ECE 802-604: Nanoelectronics

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ECE 802-604Nanoelectronics

• Prof. Virginia Ayres
• Electrical Computer Engineering
• Michigan State University
• ayresv_at_msu.edu

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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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Goal polyacetylene
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l
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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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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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-1
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Will come back to this step
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Plot E and E- versus k. What is the range of k?
Symmetric
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a
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
H
H
H
A
c
c
c
c
c
B
H
H
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TYPO
CORRECT
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Back to this step
Transfer matrix
Overlap matrix
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Quick review of what the F are Bloch
wavefunctions
a here is the distance between identical
situations.
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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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atoms
Because polyacetylene has inequivalent C atoms
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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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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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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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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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Where are the nearest neighbor A to A C-atoms?
a
H
H
H
A
c
c
c
c
c
B
H
H
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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
a
a
H
H
H
H
A
c
c
c
c
c
c
c
B
H
H
H
-a
a
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?
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Need to model the wavefunctions 2pxgt Model
pgt orbital
ECE 802-604 Use this result, p.24 t -1.0 s
0.2 e2p 0.0
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Rules for finding the electronic structure (p.
21)
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Find Unit cell a
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Find k
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Find H and S elements
Det H SI 0
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Solve for E(k)