But, where is the magnetic flux?

𝑉=𝑀 (𝑡 ) 𝐼

Chua, 1971Strukov et al, 2008

The Mem-Con model is based on calculating the MAGNETIC FLUX of the IONS for several reasons:

• The IONS are the memory property, i.e. they hold the state of the memristor

• The IONS move slower than the electrons and it is this that causes both the lag (hysteresis) and frequency response

• The ION mobility, , is the physical quantity that controls the dynamics of the system

Therefore, using magnetostatics to calculate the relationships between the ionic magnetic flux and charge we will arrive at a formula for memristance that satisfies Chua’s definition

CALCULATING THE CHUA MEMRISTANCE

Mem-Con Theory

𝑞 ↔ 𝑀(𝑞) ↔ 𝜑 ↑ 𝑉 ↔ 𝑅𝑡𝑜𝑡(𝑡) ↔ 𝐼

Ionic Electronic

Gale, The Missing Magnetic Flux in the HP Memristor Found, Submitted, 2011

EXTENDING THE MEM-CON THEORY TO FILAMENTS

SHAPE OF THE FILAMENT

EQUIVALENT CIRCUIT DIAGRAM TO THE DEVICE

CHEMISTRY

𝑅𝑇𝑜𝑡(𝑡 )=1

1(𝑅𝑢+𝑀𝑒 (𝑡 )+𝑅𝑜𝑓𝑓 (𝑡 ) )

+2𝐻 (𝑤−𝐷 ) 1𝑅𝑓𝑖𝑙

𝑀𝑒𝑅𝑂𝑓𝑓

𝑅𝑢

e

• Memristance based on • Due to the shape, varies with

M: TIME-DEPEDENDANT EXPRESSION FOR THE VOLUMES


𝑅𝑢

Vacancy Magnetic Field

G can be solved by

where we useand



𝑅𝑢

Cuboid of

is the surface normal for area infinitesimal

WbFor Strukov’s device:

b [1]

As [2]

,and

MEMORY FUNCTION

1. Gale, The Missing Magnetic Flux in the HP Memristor Found, Submitted, 20112. Chua, Memristor: The Missing !!!

Not as easy as it looks.RESISTANCE OF A CONICAL

RESISTOR

𝑅 𝑓𝑖𝑙=𝑟1−𝐷 𝑓 +1

FILAMENT RESISTANCE


𝑅𝑢

Ref?

EQUIVALENT CIRCUIT DIAGRAM TO THE DEVICE

CHEMISTRY

𝑅𝑇𝑜𝑡(𝑡 )=1

1(𝑅𝑢+𝑀𝑒 (𝑡 )+𝑅𝑜𝑓𝑓 (𝑡 ) )

+2𝐻 (𝑤−𝐷 ) 1𝑅𝑓𝑖𝑙


𝑅𝑢

e

Experiment Theoretical Model

COMPARISON TO EXPERIMENT

• Memristance is a phenomenon associated with ionic current flow

• Therefore calculate the magnetic flux of the IONS

Vacancy Volume Current , L = eLectric field


Vacancy Magnetic Flux

Starting From The Ions…


Vacancy Volume Current

,

L = eLectric field

Calculate the Magnetic B field Associated with the

ions


𝑅𝑢

• Filamentary addition to the Mem-Con model gives

good qualitative agreement to experiment

Work out the quantitative values

Re-do derrivation allowing a back-ground bulk

memristance

Conclusions Further Work

CONCLUSIONS & FURTHER WORK

• Ben de Lacy Costello

• Andrew Adamatzky• David Howard• Larry Bull

With Thanks to

• Steve Kitson (HP UK)

• David Pearson (HP UK)

• Bristol Robotics Laboratory

• A larger study to test Georgiou et al’s model has been undertaken

• Repetition of size experiments with a different memristor at a different lab

FURTHER WORK

Influx of Ionic I

Voltage Spike

Axon:Transmission along neuron

Synapse:Transmission between

neurons

How does a Neuron Compute?

Memristive Systems to Describe Nerve Axon

Membranes

Synapse Long-Term Potentiation

• Definition based on behaviour

• UPS – Voltage polarity irrelevant

• BPS –Voltage polarity relevant

• Pinched hysteresis loop in I-V space

• Different behaviour based on forming process, complience current

• Satisfy Chua’s definition:

• Pinched hysteresis loop in I-V space

• --

ReRAM Memristor

A HUGE PROBLEM OF TERMINOLOGY

The Memristor as a Synapse

Before learning Before learning

During learningAfter learning

After learning

• Process by which synapses are potentiated

• Related to Hebb’s Rule• Possibly a cause of memory and learning• Relative timing of spike inputs to a

synapse important

Spike-Time Dependent Plasticity, STDP

Bi and Poo, Synaptic Modifications in Cultured Hippocampal Neurons: Dependence on Spike Timing, Synaptic Strength and Postsynaptic Cell Type, J. Neurosci., 1998

Memristor Structure and Function

Charge-Controlled Memristor

Flux-Controlled Memristor

Chua’s Definitions of Types of Memristors

L. Chua, Memristor – The Missing Circuit Element, IEEE Trans. Circuit Theory, 1971

φ

q

V

I

Definition of the Memristor

Resistor

Capacitor

Inductor Memristor

What the Flux?

𝑑𝜑=𝑀 (𝑞 (𝑡 ) )𝑑𝑞𝑀 (𝑞 (𝑡 ) )=𝑅𝑜𝑓𝑓−𝜇𝑣𝐷2 𝑅𝑜𝑓𝑓 𝑅𝑜𝑛𝑞(𝑡)

But, where is the magnetic flux?

𝑉=𝑀 (𝑡 ) 𝐼

Chua, 1971Strukov et al, 2008

Mem-Con Theory

𝑞 ↔ 𝑀(𝑞) ↔ 𝜑 ↑ 𝑉 ↔ 𝑅𝑡𝑜𝑡(𝑡) ↔ 𝐼

Ionic Electronic


Pictures




Memristor I-V Behaviour

To make a memristor brain

& thus a machine intelligence

Our Intent:

Connecting Memristors with Spiking Neurons to Implement STDP

1. Zamarreno-Ramos et al, On Spike Time Dependent Plasticity, Memristive Devices and Building a Self-Learning Visual Cortex, Frontiers in Neuroscience, 20110. Linares-Barranco and Serrano-Gotarredona, Memristance can explain Spike-Time-Dependent-Plasticity in Neural Synapses, Nature Preceedings, 2009

Simulation Results

Memristors Spike

Naturally!

But,

Current Spikes Seen in I-t Plots

Voltage Square Wave Current Spike Response

Spikes are Reproducible

Voltage Ramp Current Response

Spikes are Repeatable

Neuron

Memristor

Memristor Behaviour Looks Similar to Neurons

Bal and McCormick, Synchronized Oscilliations in the Inferior Olive are controlled by the Hyperpolarisation-Activated Cation Current Ih, J. Neurophysiol, 77, 3145-3156, 1997

SPIKES SEEN IN THE LITERATURE

Pershin and Di Ventra, Spin Memristive Systems: Spin Memory Effects in Semi-conductor Spintronics, Phys. Rev. B, 2008

Spintronic Memristor Current Spikes

• Direction of Spikes is related to not V

• The switch to 0V has a associated current spike

• Spikes are repeatable• Spikes are reproducable• Spikes are seen in bipolar switching

memristors/ReRAM• Spikes are not seen in unipolar

switching, UPS ReRAM type memristors

Properties of Spikes

Where do the Spikes Come From?

Does Current Theory Predict Their Existence?

q φI V

q φV I

Neurons Memristors

Mem-Con Model Applied to Memristor Spikes

• Dynamics related to min. response time, τ, related to speed of ion diffusion across membrane

• Memory property = ???• Neuron operated in a

current-controlled way

• Dynamics related to τ, which is related to

• Memory property = qv

• Memristor operated in voltage controlled way

Neuron Voltage Spikes Memristor Current Spikes

In Chua’s Model

• More complex system than a single memristor

• Short-term memory associated with membrane potential

• Long term memory associated with the number of synaptic buds

What is the Memory Property of Neurons?

Sol-Gel Memristor Negative V

Sol-Gel Memristor Positive V

Memristor Models Fit the Data

Memristor Model Fits the PEO-PANI Memristor

Al-TiO2-Al Sol-Gel Memristor

Time & Frequency Dependence of Hysteresis for Al-TiO2-Al

Au-TiO2-Au WORMS Memory

I-t Response to Stepped Voltage

Time Dependent I-V

Au-TiO2-Au WORMS Memory

Voltage Ramp Current Response

Al-TiO2-Al Current Response to Voltage Ramp

Neurology:• Modelling Neurons with the Mem-Con

Theory to prove that they are Memristive• Investigate the Memory Property for

neurons

Unconventional Computing:• Further Investigation of memristor and

ReRAM properties• Attempt to build a neuromorphic control

system for a navigation robot

Further Work

• Neurons May Be Biological Memristors• Neurons Operate via Voltage Spikes• Memristors can Operative via Current

Spikes• Thus, Memristors are Good Candidates for

Neuromorphic Computation• A Memristor-based Neuromorphic

Computer will be Voltage Controlled and transmit data via Current Spikes

Summary

the effect of electrode size on memristor properties: an experimental and theoretical study

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