Mechano-electrical properties of metallic
carbon nanotubes
An Honors Thesis by:
Evan Wilson
Advisor:
Dr. Ronald Cosby
Ball State University
Muncie, Indiana
Expected Graduation: May 2010
Mechano-electrical properties of metallic
carbon nanotubes
An Honors Thesis by:
Evan Wilson
(HONRS 499)
Advisor:
Dr. Ronald Cosby
f<~d!i2 JrC~
May 2010
Ball State University
Muncie, Indiana
Expected Graduation: May 2010
'.-'((
Mechano-e'lectrical properties of metallic carbon nanotubes
Abstract
Single-walled metallic (4,4) nanotubes are simulated using proprietary density functional theory
software. Current voltage curves and conductance are calculated for varying axial strains.
Density of states calculations are then made in an attempt to understand the strain-conductance
relationship.
Acknowledgements
This study would not have been possible without the tutelage and knowledge of Dr.
Ronald Cosby. His exactitude and advise have been a constant benefit. I would like to
acknowledge the Ball State Honors College for their financial support via the Undergraduate
Fellowship. I would also like to thank Feras Alzubi for his advice when I was beginning my
research.
Introduction
occur when O<m<n [1]. This may seem a
Carbon nanotubes (CNT's) are allotropes of
bit abstruse. However, one can think of
carbon that have received a great deal of
creating a nanotube by taking a sheet of
attention in the past two decades because of
graphene, which looks a bit like chicken-
their dizzying array of practical applications.
wire, and cutting out a section. It is easy
CNT's are so highly considered because
then to visualize rolling this chicken-wire
they have some extraordinary properties.
into a tube by joining the comers of our
They can be semiconducting, or metallic,
section. This yields an armchair chicken-
and have high flexibility and high strength.
wire tube. If one were to now put a twist on
While all nanotubes are all essentially tiny
the mesh before rolling it up, such that the
cylinders of carbon, they can have
comers were joined to vertices other than
remarkable variability in their structure.
the corresponding comers, a tube with a
The most grossly visible property is that
different chirality would be obtained. It
CNT's come in single-walled and multi-
should be noted that this is not in fact how
walled varieties. In this study the nanotubes
CNT's are constructed; rather it is a useful
in question were single-walled. The next
important structural variable in CNT's is
known as the chirality. Chirality in the
context of carbon nanotubes is described by
11,0) zigzag
the chiral vector (n,m). Three separate
classes of nanotube are defined in this way.
Armchair nanotubes have n=m, zig-zag
nanotubes are m=O and chiral nanotubes
(IIJI) armchair
Figure 1: Shows the chiral vector and the chickenwire like molecular lattice of graphene [2J
2
mental artifice used for visualizing the
produced by the Quantum Wise company.
difference in structure that occurs in these
This software is run on the Ball State
nanotubes. However, the chirality is not
University College of Science and
merely a matter of the geometry ofthe
Humanities Computing Cluster. ATK
constituent atoms. It also determines the
comes with a native graphical user interface
way in which the nanotube conducts
for generating scripts to run called Virtual
electrical currents. Armchair nanotubes are
NanoLab(VNL). It can be used to develop
exclusively metallic [3]. This study solely
whole systems for investigation or
examines single-walled, (4,4), metallic
generating useful bits of codes for inclusion
nanotubes.
in one's own scripts. ATK can simulate
The purpose of this study is to
semi-infinite bulk systems, molecules or two
examine the electrical properties of single
probe systems. It can also perform
walled armchair, (4,4) nanotubes under axial
complicated transport calculations to
compression. Using proprietary modeling
determine the electrical properties of a
software, we examine the conductance-
system. To do this, ATK implements two
strain relationship for a 48-atom carbon
basic techniques, Density Functional Theory
nanotube segment between copper contacts.
(DFT) and Non-Equilibrium Green's
We also present density of states data in an
Functions (NEGF) [4].
attempt to fundamentally describe the
A central problem of quantum
conductance behavior.
chemistry and condensed matter physics is
Methods and Techniques
that the equation which governs objects at
The software utilized for this study is
the Atomistix ToolKit (ATK) that is
the scale of molecules and smaller, the
Schroedinger Equation is intractable when
3
dealing with a system of more than three or
four particles. A solution in these cases is
not difficult to achieve, it is impossible.
Therefore several models and
approximations exist to try to escape this
Figure 2: A 10 atom linear carbon chain
difficulty. One such method is DFT. DFT
uses the real space density of electrons in the
ground-state in place of the many-body
problem ofN electrons and 3N coordinates.
The downside of this approach is that the
energies that result from this calculation are
dependent on the electron density and the
electron density is dependent on the energy
configuration of the system. Therefore, the
accuracy of the calculation depends on the
degree to which this self-consistent
calculation converges. A TK uses Density
Functional Theory to calculate energies,
electron densities, atomic forces and other
properties. NEGF methods are the standard
when constructing models of transport in
systems where quantum effects must be
accounted for. It is used in ATK to calculate
properties like current. It too results in a
self-consistent calculation of the electron
density that must be iterated until it
converges to some desired tolerance. A
linear carbon chain was chosen as a test
system from parameterization purposes
because it was small, undemanding in terms
of computation time, and bears a superficial
resemblance to the carbon nanotube system
that is of true interest. The specific
parameters that the linear chain is used to
probe are called the mesh cutoff, the energy
shift, and the self-consistent field tolerance
[4]. The mesh cutoff determines the fineness
of the real space grid of points that ATK
samples to perform the iterative selfconsistent calculation. A larger mesh cutoff
indicates a finer grid, requiring ATK to
4
calculate electron densities at more points
consistent field tolerance was found to have
and taking longer to calculate, but providing
a negligible effect on the value of total
more accurate values for quantities like
energy. Complete parameter data is
structure and energy. The energy shift
included in the appendix.
The primary
determines the cutoff radius of electron
wavefunctions. True wavefunctions extend
system we are
outwards to infinity, though the magnitude
examining is a small
of the wavefunction approaches zero.
segment of nanotube
Practically we must designate a cutoff radius
between copper
to avoid calculations that are not only time-
contacts with three
consuming, but useless. The energy shift is
scattering layers.
inversely related to this radius, smaller
energy shifts indicate larger radii. This costs
calculation time, but can capture the effects
of longer range, non-nearest-neighbor
interactions. The self-consistent field
tolerance is the degree of accuracy that we
ask the self-consistent calculation to
converge to [4]. To investigate the
parameter space a script was written to
calculate the total energy of the system as a
function of the mesh cutoff and energy shift.
During the investigation process the self-
The segment is
unrealistically small,
Figure 3: A 3 unit cell carbon
nanotube segment between
two copper electrodes with
three scattering layers
only 48 atoms. This
was chosen to reduce computation time to
manageable levels.
Goals
If nanotubes become widespread as
electrical components in devices, it is likely
that the nanotubes will at some point be
deformed or strained. Therefore, it makes
good sense to understand the effects this will
have on the device. This understanding may
5
Compressive Strain
160
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also lead to knowledge that would allow the
electrical properties of carbon nanotubes to
8
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be fine-tuned.
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Results
Current-voltage (I-V) curves were
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segtne3.: ber;ve-~ copper ccnta:-t\
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calculated for varying strain. The slopes of
2
4
6
R
Strain (%)
Figure 4
these curves are the conductances of the
conductance tends to go up because we are
nanotube under these strains. Figure 5 is an
reducing the distance electrons must travel
example I-V curve. It is linear and therefore
under a given voltage. One may think of a
Figure 5: Example I-V curves
classical wire: the resistance is inversely
has a simply defined conductance.
These conductance results were collected
and are presented versus strain in Figure 7.
related to the length of the wire. Second,
compressing the tube introduces a bandgap
as has been experimentally shown by Minot
The results were a bit unexpected. The
conductance increases until 6% and then
et al [4]. It is worth noting that strains of
more than a few percent may be unphysical
8
as buckling and breakage may occur before
,--.,
1
6
then.
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Example
I-V Data
--+6% strain
_ _ _ _ 10·0
2
°0
10
20
30
40
50
Voltage (mV)
decreases. We believe this is due to two
competing phenomena. First, the
-
10
6
Density of States with Zero Strain
14000
illuminating. To better
12000
10000
Difference in DOS: 10% and 0%
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Density of States for CNT Segment with 10% Strain
.. ..
12000
10000
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Energy (eV)
Figure 6
.. ..
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-500
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DOS
Figure 8
l-
see these changes we took the differences
8000 l6000 l-
between the density of states for 10% and
4000 l-
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Gooe·.ooooooo
0
-2000
2.5
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7.5
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6% and 10% and 0% respectively. These
are shown as Figures 8
Energy leV)
Difference in Density of States: 10% and 6%
Figure 7
.
We have attempted to verify that this
is indeed the mechanism by which the
2000
1500 l-
.
0
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1000 I-
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. .. .. e .
.
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conductance decreases by examining the
500 l-
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density of states (DOS) of the nanotube
- rJ'.
.
.
-500
-1000
••...1.
-"-
10
5
o·
0
0
under zero and 10% strains. These DOS
plots (Figures 8 and 9) are not particularly
-"-
_50
0
~
Energy (eV)
.
Figure 9
and 9. The data is very scattered. The
difference between from 10% to 0% seems
superficially larger than the difference
between 10% and 6%. However, no strong
conclusions can be drawn from this. We
7
now believe that perhaps the total density of
until local density of states data can be
states of the entire system too closely
calculated.
resembles that of the copper scattering
Acknowledgements
layers rather than that of the CNT segment.
This study would not have been
To find out if the compressive strain is in
possible without the tutelage and knowledge
fact altering the density of states of the
of Dr. Ronald Cosby. His exactitude and
CNT, a local density of states calculation
advise have been a constant benefit. I would
will be required.
like to acknowledge the Ball State Honors
Conclusions
College for their financial support via the
Conductance-strain data for the 48 atom
Undergraduate Fellowship. I would also
CNT between copper contacts was
like to thank Feras Alzubi for his advice
presented. Our hypothesis about bandgap
when I was beginning my research.
formation via strain remains unsupported
8
[1] M. S. Dresselhaus, G. Dresselhaus, Ph. Avouris, Carbon Nanotubes Synthesis, Structure,
Properties, and Applications, Springer (2001).
[2] Image from Wikimedia Commons. File is a part of the public domain.
[2] S. Reich, C. Thomsen, 1. Maultzsch, Carbon Nanotubes: Basic Concepts and Physical
Properties, Wiley-VCH (2004)
[3]Atomistix ToolKit version 8.1, Quantum Wise A/S (www.quantumwise.com).
[4] E. D. Minot, Y. Yaish, V. Sazonova, 1. Y. Park, M. Brink, P. L. McEuen Phys. Rev. Lett.
[5] R. Saito, G. Dresselhaus, M.S. Dresselhaus, Physical Properties of Carbon Nanotubes,
Imperial College Press (1998).
9
Appendix: Parameterizations
Com~utational
Parameters Used
The results of the parameterizations using the
Mesh Cutoff - 150 Rydbergs
linear carbon chain test case are shown below in
Energy Shift - 0.001 Rydbergs
Figure 10. The data is well converged in total
Exchange Correlation - Local Density
Approximation.PZ
energy except for a large jump at approximately
0.0006 Rydbergs. At this point, the jump in total
SCF Tolerance - 10-4
Basis Set - C -Double Zeta Polarized
energy is unexplained. However, this anomaly
Basis Set - Cu - Single Zeta
does not appear to affect the calculated quantities
Electron Temperature - 300k
Total Energy vs. Mesh Cutoff and Energy Shin
-1556.5
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-1558.0
220
200
180
160
Mesh Cutoff (Ryd) 140
-1558.5
0.0002
0.0000
120
100
0.0010
0.0008
Figure 10: The complete parameter space for the linear carbon chain.
Energy Shift {Ryd}
10
-------- - - - -
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Total Energy vs. Lattice Parameter for varying Energy Shifts
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1.21
1.22
1.23
1.24
1.25
1.26
1.27
1.28
1.29
1.30
1.31
1.32
1.33
1.34
1.35
Lattice Parameter (Angstroms)
Figure 7: Multiple curves corresponding to multiple energy shifts; all have the same equilibrium lattice
parameter
that are of interest. Figure 11 shows multiple curves of lattice parameter versus total energy.
Again the jump is visible, but each curve has the same total energy minimum and thus the same
equilibrium lattice parameter. Therefore, as long as we avoid using the value of 0.0006, we feel
that this anomaly will not affect results.