Handout 31
Carbon Nanotubes: Physics and Applications
In this lecture you will learn:
• Carbon nanotubes
• Energy subbands in nanotubes
• Device applications of nanotubes
Sumio lijima
(Meijo University, Japan))
Paul L. McEuen
(Cornell University)
Mildred Dresselhaus
(MIT)
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Another Look at Quantum Confinement: Going to Reduced
Quantum Well
Dimensions by Band Slicing Quantum Wire
E c  
y
y
x
z
E c  p, k x , k z   E c 1  E p 
E
x
z
 2 k x2  2k z2

2me 2me
kx 
E c  p, k z   E c 1   p 
E

L
kx  2
L
 2 k z2
2me

L
E c 1  E1
Ec1
kx
kz
kz
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
1
Graphene and Carbon Nanotubes
y
a = 2.46 A
a
Single wall carbon
nanotube (SWNT)
x
Multi wall carbon
nanotube (MWNT)
a
a
3
2a
3
• Carbon nanotubes are rolled up graphene sheets
• Graphene sheets can be rolled in many different
ways to yield different kinds of nanotubes with very
different properties
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Graphene: -Energy Bands
Energy
Recall the energy bands of graphene:
ky
K
2
3a
K’

M
2
3a
K’
M
4
3a
kx
FBZ
K
K
FBZ
K’




E k  E p  Vpp f k
 
 
 

f k  e ik .n1  e ik .n2  e ik .n3


 

 n,k r   e ik .r un,k r   e

i k x x ky y
u

n ,k

r 
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
2
Graphene Edges
Armchair edge
Zigzag edge
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Rolling Up Graphene
Zigzag
nanotube
Armchair
nanotube
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
3
Zigzag Nanotubes: Crystal Momentum Quantization
y
L
Primitive
cell
L  C
C
Circumference of the zigzag nanotube:
m  2,3,4.......
C  ma
Boundary condition on the wavefunction:

 n,k r   e

i k x x k y y
u

n,k
a

r 
x
The wavefunction must be continuous along the
circumference after one complete roundtrip:
3a
 n,k  x , y  C, z    n,k  x , y , z 


ik y C
1
2 n
ky 
C
e
n  integer, range?
The crystal momentum in the y-direction (in
direction transverse to the nanotube length)
has quantized values
3a
Periodicity in the x-direction:
Number of atoms in the
primitive cell: 4 m
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Zigzag Nanotubes: 1D Energy Subbands
Energy
Obtain all the 1D
subbands of the
nanotube by taking
cross sections of the
2D energy band
dispersion of
graphene
K
2
3a
FBZ


E k  E p  Vpp f k

ky
FBZ

2
3a
K’
K’
M

M
K
2
C
K


3a
K’
 kx 
4
3a
kx

3a
One will obtain two subbands (one from the conduction and one from the valence
band) for each quantized value of k y
But number of bands = number of orbitals per primitive cell = 4 m
 Number of distinct quantized k y values must equal 2m
ky 
2 n
C
n  m  1,.......  1,0,1,......., m
C  ma
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
4
Zigzag Nanotubes: 1D Energy Subbands
Energy
ky
FBZ
2
3a
K
2
3a
K’
K’
M

M
K
K’
FBZ


3a
4
3a
kx
2
C
K

 kx 
3a
Suppose C = 4a (i.e. m = 4)
2 n  n

C
2a


E k  E p  Vpp f k

Bandgap!
ky 

n  3,2,1,0,1,2,3,4

16 1D subbands
total
Lower 8 subbands will be completely full at T=0K
The nanotube is a semiconductor!
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
ky
FBZ
Zigzag Nanotubes: 1D Energy Subbands
2
3a
K
2
3a
K’
K’
M

M
kx
2
C
K
4
3a
K
K’


3a
 kx 

3a
The bandgap appears because the quantized
ky value is such that the “green line” misses
the K-point
k y  K 'y
K'
k x  K 'x
Bandgap!
When: R  a
Eg 
(R = radius of nanotube)
2v
1

R
3R
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
5
Zigzag Nanotubes: Semiconductor and Metallic Behavior
ky
Suppose C = 6a (i.e. m = 6)
FBZ

K
4
3a
ky 
2 n  n

C
3a
n  5,....  1,0,1,......6
Two lines for n=4 pass through the Dirac points
K’
K’
M
K

M
2
C
K
kx
K’


3a
 kx 

3a
24 1D subbands total, 12 lower ones will be completely
filled at T=0K, and there is no bandgap!
• All zigzag nanotubes for which m = 3p (p any integer) will have a zero bandgap
 All zigzag nanotubes with radius R = C/2= 3pa/2 (p any integer) will have a
zero bandgap
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Motion of Conduction Band Bottom Electrons in Zigzag
Nanotubes


i k x  k y y 
 n ,k r   e x
un ,k r 
ik C
 e y 1
2 n
 ky 
C
n  m  1,.......  1,0,1,......., m
For ky – K (K’) > 0
y
• The electrons coil around the nanotube
as they move forward
x
• The direction of coiling can be given
by the right hand rule:
Direction of
propagation
y
For ky – K (K’) < 0
x
or by the left hand rule
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
6
Armchair Nanotubes: Crystal Momentum Quantization
Primitive
cell
y
L
L  C
C
Circumference of the armchair nanotube:
m  2,3,4.......
Cm 3a
a
Boundary condition on the wavefunction:

 n,k r   e

i k x x ky y
u

n ,k
x
3a

r 
The wavefunction must be continuous
along the circumference

e ik x C  1

kx 
2 n
C
n  integer, range?
The crystal momentum in the x-direction (in
direction transverse to the nanotube length)
has quantized values
Periodicity in the y-direction: a
Number of atoms in the
primitive cell: 4 m
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Armchair Nanotubes: 1D Energy Subbands
Energy
Obtain all the 1D
subbands of the
nanotube by taking
cross sections of the
2D energy band
dispersion of
graphene
FBZ
ky
K
K’
M
K’
2
3
a
M
kx

K
FBZ


E k  E p  Vpp f k




a
 ky 
2
3a

a
K’
2
C
K
One will obtain two bands for each quantized value of k x
But number of bands = number of orbitals in the primitive cell = 4m
 Number of distinct quantized k x values must equal 2m
kx 
2 n
C
n  m  1,.......  1,0,1,......., m
C  ma
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
7
Armchair Nanotubes: 1D Energy Subbands
Energy
ky
FBZ
K
2
3a
K’
K’
M

M
kx
K

FBZ

a
 ky 
K

K’
a
2
C
Suppose C = 4√3 a (i.e. m = 4)
 n
2 n

2 3a
C


E k  E p  Vpp f k

kx 

n  3,2,1,0,1,2,3,4

16 1D subbands
total
Lower 8 subbands will be completely full at T=0K
The nanotube has a zero bandgap!
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Armchair Nanotubes: Metallic Behavior
Energy
ky
FBZ
K
K’
K’
M

M
kx
K
FBZ


a
 ky 

a
2
3a
K
2
C
K’
Armchair nanotubes always have a zero bandgap
Proof:
Suppose C = m√3 a
2 n
2 n
n  (m  1),......  1,0,1,........, m

C
m 3a
2
For n = m : k x 
and the line passes through the Dirac points
3a

kx 
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
8
Carbon Nanotubes: Applications
CNT
AFM Image
Nanotube PN Diode
(McEuen et. al.)
CNT MEMs
CNT field
emission tips
for electron
guns
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Carbon Nanotubes: Applications
Carbon Nanotube FET (IBM)
Carbon Nanotube LEDs (IBM)
Carbon Nanotube FET (Burke et. al.)
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
9
Carbon Nanotubes: Applications
One main obstacle to making a space
elevator is finding a material for the cable
that is strong enough to withstand a huge
amount of tension. Some scientists think
that cables made from carbon nanotubes
could be the answer……
Carbon Nanotube Space Elevator !!
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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