Numerical study of transport properties
of carbon nanotubes
• Dhanashree Godbole
• Brian Thomas
Summer Materials Research Training
Oakland University 2006
Overview
 CNT: The material of the future
 Structure of Graphene and CNT’s
 Carbon nanotubes: bands and DOS
 Infinite CNT’s to quantum dots
 Quantum dots: what and why
 Numerical results:
• Coulomb Blockade
• Kondo effect
Why Carbon nanotubes?
 Amazing electronic and physical
properties
• Space and mass saving
• Strength and durability
The Graphene Layer
 The graphene sheet and its band structure
(n,m)
CNT: band structure and DOS
1/ 2
a
Eq

 q   ka 
2  ka  
(k )  t 1  4 cos
 cos   4 cos  
 n   2
 2 

   ka    , q  1,...,2n
,
DOS
An Infinite CNT to finite CNT
Finite CNT to Quantum Dot
 Artificial atoms
 Transfer of a single electron charge
Coulomb Blockade
 Increased resistance
No conduction
Conduction
Kondo Effect
 Low-temp. increase in resistance
 Kondo resonance creates existence of a
new state
Numerical Results: One Level QD
-2
0
1
2
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
2
for the electron
G(e /h)
 One available state
-1
-2
-1
0
1
2
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
1.0
1.0
0.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
-2
-1
0
1
2
2
G(e /h)
Vg
-3
0.0
0
Vg
1
-1
0
1
2
2
2
-1
G(e /h)
-2
-2
0.0
0.0
-3
-2
-1
0
Vg
1
2
CNT as a Quantum dot
 Carbon nanotubes have two available
states for electrons to propagate in
2.0
2.0
1.8
1.6
1.5
1.2
2
G(2e /h)
G(2e2/h)
1.4
1.0
0.8
1.0
0.5
0.6
0.4
0.2
0.0
0.0
-1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2
Vg
-3
-2
-1
Vg
0
1
CNT as Quantum dot
 B field kills Kondo effect…why?
1.0
2
G(2e /h)
2
G(2e /h)
1.0
0.5
0.0
0.5
0.0
-3
-2
-1
Vg
0
1
-3
-2
-1
0
Vg
1
2
Experimental Results
b
0.0
U=U'=0.5
t'=0.2
t"=0.0
orb=0.2
a
2.0
1.5
1.0
1.0
1.5
3.0
B
0.5
G(2e2/h)
sp=0.04
Esp
U=U'=0.5
t'=0.2
t"=0.0
E=0.035
Eorb
orb=0.2
0.5
sp=0.04
-2.5 -2.0 -1.5 -1.0 -0.5
0.0
0.5
0.0
1.0
Vg
Delft University - Netherlands
Prof. Leo Kouwenhoven group
-2.0 -1.5 -1.0 -0.5 0.0
Vg
0.5
1.0
1.5
Conclusion
 What we learned
• Basics of quantum mechanics and its
applications in condensed matter physics
• First time using computer code and
programming with FORTRAN
• Energy dispersion relations and their use in
research of CNTs
• Quantum Dots and the use of CNTs as QDs
• Modeling and using numerical operations to
represent real systems.