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

A.

Which Model of

Memristance Works

Best

B.

What Effect

Electrode Size has

on Memristor

Properties

M

= memristance

q

= charge

φ

= magnetic flux

CHUA’S

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. Phenomenological

Model

𝑀

(

(

)

)

𝐷2 𝑅off 𝑅on 𝑞(𝑡 )

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

= ionic mobility of the O+

vacancies

R_off = resistance of TiO₂

R_on = resistance of TiO_(2-x)

•

Rewrote Strukov et al’s model as

Bernoulli Equations

•

Gained Some Analytical Solutions

•

Predicts the Size of the Hysteresis, ,

in Memristor I-V curves

= ‘Dimensionless Lumped Parameter’

Contains:

• _{‘all’ physical dimensions of} device

• _{all parameters of experiment}

is related to

is related to

2. GEORGIOU ET AL’S MODEL

~

𝛽

=

2

𝛽

=

2

𝑉

𝜔

𝑅

𝜇

(

𝑅

𝐷

)

2

(

𝑅

𝑅

−

1

)

•

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

𝑀

𝑒

=

𝐶

𝑀

∙

𝑀

+

𝐶

2

𝑅

𝐶𝑜𝑛

=

(

𝐷

−

𝑤

(

𝑡

)

)

𝜌

𝑜𝑓𝑓

𝐸𝐹

Memory Function Conservation Function

MEM-CON MODEL

Goal: To Investigate Which Theoretical Model Works Best Method:

A. Spatial Dimension Effects (Strukov and Mem-Con)

B. Test Hysteresis Predicitons (Georgiou)

• 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

Our Memristors

•

Crossed

Aluminium

electrodes

•

Thin-film (40nm)

TiO

sol-gel layer

•

E = 4mm

•

F = 1, 2, 3, 4 or

5mm

1. Gergel-Hackett et al, A Flexible Solution Processed Memristor, IEEE Elec. Dev. Lett., 2009

Pictures

Curved (BPS -like) Memristors

Triangular (UPS -like) Memristors

The Effect of

Varying Electrode Size

As , ,

Fit Memory Function to as a function of

Memory function Describes

’s variation with F

CONSERVATION FUNCTION DESCRIBES ’S VARIATION WITH F

• 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

Is the Hysteresis Related

to the ‘dimensionless

lumped parameter’, ?

THE EXAMPLE GIVEN IN GEORGIOU ET AL’S PAPER

• 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)

Ella Gale, Ben de Lacy

Costello and Andrew

Adamatzky

FILAMENTARY EXTENSION OF THE

MEM-CON THEORY OF

MEMRISTANCE AND ITS

Pictures

Curved (BPS -like) Memristors

Triangular (UPS -like) Memristors

Extend the Mem-Con

Model to Describe

Filamentary

(Triangular)

Memristors

φ

q

V

I

What the Flux?

𝑑 𝜑

=

𝑀

(

𝑞

(

𝑡

)

)

𝑑𝑞

𝑀

(

𝑞

(

𝑡

)

)

=

𝑅

−

𝜇

𝐷

𝑅

𝑅

𝑞

(

𝑡

)

But, where is the magnetic flux?

𝑉

=

𝑀

(

𝑡

)

𝐼

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

Mem-Con Theory

𝑞

↔

𝑀(𝑞 )

↔

𝜑

↑

𝑉

↔

𝑅

𝑡𝑜𝑡

(𝑡)

↔

𝐼

Ionic

Electronic

EQUIVALENT CIRCUIT DIAGRAM TO THE DEVICE

CHEMISTRY

𝑅_𝑇𝑜𝑡(𝑡)= 1

1

(

)

+2 𝐻 (𝑤− 𝐷) 1

𝑅_𝑓𝑖𝑙

𝑀𝑒

𝑅

𝑅

• Memristance based on

• Due to the shape, varies with

M: TIME-DEPEDENDANT EXPRESSION FOR THE VOLUMES

𝑀𝑒

𝑅

Vacancy Magnetic Field 

G can be solved by

where we use and

Vacancy Magnetic Field

𝑀𝑒

𝑅

𝑅

is the surface normal for area infinitesimal

Wb

For Strukov’s device: b [1]

As [2]

, and

MEMORY FUNCTION

Not as easy as it looks.

𝑅

𝑓𝑖𝑙

=

𝑟

1

−

𝐷

𝑓

+

1

FILAMENT RESISTANCE

𝑀𝑒

𝑅

𝑅

EQUIVALENT CIRCUIT DIAGRAM TO THE DEVICE

CHEMISTRY

𝑅_𝑇𝑜𝑡(𝑡)= 1

1

(

)

+2 𝐻 (𝑤− 𝐷) 1

𝑅_𝑓𝑖𝑙

𝑀𝑒

𝑅

𝑅

Experiment Theoretical Model

• 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 Field 

Vacancy Magnetic Flux 

Starting From The Ions…

Vacancy Volume Current 

,

L = eLectric field

Calculate the Magnetic B

field Associated with the

ions

𝑀𝑒

𝑅

• 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 Field 

Vacancy Magnetic Flux 

Starting From The Ions…

• 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

•

Ben de Lacy Costello

•

Andrew Adamatzky

•

David Howard

•

Larry Bull

With Thanks to

•

Steve Kitson (HP

UK)

•

David Pearson (HP

UK)

•

Bristol Robotics

• 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

Influx of Ionic I Voltage Spike Axon: Transmission along neuron Synapse: Transmission between neurons

Memristive Systems to

Describe Nerve Axon

• 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 Field 

Vacancy Magnetic Flux 

Starting From The Ions…

• 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

The Memristor as a Synapse

Before learning Before learning

During 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:

Charge-Controlled

Memristor

Flux-Controlled

Memristor

Chua’s Definitions of Types of

Memristors

φ

q

V

I

What the Flux?

𝑑 𝜑

=

𝑀

(

𝑞

(

𝑡

)

)

𝑑𝑞

𝑀

(

𝑞

(

𝑡

)

)

=

𝑅

−

𝜇

𝐷

𝑅

𝑅

𝑞

(

𝑡

)

But, where is the magnetic flux?

𝑉

=

𝑀

(

𝑡

)

𝐼

• 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 Field 

Vacancy Magnetic Flux 

Mem-Con Theory

𝑞

↔

𝑀(𝑞 )

↔

𝜑

↑

𝑉

↔

𝑅

𝑡𝑜𝑡

(𝑡)

↔

𝐼

Ionic

Electronic

Pictures

Curved (BPS -like) Memristors

Triangular (UPS -like) Memristors

To make a

memristor brain

& thus a machine

intelligence

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, 2011

0. Linares-Barranco and Serrano-Gotarredona, Memristance can

explain Spike-Time-Dependent-Plasticity in Neural Synapses, Nature Preceedings, 2009

Memristors

Spike

Naturally!

Voltage Square Wave

Voltage Ramp

Current Response

Neuro n

Memrist or

Memristor Behaviour Looks

Similar to Neurons

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

•

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

Curved (BPS -like)

Memristors

Triangular (UPS

-like) Memristors

Where do the Spikes

Come From?

q

φ

I

V

q

φ

V

I

Neurons

Memristors

Mem-Con Model Applied to

• _{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 =

q

•

Memristor operated

in voltage controlled

way

Neuron Voltage Spikes

Memristor Current

Spikes

•

More complex system than a single

memristor

•

Short-term memory associated with

membrane potential

•

Long term memory associated with

the number of synaptic buds

Sol-Gel Memristor

Negative V

Sol-Gel Memristor

Positive V

I-t Response to

Stepped Voltage

Time Dependent

I-V

Voltage Ramp

Current Response

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

•

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