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The Effect of Electrode Size on
Memristor Properties: An Experimental
and Theoretical Study
Ella Gale, Ben de Lacy Costello
and Andrew Adamatzky
We Want To Know…
A. Which Model of
Memristance Works Best
B. What Effect Electrode
Size has on Memristor
Properties
THEORIES OF MEMRISTANCE
CHUA’S PHENOMENALOGICAL
DEFINITION
 =   
M =
q =
φ =
memristance
charge
magnetic flux

There Are Three Theoretical Memristor Models
1. Strukov et al’s Phenomenalogical Model
2. Georgiou et al’s Bernoulli Equations
3. Mem-Con Model
1. Strukov et al
2. Georgiou et al
3. Gale,
1. Phenomenological Model
  

= off − 2 off on ()

This is a 1-D model
 = ionic mobility of the O+
vacancies
Roff = resistance of TiO2
Ron = resistance of TiO(2-x)
Strukov et al, The Missing Memristor Found, Nature, 2008
2. GEORGIOU ET AL’S 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
The Model
Predictions
= ‘Dimensionless Lumped Parameter’
 is related to 
 is related to 
Contains:
•
‘all’ physical dimensions
,  ,  ,  of device
•
all parameters ( , 0 , 0 ) of
experiment
 = 2 =
This is a 1-D model
2 
0 0
2



2

−1

3. Memristance, as Derived from Ion Flow
  
• Universal constants:
0
4
0
=
(())
4
This is a 3-D model
•  =  ∙  ∙ , Experimental constants: product of surface
area () and electric field (),
• , Material variable, =  (  ), where  (  , , )
Gale, The Missing Magnetic Flux in the HP Memristor Found, 2011
MEM-CON MODEL
Memor y Function
 =  ∙  + 2
C o n s e r va t i o n F u n c t i o n

 −   
=

  =  + 
OUR PREMISE
Goal: To Investigate Which Theoretical Model Works Best
Method:
A. Spatial Dimension Effects (Strukov and Mem-Con)
B. Test Hysteresis Predicitons (Georgiou)
SIZE PREDICTIONS
• 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)
TiO2 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
2. Gale et al, Aluminium Electrodes Effect the Operation of Titanium Dioxide Sol-Gel
Memristors, Submitted 2012
Two Different Types of Memristor
Behaviour Seen in Our Lab
C ur ved (BP S - li ke)
M em ri s to r s
Pictures
Tri a n gu la r ( U P S - li ke)
M em ri s to r s
TEST 1
The Effect of
Varying Electrode Size
CURVED SWITCHING
MEMRISTORS
As  → 0 ,
 →  ,
 → 
Fit Memory Function to  as a function of 
 =   =  ∙ ()  + 2
Fit Conservation Function to  as a function of F
 =  () 
Memory function Describes
 ’s variation with F
Only 1 fitting parameter needed:  = 2.09 × 1025 (2 = 1.25 × 10−20 , ~ 1 × 10−23 %)
CONSERVATION FUNCTION
DESCRIBES  ’S VARIATION WITH F
One Fitting Parameter,  , = 6.82 × 1010 Ωm-1
(Bulk value: 1012 Ωm-1)
SO,
• 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
TEST 2
Is the Hysteresis Related to the
‘dimensionless lumped
parameter’,  ?
THE EXAMPLE GIVEN IN
GEORGIOU ET AL’S PAPER
Voltage Source
Waveforms:
• Green: Bipolar
Piece-Wise
Linear
(analytically
calculable)
• Red: Sinusoidal
(not analytically
calculable)
• Blue: Triangular
(analytically
calculable)
Ref…
Simulated Result
MEASURED HYSTERESIS VERSUS
EXPERIMENTAL VALUES OF 
DOES GEORGIOU ET AL’S PREDICTED
 RELATE TO MEASURED ?
HYSTERESIS SIZE DEPENDS ON F
AND RON
SUMMARY
• 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)
FILAMENTARY EXTENSION OF THE MEMCON THEORY OF MEMRISTANCE AND ITS
APPLICATION TO TITANIUM DIOXIDE SOLGEL MEMRISTORS
Ella Gale, Ben de Lacy Costello
and Andrew Adamatzky
Two Different Types of Memristor
Behaviour Seen in Our Lab
C ur ved (BP S - li ke)
M em ri s to r s
Pictures
Tri a n gu la r ( U P S - li ke)
M em ri s to r s
Memristor Structure and Function
SHAPE OF THE FILAMENT
We Want To…
Extend the Mem-Con Model
to Describe Filamentary
(Triangular) Memristors
THE MEM-CON THEORY
Definition of the Memristor
Inductor
Memristor
Resistor
Capacitor
What the Flux?
But, where is the magnetic flux?
Strukov et al, 2008
  
Chua, 1971

=  − 2   ()

=  
 =   

CALCULATING THE CHUA
MEMRISTANCE
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
•  =  + 
•  =    
•  =
(−  )TiO2

 ↔ () ↔  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



e
 () =
1
1
 +   +  
1
+ 2  −  

M: TIME-DEPEDENDANT
EXPRESSION FOR THE VOLUMES
•
•
Memristance based on  
Due to the shape, () varies with ()
 = 1 3   (1 2 + 21   + ()2 )
 = 1 3   (()2 + 2  2 + 2 2 )
 
1
=
2
−21 − 22 + 22 +
 2 + 4 1 2 − 1 2 + 1 2 + 2 2
 = 21 − 22 + 22
  =
2  + 2 



Vacancy Magnetic Field
Vacancy Magnetic Field 
0  × 
 =    =

4
||


G can be solved by
  = 

  
where we use
=1
 2 + 2 + 2
 =  × 
3
2
and
Cuboid of  ∙ 2 ∙ 2
 ≡ { ,  ,  }
Vacancy Magnetic Field
 =     0, −  ,   
∙=0
MEMORY FUNCTION
+
+
=
 ∙  
−
−
 is the surface normal for area
infinitesimal 
 =     
 = 6.52 × 10−30 Wb
For Strukov’s device:
 = ‼! Wb
As
 =     [2]
  
=   () ,
and
 = 
[1]
1. Gale, The Missing Magnetic Flux in the HP Memristor Found, Submitted, 2011
2. Chua, Memristor: The Missing !!!
RESISTANCE OF A CONICAL
RESISTOR
Not as easy as it looks.


=
2()2
  − 2
−
2
2
+   − 2
2
+ ( − )
FILAMENT RESISTANCE
 = 1 − +1



Ref ?
EQUIVALENT CIRCUIT DIAGRAM
TO THE DEVICE CHEMISTRY



e
 () =
1
1
 +   +  
1
+ 2  −  

COMPARISON TO EXPERIMENT
Experiment
Theoretical Model
Starting From The Ions…
• Memristance is a phenomenon associated with ionic current flow
• Therefore  calculate the magnetic flux of the IONS
Vacancy Volume Current   =
   
(2−)
Vacancy Magnetic Field   =
Vacancy Magnetic Flux  =
, L = eLectric field


×
||

0
||  (
4
 )
Gale, The Missing Magnetic Flux in the HP Memristor Found, Submitted, 2011
Calculate the Magnetic B field
Associated with the ions
Vacancy Volume Current 
=
   
,
 (  ,  )
L = eLectric field



Starting From The Ions…
• Memristance is a phenomenon associated with ionic current flow
• Therefore  calculate the magnetic flux of the IONS
Vacancy Volume Current   =
   
(2−)
Vacancy Magnetic Field   =
Vacancy Magnetic Flux  =
, L = eLectric field


×
||

0
||  (
4
 )
Gale, The Missing Magnetic Flux in the HP Memristor Found, Submitted, 2011
CONCLUSIONS & FURTHER WORK
Conclusions
F u r t h e r Wo r k
• Filamentary addition to the MemCon model gives good qualitative
agreement to experiment
Work out the quantitative values
Re-do derrivation allowing a background bulk memristance
With Thanks to
• Ben de Lacy Costello
• Andrew Adamatzky
• David Howard
• Larry Bull
• Steve Kitson (HP UK)
• David Pearson (HP UK)
• Bristol Robotics
Laboratory
FURTHER WORK
• 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
How does a Neuron Compute?
Influx of
Ionic I
Voltage Spike
Axon:
Transmission along neuron
Synapse:
Transmission between
neurons
Memristive Systems to Describe
Nerve Axon Membranes
Synapse Long-Term Potentiation
Starting From The Ions…
• Memristance is a phenomenon associated with ionic current flow
• Therefore  calculate the magnetic flux of the IONS
Vacancy Volume Current   =
   
(2−)
Vacancy Magnetic Field   =
Vacancy Magnetic Flux  =
, L = eLectric field


×
||

0
||  (
4
 )
Gale, The Missing Magnetic Flux in the HP Memristor Found, Submitted, 2011
A HUGE PROBLEM OF
TERMINOLOGY
ReRAM
•
•
•
•
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
Memristor
• Satisfy Chua’s definition:
 =   
V=  
• Pinched hysteresis loop in I-V
space
• --
The Memristor as a Synapse
Before learning
Before learning
After learning
After learning
During learning
Spike-Time Dependent Plasticity, STDP
•
•
•
•
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
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
Chua’s Definitions of Types of Memristors
Charge-Controlled
Memristor
  =    ()
()
  ≡

F lux-Controlled
Memristor
  =  
()
()
() ≡

L. Chua, Memristor – The Missing Circuit Element, IEEE Trans. Circuit Theory, 1971
Definition of the Memristor
Inductor
Memristor
Resistor
Capacitor
What the Flux?
But, where is the magnetic flux?
Strukov et al, 2008
  
Chua, 1971

=  − 2   ()

=  
 =   

Starting From The Ions…
• Memristance is a phenomenon associated with ionic current flow
• Therefore  calculate the magnetic flux of the IONS
Vacancy Volume Current   =
   
(2−)
Vacancy Magnetic Field   =
Vacancy Magnetic Flux  =
, L = eLectric field


×
||

0
||  (
4
 )
Mem-Con Theory
•  =  + 
•  =    
•  =
(−  )TiO2

 ↔ () ↔  Ionic
↑
 ↔  () ↔  Electronic
Gale, The Missing Magnetic Flux in the HP Memristor Found, Submitted, 2011
Two Different Types of Memristor
Behaviour Seen in Our Lab
C ur ved (BP S - li ke)
M em ri s to r s
Pictures
Tri a n gu la r ( U P S - li ke)
M em ri s to r s
Memristor I-V Behaviour
Our Intent:
To make a memristor
brain
& thus a machine
intelligence
Connecting Memristors with Spiking Neurons to
Implement STDP
Simulation Results
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-TimeDependent-Plasticity in Neural Synapses, Nature Preceedings, 2009
But,
Memristors Spike
Naturally!
Current Spikes Seen in I-t Plots
Spikes are Reproducible
Volta ge Square Wave
Cur rent Spike Response
Spikes are Repeatable
Voltage Ramp
Current Response
Memristor Behaviour Looks Similar to
Neurons
Memristor
Neuron
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
Spintronic Memristor Current Spikes
Pershin and Di Ventra, Spin Memristive Systems: Spin Memory Effects in Semi-conductor
Spintronics, Phys. Rev. B, 2008
Properties of 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
•
•
•
•
•
Two Different Types of Memristor Behaviour
Seen in Our Lab
Cur ved (BPS-like)
Memristor s
Triangular (UPS -like)
Memristor s
Where do the Spikes Come
From?
Does Current Theory Predict Their Existence?
Mem-Con Model Applied to Memristor Spikes
Neurons
Memristors
q
φ
q
φ
I
V
V
I
In Chua’s Model
Neuron Volta ge 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
Memristor Cur rent Spikes
∆ =    ∆
• Dynamics related to τ,
which is related to 
• Memory property = qv
• Memristor operated in
voltage controlled way
What is the Memory Property of Neurons?
• More complex system than a single memristor
• Short-term memory associated with membrane
potential
• Long term memory associated with the number
of synaptic buds
Memristor Models Fit the Data
Sol-Gel Memristor
Negative V
Sol-Gel Memristor
Positive V
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
Au-TiO2-Au WORMS Memory
I-t Response to
Stepped Voltage
Time Dependent I-V
Al-TiO2-Al Current Response to Voltage Ramp
Voltage Ramp
Cur rent Response
Further Work
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
Summary
•
•
•
•
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

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