1. ECE 802-604: Nanoelectronics
Prof. Virginia Ayres
Electrical & Computer Engineering
Michigan State University

2. Lecture 21, 07 Nov 13 Carbon Nanotubes and Graphene
CNT/Graphene electronic properties
sp2: electronic structure
Find E-k relationship/graph for polyacetylene
Find E-k relationship/graph for graphene
R. Saito, G. Dresselhaus and M.S. Dresselhaus
Physical Properties of Carbon Nanotubes
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3. Goal: polyacetylene:
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6. l
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13. All Structure: s-bonds Electronic: p-bonds
Lec 20: Division of structural and electronic properties in sp2:
This is for graphene but it shows you the valence and conduction energy states from both ss and ps
All
Structure:
s-bonds
Electronic:
p-bonds
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14. All Structure: s-bonds Electronic: p-bonds
Lec 20: Division of structural and electronic properties in sp2:
All
Structure:
s-bonds
p valence and conduction energy levels for polyacetylene
Electronic:
p-bonds
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21. -1
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22. Will come back to this step
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25. Plot E+ and E- versus k. What is the range of k?
Symmetric:
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27. Can graph this way if you acknowledge that you have two inequivalent carbons: go up using a1 to get to “A” types and down by a2 to get to “B” types a H H H
“A” c c c c c
“B” H H
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31. TYPO
CORRECT:
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33. Back to this step Transfer matrix Overlap matrix
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35. Quick review of what the F are: Bloch wavefunctions:
“a” here is the distance between identical situations.
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36. “a” here is the distance between identical situations.
From previous slide 30, for polyacetylene:
“a” is the size of our Unit cell.
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37. Because: polyacetylene has inequivalent C atoms
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38. a H H H
“A” c c c c c
“B”
More simply: where is C-atom “B” relative to C-atom “A”? H H
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39. a H H H
“A” c c c c c
“B”
Question: where is C-atom “B” relative to C-atom “A”?
Answer: +a/2 H H
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40. a H H H
“A” c c c c c
“B”
Where are the nearest neighbor C-atoms “B” relative to “A”? H H
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41. a H H H
“A” c c c c c
“B”
Where are the nearest neighbor C-atoms “B” relative to “A”?
Answer: +a/2 and –a/2 H H
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42. Where are the nearest neighbor “A” to “A” C-atoms?
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43. This is the summation from “A” to all “A” type carbon atoms:
Where are the nearest neighbor “A” to “A” C-atoms?
Answer: +a and -a
This is the summation from “A” to all “A” type carbon atoms: a a H H H H
“A” c c c c c c c
“B” H H H -a +a
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45. ✔ ✔
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48. ✔ ✔ ✔ ✔ ✔ ✔
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49. Need to model the wavefunctions: |2px>
Model |p> orbital
ECE : Use this result, p.24:
t = -1.0
s = +0.2
e2p = 0.0
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50. 1 Find Unit cell “a” 2 Find k: 3 Find H and S elements 4
Rules for finding the electronic structure (p. 21): 1
Find Unit cell “a” 2
Find k: 3
Find H and S elements
Det [H – SI] =0 4
Solve for E(k)
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