Percolation Effects on Electrical
Resistivity and Electron Mobility
By: Jared Weddell
University of Illinois at Chicago
Department of BioEngineering
[Mott Corporation, 2009]
Overview
• Introduction
– Motivation
•
•
•
•
Project Overview
Methods
Results
Conclusion
• Percolation : Study of the movement of mass across a porous
material.1
– How a porous metal allows flow of electrons across surface
Example of a network used in percolation studies
[Wikipedia, 2006]
1: B. Last and D. Thouless: Physical Review Letters, 1971, 27, 1719 – 1722.
Project Overview
• To determine electrical resistance as a
function of porous cuts in a conductive sheet
• Determination of percolation threshold - area
when resistance becomes infinite
• Establish an experimental method for
performing percolation studies
• Application: To describe transport quantities
such as thermal conductivity, elastic module,
and diffusion constant
• Matlab creates the random system of networks to
analyze. AutoCAD visualizes the network and
communicates with a laser that makes the cuts.
Random ellipse network created by Matlab. The left section is the
percolation sheet of interest while the right displays how much area
remains in the left section.
• Percolation threshold found by comparing current of
the left and right side as pores are being cut.
50
45
40
Current (mA)
35
30
25
20
15
10
5
0
0
500
1000
1500
2000
Time (s)
The current vs. time of the percolating sheet (blue) and equivalent area sheet (red).
The black line shows the area threshold is found by observing the equivalent area
current at the point where the percolation current becomes zero.
• Ellipses are the major pore type of interest in this
study. The threshold will change based on aspect
ratio b/a.2
[Wikipedia, 2009]
Ellipse with major axis a and minor axis b. The resistance will
vary depending on aspect ratio b/a used.
2: S. Vellaichamy, B. Prakash, and S. Brun. Computers & Structures, 1990, 37, 241-246.
• Comparing experimental ellipse data to
theoretical results from Xia and Thorpe.3
• Find the percolation threshold of a trial using
elliptical pores with a certain aspect ratio and
compare against other aspect ratios.
• The number of elliptical pore cuts is held
constant across all trials.
0.7
0.15
3: W. Xia, and M. F. Thorpe: Physical Review A, 1988, 38, 2650 – 2656.
0.0025
• Comparing experimental ellipse data to theoretical
results.
Aspect Ratio
Theoretical
Threshold3
Experimental
Threshold
1.0000
0.33
0.351
0.7000
0.34
0.392
0.6000
0.35
0.414
0.4000
0.41
0.523
0.2500
0.50
0.588
0.1500
0.62
0.622
0.1000
0.70
0.723
0.0500
0.83
0.856
0.0250
0.91
0.912
0.0125
0.949
0.923
0.7
0.15
Uncertainty for all experimental thresholds is +0.0011
3: W. Xia, and M. F. Thorpe: Physical Review A, 1988, 38, 2650 – 2656.
0.0025
• Experimental ellipse percolation thresholds vs.
theoretical values3
1
0.9
Percolation Threshold
0.8
0.7
0.6
0.5
0.4
Theoretical
Experimental
0.3
0.2
0.1
0
0.0125 0.025
0.05
0.1
0.15
0.25
0.4
Ellipse Aspect Ratio
3: W. Xia, and M. F. Thorpe: Physical Review A, 1988, 38, 2650 – 2656.
0.6
0.7
1
Kerf Measurement
•
•
•
•
Kerf is the width of the cut made by the laser
Approximated kerf was 0.0020 inches (50 µm)
Measured kerf was 0.0071 inches (180 µm)
This difference leads to an uncertainty of error
of 0.11% of the total area
– Giving uncertainty +0.0011 to thresholds
Conclusion
• As the aspect ratio approaches zero, the
percolation threshold approaches one.
• Experimental and theoretical results are in
good agreement.
• Variation comes from random nature of which
the pores are cut.
Summary
• Percolation studies useful for determining
transport properties of materials.
• Matlab and AutoCAD used to create
percolating systems.
• Experimental results agree with theoretical.
Ellipses have higher percolation threshold at
lower aspect ratios.
Acknowledgments
• The financial support from the National
Science Foundation, EEC-NSF Grant # 1062943
• Dr. Feinerman
• Dr. Takoudis and Dr. Jursich