ECE 802-604:
Nanoelectronics
Prof. Virginia Ayres
Electrical & Computer Engineering
Michigan State University
ayresv@msu.edu
Lecture 23, 19 Nov 13
Carbon Nanotubes and Graphene
CNT/Graphene electronic properties
sp2: electronic structure
E-k relationship/graph for polyacetylene
E-k relationship/graph for graphene
E-k relationship/graph for CNTs
R. Saito, G. Dresselhaus and M.S. Dresselhaus
Physical Properties of Carbon Nanotubes
VM Ayres, ECE802-604, F13
Polyacetylene E-k:
E
-kx
kx
VM Ayres, ECE802-604, F13
To really finish:
Need to model the wavefunctions:
Could let f = |2px>
Could let f = |p>
(f = |sp2> is the s-bond)
ECE 802-604: Use this result, p.24:
t = -1.0
s = +0.2
e2p = 0.0
VM Ayres, ECE802-604, F13
Real space
a
H
c
H
“A”
c
H
c
“B”
H
c
H
c
VM Ayres, ECE802-604, F13
Two different spring constants:
tighter k1 (double bond) and looser k2 single bond
a
H
c
H
k1
“A”
c
H
k2
c
“B”
H
c
c
H
VM Ayres, ECE802-604, F13
Graphene E-k:
VM Ayres, ECE802-604, F13
e2p
A
To really finish:
Need to model the wavefunctions:
Could let f = |2px>
Could let f = |p>
(f = |sp2> is the s-bond)
ECE 802-604: Use this result, p.27:
t = -3.033 Units
s = 0.129 Units
e2p = 0.0 Units
Example: what are the Units?
VM Ayres, ECE802-604, F13
e2p
A
To really finish:
Need to model the wavefunctions:
Could let f = |2px>
Could let f = |p>
Answer:
(f = |sp2> is the s-bond)
ECE 802-604: Use this result, p.27:
t = -3.033 eV
s = 0.129 pure number
e2p = 0.0 eV
VM Ayres, ECE802-604, F13
Graphene E-k:
VM Ayres, ECE802-604, F13
Bottom of the conduction band: the 6 equivalent K-points
E
ky
kx
VM Ayres, ECE802-604, F13
What you can do with an E-k diagram:
Example: What is “k” in 2D? In 1D?
VM Ayres, ECE802-604, F13
What you can do with an E-k diagram:
Answer:
VM Ayres, ECE802-604, F13
Lecture 23 & 24, 19 Nov 13
Carbon Nanotubes and Graphene
CNT/Graphene electronic properties
sp2: electronic structure
E-k relationship/graph for polyacetylene
E-k relationship/graph for graphene
E-k relationship/graph for CNTs
R. Saito, G. Dresselhaus and M.S. Dresselhaus
Physical Properties of Carbon Nanotubes
VM Ayres, ECE802-604, F13
Graphene:
the 6 equivalent K-points  Bottom of the conduction band
the 6 equivalent K-points  metallic
E
ky
kx
Therefore: CNTs are metallic at the conditions for the K-points of graphene
VM Ayres, ECE802-604, F13
Rules for finding the electronic structure (p. 21):
1
Use to find k
2
Find k
3
Find HAA, HAB, SAA, SAB
Find det|H – ES| = 0 => E = E(k)
4
Plot E versus k
VM Ayres, ECE802-604, F13

sp2 electronic structure: CNTs
– Real space (Unit cell)
– Reciprocal space
– Use Real and Reciprocal space to find E
VM Ayres, ECE802-604, F13
CNT Unit cell in green:
C h = n a1 + m a2
|Ch| = a√n2 + m2 + mn
dt = |Ch|/p
cos q = a1 • Ch
|a1| |Ch|
T = t1 a1 + t2 a2
t1 = (2m + n)/ dR
t2 = - (2n + m) /dR
dR = the greatest common
divisor of 2m + n and 2n+ m
N = | T X Ch |
| a1 x a2 |
= 2(m2 + n2+nm)/dR
VM Ayres, ECE802-604, F13
Example: Evaluate K1 for a (4,2) CNT:
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In class:
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In class:
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Reciprocal space
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VM Ayres, ECE802-604, F13
Example: Evaluate K2 for a (4,2) CNT:
VM Ayres, ECE802-604, F13
VM Ayres, ECE802-604, F13
VM Ayres, ECE802-604, F13
VM Ayres, ECE802-604, F13
Example: add a set of axes
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Answer:
ky
kx
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VM Ayres, ECE802-604, F13
0 through 27  28 of these:
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ECNT is quantized
VM Ayres, ECE802-604, F13
VM Ayres, ECE802-604, F13
Example:
For a (4,2) CNT evaluate:
Ch,|Ch|, T, |T|, K1, K2, |K1|, |K2|
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VM Ayres, ECE802-604, F13
VM Ayres, ECE802-604, F13
VM Ayres, ECE802-604, F13
VM Ayres, ECE802-604, F13
VM Ayres, ECE802-604, F13
Example:
Compare |b1|, |b2|, with |K1|, |K2| for a (4, 2) CNT
VM Ayres, ECE802-604, F13
VM Ayres, ECE802-604, F13
VM Ayres, ECE802-604, F13
CNT E-k; Energy dispersion relations (E vs k curves):
Quantization of Energy E
is here
K1 is quantized by m
in Ch direction
K2 = k is continuous
in T direction
VM Ayres, ECE802-604, F13