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 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
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Graphene
Real space
Reciprocal space
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CNT Unit cell in green
Ch n a1 m a2 Ch avn2 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
n2nm)/dR
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For a (4,2) CNT
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Real space
Reciprocal space
Graphene
(4, 2,) CNT
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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
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CNT Unit cell in green
Ch n a1 m a2 Ch avn2 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
v 3(m2 n2nm)/dR v 3Ch/dR N T X Ch
a1 x a2 2(m2 n2nm)/dR
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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 e- wavelengths around the
diameter Can only fit an integral number of e-
wavenumbers K1 around the diameter
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(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 e-wavefunction 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)
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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
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(n, n) armchair CNTs are all metallic
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Consider a (n, 0) zigzag CNT
(n, 0) zigzag CNTs are metallic when n is a
multiple of 3.
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Consider a (n, 0) zigzag CNT
Example metallic or semiconducting? (9,0)
? (10,0) ?
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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 eqn 2.29. That leaves just ky as open, MD
calls it just k.
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Consider an (n, 0) zigzag CNT. You can write a
periodic boundary condition on ky and substitute
into eqn 2.29. That leaves just kx as open, MD
calls it just k.
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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