NCN@Purdue-Intel Summer School
Notes on Percolation Theory
Lecture 1
Percolation in Electronic Devices
Muhammad A. Alam
Electrical and Computer Engineering
Purdue University
West Lafayette, IN USA
NCN
www.nanohub.org
1
plan for the lecture series
1)  Percolative transport in electronic devices
2)  Basic concepts: threshold, island sizes, fractal dimensions
3)  Electrical conduction in random media
4)  Theory of stick percolation: application to nanonet transistors
5)  2D nets in 3D world: sensors, solar cells, super-capacitors
Acknowledgement:
D. Varghese, P. Nair, and E. Islam.
2
outline of lecture 1
1)  Order is an anomaly … randomness rules
1)  Randomness in electronics.
2)  Randomness in nature.
2)  Why did we not hear about it
1)  Averaging over large numbers
2)  Quantization and coherence.
3)  Approximate randomness at your own peril
1)  Weibull distribution vs. Gaussian distribution.
2)  Poisson vs. Fish arrival distributions
4)  Conclusions
3
randomness in electronics …
Dielectric BD
Quantum
transport
high
Poly-Si
a-si, poly-Si TFT
Organic transistors
Solar cells
Fuel cells
Super-capacitors
small
percolation
NC Flash
X-Si CMOS
logic/memory
low
ballistic
Random Dopants
medium
Performance
1947 ….
medium
NanoNet
Biosensors
large
Area
super-capacitors
4
random dopant fluctuation
side view
@1e18/cm3
100 nm --- 1000 dopants
20 nm --- 40 dopants
10 nm --- 10 dopants
model
top view
5
NBTI induced interface de-passivation
Si-H bonds @5e12 cm-2
H
H
H
H
H
Si Si Si Si Si
Top view
100 nm --- 500 states
20 nm --- 20 states
10 nm --- 5 states
model
6
Flash
vs.
1 to 0
Nanocrystal Flash
Anomalous
leakage
model
Top view
7
solar cells and display electronics
Key issues:
Transport through barriers
created by grain boundaries
Device/device fluctuation
8
Flexible nanonet electronics
so
ce
ur
in
ra
D
StickModel
percolation
Heterogeneous
?percolation
Cao, Nature, 2008
9
solar cells
side view
Band-diagram
Exciton recombination before dissociation
at the junction makes it a poor cell …
10
nano-structured solar cells
mixed layers
heterogeneous
percolation
How do we describe exciton dissociation/charge collection
11
flexible electrodes
ITO electrode
(expensive)
Transparent
nanonet electrodes
Standard solar cells
12
super-capacitors
+ve
+
d
-ve
Parallel plate
capacitor
Electrolyte
capacitor
Super-capacitor
13
biosensors …
An array …
Individual sensor
Capture Probe
Q1
C
A
G
A
Q1
C
G
A T
G C
Q2
A T
T A
Conductance (G)
T
ts response time
ΔG
G0
Time
14
diffusion towards disordered biosensors…
Key issues:
density dependent response time
conductivity, transfer resistance
or substrate dependence
  Channel length scaling


15
chemical sensors and e-nose
sensor
Organic substrate
After expansion
16
randomness is the rule, not the exception ….
Cluster sizes
Oil fields, NC Flash
In plane transport
epidemics, forest fire, Out of plane transport:
Aerosol, paper, sensors
telecom grid, www,
Nanonets, photovoltaics
17
outline of lecture 1
1)  Order is an anomaly … randomness is the rule
2)  Why did we not hear about it
3)  Approximate randomness at your own peril
4)  Conclusions
18
Why did I not hear about it (1) ?
1 mm3 ~ 1017 molecules
approximate theory
correct theory
R
impurities
Given traces of impurity, change in property (e.g.
reflection) is easily predicted.
Fluctuation in properties of large system is small.
19
Why did I not hear about it (2) ?
Some small systems have
unusually robust properties,
(e.g. quantum Hall effect)
and physicists often focus
on those extra-ordinary
aspects of small systems …
20
mean and deviation
Bottom up
PDF
Top down
G
pC
p
…. wrong on both counts.
G
21
effective media approach ….
p=0.8
p=0.59
p=0.3
G
p
… that’s what textbooks say!
22
basics of percolation: averaging matters
p=0.8
p=0.59
p=0.3
Consequences of adding a
new disk depends on
existing configuration ...
G
pC
p
23
… and so does the fluctuation
p=0.8
p=0.8
PDF
p=0.8
G
pC
p
G
May look the same, but have very different implications
24
current approach: transistor design
I
pC
p
Is Gaussian distribution appropriate ?!!
25
current approach: molecular conduction
X
NH2
NH2
I
26
current approach: thermal conduction
27
outline of lecture 1
1)  Order is an anomaly … randomness is the rule
2)  Why did we not hear about it
3)  Approximate randomness at your own peril
4)  Conclusions
28
warranty, product recall and other facts
of life …
A manufacturer bets
the company of the
physics of reliability …..
… because the ICs
operate in incredibly
harsh conditions, turning
on and off trillions of time
during its lifetime ….
… because the lines could
open, the source/drain
can be shorted, the gate
oxide can break ….
29
oxide degradation and breakdown
Gate Current
Breakdown
I
time
pC
p(t)
30
(simple) Theory of breakdown
M
N
If the bottom up view is correct, then we will have a
straight-line in a Weibull plot and slope proportional to thickness
31
bottom-up prediction for oxide scaling
Measurement
Theory
Tox
Thin oxide breaks much faster than thick oxide due to percolation,
process-improvement can not solve this problem
32
very different lifetime projection …
1 CPU ~ 109 Transistors
G
pC
p(t)
When one fails, so does the
whole. Mean is not enough
….
33
example 2: arrival time distribution
Mountain
Fishes dropped at x0 at t=0
waterfall
Find the arrival time distribution at the waterfall.
1D model for …..
  field-return
of components
loss in Nanocrystal Flash
  release of proteins from inside the cells, etc.
  Drug release from capsules, etc.
  charge
34
approximation by classical distributions
Monte
Carlo
1000,
10000,
50000
1000
Samples
Gamma
distribution
Log-normal
distribution
35
Derivation of “Fishy” (or BFRW) distribution
36
long tail of a distribution
Gamma Distribution
Log-normal Distribution
Exact solution
37
so percolation theory is indeed necessary …
NC Flash
Poly-Si
X-Si CMOS
logic/memory
a-si, poly-Si TFT
Organic transistors
Solar cells
Fuel cells
Super-capacitors
medium
high
Random Dopants
low
Performance
Dielectric BD
small
medium
NanoNet
Biosensors
large
Area
super-capacitors
38
… but classical percolation is not enough
  Large
system in thermodynamic limit
Not explicitly concerned with fluctuation
  Disk
percolation as a central paradigm
  Linear
response theory
  Heterogeneous
  Transport
  Primary
percolation is seldom used
on plane or a volume
interest in steady state systems
See you in lecture 2 then!
39
References/credits
(4) Oxide breakdown: Pey and Tung. 200 http://spie.org/x17096.xml
Random dopants: http://www.nature.com/nature/journal/v437/n7062/fig_tab/nature04086_F4.html
Super-capacitor: Joel Schindall IEEE Spectrum, 2007. The charge of ultra-capacitors
(7) Computer: www.dell.com
Solar cell panels: ???
(9) Ref. Flexible Circuits, Cao, Nature, 2008.
(11) solar cell: Peter Peumans, Ph.D. Thesis, 2003.
(16) Porous aluminum: Fobes, 2008. Super-capacitor: Joel Schindall IEEE Spectrum, 2007. The charge of ultra-capacitors
(17) Oil well: www.ambercore.com/Petroleum_iQ.php
Aerosol, and paper: Random walk in Fractal dimensions, 2002.
(20) Quantum Hall effect: Google image. http://nobelprize.org/nobel_prizes/physics/laureates/1998/illpres/practice.html
(25) Random dopant: Roy, Cheng, and Asenov. http://userweb.elec.gla.ac.uk/s/sroy2/circuits.html
(26) Molecular conduction: Columbia work.
(27) Thermal conduction: Arun Majumdar, 2007.
Long tail: Chasing the long tail. http://www.leftclick.com/blog/chasing-the-long-tail
Water drop --- scubasocal.wordpress.com/2007/08/ micro.magnet.fsu.edu/.../6x86polylarge.html
40