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ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Handout 31
Carbon Nanotubes: Physics and Applications
In this lecture you will learn:
• Carbon nanotubes• Energy subbands in nanotubes• Device applications of nanotubes
Paul L. McEuen (Cornell University)
Sumio lijima (Meijo University, Japan)) Mildred Dresselhaus
(MIT)
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Another Look at Quantum Confinement: Going to Reduced Dimensions by Band SlicingQuantum Well Quantum Wire
zk
xk
E
e
z
e
xpczxc m
kmk
EEkkpE22
,,2222
1
x
y
z
L
Lkx
2
Lkx
e
zpczc m
kEkpE
2,
22
1
cE
zk
E
1cE11 EEc
x
y
z
2
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Single wall carbon nanotube (SWNT)
Graphene and Carbon Nanotubes
a
a
x
y
a = 2.46 A
• 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
32a
3a
Multi wall carbon nanotube (MWNT)
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
K
K’
K
K’
K’
K
M
FBZM
kfVEkE ppp
xk
yk
Graphene: -Energy BandsEnergy
FBZ
rueruer knykxki
knrki
knyx
,,
.,
321 ... nkinkinki eeekf
a32
a32 a3
4
Recall the energy bands of graphene:
3
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Armchair edge
Zigzag edge
Graphene Edges
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Zigzag nanotube
Armchairnanotube
Rolling Up Graphene
4
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Zigzag Nanotubes: Crystal Momentum Quantization
a
Circumference of the zigzag nanotube:
.......4,3,2 mmaC
ruer knykxki
knyx
,,
Boundary condition on the wavefunction:
x
y
The wavefunction must be continuous along the circumference after one complete roundtrip:
range? integer,2
1
,,,, ,,
nC
nk
e
zyxzCyx
y
Cik
knkn
y
The crystal momentum in the y-direction (in direction transverse to the nanotube length) has quantized values
C
L
CL
Periodicity in the x-direction: a3
a3
Primitive cell
Number of atoms in the primitive cell: m4
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Zigzag Nanotubes: 1D Energy SubbandsEnergy
FBZ
K
K’
K
K’
K’
K
M
FBZ
Mxk
yk
a32
a32 a3
4
C2
ak
a x 33
Obtain all the 1D subbands of the nanotube by taking cross sections of the 2D energy band dispersion of graphene
kfVEkE ppp
One will obtain two subbands (one from the conduction and one from the valence band) for each quantized value of But number of bands = number of orbitals per primitive cell =
Number of distinct quantized values must equal 2m
ykm4
maC
yk
mmnC
nky ,.......,1,0,1,.......1
2
5
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Zigzag Nanotubes: 1D Energy SubbandsEnergy
FBZ
K’
K
K’
K’
K
M
FBZ
Mxk
yk
a32
a32 a3
4
C2
ak
a x 33
kfVEkE ppp
Suppose C = 4a (i.e. m = 4)
4,3,2,1,0,1,2,32
2 n
an
Cn
ky
16 1D subbands total
Lower 8 subbands will be completely full at T=0KThe nanotube is a semiconductor!
Bandgap!
K
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Zigzag Nanotubes: 1D Energy Subbands
'yKyk
'xKxk
Bandgap!
K'
The bandgap appears because the quantized ky value is such that the “green line” misses the K-point
When: (R = radius of nanotube)aR
RRv
Eg1
32
K’
K
K’
K’
K
M
FBZ
Mxk
yk
a32
a32 a3
4
C2
ak
a x 33
K
6
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Zigzag Nanotubes: Semiconductor and Metallic Behavior
K
K’
K
K’
K’
K
M
FBZ
Mxk
yk
a34
ak
a x 33
Suppose C = 6a (i.e. m = 6)
6,......1,0,1,....53
2 n
an
Cn
ky
Two lines for n=4 pass through the Dirac points
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
C2
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Motion of Conduction Band Bottom Electrons in Zigzag Nanotubes ruer kn
ykxkikn
yx
,,
1
Cikye
x
• The electrons coil around the nanotube as they move forward
• The direction of coiling can be given by the right hand rule:
or by the left hand rule
Direction of propagation
mmnC
nky ,.......,1,0,1,.......1
2
For ky – K (K’) > 0
x
For ky – K (K’) < 0
y
y
7
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Armchair Nanotubes: Crystal Momentum Quantization
a
Circumference of the armchair nanotube:
.......4,3,23 mamC
ruer knykxki
knyx
,,
Boundary condition on the wavefunction: x
y
The wavefunction must be continuous along the circumference
range? integer,2
1
nC
nk
e
x
Cikx
The crystal momentum in the x-direction (in direction transverse to the nanotube length) has quantized values
C
L
CL
Periodicity in the y-direction: a
a3
Primitive cell
Number of atoms in the primitive cell: m4
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Armchair Nanotubes: 1D Energy SubbandsEnergy
FBZ
K
K’
K
K’
K’
K
M
FBZ
Mxk
yk
a32
a32
C2
ak
a y
Obtain all the 1D subbands of the nanotube by taking cross sections of the 2D energy band dispersion of graphene
kfVEkE ppp
One will obtain two bands for each quantized value of
But number of bands = number of orbitals in the primitive cell =
Number of distinct quantized values must equal 2m
xk
m4
maC
xk
mmnC
nkx ,.......,1,0,1,.......1
2
8
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Armchair Nanotubes: 1D Energy SubbandsEnergy
FBZ
kfVEkE ppp
Suppose C = 4√3 a (i.e. m = 4)
4,3,2,1,0,1,2,332
2 n
an
Cn
kx
16 1D subbands total
Lower 8 subbands will be completely full at T=0KThe nanotube has a zero bandgap!
K
K’
K
K’
K’
K
M
FBZ
Mxk
yk
a32
C2
ak
a y
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Armchair Nanotubes: Metallic BehaviorEnergy
FBZ
Suppose C = m√3 a
mmnam
nC
nkx ,........,1,0,1),......1(
322
K
K’
K
K’
K’
K
M
FBZ
Mxk
yk
a32
C2
ak
a y
Armchair nanotubes always have a zero bandgap
For n = m : and the line passes through the Dirac pointsa
kx 32
Proof:
9
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Carbon Nanotubes: Applications
CNT
Nanotube PN Diode(McEuen et. al.)
CNT MEMs
AFM Image
CNT field emission tips for electron guns
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Carbon Nanotubes: Applications
Carbon Nanotube LEDs (IBM)
Carbon Nanotube FET (IBM)
Carbon Nanotube FET (Burke et. al.)
10
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Carbon Nanotubes: Applications
Carbon Nanotube Space Elevator !!
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……
