ECE 802-604:
Nanoelectronics
Prof. Virginia Ayres
Electrical & Computer Engineering
Michigan State University
ayresv@msu.edu
Lecture 24, 21 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
VM Ayres, ECE802-604, F13
VM Ayres, ECE802-604, F13
VM Ayres, ECE802-604, F13
Graphene:
Real space
Reciprocal space
VM Ayres, ECE802-604, F13
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
VM Ayres, ECE802-604, F13
VM Ayres, ECE802-604, F13
For a (4,2) CNT:
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Reciprocal space
(4, 2,) CNT
Graphene
Real space
T
C
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K1 is in same direction as Ch
Specify direction of Ch using choral angle
K2 is in same direction as T
Ch is the diameter direction
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
|T| = √ 3(m2 + n2+nm)/dR = √ 3|Ch|/dR
N = | T X Ch |
| a1 x a2 |
= 2(m2 + n2+nm)/dR
VM Ayres, ECE802-604, F13
K1 is in same direction as Ch
Specify direction of Ch using chiral angle
K2 is in same direction as T
Ch is the diameter direction
Can only fit an integral number of ewavelengths around the diameter
Can only fit an integral number of ewavenumbers K1 around the
diameter
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VM Ayres, ECE802-604, F13
(4,2): 0 through 27  28 of these:
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In blue box: a unit vector in the K2 direction
The K2 direction is the same as T: along the length of the CNT
k is continuous so this is an E-k relationship.
k depends on (n,m)
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CNT E-k; Energy dispersion relations (E vs k curves):
Quantization of Energy E
is here
Quantization of ewavefunction by m
in Ch / K1 direction
k is continuous
In T/ K2 direction
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CNT E-k; Energy dispersion relations (E vs k curves):
VM Ayres, ECE802-604, F13
Energy dispersion relations (E vs k curves) for a CNT:
Example: How many E vs k curves are
there for the (4,2) CNT?
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Energy dispersion relations (E vs k curves) for a CNT:
Answer: 2N = N x E+(k) + N x E-(k)
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Graphene:
the 6 equivalent K-points  Bottom of the conduction band
the 6 equivalent K-points  metallic
E
ky
kx
This factor slices the graphene Eg2D
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Consider (4,2) CNT:
m=9
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Near a K- point = metallic:
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Condition for hitting a K-point:
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Consider a (n, n) armchair CNT:
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Consider a (n, n) armchair CNT:
(n, n) armchair CNTs
are all metallic:
n
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Consider a (n, 0) zigzag CNT:
(n, 0) zigzag CNTs
are metallic when n
is a multiple of 3.
0
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Consider a (n, 0) zigzag CNT:
Example: metallic or
semiconducting?
(9,0) = ?
(10,0) =?
0
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Consider a (n, 0) zigzag CNT:
Answer:
(9,0) = metallic
(10,0)
= semiconducting
0
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General Energy dispersion relations
(E vs k curves) for a CNT:
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Consider an (n, n) armchair CNT.
You can write a periodic boundary condition on kx and
substitute into eq’n 2.29. That leaves just ky as open, MD
calls it just k.
VM Ayres, ECE802-604, F13
VM Ayres, ECE802-604, F13
Consider an (n, 0) zigzag CNT.
You can write a periodic boundary condition on ky and
substitute into eq’n 2.29. That leaves just kx as open, MD
calls it just k.
VM Ayres, ECE802-604, F13
VM Ayres, ECE802-604, F13
Example: Zigzag: (n,0)
What is the condition on
n from this E-k diagram?
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Answer: Zigzag: (n,0)
Diagram shows its
metallic: n is a multiple
of 3
VM Ayres, ECE802-604, F13