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 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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0
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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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Graphene Unit cell
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Real space unit vectors and Unit cell
Area of Unit cell a1 x a2 same a one on the
hexagons
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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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Simple way
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Can guess what the terms in brackets the have to
be
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Why this criteria
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Real space
Reciprocal space
Similar to lt111gt 0 lt100gt in a cubic system
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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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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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Graphene HAB types find the graphene factor
60º
60º
Compare with polyacetylene Not the same
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Graphene factor HAB types
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Graphene factor HAB types
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Graphene factor HAB types
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Graphene factor HAB types
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Graphene factor HAB types
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Graphene factor HAB types
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Graphene factor HAB types
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Graphene factor HAB types f(k)
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Graphene HAB types
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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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Find E-k
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Find E-k
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Find E-k
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Find E-k
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2D graphene kx and ky dependence
1D polyacetylene kx dependence
This picture is a slice
This picture isnt
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2D graphene kx and ky dependence
Example label the axes for this figure.
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2D graphene kx and ky dependence
Answer
E
ky
kx
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• 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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• 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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30o
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Example how many equivalent K points are
there? How would you get to them?
30o
40
Answer 6 equivalent K-points
E
ky
kx