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

2. Lecture 22, 12 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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4. 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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5. Graphene Unit cell:
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7. Real space unit vectors and Unit cell:
Area of Unit cell = a1 x a2 = same a one on the hexagons
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8. 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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10. Simple way:
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11. Can guess what the terms in brackets the have to be:
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14. Why this criteria:
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15. Similar to <111> 0 <100> in a cubic system
Real space
Reciprocal space
Similar to <111> <100> in a cubic system
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16. 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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17. Graphene versus polyacetylene:
Similar: 2 inequivalent C-atoms
Self energy HAA and HBB terms = same = e2p
Self energy SAA and SBB terms = same = 1
Dissimilar: 3 nearest neighbor C-atoms at 120º not 2 nearest neighbor atoms at 180º
Nearest neighbor HAB and HBA terms = same = (graphene factor) t
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18. Graphene: HAB types: find the graphene factor:
60º
60º
Compare with
polyacetylene:
Not the same
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19. Graphene factor: HAB types:
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20. Graphene factor: HAB types:
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21. Graphene factor: HAB types:
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22. Graphene factor: HAB types:
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23. Graphene factor: HAB types:
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24. Graphene factor: HAB types:
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25. Graphene factor: HAB types:
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26. Graphene factor: HAB types: f(k):
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27. Graphene: HAB types:
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28. 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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29. Find E-k:
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30. Find E-k:
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31. Find E-k:
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32. Find E-k:
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33. 2D graphene 1D polyacetylene
kx and ky dependence
1D polyacetylene
kx dependence
This picture is a slice
This picture isn’t
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34. 2D graphene kx and ky dependence
Example: label the axes for this figure.
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35. 2D graphene kx and ky dependence E ky kx Answer:
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36. Three points about E-k  E-kx & ky:
Smallest (best: metallic) bandgap between E+ and E- at K.
Biggest (worst) bandgap between E+ and E- at G.
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37. Three points about E-k  E-kx & ky:
Smallest (best) bandgap between E+ and E- at K: Example where are K?
Biggest (worst) bandgap between E+ and E- at G: at kx = ky = 0
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38. 30o
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39. Example: how many equivalent K points are there?
How would you get to them?
30o
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40. Answer: 6 equivalent K-pointsky kx
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