ELECTRONIC STRUCTURE CALCULATIONS OF
DOUBLE WALLED CARBON NANOTUBES
USING LOCAL DENSITY
FUNCTIONAL THEORY
By
BENJAMIN A. LANDIS
Bachelor of Science
Emporia State University
Emporia, KS
2001
Submitted to the Faculty of the
Graduate College of the
Oklahoma State University
in partial fulfillment of
the requirements for
the Degree of
MASTER OF SCIENCE
December, 2005
ii
ELECTRONIC STRUCTURE CALCULATIONS OF
DOUBLE WALLED CARBON NANOTUBES
USING LOCAL DENSITY
FUNCTIONAL THEORY
Report Approved:
John Mintmire
Thesis Adviser
Bret Flanders
Bob Hauenstein
A. Gordon Emslie
Dean of the Graduate College
iii
TABLE OF CONTENTS
Chapter Page
I. Introduction ................................................................................................................1
II. Approach ...................................................................................................................3
III. Results....................................................................................................................12
IV. Summary................................................................................................................18
V. Bibliography............................................................................................................19
iv
LIST OF FIGURES
1. Carbon Nanotube Lattice With example Rollup Vector.......................................4
2. Band Structure of Double Walled Nanotube (26, 21) (33, 1).............................13
3. Band Structure of Outer Nanotube (26, 21)........................................................14
4. Band Structure of Inner Nanotube (33, 1) ..........................................................15
5. Unstrained Binding Energy of the Double Walled Nanotubes...........................16
6. Strained Binding Energy of the Double Walled Nanotubes ...............................17
v
LIST OF TABLES
1. Calculated Double Walled Nanotubes................................................................20
1
I. Introduction
Carbon nanotubes have been a topic of much interest since their discovery by
Iijima [1]. In the years following, a great deal of research has been done on carbon
nanotubes. The physical properties of carbon nanotubes are of particular interest as some
predictions for carbon nanotubes have large values for the elastic constants. Research on
carbon nanotubes differs significantly between experimental and theoretical approaches.
Synthesis methods for carbon nanotubes cannot yet produce specific tubes;
therefore, experimental research is carried out on new tube formation methods or on the
types of nanotubes current methods create. Yu et al. [2] stretched multi-walled carbon
nanotubes inside an SEM by applying a solid carbonaceous deposit to the ends of the
nanotube on two opposing AFM probes. The tubes were then pulled apart to measure
tensile failure of the tubes. The broken tubes were then removed and examined by TEM.
Williams et al. [3] attached a multi-walled nanotube to a silicon wafer using metal
deposition and create a paddle of metal attached to the center of the tube. Using AFM a
force is applied to the paddle causing a torsional stain to the tubes. This is used to
determine the torsional stain properties of the nanotube.
Theoretical studies, however, focus on characterizing the properties of specific
carbon nanotubes. The physical and electrical properties of single wall carbon nanotubes,
SWNTs, have been determined by many groups. However, multi-walled carbon
nanotubes, MWNTs, are also a product of the normal synthetic approaches for SWNTs.
2
Computational studies have been slower to analyze MWNTs because the multi-walled
calculations take much larger amounts of computer time.
Multi-walled nanotubes have not been characterized as well as SWNTs. Most
theoretical groups use numerical calculations to investigate nanotubes. Using an elastic
force model, Lu [4] investigated the elastic moduli of MWNTs, finding that the elastic
moduli are unaffected by number of walls. Lu also reported a large anisotropy in the
properties of SWNTs and MWNTs. Zhao et al. [5] used quantum simulations to estimate
the elastic strength of carbon nanotubes. Their results show that the formation of defects
in the hexagonal structure of nanotubes is similar to that of a graphite sheet. Furthermore
the chirality of the tube affects when the defects become energetically favorable, leading
to the claim that nanotubes may be the strongest known material. Sudak [6] examines the
buckling of MWNTs by using nonlocal continuum mechanics. His findings conclude
that the buckling of MWNTs cannot use the more traditional beam bending model
because of the discrete layers and finite van der Waals interaction. Liu et al. [7] used
reactive empirical bond-order and four different van der Waals potentials to study both
uniaxial tensile behavior and the bending characteristics of double walled nanotubes,
DWNTs, claiming these results to be insensitive to the van der Waals potentials that were
used. The axial elastic modulus decreases for both armchair and zigzag tubes with
increasing radius but the armchair nanotubs have much larger values than zigzag. The
tensile strength is unaffected by diameter but again armchair nanotubes have larger
values. Shen and Li [8] used an energy approach in the framework of molecular
dynamics to determine closed-form solutions to the five independent elastic moduli of
SWNTs. In a second paper the work is extended to encompass MWNTs [9].
3
In this report, DWNTs are studied by using electronic structure simulations using
local density functional theory. The calculations will show that applying small strains to
SWNTs in order to form DWNTs has only a minimal effect on the nanotubes. The
binding energies of the DWNTs will allow an estimation of the minimum inter tube
separation distance.
II. Approach
First, a brief introduction to the terminology used in discussing carbon nanotubes
is necessary. A carbon nanotube can be considered to be a rolled graphite sheet. This
rolling can be characterized by the graphite lattice vector which wraps a round the
circumference of the tube, which will be called the rollup vector. The tube will then be
named by the integer indices of the two primitive lattice vectors that make up this rollup
vector, R R1 R2
r r r
= n1 + n2 , in general forming a (n1, n2) tube. The n2 index will always be
less than or equal to the n1 index or the tube is equivalent to another tube having other
indices. Figure 1 shows the carbon lattice and an example rollup vector for a (6, 5)
nanotube. From these two indices and the atomic spacing of the carbon atoms we can
determine the coordinates of all atoms in the tube; however, due to the symmetry in
carbon nanotubes, only the coordinates of atoms in the unit cell are necessary. All carbon
nanotubes of infinite length have lattice translational symmetry along the tube axis. The
coordinates of all atoms within the unit cell of the tube along with periodic boundary
conditions corresponding to the symmetry are sufficient to define the entire lattice. Using
a helical symmetry instead of a straight translation reduces the size of the unit cell. A
helical unit cell will contain a number of carbon atoms equal to twice the greatest
common factor of the two indices [10]. A helical unit cell can be described as a
4
Figure 1 A carbon nanotube lattice with an example (6, 5) rollup vector.
translation along the tube axis followed by a rotation about the axis or more simply by the
vector of this change. The helical vector is found by solving ± N = n1m2 n2m1 for
integer solutions of m1 and m2, where N is the greatest common factor of n1 and n2. The
helical vector has the formH R1 R2
v r r
= m1 + m2 .
Before any calculations on double walled nanotubes could be started, first a list of
DWNTs needed to be compiled. The helical vectors of both the outer nanotube and the
inner nanotube need to have the same periodicity. This is not a requirement of the
nanotubes themselves but rather a limitation of the band structure approach the program
uses to run the simulations. Forming the list was accomplished by creating all SWNTs
starting from the (3, 0) nanotube and increasing both n1 and n2 until all nanotubes with
indices less than the (37, 37) nanotube had been formed. Determining the acceptable
DWNTs was accomplished by comparing all the nanotubes on this list were to every
5
other nanotube on the list by comparing the vector of helical twist. The comparison did
not check for an exact match but only nanotubes with very similar vectors that can be
adjusted to a match causing as little change as possible. This is a necessary
approximation to form the list of DWNTs. This helical vector comparison included
integer multiples of the helical vectors, limiting the cell size of a SWNT to forty-eight
atoms. After this complete list of DWNTs was formed, the list was trimmed limiting the
total size of the DWNT to fifty-four atoms or less. The list was also reduced by limiting
the inter-tube separation distance to be between 2.8 - 4 Å, a distance roughly centered
about the layer spacing of graphite. The final list of DWNTs is contained in Table 1.
The actual helical vectors must be identical for the calculation to run properly.
If no strain is applied, the two tubes would need to be an exact helical match or
one of the tubes will be mismatched at the helical boundary. Applying slight strains to
the tubes forces a match to any tubes determined to approximately match. There are
three strains of interest to consider in causing the nanotubes to match symmetry with one
another. The first of these strains is the longitudinal strain corresponding to a change in
the length of the tube along its axis. The shear strain is the strain causing a twist in the
tube. The third strain referred to as the transverse strain corresponds to an increase in the
radius of the tube. However, the radius of the tube is not strained for these calculations.
In calculating the proper strain the elastic moduli within a graphite sheet were used as the
elastic moduli for the carbon nanotubes. The longitudinal strain is related to C11 in
graphite and the shear strain to C66. To actually apply strain or calculate strain to be
applied it will be convenient to find the coordinates of the atoms.
6
The coordinates will therefore be created for the two SWNTs matched by helical
symmetry to form the DWNT. The standard graphite unit cell has two atoms separated
by a bond of length 1.42 Å. The helical unit cell of a single walled nanotube will use this
graphitic cell as a basis. However, previous optimizations of the bond length in (5, 5)
nanotube reveals a slightly larger bond length of 1.44 Å which will be used for these
calculations. Depending on the individual nanotube, more graphitic cells may be
required for a single helical cell. In a graphitic unit cell located at the origin, one atom is
placed at half the bond distance above the y-axis the second at the same distance below.
This gives the basis for the unit cell of the tube but in planar form, which will later need
to be transformed into a tube. The first transformation takes each vector, the helical, and
two vectors pointing to the location of the atoms, and uses the transformation equation
+
=
R
V R V R
R
V R V R
V
r
r
r
x y y x
x x y y
,
putting these vectors in the frame of the tube’s rollup vector. It is after this
transformation that the tubes are most easily strained. If the required strains have been
determined, the nanotubes can be strained by the following equations
( ) Vy = 1+ l Vy
Vx = Vx + sVy .
Here it is necessary to computeVy before Vx because Vy is used in its calculation. The
variables l
and s are the longitudinal strain and shear strain to be applied. Finally the
7
sheet coordinate vectors are ready to be transformed into cylindrical coordinates. The
radial coordinate of the nanotube is
2
V r
= .
Both of the carbon atom vectors yield identical results for this. The other two
coordinates, found separately for each atom are
V
V r
= 2 x
z = Vy .
At this point if the greatest common factor of the nanotube’s indices, N, is greater than
one, additional atoms need to be added, so that the total number of atoms is 2N. The
radius is unaffected by the addition of atoms to complete the helical cell. The new
terms are N i i / 2 2 + = + , where 1
and 2
are the originally calculated values of the
two atoms, and the new z terms are zi+2 = zi using the values of the first two atoms again.
When the helical vector undergoes this transformation its z coordinate becomes the step
distance, and its coordinate becomes the twist angle. After the unit cell has been created
any super-cell required for the match is created similarly to the previous procedure but
with the coordinates , , and z instead of the helical vector. With all this accomplished
we can calculate the strain we need to apply to each tube.
To calculate the strain for each tube it is assumed that the elastic constants of
graphite are the same as those of the rolled nanotube. Previous calculations have verified
that in a SWNT, a small strain away from the equilibrium position has a harmonic
potential. The amount of energy required to strain a nanotube can thus be estimated from
8
the experimentally-determined elastic constants of graphite. To reduce total energy
required to strain both nanotubes, the derivative of total energy used to strain each
nanotube is set to zero. As the elastic constants are the same for each nanotube the
minimum energy occurs when the energy used to strain one nanotube is equal to the
energy to strain the other nanotube. While the energy required to strain each nanotube is
the same the actual strain applied to either nanotube will be different in both direction
and magnitude. It must be noted that the longitudinal strain and shear strain are
calculated independently.
The longitudinal strain is found starting with the equation 11
2E 2 VC xx = derived
from graphite, the total energy for straining the two nanotubes can be found as
( ) 2,
2,
2 11
0 0 1
Etotal = Eouter + Einner + VC Souter l outer + Sinner l inner
where Etotal is the energy required to strain both nanotubes, the E0 terms are the rest
energies of each nanotube, V is the volume of the graphitic cell, C11 is an elastic modulus
in graphite, and S is the number of super cells in each nanotube. Using Z as the
difference in step distance of the helical cell from each nanotube it can be shown that the
change on the inner nanotube is zinner = Z + zouter ; this is a plus sign as the nanotubes
must be strained in opposite directions to match with the lowest strain energy. The strain
equation ( ) Vy = 1+ l Vy placed in its cylindrical form becomes
zstrained = zoriginal + z = zoriginal + zoriginal l
from which it is found z = z l . Substituting and solving for l,inner gives
inner
outer l outer
l inner z
Z z ,
,
= +
9
and the total energy in terms of l,outer becomes
( )
= + + + l outer
inner
inner outer
inner
inner
total outer inner z
ZS z
z
E E E VC Z S 2 2 ,
2
2 11
0 0 1 2
+ + 2,
2
2
2,
l outer
inner
inner outer
outer l outer z
S S z
Finding the optimal strain can be accomplished by
2 , 0
2
11 2 11
,
=
= +
+
l outer
inner
inner outer
outer
inner
inner outer
l outer
total
z
VC S S z
z
E VC ZS z
Solving for the strain we obtain
, 2 2
inner outer outer inner
inner outer
l outer S z S z
ZS z
+
=
, 2 2
inner outer outer inner
outer inner
l inner S z S z
ZS z
+
= .
Similarly for the shear strain is determined by the equation E xy C66V
2 2 = from
which follows the equation
( 2 )
,
2
2 66 ,
0 0 1
Etotal = Eouter + Einner + VC Souter s outer + Sinner s inner .
The change in the twist angle on the inner nanotube is inner = + outer and the
equation for strain Vx = Vx + sVy when viewed from cylindrical coordinates is
r
zstrained s
strained original original
= + = + .
From which it is seen
r
= zstrained s substituting into inner = + outer leads to
outer
strained s outer
inner
strained s inner
r
z
r
z , = + , ,
10
here zstrained must be the same for both nanotubes and the subscript will henceforth be
dropped. The strain in the inner nanotube is found to be
s outer
outer
inner inner
s inner r
r
z
r
, , = + .
Upon substitution we find the total strain energy to be
( )
= + + + s outer
outer
inner inner inner inner
total outer inner r z
S r
z
E E E VC S r ,
2
2
2 2
2 66
0 0 1 2
+ + 2
2 ,
2
2
, s outer
outer
inner inner
outer s outer r
S S r
Setting the derivative of strain energy equal to zero
2 , 0
2
66
2
66
,
=
= +
+
s outer
outer
inner inner
outer
outer
inner inner
s outer
total
r
VC S S r
r z
E VC S r
the solution for the strains can be found in the form
S r z S r z
S r r
outer outer inner inner
inner inner outer
s outer 2 2
2
, +
=
S r z S r z
S r r
outer outer inner inner
outer outer inner
s inner 2 2
2
, +
= .
Now that the necessary strains have been found, the nanotubes can be created and
calculations begun. The strains involved are very small. To show that the strain has
minimal effect for each DWNT calculation, both SWNTs were calculated as both
strained and unstrained systems.
Our electronic structure simulations were based on a local density functional
approach using linear combinations of Gaussian orbitals. Using a one-dimensional band
11
structure approach our method utilizes the helical symmetry inherent in the nanotubes
and calculates the electronic structure. Below is a brief description of the approach, with
full details available elsewhere [11]. The screw operator is a rotation of the unit cell
about the axis of the nanotube, J, along with a translation, h, along the axis. The screw
operator is isomorphic with the one dimensional lattice translation group, so Bloch’s
theorem can be generalized to
S (r; ) i (r; )
i m
i
m = e .
The quantity K is dimensionless and usually limited to the range, < , the first
Brilloun zone. The one electron wavefunctions, i
, are formed from a linear
combination of Bloch functions j
, which are in turn constructed from a linear
combination of nuclear-centered Gaussian-type orbitals j (r )
r
( ) = ( ) ( )
j
i r; c ji j r;
( ) ( ) =
m
j
i m m
j r e S r ;
The one-electron density matrix is then given by
( ) ( ) ( ) ( )
=
i
d ni i i k
; ;
2
r;r 1 * r r
(r ) ' (r)
' '
'
' '
mj
jj m
m mj
m
m
= Pj j +
where ( ) r
ni are the occupied numbers of the one-electron states, mj
denotes S j (r)
m ,
and mj
Pj are the coefficients of the real lattice expansion of the density matrix given by
( ) ( ) ( )
=
i
i m
i j i ji
m
Pj j d n c c e
*
' 2 '
1 .
12
The total energy of the system is given by
[ ( )]
! "
+
# $ %
! "
= & +
'' '
2 0 0 ' '
2
1
2
1
m n n
n n
ij m
xc j
m
j i
m
i
m
ij
E P Z Z m''
n'
0n
R R
r
[ ] #
$ %
+
+
n
j
m n
i
i j m
mj
m m
j i
m
i
mj
i
ij m
m
ij
P P Z 0
' ' '
' '' ''
'
' 0
' ' 2 m''
r Rn
where Zn and Rn denote the nuclear charges and coordinates within a single unit cell,
m
Rn denotes the nuclear cell m( ) n
m mn
R ' S R , and [ ] 1 2 denotes an electrostatic
interaction integral
[ ]= ( ) ( )
12
1 1 2 2
2
3
1
3
1 2 r
d r d r r r .
Rather than solve directly for total energy, we fit the exchange-correlation potential and
the charge density to a linear combination of Gaussian-type functions. All the
calculations were done using the C7s, 3p Gaussian basis set. All of the nanotubes both
single and double were treated with periodic boundary conditions. It is important to note
the individual carbon-carbon bond lengths were not optimized but held at a bond length
of 1.44 N, the bond length previously found to be the optimum in the (5, 5) carbon
nanotube.
III. Results
All of the calculations were done using the polyxO code. For each DWNT, five
sets of calculations were performed. These include: the two SWNTs; the two SWNTs
strained to have matching helical vector; and, the DWNT itself. The results of the
unstrained SWNTs calculations were compared to the strained SWNTs results.
13
Figure 2 Calculated band structure of an example double walled nanotube.
a) (26, 21) strained outer nanotube b) (33, 1) strained inner nanotube
c) (26, 21) (33, 1) double walled nanotube
Figure 2 is a representative sample of the calculated band structure. It can be seen that the
band structure of the DWNT has the same band structure as each of the single walled
tubes upon which it is based, showing little coupling between the nanotubes.
14
Figure 3 Comparison band structure of the outer single walled nanotube.
a) (26, 21) nanotube b) strained (26, 21) nanotube
In Figure 3 the band structure of the outer SWNT in both the original state, on the left,
and strained configuration, on the right, can be seen.
15
Figure 4 Comparison band structure of the inner single walled nanotube.
a) (33, 1) nanotube b) strained (33, 1) nanotube
Figure 4 shows the original and strained band structure of the inner SWNT. The band
structures are visually identical confirming the strain necessary to alter the nanotubes to
fit the helical symmetry is low enough to have a negligible effect on the calculations.
16
Figure 5 The unstrained binding energy of the double walled nanotubes vs. radial difference of the
nanotubes.
“A central quantity in characterizing graphite cohesion is the interplanar binding
energy per atom Ei [12].” As carbon nanotubes are essentially rolled graphite sheets this
quantity can be found as the change in energy per atom between the SWNTs and DWNT
and is useful in determining the stability of calculated DWNTs. Figure 5 shows the
binding energy of the unstrained nanotubes tubes and Figure 6 shows the binding energy
of the single tubes in their strained configurations. Each of these figures has been fitted
with a Morse potential. When the radial difference is low the binding energy shows
17
Figure 6 The strained binding energy of the double walled nanotubes vs. radial difference of the
nanotubes.
unfavorable energies as expected. This is because the R electrons of the two nanotubes
are being forced into the same region. As the radial difference increases a minimum in
the binding energy is found around 3.3 Å. There is an unfortunate gap the compiled list
of double walled nanotubes with radial difference between 3.18 Å and 3.31 Å as it is
possible the minimum would be shifted by data in this region. This minimum is lower
than had been expected. The inter-tube separation was expected to be similar to the
18
interlayer spacing of graphite or 3.35 Å. Charlier and Michenaud [13] found the ideal
interlayer distance of 3.39 Å in their calculations. Their experiment, however, only
contains three data points in range of radial separation in the present calculations, with
radial differences at 2.71 Å, 3.39 Å, and 4.07 Å. The minimum calculated in this study is
between two of their data points and accurate comparison is impossible. The trend in my
calculations does not show a negative binding energy for nanotubes with radial
separations as low as 2.71 Å as their calculations indicate. In the calculations of both Lu
[4] and Liu et al.[7] the accepted value of tube separation was 3.4 Å. The calculations of
Okada and Oshiyama [14] show a minimum tube separation of 3.52 Å and estimate the
ideal is slightly higher at 3.56 Å. Ren and Cheng [15] observed their synthesized double
walled carbon nanotubes to have a mean diameter difference of 7.4 Å corresponding to
an interlayer spacing of 3.7 Å by using Raman spectroscopy.
IV. Summary
Our results seem to show that the approximation of applying a small strain to two
single-walled nanotubes in order to match helical symmetry and form a double walled
nanotube usable by our code is physically acceptable. The change in each single walled
nanotube is minimal. The effect of small strains on the double walled nanotubes should
be just as small. The binding energies of the double walled nanotubes show an inter tube
separation distance that is somewhat lower than reported elsewhere.
19
V. Bibliography
[1] S. Iijima, Nature 354, 56 (1991)
[2] M. Yu, O. Lourie, M. J. Dyer, K. Moloni, T. F. Kelly, and R. S. Ruoff, Science 287,
637 (2000)
[3] P. A. Williams, S. J. Papadakis, A. M. Patel, M. R. Falvo, S. Washburn, and R.
Superfine, Phys. Rev Lett. 89, 255502 (2002)
[4] J. P. Lu, Phys. Rev. Lett. 79, 1123 (1997)
[5] Q. Zhao, M. B. Nardelli, and J. Bernholc, Phys. Rev. B 65, 144105 (2002)
[6] L. J. Sudak, Journal of Applied Physics 94, 7281 (2003)
[7] P. Liu, Y. W. Zhang, C. Lu, and K. Y. Lam, J. Phys. D: Appl. Phys. 37, 2358 (2004)
[8] L. Shen and J. Li, Physical Review B 69, 045414 (2004)
[9] L. Shen and J. Li, Physical Review B 71, 035412 (2005)
[10] C. T. White and J. W. Mintmire, J. Phys. Chem. B, 109, 52 (2005)
[11] J. W. Mintmire, in Density Functional Methods in Chemistry, edited by J. K.
Labanowski (Springer-Verlag, Berlin, 1990), pp.125-138.
[12] S. B. Trickey, F. Müller-Plathe, G. H. F. Diercksen, and J. C. Boettger, Phys. Rev.
B, 45, 4460 (1992)
[13] J. –C. Charlier and J. –P. Michenaud, Phys. Rev. Lett., 70, 1858 (1993)
[14] S. Okada and A. Oshiyama, Phys. Rev. Lett., 91, 216801 (2003)
[15] W. Ren and H. Cheng, J. Phys. Chem. B, 109, 7169 (2005)
20
Table 1 A list of all calculated double walled nanotubes.
Outer Tube # Unit Cells Inner Tube # Unit Cells Radial Difference # of Atoms
(16, 1) 5 ( 6, 1) 2 3.9558 14
(11, 7) 6 ( 5, 4) 3 3.1383 18
(31, 9) 5 (23, 9) 4 3.0813 18
(34,11) 5 (25,11) 4 3.4454 18
(37,13) 5 (27,13) 4 3.8076 18
(13, 8) 7 ( 7, 5) 4 3.1428 22
(26,21) 6 (33, 1) 5 2.8854 22
(28,27) 6 (37, 4) 5 3.3664 22
(35,13) 6 (27,13) 5 3.0402 22
(37,11) 6 (29,11) 5 3.0818 22
(18,17) 7 (13,12) 5 3.4371 24
(31, 1) 7 (22, 1) 5 3.5707 24
(21,19) 8 (16,14) 3 3.4358 28
(13, 4) 10 ( 7, 1) 5 3.1141 30
(16,11) 9 (10, 8) 3 3.1342 30
(25, 3) 9 (16, 3) 6 3.5469 30
(13, 7) 10 ( 7, 5) 6 2.8335 32
(14, 1) 11 ( 6, 1) 5 3.1631 32
(23,22) 9 (18,17) 7 3.4374 32
(34,11) 9 (24,11) 7 3.8237 32
(15, 7) 11 ( 7, 5) 6 3.5836 34
(17, 1) 11 ( 9, 1) 6 3.1685 34
(28, 3) 10 (19, 3) 7 3.5528 34
(26,24) 5 (21,19) 8 3.4366 36
(14, 9) 12 ( 9, 4) 7 3.3909 38
(19,14) 11 (13,11) 8 3.1278 38
(21, 1) 12 (12, 1) 7 3.5676 38
(31, 3) 11 (22, 3) 8 3.5569 38
(27, 4) 12 (17, 4) 8 3.9271 40
(28,27) 11 (23,22) 9 3.4375 40
(15,11) 13 ( 9, 7) 8 3.4586 42
(17, 5) 13 (11, 2) 8 3.1164 42
(34, 3) 12 (25, 3) 9 3.5599 42
(17, 9) 13 (11, 7) 9 2.8394 44
(23, 4) 13 (15, 4) 9 3.1318 44
(19, 9) 14 (11, 7) 9 3.5895 46
(23,16) 13 (17,13) 10 3.1353 46
(25,19) 13 (26, 5) 10 3.7298 46
(27,20) 13 (28, 5) 10 3.9884 46
(37, 3) 13 (28, 3) 10 3.5620 46
(11, 6) 17 ( 4, 3) 7 3.5132 48
(20, 3) 15 (11, 3) 9 3.5287 48
(25, 4) 14 (17, 4) 10 3.1392 48
(33,32) 13 (28,27) 11 3.4376 48
(34,31) 13 (29,26) 11 3.4363 48
(17,13) 15 (11, 9) 10 3.4568 50
(20, 1) 16 (11, 1) 9 3.5669 50
(25,17) 14 (19,14) 11 3.1377 50
(26, 5) 15 (16, 5) 10 3.9009 50
(29,12) 14 (24, 7) 11 3.3146 50
(34, 5) 14 (26, 5) 11 3.1474 50
(19,11) 15 (13, 9) 11 2.8302 52
(37, 1) 15 (27, 1) 11 3.9681 52
(25, 1) 16 (17, 1) 11 3.1730 54
VITA
Benjamin Arthur Landis
Candidate for the Degree of
Master of Science or Arts
Thesis: ELECTRONIC STRUCTURE CALCULATIONS OF DOUBLE WALLED
CARBON NANOTUBES USING LOCAL DENSITY FUNCTIONAL THEORY
Major Field: Physics
Biographical:
Personal Data: Born March 15, 1979, the son of Arthur and Linda Landis
Education: Graduated from Emporia High School in Emporia KS in May 1997.
Received Bachelor of Science degree in Physics form Emporia State
University in May 2001.
Experience: Employed by Oklahoma State University, Department of physics
as a graduate teaching assistant and a graduate research assistant August
2001 to present.
Professional Memberships: None
Name: Benjamin Landis Date of Degree: December, 2005
Institution: Oklahoma State University Location: Stillwater, Oklahoma
Title of Study: ELECTRONIC STRUCTURE CALCULATIONS OF DOUBLE
WALLED CARBON NANOTUBES USING LOCAL DENSITY
FUNCTIONAL THEORY
Pages in Study: 21 Candidate for the Degree of Master of Science
Major Field: Physics
Scope and Method of Study:
Findings and Conclusions: Electronic structure calculations using local density
functional theory are used to study double walled carbon nanotubes. The
calculations show that applying small strains to single walled nanotubes in order
to form double walled nanotubes has only a minimal effect on the nanotubes. The
binding energies of the double walled nanotubes allows an estimation of the
minimum inter tube separation distance.
ADVISER’S APPROVAL: John Mintmire