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9 058 9986
This is to certify that the
dissertation entitled
EQUILIBRIUM GEOMETRY AND ELECTRONIC PROPERTIES
OF NANOSTRUCTURES
presented by
Young-Kyun Kwon
has been accepted towards fulfillment
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Ph ..D degree in Physics
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EQUILIBRIUM GEOMETRY AND ELECTRONIC PROPERTIES OF
NANOSTRUCTURES
By
Young-Kyun Kwon
A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Physics and Astronomy
1999
ABSTRACT
EQUILIBRIUM GEOMETRY AND ELECTRONIC PROPERTIES OF
NANOSTRUCTURES
By
Young-Kyun Kwon
Nanostructures are one of most exciting and interesting research topics because
of their anomalous properities and small sizes. I use ab initio and parametrized
linear combination of atomic orbitals (LCAO) formalism to study growth mechanism,
structural and electronic properties of nanostructures. To investigate microscopic
dynamic behaviors, such as melting transitions, and their usefulness as electronic and
mechanical devices of nanostructures, I use molecular dynamics and Monte Carlo
simulations. I also calculate their ballistic quantum conductance using Landauer-
Biittiker formalism.
I find that strong covalent bonds connecting the exposed edges of adjacent walls
stabilize open-ended structures at the growing edge against dome closure in multi-wall
carbon nanotubes. Due to the small inter-tube interaction and lattice frustration, I
predict a very soft libration mode to occur in interacting nanotube systems such as
single-wall nanotube “ropes” and multi-wall nanotubes. These modes are expected
to disappear above the orientational melting temperature, T 0 M2160 K, at which the
onset of free “diffusion” of orientational dislocations or twistons formed during the
synthesis, corresponding to orientational melting, occurs. In addition, I investigate
thermal behaviors of carbon nanotubes. I find that structural melting transition
occurs at TSMzleOO K similar to other sp2 bonded carbon structures and nanotubes
have negative thermal expansion coefficients.
In an ordered bundle of (10, 10) carbon nanotubes, inter-tube coupling and sym-
metry lowering opens a pseudo-gap near the Fermi level, which is almost independent
of tube orientation. In a (5, 5)@(10, 10) double-wall nanotubes, on the other hand,
four pseudo-gaps open and close periodically near Ep during the soft librational
motion in a (5, 5)@(10, 10) double-wall nanotubes. Such inter-wall interactions and
structural inhomogeneity in multi-wall nanotubes not only block some of the quan-
tum conductance channels, but also redistribute the current non-uniformly over the
individual tubes across the structure, and thus result in fractional quantum conduc-
tance.
I propose two possible applications of nanotubes and related fullerene structures.
I show curved nanotubes with pentagon—heptagon defect pairs can be utilized as a
micro—fastening system, which forms permanent, extremely strong, yet self-repairing
bonds. I also show “bucky-—shuttle” structure to be a tunable two-level system, where
transition between the two states can be induced by applying an electric field, and
discuss its potential application as a non-volatile memory element.
I also determine the equilibrium geometry, electronic and magnetic structure for
aggregates containing up to tens of iron atoms. Same as bulk iron, I find that many
aggregates favor reduced atomic packing, resulting in narrower bands and a higher
level density near the Fermi level. This in turn leads to a substantial gain in exchange
energy, at the cost of a reduction in the band-structure energy. Most stable structures
and novel magic numbers in iron clusters result from the non-trivial inter-play between
geometry and magnetism in these systems.
To my lovely wife, Heejeong
iv
ACKNOWLEDGMENTS
First of all, I thank God for all the blessings He has bestowed on me.
I would like to express my deep gratitude to my advisor, Professor David Tomanek.
Without his constant support and guidance, I would not be at this position. Through-
out my Ph. D. work, he shared with me his insight, inspiration and enthusiasm for
physics. I cannot forget our many stimulating conversations on science as well as on
diverse aspects of life including philosophy, literature, art, music and politics. He al-
ways guided me with love and confidence. He was, is, and will be my advisor, teacher
and friend.
I am also very grateful to the members of my Guidance Committee: Professors
. Jack Hetherington, S.D. Mahanti, Norman Birge, Michael Thoennessen and Richard
Enbody. At various stages of my Ph. D. studies, they offered me timely help and
unfailing support.
I owe great thanks to Dr. Seong-Gon Kim, Prof. Young Hee Lee, Prof. Susumu
Saito, Dr. Sumio Iijima, Dr. Stefano Sanvito, Prof. Colin J. Lambert, Mr. Pedro
C. Serra, Prof. Jorge M. Pacheco, Mr. Savas Berber, Mr. Mark Brehob, Prof. Richard
J. Enbody, Prof. Jean S. Chung, Prof. Jisoon Ihm, Dr. Philippe Jund, Prof. Kee Hag
Lee, Prof. Richard E. Smalley, with whom I have had fruitful discussions on various
topics in my dissertation.
I also thank my friends — too many to go through the list — who made my life
in East Lansing much more enjoyable and pleasant.
I acknowledge financial support from the Center for Fundamental Materials Re-
search, the Office of Naval Research and DARPA without which my Ph. D. work
could not have been completed.
I would like to express my gratitude to my parents for their prayers and to my
parents—in—law for their encouragement over my Ph. D. years. Without their love, I
would not be where I am today. I also thank my grandmother, my brother and his
family, and my two sisters and their families for their moral support.
No matter how much I appreciate the love and support I received from all those
above, my deepest gratitude goes to my lovely wife Heejeong; she has made countless
sacrifices for me and has always been my cheerleader for the years we have spent
together. With love and gratitude, I dedicate this dissertation to her.
vi
Contents
LIST OF TABLES ix
LIST OF FIGURES x
1 Introduction 1
1.1 Classification of single-wall nanotubes .................. 3
1.2 Electronic structure of single-wall nanotubes .............. 5
1.3 Outline of the dissertation ........................ 7
2 Growth Mechanism of Multi-Wall Nanotubes 12
3 Equilibrium Structure 25
3.1 Single-wall nanotube ropes ........................ 25
3.2 Multi-wall nanotubes ........................... 33
4 Structural Properties 39
4.1 Structural melting transition ....................... 39
4.2 Orientational melting transition ..................... 45
4.3 Thermal contraction ........................... 56
5 Electronic Structure 61
5.1 Single-wall nanotube ropes ........................ 61
5.2 Multi-wall nanotubes ........................... 69
6 Transport 75
7 Applications 87
7.1 Nanotube—based micro—fastening system ................ 87
7.2 Nanotube-based memory device ..................... 94
vii
8 Magnetic Iron Clusters
A Total Energy Calculation Based on Recursion Technique
A.l Local repulsive energy ..........................
A.2 Local band structure energy .......................
A.3 Force Calculation .............................
B General Scattering Technique
Bibliography
viii
105
120
122
124
131
132
137
List of Tables
4.1
5.1
8.1
A.1
A2
A3
A.4
Parametrization of the inter-tube interaction .............. 49
Energy difference between Van Hove singularities in isolated and bun-
dled (n, n) single-wall nanotubes ..................... 68
Calculated properties of small iron clusters ............... 108
Parametrization of the local atomic density in carbon systems . . . . 123
Parametrization of the repulsive energy in carbon systems ...... 123
Calculated recursion coefficients for a graphene sheet ......... 127
Parametrization of the band-structure energy at large separations . . 130
ix
List of Figures
1.1
1.2
2.1
2.2
2.3
2.4
3.1
3.2
3.3
3.4
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
5.1
5.2
5.3
5.4
5.5
Unit cell of a nanotube projected on the unrolled graphene sheet . . .
Allowed wave vectors of nanotubes ...................
Relaxed geometry at the growing edge of double-wall tubes ......
Color-coded binding energy of individual atoms ............
Optimum geometry as a function of edge coverage 9 .........
Edge energy and total energy gain as a function of coverage .....
Density functional results for the total energy of rope and hexagonal
graphite ..................................
Dependence of the rope energy on tube orientation ..........
Structure of a double-wall nanotube ...................
Dependence of double-wall tube energy on radial and angular displace-
ment ....................................
Thermodynamical behavior of structural melting in nanotubes . . . .
Evolution of nanotube structures during the melting process .....
Interaction energy between a pair of aligned (10,10) nanotubes . . . .
Equilibrium structure of a (10,10) tube rope ..............
Torsion energy of an individual (10, 10) nanotube ...........
Orientational melting in a nanotube rope composed of non-interacting
“disks” ...................................
Orientational melting in a nanotube rope ................
Negative thermal expansion of nanotubes ................
Electronic band structure of a nanotube rope ..............
Density of states of a (10, 10) nanotube rope ..............
Densities of states of isolated and bundled (72,71) single-wall carbon
nanotubes .................................
Electronic band structure of aligned single- and double-wall nanotubes
Density of states of aligned single— and double-wall nanotubes
14
17
20
21
27
29
34
36
42
43
48
49
51
52
54
59
63
65
67
71
72
6.1
6.2
6.3
6.4
6.5
6.6
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
7.10
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
B.1
Density of states and conductance of a (10, 10) nanotube ....... 77
Density of states and conductance of double— and triple-wall nanotubes 78
Partial conductance of the constituent tubes .............. 79
Dependence of conductance on nanotube immersion depth in liquid Hg 81
Comparison of the calculated conductance with experimental results . 82
Alternate scenario for a conductance measurement ........... 84
Possible structures of micro—fastening elements ............. 89
Opening mechanism of the micro—fastening system ........... 91
Closing mechanism of the micro—fastening system ........... 92
TEM images of “bucky-shuttle” structures ............... 95
Comparison of a structural model for a “bucky-shuttle” with a TEM
image ................................... 96
Potential energy of K@Cg§, as a function of its position ........ 96
Schematic of a high-density memory board ............... 99
Kinetics of the “bucky—shuttle” ..................... 101
Time dependence of the potential and kinetic energy in the “bucky-
shuttle” .................................. 101
Vibrational temperature of constituents of “bucky-shuttle” ...... 102
Atomic binding energy and magnetic moment of Fe; .......... 109
Optimized geometries of Fe;,, Fe4, and Fe5 ............... 110
Dependence of binding energy and magnetic moment on cluster size . 111
Dependence of electronic and magnetic energies, and ratio of magnetic
energy to electronic energy on cluster size ............... 112
Dependence of average bond length and number of bonds, and com-
pactness on cluster size .......................... 114
Characterization of F83 cluster ...................... 115
Characterization of Fem cluster ..................... 116
Characterization of F818 cluster ..................... 116
Predicted abundance spectra in iron clusters .............. 117
Scheme of the scattering calculation ................... 133
xi
Chapter 1
Introduction
This dissertation is devoted to discussions of physical prOperties of nanostructures,
focusing on the recently discovered carbon nanotubes [Iiji91] and briefly addressing re-
lated systems including magnetic clusters. Nanostructures are entities that have sizes
typically ranging from few to hundreds of nanometers. N anostructures have attracted
attentions from many scientists and engineers, not only because of their small size,
but also due to their unusual properties and their potential use as nanometer—scale
devices. The science devoted to nanostructures is called “nanotechnology”.
Development of new instruments like the scanning tunneling microscope, the
atomic force microscope, and the tunneling electron microscope enables the mea-
surement and the manipulation of nanostructures. In addition, recent surprising
improvement in the capability of computers and in computational techniques enables
sophisticated computer simulations of nanostructures. These new technologies pos-
sess the potential to study previously unknown behavior at the nanoscale.
Along with the cutting—edge technologies, predicting the behavior of nanostruc-
tures also calls for the development of new models and theories. Traditional models
and theories for most bulk material properties involve assumptions leading to “criti-
cal scale lengths” that are frequently larger than 100 nm. As the size of a structure
reduces, critical length scales of physical phenomena become comparable to or even
larger than the size of the structure. Such models and theories are no longer suitable
for describing novel phenomena. The unknown behavior of nanostructures can be
described by the new phenomena based purely on quantum mechanics, such as size
confinement and predominance of surface phenomena, as well as by size reduction.
Nanostructures, such as thin films, quantum dots and carbon nanotubes, which
show very interesting and novel properties, are led by reducing their dimensions.
Although many research groups have focused their studies on these systems, nanos-
tructures are still new and fascinating systems. If their underlying principles can be
discovered, understood, and fully utilized, these nanostructures will open a completely
new age for science and technology.
Furthermore, nanostructures offer a new paradigm for manufacturing materials
by assembling —— ideally utilizing self—organization and self—assembly — to create an
entity rather than the laborious chiseling away from a larger structure.
Among nanostructures, carbon nanotubes are one of the most exciting and fasci-
nating materials. Carbon nanotubes [Iiji91, Dre596], which are narrow, seamless and
atomically perfect graphitic cylinders, show an unusual combination of nanometer—
size diameter and millimeter—size length. This unusual quasi-one dimensional t0pol-
ogy of nanotubes, combined with the absence of defects on a macroscopic scale
[Wild98, Odom98], gives rise to uncommon electronic properties of individual single-
wall nanotubes [Iiji93, Beth93], which —— depending on their diameter and chirality
—— can be either conducting or insulating [Mint92, SaiR92, Hama92]. Electrical trans-
port measurements for individual nanotubes indicate that these systems may behave
as genuine quantum wires [Tan397], nonlinear electronic elements [Bock97], or tran-
sistors [Tans98].
O
o
a
8
93
0‘0
.0
cc
Figure 1.1: A nanotube can be constructed by connecting site 0 to site A and site
B to site B’. This nanotube is (6, 3) (see the text for the tube classification). The
chiral vector Ch and the translational vector T of the nanotube are represented by
arrow lines of 0A and OB, respectively. The rectangle OAB’ B defines the unit cell
of the nanotube.
While related to carbon fibers, nanotubes are free of atomic scale defects, which
accounts for their high tensile strength, as compared to that of the strength of in-
dividual graphite layers. Like graphite and fullerenes, carbon nanotubes exhibit 3p2
bonding which gives rise to a relatively high degree of flexibility and resilience.
1.1 Classification of single-wall nanotubes
Carbon nanotubes are classified primarily into achiral and chiral nanotubes [Dre396,
SaiR98]. An achiral nanotube exhibits a mirror symmetry on the plane normal to
the tube axis whereas a chiral one shows a spiral symmetry. There are only two
types of achiral nanotubes that show higher symmetry than chiral tubes. The one is
an “armchair” type and the other a “zigzag”, as discussed below. The structure of a
nanotube is more specified by the orientation of hexagonal carbon rings on cylindrical
graphene sheet with respect to the tube axis. This orientation is characterized by the
chiral index (n, m) defined by the chiral vector Ch
Ch 2 nal + mag, (1.1)
where a,(z' = 1,2) are real space unit vectors of the hexagonal lattice. This chiral
vector, as shown in Figure 1.1, connects two equivalent sites 0 and A on a graphene
sheet. Its magnitude Ch represents a circumferential length of a nanotube being
characterized by Ch. The direction perpendicular to Ch becomes a tube axis. Pairs
of integers (n, m) in Eq. (1.1), specifying all possible chiral vectors, defines a different
way of rolling the graphene sheet to form a nanotube. Zigzag nanotubes, which have
the zigzag shape of the cross-sectional ring, and armchair nanotubes, which have the
armchair shape, are denoted by the vectors (n, O) and (n,n), respectively.
Tube diameter dt is given by
d; : Ch/W
Vida/(n? +nm +m2)1/2/7r (1.2)
where deg is the nearest-neighbor distance between two carbon atoms (in graphite
doc = 1.42 A). And the chiral angle 0, defined as the angle between the chiral vector
Ch and the lattice vector 31, is given by
Ch ‘ a1
Cha
c080 =
_ 2n+m (13)
— 2x/n2+nm+m2' .
The chiral angle 6’ is just in the range of 0 _<_ WI 3 30°, because of the hexagonal
symmetry of the graphene sheet. Armchair nanotubes, in particular, correspond to
9 = 30° and zigzag ones 0 = 0°.
The vector T, called the translation vector, is parallel to the tube axis, i.e.
perpendicular to the chiral vector Ch. Using the relation of Ch - T, the vector T,
which becomes the lattice vector in 1D tube unit cell, can be expressed in terms of
the basis vectors a,- as
1
T:—
dn
—(n + 2m)a1 + (2n + "032], (1.4)
where d}; is the greatest common divisor of (n + 2m) and (277. + m). Furthermore, (in
can be expressed in term of the greatest common divisor d of n and m. If n — m is a
multiple of 3d, d}; = 3d, and otherwise d R = d. Note that the bigger dig, the smaller
the length of T is. For example, T = ——a1 + a2 for any (n,n) armchair nanotubes
((1,; = 3d = 3n) and T = ——a1 + 2a2 for any (71,0) zigzag nanotubes (d3 = n). A
(6,3) nanotube (d = cl}; = 3) shown in Figure 1.1 has T = —4 a1 + 5 a2.
1.2 Electronic structure of single-wall nanotubes
Depending on their chirality and diameter, single-wall nanotubes can be either metal-
lic or semiconducting [Mint92, SaiR92, Hama92]. In this section, I review this unique
electronic structure of single-wall carbon nanotubes.
The electronic band structure of a nanotube somewhat resembles that of a gra-
phene sheet. The difference between two cases comes from the periodic boundary
conditions. A graphene sheet is regarded as an infinitely extended two-dimensional
plane, whereas a nanotube is one—dimensional structure which is infinitely long along
the tube axis. However, the periodic boundary condition, Ch. k = 27rl, where l is
an integer, is imposed for a finite period along the circumference. This results in
Bloch wave functions with discretely selected wave vectors, which are shown by the
straight lines in Figure 1.2, in the first Brillouin zone of a graphene sheet. Note that
the bonding and anti-bonding 7r bands, which are located near the Fermi level, are
(a) I l (b) 9 """""""""" ”PE
‘0"
Figure 1.2: Due to the periodic boundary condition along the circumference, only
discrete wave vectors are allowed on the Brillouin zone of a graphene sheet. The
allowed wave vectors are shown in (a) for a (4, 4) armchair nanotube and in (b) for
a (6, 0) zigzag nanotube. Open circles show the degenerate point shifted from the K
point.
degenerate at the K point in the Brillouin zone of a graphene sheet, while sp20 bands
are located far from the Fermi level. Therefore, if at least one of the selected wave
vectors crosses the K point, the nanotube becomes a metal. The degenerate point,
however, in the nanotube does not exactly corresponds to the K point, because 7r
atomic orbitals, which are pointed in the radial direction, are not exactly parallel to
each other and thus the a—bond components are incorporated, an additional electron
transfer is produced between those orbitals. Hence metallic nanotubes classified in
this way become small band—gap semiconductors with an exception of (n, n) armchair
nanotubes, which are truly metallic regardless of their curvature, as discussed below.
Figure 1.2 explains this situation for achiral nanotubes. The vertical lines in
Figure 1.2(a) shows the set of allowed wave vectors of a (4,4) armchair nanotube
on the first Brillouin zone of a graphene sheet, which is surrounded by the dashed
line in the figure. Although the degenerate point is shifted from K point to the
positions shown by open circles, by the enhancement of the electron transfer along
the tube circumference, it is always on the P—K line, which exactly overlap with
one of allowed wave vectors. Therefore, the nanotube becomes truly metallic. This
feature is essential for any (n, n) armchair nanotubes.
The set of allowed wave vectors of a (6,0) zigzag nanotube is shown in Fig-
ure 1.2(b). Although one of the allowed wave vector crosses the K point, the shifted
degenerate point shown by open circles is not crossed by any of allowed wave vectors.
Therefore, this tube is not metallic, but narrow band gap semiconducting. For gen-
eral (n, 0) zigzag nanotubes, the allowed wave vectors cross the points which divide
the doubled F-M line into 11. parts. If n is a multiple of 3, one of the allowed wave
vectors crosses the K point, which is close to the degenerate point, like in a (6,0)
nanotube and thus the (n, 0) tube will be a narrow band gap semiconductor. If n is
not a multiple of 3, then the tube will be a moderate or wide band gap semiconductor.
By using periodic boundary conditions, the condition for general (72, m) nan-
otubes can be easily derived. The condition is
= 0 for metallic nanotubes
n — m = 3q for narrow band gap semiconducting nanotubes (1.5)
75 3q for wide band gap semiconducting nanotubes,
where q is an integer.
1.3 Outline of the dissertation
In the following chapters, I will make use of ab initio density functional theory
[Hohe64, Kohn64] and parametrized linear combination of atomic orbitals (LCAO)
formalism to investigate structural and electronic properties of nanostructures. These
total energy calculations are combined with molecular dynamics, conjugate gradient,
and Monte Carlo techniques to search for the equilibrium structures and to study
dynamical properties and thermodynamical phase transitions. I also use Landauer—
Biitikker formalism based on LCAO technique to calculate quantum conductance.
This dissertation begins in Chapter 2 with a discussion of the growth mechanism
of multi-wall carbon nanotubes. I study the morphology and structural stability
at the growing edge of multi-wall carbon nanotubes. I find that these open-ended
structures are stabilized against dome closure by strong covalent bonds connecting
the exposed edges of adjacent walls. Growth at the open edge involves rearrangement
of these bonds, which are mediated by carbon atoms bridging the gap and change the
tip morphology significantly. Presence of a strong “lip—lip” interaction can explain
formation of carbon nanotubes under annealing conditions.
In Chapter 3, I calculate the equilibrium structure and the low-frequency vibra-
tional modes of an ordered “bundle” of single-wall carbon nanotubes and multi-wall
tubes. Due to the small inter-tube interaction and lattice frustration, a very soft
I
libration mode is predicted to occur at V2312 cm‘ in single-wall (10,10) nanotube
“ropes” (Section 3.1). A similar libration mode is predicted to occur at 11,330 cm"1
in a (5,5)@(10,10) double-wall carbon nanotube (Section 3.2). These modes are ex-
pected to disappear above the orientational melting temperature which marks the
onset of free tube rotations about their axis.
Chapter 4 is devoted to investigate two types of “melting transitions” that may
occur in systems consisting of quasi—one-dimensional nanotubes of carbon, and ther-
mal expansion for carbon nanotubes. In analogy to many other 3192 bonded carbon
structures, such as graphite and fullerenes, I find that nanotubes also show a struc-
tural melting transition at TSMz4000 K (Section 4.1). Results of molecular dynamics
Simulations indicate that multi-wall nanotubes start melting from the edge and trans-
form into “monatomic chain” structure. These results provide further insight into the
process of nanotube formation from the gas phase.
Similar to solid C60 that shows an orientational melting transition at room tem-
perature or below, I investigate the possibility of an orientational melting transition
within a “rope” of (10,10) carbon nanotubes in Section 4.2. As twisting nanotubes
bundle up to a rope during the synthesis, orientational dislocations or twistons form
along the tubes. These dislocations result from a competition between the anisotropic
inter-tube interaction, tending to align neighboring tubes, and the torsion rigidity of
individual tubes that tends to keep them straight. In Monte Carlo simulation of the
orientational melting process, I first map the energetics of a rope with twistons onto
a lattice gas model. Results suggest that the onset of free “diffusion” of twistons,
corresponding to an “orientational melting” transition, occurs at To [”2160 K.
In Section 4.3, I describe the thermal behavior of carbon nanotubes. They exhibit
thermal contraction across the broad range of temperature. According to the analysis
of the relative importance of the particular modes on the thermal contraction, a much
larger contribution results from transverse acoustic modes causing a snake—like motion
of the nanotube.
Chapter 5 is devoted to calculate the electronic properties of an ordered bundle of
(10,10) carbon nanotubes and multi-wall carbon nanotubes. Results for bundled tubes
(Section 5.1) indicate that inter-tube coupling causes an additional band dispersion
of $0.2 eV and opens up a pseudo-gap of the same magnitude at Ep. Whereas the
density of states near Ep increases by 7% due to inter-tube coupling and by one order
of magnitude due to K doping in KCg, these states do not couple to tube librations.
In the case of multi-wall carbon nanotubes (Section 5.1), I find that the weak inter-
wall interaction and changing symmetry cause four pseudo-gaps to Open and close
periodically near the Fermi level during the soft librational motion. This electron-
libration coupling, absent in solids composed of fullerenes and single-wall nanotubes,
may yield superconductivity in multi-wall nanotubes.
Chapter 6 studies the ballistic quantum conductance of multi-wall carbon nan-
otubes. I find that inter-wall interactions not only block some of the quantum con-
ductance channels, but also redistribute the current non-uniformly over the individ-
ual tubes across the structure. Results provide a natural explanation for the unex-
pected integer and non—integer conductance values reported for multi-wall nanotubes
in Ref. [Fran98].
In Chapter 7, I propose the possible applications of carbon nanotubes and related
fullerene structures. In analogy with the “velcro”TM fastening system, I postulate the
possibility to establish very strong bonds connecting two surfaces covered by curved
nanotubes with pentagon-heptagon defect pairs in Section 7 .1. Results indicate that
the force needed to open an assembly of two ”hooks”, each consisting of a (7,0)
nanotube terminated by a half—torus, is very large.
In Section 7.2, I investigate the internal dynamics of a related model system,
consisting of a K@Cg’O endohedral complex enclosed in a C480 nano—capsule. I show
this to be a tunable two—level system, where transitions between the two states can
be induced by applying an electric field between the C480 end caps, and discuss its
potential application as a non-volatile memory element.
Finally, in Chapter 8, I determine the equilibrium geometry, electronic and mag-
netic structure of iron clusters combining a parametrized one—electron Hamiltonian
with the Stoner model of itinerant ferromagnetism. I find that many aggregates favor
reduced atomic packing, resulting in narrower bands and a higher level density near
the Fermi level. This in turn leads to a substantial gain in exchange energy, at the
10
cost of a reduction in the one—electron band-structure energy.
11
Chapter 2
Growth Mechanism of Multi—Wall
Nanotubes
The following discussion of the growth mechanism of multi—wall carbon nanotubes
follows that presented in Ref. [Kwon97].
Since the first observation of nanotubes in the carbon arc [Iiji91, Ebbe96], their
formation mechanism has been traditionally associated with external factors such as
strong electric fields [Iiji91, Ebbe96, Sma193, SaiY93, Ebbe94, Mait94], presence of
hydrogen atoms [End093, Hatt94, Howa94], catalytic metal particles [Iiji93, Beth93,
Ame194, Ivan94, Yaca93, Kian94, The396, LeeY97a], or a surface at low temperature
[GeM94, Cher94].
Successful synthesis of nanotubes by laser vaporization of carbon [GuoT95, The896],
raises the question, why such tubular structures often prevail over their more stable
spherical counterparts [Ugar92, Adam92, Toma93, Mait93, Robe92, Zhan93, Ame195].
These nanotubes, even though grown under annealing conditions, are very long and
defect-free, appear to be rather inert. It is furthermore intriguing that only single-wall
nanotubes have been synthesized in the presence of metal particles (e.g. Ni, Co, Fe)
[The396], whereas only multi-wall nanotubes have been obtained from pure carbon
vapor [GuoT95].
12
Observation of the rather unstable carbon tubes under annealing conditions
suggests the presence of an efficient mechanism that prevents their closure by a
dome [GuoT95, Toma96]. In the case of single-wall nanotubes, metal atoms are
believed to perform this task by catalytically removing pentagon defects at the edge
[The396, LeeY97a]. In absence of such a catalyst or spatial anisotropy (e.g. due to an
electric field [Iiji91]), formation of pentagon defects must be prevented at the growing
edge of a multi-wall tube [Kwon97].
In the following, I present the first microscopic study of the preferential growth
mechanism of multi-wall nanotubes [Kwon97]. Results elucidate not only the stable
morphologies at the reconstructed edge of growing multi-wall “armchair” and “zig-
zag” tubes, but also the detailed dome closure mechanism terminating the growth.
I find that carbon atoms adsorbing at the growing edge often prefer to bridge the
gap between adjacent wall edges by covalent bonds, rather than to saturate dangling
bonds at the edge of individual walls.
The ab initio calculations are performed using the cluster code DMol1 [De1190],
based on the Density Functional formalism within the Local Density Approximation
(LDA). In the double-numerical basis set [De1190], the C23 and C2p orbitals are
represented by two wave functions each, and 3d type wave functions are used to
describe polarization. The frozen core approximation is used to treat the inner core
electrons and the von Barth and Hedin exchange-correlation potential [Bart72]. Since
a full structure optimization of long open-ended nanotubes is not tractable using
ab initio techniques, I guided my search by a parametrized Linear Combination of
Atomic Orbitals (LCAO) method. This computationally efficient 0(N) approach
[Zhon93b] discussed in Appendix A, has been previously used with success to describe
the formation energetics [Toma91] and disintegration dynamics [Kim894] of fullerenes.
lDMol is a registered software product of Molecular Simulations Inc.
13
ZZ—O ZZ—l
Figure 2.1: Relaxed geometries at the growing edge of achiral double—wall carbon
nanotubes. (a) The (5,5)@(10,10) armchair double tube, with no lip—lip interaction
(structure AA-O, in perspective and end-on view), and with lip-lip interaction (struc-
tures AA—l and AA—2). (b) The (9,0)@(18,0) zig-zag double tube, with similar radii
as the armchair system. Shown are structures with no lip-lip interaction (ZZ-O, in
perspectivic and end-on view), and with lip—lip interaction (ZZ-l, ZZ—2, and ZZ-
3). Inner-tube atoms are shown in green (bright color), exposed edge atoms in blue
(black), and the added atoms in red (gray).
14
Here, I consider two examples of achiral double-wall nanotubes, shown in F ig-
ure 2.1. The (5,5)@(10,10) tube [Figure 2.1(a)] consists of concentric tubes with
armchair edges, and the (9,0)@(18,0) tube [Figure 2.1(b)] contains tubes with zig-
zag edges [Dre896]. The equilibrium C-C distance in these tubes is very close to the
graphite value dag = 1.42 A. The radii ofthe inner tubes, R(5, 5) = 3.42 A z R(9, 0) z
3.57 A, and the outer tubes, R(10, 10) = 6.78 A z R(18, 0) = 7.06 A, are very similar
for the armchair and the zigzag systems. The inter-wall distance of z3.4 A is close
to the graphite value [Iiji91, Ebbe96].
During the growth process, carbon atoms can adsorb at the open edge of either
tube, or connect these edges by a bridging covalent bond. The structure can be
characterized by the type of the inner and outer tube, the mutual tube orientation,
and the arrangement of the adatoms. Let me first consider the exposed edge of
the double-armchair (AA) (5,5)@(10,10) nanotube. Selected Optimized tube-edge
structures are shown in Figure 2.1(a). In order to minimize the finite-size effect on
the geometry and energetics, I consider double-wall tubes containing 210 atoms in
seven carbon ring layers. Different structures at the tip are generated by adding
Nadd = 10 carbon atoms in various positions at both tube ends. A global unrestricted
optimization has been performed using the LCAO method. Only the topmost two
layers, in addition to the adatoms, are relaxed in the corresponding LDA calculations.
When constructing the reference structure AA-O, I let the extra carbon atoms
saturate the dangling bonds at the inner tube. There is no interaction between the
adjacent walls in this open structure. Placing the extra atoms radially between the
two exposed edges in the AA-I structure gains the system AE = —0.42 eV per extra
carbon atom when the structure is fully relaxed. Stabilizing “lip-lip” interactions have
been previously postulated based on chemical intuition supported by parametrized
calculations [GuoT95, Toma96], and on molecular dynamics simulations [Char97].
15
The large energy gain in the AA-l structure and the short bond length do = 1.33 A
(1.31 A from LDA) of the dimer bridging the gap indicate that the “lip-lip” interac-
tion is mediated by a strong covalent bond. The AA-l structure shows an outward
distortion of the inner lip by @005 A and an inward distortion of the outer lip by
no.5 A. The energy associated with this strain, 7:56 eV at each end of the tube, is
more than outweighed by the energy gain of 0.8 eV for each of the 10 added atoms
due to the formation of covalent bonds connecting the tube edges. Restricting the
relaxation to the two topmost layers at the tube end reduced the energy gain per
extra carbon atom to AE = —0.37 eV. This energy difference is mainly associated
with the relaxation of tube walls from a cylindrical shape to a faceted structure im-
posed by the tube end, and very close to the LDA value AE = —0.25 eV. Also the
geometry optimized within LDA is virtually indistinguishable from that based on the
LCAO calculation. Note that the added atoms prefer to bind asymmetrically to the
“arm” rather than the “seat” site of the inner tube. In this way, the edge morphology
imposed by the “lip-lip” interaction prevents pentagons from forming at the inner
tube, which would otherwise initiate a premature dome closure.
I find that not all possible covalent bonds between adjacent tube edges stabilize
the system with respect to the AA—O structure. The energy difference per added
atom AE = +3.52 eV for the AA-2 structure, shown in Figure 2.1(a), suggests that
this structure is unlikely to form. To understand the stability gain of the AA—l and
stability loss of the AA-2 double-tubes with respect to the AA-O structure, I show the
color-coded binding energies of individual atoms at the relaxed edge in Figure 2.2.
Bright colors indicate that it is only atoms at the outermost tube edge that are less
bound than atoms in the tube interior, which are nearly as stable as in graphite.
Energy gain of the AA—l structure results from one of the added atoms closing a
pentagon at the outer edge, thus gaining stability for itself and its neighbors at the
16
K
v
A
V
x
V
a
v
4
\4 \4 54 \4 H
.4
-7 eV —5 eV
Figure 2.2: Color-coded binding energy of individual atoms in the optimized AA—O,
AA—l, and AA-2 nanotubes, described in Figure 2.1, in end-on and side view.
17
outer edge. I find that also the inner lip atoms are stabilized, but to a lesser degree.
It is interesting to find that allowing the inner tube to rotate in order to form a
pentagon also at the inner lip is energetically unfavorable with respect to the AA-l
topology. Figure 2.2 also indicates that it is the unstable triangles at the edge which
destabilize the AA-2 structure.
Qualitatively similar results are obtained for the double—zig-zag (ZZ) (9,0)@(18,0)
nanotube. The double tube I consider contains 216 atoms in eight layers. Different tip
morphologies are generated by placing 18 extra carbon atoms in different positions
at either tube end. The geometries of globally optimized structures are shown in
Figure 2.1(b).
In analogy to the armchair case, I construct the reference structure ZZ-O by plac-
ing extra carbon atoms only at the edge of the inner tube, as shown in Figure 2.1(b).
While there is no connection between adjacent tube edges in the ZZ-O structure, such
an interaction can be established by placing extra carbon atoms in-between the two
exposed edges. As in the case of armchair double-wall tubes, not all such “lip-lip”
interaction topologies are energetically favorable. In the ZZ-l structure, radially ar-
ranged adatoms saturate the dangling bonds at the exposed edges, thereby gaining
the system AE 2: —O.25 eV per extra carbon atom with respect to the reference
structure ZZ-O. Similar to the situation at the AA-l tube end, I find the added dimer
to connect the edges by a strong covalent bond of length d, = 1.28 A (1.26 A from
LDA). In the even more favorable edge morphology ZZ—2, with an azimuthal arrange-
ment of the extra atoms, the system gains AE = -0.54 eV per extra carbon atom
with respect to ZZ-O when the structure is fully relaxed. The azimuthal orientation
of the dimer allows it to saturate dangling bonds at both edges. The dimer retains
a strongly covalent character and a short bond length of 1.36 A (1.41 A from LDA).
Even though the gap between the exposed edges is bridged by a single adatom that is
18
connected to the edges by strong single bonds of lengths 1.50 A (1.51 A from LDA)
and 1.43 A (1.43 A from LDA), which impose strain in the tubes by bending the
outer-lip atoms inward and the inner-lip atoms outward, this appears to be the most
stable “lip-lip” interaction morphology. In contrast to the favorable ZZ—l and ZZ-2
morphologies, the ZZ-3 structure is less stable by AE = +0.67 eV than the ZZ-O
reference structure, since it contains squares at the outer edge. Due to energetic pref-
erence of rigid covalent bonds connecting exposed edges of adjacent walls, individual
tubes are not expected to rotate independently in nested multi-wall nanotubes.
To analyze the growth process of multi-wall tubes, I extend two concepts used in
discussions of crystal surface growth to nanotube edges. For a growing nanotube, the
edge coverage is defined by O = Nada/Ne. Here, Nadd is the number of added atoms
and N, is the number of edge atoms, which is the same as the number of atoms in
any layer of the nanotube. Next, I define the edge energy by
E3 = [Eb,tot(CN nanotube) — NE(,(C0o nanotube)] /Ne , (2.1)
which is independent of coverage. In this equation, Eb,tot(CN nanotube) is the total
binding energy of the truncated nanotube, and Eb(Coo nanotube) is the binding energy
per atom in an infinite nanotube. Hence, low E, values correspond to stable edge
morphologies.
Optimum edge morphologies and the associated energetics2 for a growing double-
wall (5,5)@(10,10) nanotube are shown in Figure 2.3 and Figure 2.4. The stable AA-l
structure, shown in Figure 2.1(a) and discussed above, corresponds to a coverage of
O = 1 / 3 and an edge energy of E8 = 0.80 eV. This structure can be further stabilized
by adsorbing extra five carbon dimers at either tube end, as shown in Figure 2.3.
2Since there is no absolute guarantee that a given structure is the globally most
stable tube isomer, the presented edge energies should be considered as upper bounds
for E8.
19
Figure 2.3: Optimum geometry at the growing edge of the (5,5)@(10,10) nanotube,
as a function of edge coverage 6. The color code is the same as in Figure 2.1.
This increases the coverage to O = 2 / 3, but reduces the edge energy to E6 = 0.56 eV,
which is significantly lower than the value E8 = 0.92 eV in absence of the “lip-
lip” interaction. Total absence of dangling bonds in this structure and essentially
no additional distortion of the tube ends both give rise to a considerable energy
gain AE = —0.53 eV per extra carbon atom by partially relaxing the topmost two
layers, when compared to a structure with no “lip-lip” interaction and all extra atoms
adsorbed on the outer tube edge. Structures optimized for a given coverage of the
growing double-tube are shown in Figure 2.3 and the corresponding edge energies in
Figure 2.4(a). Note that the most stable structures contain hexagons and pentagons,
but no squares or triangles. Since the function of the covalent “lip-lip” interactions is
to saturate dangling bonds at the edge, the bridging atoms continuously move with
the growing edge. As the coverages O =- 0 and O = 2 correspond to the same tube,
all information relevant for continuous growth is contained in the coverage interval
0 < 632 described here.
Tube growth is driven by the total energy gained when carbon from the atmo-
20
1.0b ""‘~:n- ’,—--‘ _'
{fix I” 0~\‘\‘ °°°°°°° 4h ....... . ' l 3“
\‘o X? \o I,’%”',".'--. ‘2
S. \\\‘ I,’ \‘\\ ’/:., ’ -*.; a}. \x‘
‘9" 0.5 '- 6- ' o -(l- '—'—' ‘52:?“ I
u? — + - tube growth “4‘
.. + -- dome closure (no L-L) be
0 - -0- - tube growth - -B - - dome closure (with L-L)
'00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
O 9
(b) O 0. l r l I d
\ v\
\0 \
'100 ’ ”'0‘- —o~ ‘ ‘
‘ *0
G) ‘x-
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Lg VL“ . ‘l;
.. . 9.? ‘
-3oo - n.1,, -
- -°- - tube growth “92$
-400 .- - -+ - - dome closure (no L-L) “it? -
l 1 l 1 ‘.
0 1 2 3 4 5
9
Figure 2.4: (a) Edge energy E, as a function of coverage for the energetically favored
growth and dome closure mode of the nanotube. (b) Total energy gain AE for the
nanotube coexisting with an infinite carbon chain, during its growth and dome closure
process with and without “lip-lip” (L-L) interactions.
21
sphere adsorbs at the growing tube end. Since under synthesis conditions [GuoT95]
the carbon atmosphere consists to a large degree of linear structures, the total en-
ergy gain AE(O) is associated with the conversion of carbon atoms in chains, with
a binding energy E, = ——6.05 eV [Zhon93b], to a coexisting, growing (5,5)@(10,10)
double-tube. The result, shown in Figure 2.4(b), indicates that in spite of edge energy
fluctuations depicted in Figure 2.4(a), tube growth is a strongly exothermic process.
Finally I consider the energetics of dome closure that terminates the growth
process. The inner and outer tubes of the (5,5)@(10,10) system can be closed perfectly
by hemispheres of the C60 and C240 fullerenes, respectively. Starting from the double-
tube structure described by O = 2, complete dome closure requires a coverage increase
by A822§ to O = 4% in this case. The edge energy for the case that the inner and
outer tubes begin closing by non-interacting domes (i.e. no “lip-lip” interaction) is
shown by + and the dotted line in Figure 2.4. Only selected structures of the closing
dome, such as for O = 4%, can be further stabilized by a “lip-lip” interaction, which
lowers the edge energy by as much as AEe = —0.13 eV to Ee(O = 4%) = 0.53 eV
(D and dash-dotted line). As the stabilizing “lip-lip” bonds have to be disrupted
completely for the dome closure to continue, the system encounters substantial energy
barriers during this process. Since the edge energy is higher for a closing dome than for
a continually growing tube (0 and dashed line), especially in the range 2.5 < O < 3.0
corresponding to initiation of the dome closure, spontaneous termination of growing
tubes is unlikely to occur.
These results have multiple, experimentally verifiable consequences. The main
claim is that sustained growth of defect-free carbon nanotubes is closely linked to
efficiently preventing formation of pentagon defects which would cause a premature
dome closure. For one, this is due to the covalent connection between adjacent nan-
otube walls at the growing edge which reduces the likelihood of such defects forming
22
at either edge. Another reason is that saturation of dangling bonds by “lip-lip” inter-
actions at the growing open edge should substantially reduce the growth rate, thus
leaving more time for defects to heal out and yielding perfect tubes.
One immediate consequence of postulating covalent “lip-lip” interactions as in-
dispensable for tube growth in a pure carbon atmosphere is that all such nanotubes
should have multiple walls [GuoT95]. Indeed, only multi-wall nanotubes have been
synthesized from pure carbon [GuoT95], in contrast to single-wall nanotubes asso—
ciated with metal catalyst. Without the passivating “spot-welds”, single-wall tubes
are prone to being etched away in the aggressive atmosphere under synthesis con-
ditions, which explains their absence in the tube material. Presence of covalent
“spot-welds” in chiral multi-wall tubes, on the other hand, is also expected to pre-
vent their “catastrophic burn-back” by “unraveling” in extremely high electric fields
[Rinz95, LeeY97b].
One can easily imagine other growth scenarios than the double-wall mechanism
discussed above, which yields only even-walled nanotubes. Covalent “spot-welds”,
simultaneously connecting an exposed edge to adjacent wall edges on the inner and
outer side, can yield odd-walled nanotubes. Simple bond-counting arguments suggest
that it is more difficult to saturate all dangling bonds at the edge of odd-walled tubes,
thus reducing their stability and inertness. This may explain the apparent abundance
of nanotubes with an even number of walls [GuoT95].
Since the stability of particular “lip-lip” interactions depends mainly on the local
geometry, results are expected to hold also for tubes with different radii or chiralities.
This model does not imply that the growth of multi-wall nanotubes proceeds in the
optimized and orderly fashion discussed in Figure 2.3 and Figure 2.4, or that only
specific achiral tubes may profit from the “lip-lip” interaction. It is realized that
23
the nanotube structure, whether chiral or achiral, is determined early-on during the
formation of the tube nucleus. Each favorable covalent bond connecting the exposed
tube edges, however, will lower the edge energy and hence stabilize the tip, making
it less prone to defect formation. The resulting passivation of the tube end slows
down, but does not stop the tube growth. The strained structure at the tube tip
invites preferential carbon adsorption at the growing edge. As the carbon coverage
increases, the tube end will undergo a complex, concerted exchange of atoms leading
to the most stable tip structure. Such structures have all been discussed above,
and involve covalent “lip-lip” interactions. It appears unlikely that occasional “spot-
welds” could remain intact as the tip position advances, since such defects would
locally decrease the stability and inertness of the otherwise perfect graphitic tubes.
Hence net growth of perfect tubes is expected to result from carbon accretion at
the exposed edge, involving “lip-lip” interactions only at the growing end of the
intermediate structures.
24
Chapter 3
Equilibrium Structure
The following discussion of the equilibrium structure of single-wall nanotube rOpes
(section 3.1) and multi-wall nanotubes (section 3.2) follows that presented in Refs.
[Kwon98a, Kwon98b, Kwon98c].
3.1 Single-wall nanotube ropes
Among carbon based materials, fullerenes (such as the C60 “bucky-ball” [Krot85]) and
nanotubes [Iiji91] have received much attention recently due to their high structural
stability and interesting electronic properties. Structurally rigid [Trea96] and highly
conducting [The896] “ropes” of carbon nanotubes are the molecular counterparts of
carbon fibers [Dre896].
Perfectly spherical C60 “bucky-ball” molecules [Krot85] are known to spin freely
at room temperature [Yann91, Tyck91] when crystallized to a solid [Kréit90]. One
may wonder, whether the homogeneous, perfectly cylindrical single-wall carbon nan-
otubes [LeeY97a, Thes96, Niko97] could also rotate relatively freely when forming
well-ordered bundles, the “ropes” [The596]. 13C nuclear magnetic resonance experi-
ments on solid C60 [Yann91, Tyck91] have shown that it is only below Tz260 K that
the free C60 rotation is hindered by the asphericity of the inter-molecular potential,
25
due to the discrete atomic positions. In bundles of nanotubes, the barrier for rotation
is expected to be even lower due to the frustration introduced by triangular packing
of tubes that have a D10}, symmetry and orientational dislocations caused by local
twists along the tube axes. Due to their large moment of inertia, nanotubes are not
expected to spin as fast as the C60 molecules. Nevertheless, it is useful to study
the soft librational motion of nanotubes and their transition to relatively free tube
rotations, marking the onset of orientational disorder in nanotube “ropes”.
In the following, I present theoretical evidence, supported by recent experimen-
tal data, about an orientational melting transition in “ropes” consisting of (10,10)
nanotubes. Due to the small inter-tube interaction and high degree of lattice frus-
tration, I predict a very soft librational mode to occur at uz12 cm“. This mode is
expected to disappear above the orientational melting temperature. I find that even
at relatively low temperatures, the hindrance of tube rotations due to discrete atomic
positions in the tube walls should lose its significance. This should mark the onset of
relatively free rotation or twisting motion about the tube axes.
To investigate the possibility of rotations in a bundle of nanotubes, I first opti-
mize the geometry of an ordered nanotube lattice, the “rope”, shown schematically in
Figure 3.1(a) using the density functional formalism within the local density approx-
imation (LDA). The plane-wave code [Sugi92] uses an energy cutoff of 50 Rydbergs,
describes carbon atoms using soft nonlocal pseudo-potentials [Trou91] within a sep-
arable approximation [Klei82], and uses the Ceperley-Alder exchange-correlation po-
tential [Cepe80] as parametrized by Perdew and Zunger [Perd81]. This basis had been
successfully used to optimize the lattice constant of the C60 solid [Weav91] and related
systems [Okad97]. Due to the large size of the basis set, that contains nearly 200,000
plane waves, I restrict the sampling of the irreducible part of the Brillouin zone to 4
k-points when determining the optimum inter-tube spacing and the equilibrium tube
26
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3 n 8 5
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0.4
0.2 7
0.0“ 1 1 14‘ 1 1 1 l
3.20 3.25 3.30 3.35 1.38 1.40 1.42 1.44
C [A] doe [A]
Figure 3.1: (a) Schematic end-on view of the equilibrium “rope” structure, depicting
the tube orientation angle (,0. Density functional results for the relative total energy
AE of (b) a “rope” of (10,10) carbon nanotubes as a function of the inter-tube spacing
a, (c) hexagonal graphite as a function of interlayer spacing c, and (d) hexagonal
graphite as a function of the bond length (100- All energies are given per atom.
27
geometry.
The inter-atomic distances in the tubes, optimized within the LDA, are doc =
1.397 A for bonds perpendicular to the tube axis (“double” bonds) and doc = 1.420 A
for the other (“single”) bonds. The weak inter-tube interaction in the “rope”, shown
in Figure 3.1(b), causes only very small radial deformation (“buckling”) of the tubes
with an amplitude of ARz0.03 A. As shown in Figure 3.1(b), the calculated equilib-
rium inter-tube separation ac)”, = 16.50 A lies only 2.8% below the observed value
aefixpt = 16.95i0.34 A [The596]. I also observe a significant decrease of radial tube
deformations (“buckling”) from ARz0.03 A to @0004 A as the inter-tube spac-
ing in the “rope” increases by a mere 2.8% from the theoretical equilibrium value
am}, = 16.50 A. Suppression of this radial deformation, which is likely linked to the
ability of individual tubes to spin freely, should effectively cause an increase of the
equilibrium inter-tube spacing a, at the orientational melting temperature. Hence,
the 2.8% difference between the observed and calculated value of ac may be partly
due to the fact, that the value observed at room temperature [The396] aempt was
compared to the calculated zero-temperature value aeyh.
Even though the agreement between the calculated and observed inter-tube spac-
ing is very good for a partly van der Waals system by standards of ab initio calcula-
tions, I try to understand the minor deviations by calculating the optimized geometry
0f hexagonal graphite in Figures 3.1(c) and ((1) using the very same basis. Results for
the energy dependence on the inter-layer separation 0, presented in Figure 3.1(c), show
that also this value is underestimated by 2.4% when compared to the experimental
value. Similar to the tubes, the inter-atomic distance in graphite is underestimated
by Only 0.8% with respect to the experimental value.
The k-point sampling used in the LDA calculation is sufficient to describe the
28
I I I I I I I I I I I I n
0.15 - ’,e.\ /°‘~ 1
1’ \\ ” \\
I \ ,’ \
I"I \\ II \\
S‘ 0 10 " ,IO c», /e o\ r
I \ I \
(D , \ ’ \\
IE! I, \ I” ‘\
I \
LU 0 05 L ,1, \\ I” \\ .1
Q ’ ‘
[,6 e“ [,0 e“
h I ‘\ I, \\\ I
0 00 QY’ L I I I I ‘7" I I I L I Y“
_60 _4o _20 00 20 40 60
Figure 3.2: Dependence of the “rOpe” energy AE on the orientation angle (p of
individual nanotubes. The energy is given per atom.
details of intra—tube and general features of inter-tube interactions. This interaction
depends not only on the inter-tube separation a, but also the tube orientation angle cp,
defined in Figure 3.1(a). The relatively coarse k-point grid, used in the LDA study, is
found not to be adequate to describe the minute effect of tube rotations on the inter-
tube hybridization and the density of states near Ep. Therefore, I have performed a
parametrized calculation of these quantities using 102,400 k-points in the irreducible
Brillouin zone for the “rope” lattice and 800 k-points for the tube. The tight-binding
parametrization, based on LDA electronic structure results [Toma91], has been used
successfully to describe superconductivity in bulk C60 [Sch192]. The band structure
energy functional is augmented by pairwise interactions describing both the closed-
shell inter-atomic repulsion and the long-range attractive Van der Waals interaction,
to correctly reproduce the interlayer distance and the 033 modulus of graphite.
These results, in good agreement with the LDA data presented in Figure 3.1(b),
indicate that nanotubes gain AEsz meV per atom when bunching up to a “rope”.
The calculated dependence of the binding energy AE on the tube orientation 1,? is
comparably weak, as shown in Figure 3.2. Due to the high degree of frustration in
a system of tubes with D10}, symmetry that are bundled to a triangular lattice, I
29
find AE (1,0) to be periodic in 1p, with the equilibrium tube orientation (,9, 2 0°, a
period Acp 2 6° and an activation barrier for rotation of AE§0.15 meV per atom.1
This is in good agreement with LDA results for the equilibrium tube orientation and
the rotational barrier of AEzOB meV per atom. At the relatively large equilibrium
separation dwai3.4 A between the walls of neighboring nanotubes, the repulsive part
of the interaction (originating in the kinetic energy increase of a compressed electron
gas) depends only weakly (AES0.07 meV) on the tube orientation angle (,0. The
orientational dependence of the total energy thus reflects both the changing inter-tube
hybridization and Van der Waals interaction between adjacent tubes, also discussed
in Ref. [LuJ P97].
At very low temperatures, nanotubes are expected to perform mostly librational
motion in the shallow potential wells. Tube librations or rotations within the “rope”
should be considered as a twisting motion of tube segments rather than a spinning
motion of rigid tubes. Approximating the periodic, but strongly anharmonic potential
AE(
3° results in the nanotube switching locally from one equilibrium
orientation to another. Formally, by mapping the orientational coordinate (,0 onto
the position coordinate :12, this process can be described using the Henkel—Kontorova
model used to describe dislocations in strained one-dimensional lattices. Under syn-
thesis conditions at temperatures exceeding 1,000 K, substantial finite twists is ex-
pected to occur in free nanotubes, which are associated with strictly zero energy cost
over an infinite length. Upon condensation to a bundle, in an attempt to Optimize the
inter-wall interaction while minimizing the torsion energy, orientational dislocations
are frozen in at an energy cost of only @0.17 meV per atom in a @150 A long strained
region [Kwon99b] taking into account the actual tube rigidity[Over93], as compared to
an Optimized straight bundle of nanotubes. Due to the high tube rigidity [Over93], I
expect TOM for a perfect, dislocation—free rope to be significantly higher than the few
degrees Kelvin, suggested by the 0.15 meV/ atom high activation barrier for rotation
in the unphysical zero-rigidity limit. Presence of orientational dislocations and tubes
31
Of other chiralities, on the other hand, would reduce the level of commensurability
and lower the activation barrier for tube rotations, thus resulting in a finite value of
TOM.
There are two indications that onset of orientational disorder may be signifi-
cantly below room temperature. First, low-frequency infrared modes of the “rope” at
u@15, 22,40 cm‘l, some of which may be librations, have been reported to disappear
at T@30 — 180 K [Holm98]. The second indication is the transition from nonmetal-
lic to metallic character of the nanotubes, occurring near 50 K [Fisc97], which in
my interpretation arises from subtle changes of the electronic density of states near
E: in presence of increasing orientational disorder, to be discussed in Chapter 5.
This is more closely related to the recently proposed mechanism for the tempera-
ture dependence of resistivity due to inter-tube hopping near defects [Kais98] than
temperature-induced changes in the weak localization of electrons on individual tubes
[Lang96, Fuhr99].
LDA results indicate a significant decrease of radial tube deformations (“buck-
ling”) from AR@0.03 A to @0.004 A as the inter-tube spacing in the “rope” increases
by a mere 2.8% from the theoretical equilibrium value Ge"); 2 16.50 A. Suppression of
this radial deformation, which is likely to occur in rotating tubes, should effectively
lead to an increase of the equilibrium inter-tube spacing as. The 2.8% diflerence be-
tween the observed and calculated value of a, may be partly caused by the fact that
I was comparing a room-temperature value [The896] of agent to a zero-temperature
value of aeyh. It is suggested that orientational melting of the tube lattice should
be accompanied by a significant, possibly discontinuous increase of the equilibrium
inter-tube spacing ac.
In summary, I used ab initio and parametrized techniques to determine the
32
equilibrium structure and low—frequency libration motion of an ordered “bundle”
of (10,10) carbon nanotubes. The weak inter-tube interaction and lattice frustration
results in a very soft libration mode at u@12 cm“.
Due to the small activation
barrier for tube rotations and the presence of frozen-in orientational dislocations in
the “rope”, I expect the “twisting” motion of finite tube segments to turn into an
orientational melting process within the “rope” even at very low temperatures.
3.2 Multi-wall nanotubes
Existing investigations of multi-wall nanotubes have focussed primarily on their gro-
wth mechanism [Kwon97, Char97, Nard98], as well as changes of the band structure
[SaiR93, LamP94] and total energy [Char93] as the inner tube rotates about or slides
along the tube axis. The inter-wall interaction in a multi-wall tube is very similar
to the inter-ball interaction in the C60 solid, since these structures share the same
sp2 bonding and inter-wall distance of @3.4 A found in graphite. While it is well
established that C60 molecules rotate relatively freely in the solid at room temperature
[Yann91, Tyck91], there has been little discussion of the analogous rotation in systems
consisting of nanotubes. Only recently, rotations of individual nanotubes in a rope
consisting of (10,10) tubes have been discussed [Kwon98b, Kwon98a]. Owing to the
uncommon combination of a relatively small energy barrier for rotation and a large
mass of macroscopically long nanotubes, individual tubes are unlikely to move as
rigid Objects in the rope, but rather to bend and twist locally, displacing orientational
dislocations that were frozen in during the tube synthesis.
I focus the following investigation on the (5,5)@(10,10) double-wall nanotube,
shown schematically in Figure 3.3. The individual (5,5) and (10,10) tubes are both
metallic and show the preferred “graphitic” inter-wall separation of 3.4 A when nested.
33
Figure 3.3: (a) Schematic view of a double-wall tube. The relative radial, axial, and
rotational displacement of the inner with respect to the outer tube are described by
Ar, A2, 1pm and 90mm respectively. (b) Top view and (c) perspective view of the
(5,5)@(10,10) double-wall tube with Ar = 0, A2 = 0, so," = 0 and (pm 2 0.
Assuming both tubes to be rigid cylinders with parallel axes, the double—wall tube
geometry is defined uniquely by the separation of the axes Ar, the axial offset of the
inner tube A2, the orientation of the inner tube 90,-" and the outer tube 1pm,, with
respect to the plane containing the two axes, shown in Figure 3.3(a).
To determine the structural and electronic properties of multi-wall nanotubes,
I use the parametrized tight-binding technique with parameters determined by ab
initio calculations for simpler structures [Toma91]. This approach has been found
useful to describe minute electronic structure and total energy differences for systems
with too large unit cells to handle accurately by ab initio techniques. Some of the
problems tackled successfully by this technique are the electronic structure and super-
conducting properties of the doped C60 solid [Sch192] and the Opening of a pseudo-gap
near the Fermi level in a rope consisting of (10,10) nanotubes [Kwon98b]. The band
34
structure energy functional is augmented by pairwise interactions describing both the
closed-shell inter-atomic repulsion and the long-range attractive van der Waals inter-
action, to correctly reproduce the interlayer distance and the 033 modulus of graphite.
The adequacy of this approach can be checked independently by realizing that the
translation and rotation of individual tubes are closely related to the shear motion
of graphite. The energy barrier in tubes is expected to lie close to the graphite value
which, due to the smaller unit cell, is also easily accessible to ab initio calculations
[Char94, Scha92].
I focus in my calculations on the softest vibrational modes and ignore the defor-
mation of individual tubes from a perfectly cylindrical shape. The dependence of the
total energy of the double-wall tube on the distance between the (5,5) and (10,10)
tube axes is shown in Figure 3.4(a). Even though my results suggest that the system
is coaxial in equilibrium, the potential energy is nearly flat for IArISOJ A. For a
perfectly straight tube, the shallow potential well would suggest a very soft radial
mode to occur with u = 18 cm”.
At nonzero temperatures, I do not expect the
inner tube to be coaxial, but rather to form a helix inside the outer tube. Such a
helix would maximize the inter-wall contact and also could vibrate and rotate locally
about its axis. I believe that helical or similar tube distortions of 50.1 A are real,
but probably are undetectable in Transmission Electron Microscope images.
Due to the low compressibility of tubes along their axis, axial motion of individual
nanotubes resembles that of a rigid body. Even though the activation barrier per atom
for axial displacement of one tube inside the other is only 0.2 meV [Char93], the
activation barrier to move the entire rigid nanotube is infinitely high, thus effectively
freezing in this particular degree of freedom.
Due to the relatively high symmetry of the coaxial system consisting of a D50!
35
A
93
v
1 0 3; I I I I I I I $14
\ I
0.8 P \‘ I, -
S‘ 0 0
(D \ I
E \ I
""' 0 6 P b\ Id ‘
Z \ I
E, 04 - ‘Q\ 9’ -
”0, pl
0.2 - ‘e. ,e’ -
”Os ’0’
00 -I I I tv-o_?_9—¢l’°° I I L
-0.2 -0 1 0.0 0 1 0.2
Ar [A]
1 0 I I I I I I I I I I I I I
0 8 '- ,°‘°b‘ I6 9 °\ -
a 0 6 1’0 b“ 7’0 o\
E I. / -
'2' [,0 ex“ [,0 e“
I ’ \
g 0 4 - /° ,x\ [,0 °‘. -
I0 0‘ ’0 o\
0 2 - ,9 t i” <1 _
9’0 ‘0 e?! 0‘0
0 O f? I I I I I Y I I I I I ‘7‘
-l8° -12° -6° (2; 6° 12° 18°
i
Figure 3.4: Energy dependence on (a) the separation of the tube axes Ar (for A2 = 0,
‘Pin = 0, 1,00,“ = 0), and (b) the orientation of the inner tube 1pm (for Ar = 0, A2 = 0,
99m = 0). All energies are given per atom.
36
(5,5) nanotube nested inside the D10], (10,10) nanotube, the dependence of the inter-
tube interaction on the tube orientation shows a periodicity of 18°. Results shown
in Figure 3.4(b) suggest this interaction to be harmonic, with a relatively low activa-
tion barrier for tube rotation of 0.8 meV/ atom. Assuming that the outer tube were
pinned, the inner tube would exhibit a soft libration with a frequency u,,,@31 cm‘l.
In the Opposite case of a fixed inner tube, the heavier outer wall should librate
with Vou¢@11 cm“. A free double-wall tube should librate with a single frequency
V@33 cm‘l.
As in an infinite system consisting of bundled identical nanotubes, I expect also in
an individual double-wall nanotube an orientational depinning and melting to occur at
sufficiently high temperatures. The onset of orientational melting at TOM is expected
to be qualitatively similar to that described for the infinite lattice of (10,10) nanotubes
[Kwon98b, Kwon98a]. Even though the activation barrier for rotation may be small
on a per atom basis, the relevant quantity in this case is the infinitely high barrier for
the entire rigid tube that would lock it in place. In the other extreme of a straight,
zero-rigidity tube composed of independent atoms, 0.8 meV/ atom activation barrier
for rotation would give rise to an orientational melting transition at T0M@10 K.
A more realistic estimate Of the onset of orientational disorder must consider
that multi-wall nanotubes, when synthesized, are far from being straight over long
distances. As suggested by the potential energy surface for the librational mode in
Figure 3.4(b), a local twist by (pm > 9° results in the nanotube switching locally
from one equilibrium orientation to another. Formally, by mapping the orientational
coordinate (pin onto the position coordinate :3, this process can be described using the
Frenkel-Kontorova model used to describe dislocations in strained one-dimensional
lattices. Under synthesis conditions at temperatures exceeding 1,000 K, substantial
finite twists are expected to occur in the individual tubes. Upon forming a multi-wall
37
tube, in an attempt to optimize the inter-wall interaction while minimizing the torsion
energy, orientational dislocations are frozen in at an energy cost of only @1.0 meV
per atom in a @100 A long strained region [Kwon99b], as compared to an Optimized
straight double-wall tube. Taking a straight, zero-rigidity tube as a reference, I expect
TOM to increase from the 10 K value with increasing rigidity. Presence of orientational
dislocations, on the other hand, should lower the activation barrier for tube rotations,
thus lowering TOM and possibly compensating the effect of nonzero rigidity.
In summary, I calculated the potential energy surface characterizing the inter-
action between the individual tubes in the (5,5)@(10,10) double-wall nanotube as a
function of the inter-axis distance and the mutual tube orientation. I found that the
soft librational motion is expected to occur at S. 30 cm“1 below the orientational
melting temperature, which marks the onset of relatively free rotations Of the indi-
vidual tubes, probably via axial displacement of orientational dislocations that were
frozen in during the tube formation.
38
Chapter 4
Structural Properties
The following discussion of the structural properties, such as structural and orien-
tational melting transition (sections 4.1 and 4.2), and thermal expansion behavior
(section 4.3), follows that presented in Refs. [Kwon00, Kwon99b, BaugOO].
4.1 Structural melting transition
In analogy to many other sp2 bonded carbon structure, such as graphite and fulle-
renes, carbon nanotubes show very high structural stability. Because of this, almost
no information is available about their equilibrium phases at temperatures close to
and exceeding the melting point of graphite. Molecular dynamics simulation results
of fullerenes have shown that a most dramatic transition to a “pretzel” phase with
a three-dimensional structure of multiply connected carbon rings occurs at a high
temperature of T@4000 K [KimS94]. At T2104 K, a complete conversion of com-
pact fullerenes to carbon chain fragments is driven by entropy. I expect that carbon
nanotubes may also show a similar melting transition near a similar melting temper-
ature, due to their structural similarity. I believe that the high—temperature behavior
of nanotubes is related to their formation and intermediate structures, which occur
during thermal quenching, may also be observed while heating up these structures to
39
high temperatures.
In the following, I perform a molecular dynamics study of the melting and evap-
oration process of a (5, 5)@(10, 10) double-wall nanotube, which is the structure with
a coverage of O = 1 shown in Figure 2.3 and also in Figure 4.2(a). Disintegra-
1
3
tion of this structure, which is a possible intermediate structure occurred during the
tube formation, should related to the formation of multi-wall nanotubes. In this
simulation, I investigate the response of a canonical ensemble of this double-wall
nanotube system to gradually increasing heat bath temperatures using Nosé—Hoover
molecular dynamics [Nosé84, Hoov85, Alle87]. Of course, the quality of simulation
results depends primarily on the adequacy of the total energy formalism applied to
the nanotubes. Since this simulation also addresses the fragmentation under extreme
conditions, the formalism used must accurately describe the relative stability of nan-
otubes with a graphitic structure and their fragments, mostly short carbon chains.
The corresponding transition from 3192 to 3p bonding can only be reliably described
by a Hamiltonian which explicitly addresses the hybridization between C23 and C2p
orbitals. A precise evaluation of inter-atomic forces is required to obtain the correct
long-time behavior of the system. On the other hand, the long simulation runs nec-
essary to equilibrate the small systems during the annealing process, and the large
ensemble averages needed for a good statistics, virtually exclude ab initio techniques
[Hohe64, Kohn64] as viable candidates for the calculation of atomic forces.
For this reason, the force calculation are based on a linear combination of atomic
orbitals formalism which conveniently parametrizes ab initio local density functional
results for structures as different as C2, carbon chains, graphite, and diamond [Toma91].
I use an adaptation of this formalism to very large systems, which is based on the fact
that forces depend most strongly on the local atomic environment and which has been
implemented using the recursion technique [Zhon93b]. This technique is described in
40
detail in Appendix A. The force calculation can be performed analytically to a large
degree. This not only speeds up the computations significantly, but also provides
an excellent energy conservation AE / E 510‘10 between time steps in microcanonical
ensembles, and a linear scaling of the computer time with the system size. Most im-
portant, forces on distant atoms can be efficiently calculated on separate processors
of a massively parallel computer.
In molecular dynamics simulations, I use the fifth—order Runge-Kutta interpola-
tion scheme to integrate the equations of motion over time steps At = 5 x 10‘16 3.
Each molecular dynamics run is begun by equilibrating the particular fullerene for
300 time steps at a temperature T,- = 1,200 K, after the directions of the initial
atomic velocities have been randomized. I increase the temperature of the heat bath
in steps of AT 2 400 K from T,- to the final temperature T, = 6, 800 K, and let the
double-wall nanotube equilibrate for 800 time steps at each new temperature.
In Figures 4.1 and 4.2, I present results for the molecular dynamics simulations
for melting transition of the double-wall nanotube. The temperature dependence of
the total energy E per atom is shown in Figure 4.1(a). I observe a steady increase of
E with increasing temperature, with a well-formed step at T @ 4, 000 K indicative of
a “phase transition”. To get a better understanding of the structural transformations,
a radial diffusion index 6, is defined as [Gr6n97a, Gr6n97b]
_ Zrldift) " 611(0)]2
6,. — < E (17(0)2 A (4.1)
where d,(t) is the distance of atom z' from the center of mass of the nanotube at
time t. This definition of 6, targets the onset of radial diffusion in the nanotube upon
melting, while ignoring the translation and rotation of the rigid body. This calculated
temperature dependence of the radial diffusion index 6, is shown in Figure 4.1(b).
In the solid phase, occurring at TS3,000 K and depicted in Figure 4.2(b), the
41
A
93
v
-5.0
-5.5
-6.0
-6.5
E [eV/atom]
1000 3000 5000 7000
T [K]
(b)
0.12_r-..-w.
0.08
I
1
0.04
I
I L J
I I I
I
' l
5, (diffusion index)
1000 3000 5000 7000
T [K]
Figure 4.1: Temperature dependence of (a) the total energy E per atom and (b) the
radial diffusion index defined in Eq. (4.1). The data points mark the discrete steps of
the heat bath temperature used, and the dotted lines in (a) with a slope of 31173 are
guides to the eye.
42
Figure 4.2: “Snapshots” illustrating the geometry of a (5, 5)@(10, 10) double-wall nan-
otube with lip-lip interaction discussed in Chapter 2, at temperatures corresponding
to the different “phases” discussed in the text. (a) Initial configuration corresponding
to O = 1% structure shown in Figure 2.3. Structures showing different phases (b) at
2800 K, (c) at 4400 K and (d) at 6000 K.
43
(5, 5)@(10, 10) double-wall nanotube is intact, except that it shows soft breathing and
quadrupolar deformation modes.
Above T @ 4,000 K, I observe a dramatic transition to a tangled chain phase,
which is very similar to “pretzel phase” observed during simulations of melting tran-
sition in fullerene clusters [Kim894], consisting of interconnected carbon rings and
depicted in Figure 4.2(c). In my calculation, I find this structure to be initiated by
the rupture of single bonds connecting carbon atoms at an open edge to atoms in the
bulk-like tube part, which creates a large Opening at the surface. The energetically
less favorable sp bonding of a growing number of two-fold coordinated atoms in this
phase results in the broadening of the binding energy distribution.
Tangled carbon chains and rings open up at T25, 000 K, as shown in F ig-
ure 4.2(d). The number of structural constraints in this linked chain phase decreases,
allowing individual atoms to relax. This leads to a smaller number of inequivalent
sites and more pronounced peaks in the binding energy distribution.
While time reversal of the evaporation process studied here would provide one
possible aggregation path, the general formation mechanism is clearly more complex.
I expect that many structures which occur during the fragmentation also reoccur
during the aggregation. Of course, most of the “random” aggregation paths will not
lead immediately to a perfect nanotube, yet are likely to contain relatively stable
structural elements such as chains and rings. Once such a linked chain structure is
formed, the aggregation dynamics will be governed by many unsuccessful attempts
to reversibly “roll up” the chains to a nanotube, followed by one successful attempt
to create an inert structure.
In summary, I presented molecular dynamics simulation results of the melting of
a (5, 5)@(10, 10) double-wall carbon nanotube. I found a most dramatic transition
44
to a tangled chain phase, corresponding to melting transition, to occur at a high
temperature of ”Ts/112.4000 K.
4.2 Orientational melting transition
Since their first successful synthesis in bulk quantity [The896], “ropes” Of single-wall
carbon nanotubes have been in the spotlight Of nanotube research. Recent exper-
imental data indicate that carbon nanotube ropes exhibit an unusual temperature
dependence of conductivity [LeeR97, Fisc97], magnetoresistance [KimG98], and ther-
moelectric power [Hone99, TiaM98]. Several physical phenomena have been sug-
gested to cause the intriguing temperature dependence of conductivity behavior, such
as twistons [Kane98], orientational melting [Kwon98b, Kwon98a], weak localization
[Fuhr99], and Kondo effect [KimG98]. The opening of a pseudogap near E1: in a
bundle composed of (10,10) tubes has been postulated to result from breaking the
D10); tube symmetry by the triangular lattice [Kwon98b, Dela98].
Nanotubes [Iiji91] and C60 “bucky-ball” molecules [Krot85, Kriit90] are similar
inasfar as their interaction is weakly attractive and nearly isotropic when condensing
to a solid. The small anisotropy of the C60 intermolecular potential drives the solid to
an orientationally ordered simple-cubic lattice with four molecules per unit cell at low
temperatures [Hein91]. Only at T3249 K does the C60 solid undergo a transition to a
face-centered cubic lattice. As confirmed by 13C nuclear magnetic resonance [Yann91,
Tyck91], this is an order-disorder phase transition, with C60 molecules spinning freely
and thus becoming equivalent above 249 K. Unlike the C60 solid, very little is known
about the equilibrium structure of bundled nanotubes beyond the fact that they form
a triangular lattice [The396]. In particular, nothing is known about the equilibrium
orientation of the tubes within a rope. More intriguing still is the possibility of an
45
orientational melting transition associated with the onset of orientational disorder
within a rope.
In the following, I calculate the potential energy surface and the orientational
order of straight and twisted tubes within a rope. I further postulate that realistic
nanotube ropes contain orientational dislocations that have been frozen in as the
tubes formed ropes at a finite temperature, the same way as dislocations are known
to form in crystals. I map the energetics of a rope with dislocations onto a lattice gas
model and find that the onset of orientational disorder, corresponding to a free axial
diffusion of twistons, should occur at T0 1142.160 K.
In order to determine the orientational order and the rotational motion Of tubes
in a rope, it is necessary to describe both the inter-tube interaction and the torsional
strain within the individual tubes. Since the anisotropic part of the inter-tube inter-
action in a rope is weak and local, it can be well described by pairwise inter-tube (not
inter-atomic) interactions. To describe the interaction between two neighboring tubes
as a function of their orientation, I use the parametrized linear combination of atomic
orbitals (LCAO) formalism [Toma91], described in Section 3.1. This technique has
been successfully used to explain superconductivity arising from inter-ball interac-
tions in the doped C60 solid [Sch192], the Opening of a pseudogap near Ep in a (10,10)
nanotube rope [Kwon98b, Kwon98a], as discussed in Section 5.1, and the opening and
closing of four pseudogaps during the librational motion of a (5,5)@(10,10) double-
wall tube [Kwon98c], described in Section 5.2. A smooth cutoff function1 has been
implemented to keep the total energy continuous as the neighbor topology changes
while tubes rotate. This energy functional correctly reproduces the exfoliation energy,
lFermi-Dirac expression 1+ exp(ߣ _1 is used with dc = 3.87 A and dw =
dw
0.2 A, as a smooth cutoff function of distance d for both the hopping integrals and
pairwise interactions.
46
the interlayer distance and the C33 modulus of hexagonal (AB) graphite, as well as the
energy barrier for interlayer sliding, corresponding to the energy difference between
AB and AA stacked graphite.
To determine the interaction between a pair of aligned (10,10) nanotubes, I first
define the orientational angles cpl and 902 for these tubes by the azimuthal angle of
the center of a particular bond with respect to the connection line between adjacent
nanotube axes, as shown in Figure 4.3(a). For each (aphcpg) pair, I calculate the
inter-tube interaction using a fine mesh of 800 k-points sampling the one-dimensional
irreducible Brillouin zone. Due to the high symmetry Of the system, the interaction
energy AE(<10_2
Bl Bz B3
—3.42><10_2 —9.51x10"2 1.22x10”2
01 C2 C3
2.96x10‘1 —1.12X10‘l 1.75x10'3
Figure 4.4: Top view of the equilibrium structure of bundled interacting nanotubes,
with a two-tube unit cell.
ing the total energy E with respect to the orientations of all individual tubes.2 Due to
the high level of orientational frustration in a triangular lattice of (10,10) nanotubes
with D101, symmetry, the potential energy surface E(<10‘16 s. The average
length contraction has been obtained using a simplified Fourier analysis of the total
length as a function of time, based on the average 1000-atom tube length at 0 K,
200 K, 400 K, and 800 K. Similar to the definition of B, the linear thermal expansion
coefficient is defined as
a 2 %:—§:. (4.7)
This value changes from zero to —1><10‘5 K‘1 near room temperature. It eventually
goes back to zero and turns to expansion as the anharmonic part of the inter-atomic
potential is explored at very high temperatures.
In results shown in Figure 4.8, I suggest that carbon nanotubes exhibit thermal
contraction across the temperature range studied, in agreement with the experimental
results [BaugOO]. Nanotube becomes shortest at Tz700 K as shown in Figure 4.8(a).
58
3°
(a)
0.000
-0.003 ’
00060 ‘ ‘ 500‘ ‘ {1‘01“} ((1)
T [K]
Figure 4.8: (a) Temperature dependence of ratio of change in length to length AL / L.
The most representative modes which are responsible for thermal contraction in nan-
otubes; (b) a straight tube for reference, (c) a tube showing optical quadrupolar
deformation mode, and (d) a tube showing transverse acoustic mode. Due to these
two modes, a nanotubes shows negative thermal expansion.
It is useful to analyze the relative importance of the particular modes on the thermal
contraction. At and below the temperature range of the experiment, the strong inter-
atomic bonds are within the harmonic interaction range. Consequently, bond length
fluctuations average out, resulting in a negligibly small contribution of longitudinal
acoustic modes to the length contraction. As shown in Figure 4.8(d), a much larger
contribution of close to one third of the total value results from transverse acoustic
modes causing a snake—like motion of the nanotube. The softest transverse acoustic
modes are found among the shear modes that require a minimum of bond stretching
and bending. Length contraction based on these acoustic modes, which show the
largest sensitivity to the total tube length, shows a converged value for (10, 10) tubes
with a total length of close to 100 A. The high-frequency tube torsion modes are
observed to contribute close to 10% towards tube length contraction. The most
important part of the tube formation is shown to originate in the deformation from
the circular cross-section of the tube shown in Figure 4.8(c). The most important of
these optical modes are associated with a deformation from a circular to an elliptical
59
cross-section, with the major axis changing direction along the tube. This deformation
of the cross-section of a tube with a constant surface area results in a net volume
reduction that should also occur in fullerene structures at elevated temperatures.
Not only do these optical quadrupolar modes contribute to contraction along the
tube direction, but also to that of cross section area. Using the complete elliptic
integral of the second kind E (:r), I calculate analytically the area S of an ellipse as
function of the ratio, tEb/a, of its’ major and minor axes a and b. The calculated
area,
Soct/[E(1—t2)]2
, decreases monotonically up to zero as b, thus t, goes to zero while keeping its
circumference constant. As a result, these optical modes results in volumetric thermal
contraction in carbon nanotubes.
In summary, I used canonical molecular dynamics technique to investigate ther-
mal behavior of C60 molecules and carbon nanotubes. I found that both systems
have negative thermal expansion coefficients due to Optical deformation modes and
transverse acoustic modes.
60
Chapter 5
Electronic Structure
The following discussion of the electronic structure of single-wall nanotube ropes
(section 5.1) and multi-wall nanotubes (section 5.2) follows that presented in Refs.
[Kwon98a, Kwon98b, RaoA99, Kwon98c].
5.1 Single-wall nanotube rapes
Electronic states involved in the superconducting behavior of the alkali-doped C50
solid derive from this molecule’s degenerate lowest unoccupied tlu molecular orbital
[Sch192] that extends to an z0.5 eV wide band due to the inter-C60 interactions, which
in turn depend on the molecular orientation [Sch192, SaiSQl]. Similarly, interactions
between nanotubes in pure or alkali-doped “ropes”, which depend on the mutual
orientation of adjacent tubes, are expected to affect states at the Fermi level to an
important degree. This point is especially intriguing, since recent calculations suggest
that small deformations may open up a gap at E1: in isolated conducting nanotubes
[Kane97, Roch98]. Hence, inter-tube interactions and orientational disorder may play
an important role in the conducting (and superconducting) behavior of these systems.
In the following, I present theoretical evidence that the electronic structure of
(10,10) nanotubes, especially near the Fermi level, changes significantly when the
61
tubes are bundled to “ropes”. I find that the inter-tube interaction gives rise to an
additional band dispersion of z0.2 eV near E p, which would significantly diminish the
effect of minute distortions on states near E p, predicted for individual tubes [Kane97,
Kane97]. Even though the inter-tube interaction is weak, it causes a buckling of the
Fermi surface and an z0.2 eV wide pseudo-gap to open at Ep in undoped systems.
Occurrence of this pseudo-gap in “ropes” of (10,10) nanotubes has independently
been reported in Ref. [Dela98]. Due to the inter-tube interaction, the density of
states N (E) is found to increase by z7% near the Fermi level (outside the pseudo-
gap region), almost independent of the tube orientation. Potassium doping to the
composition KCg increases N (E p) by one order of magnitude, again independent of
tube orientation, in close agreement with the observed conductivity increase by one
order of magnitude in this system upon doping [LeeR97]. As in the doped C60 solid
[Sch192], librational phonons are not expected to play an important role in BCS—based
superconductivity of nanotube “ropes”.
To investigate the effect of inter-tube coupling in a bundle of librating nanotubes
on its electronic structure, I first optimize the geometry of an ordered nanotube lattice,
the “rope”, using the density functional formalism which is described in Section 3.1.
As mentioned in Section 3.1, the relatively coarse k-point grid, used in the LDA study,
is found not to be adequate to describe the minute effect of tube rotations on the
inter-tube hybridization and the density of states near Ep. Therefore, I perform a
parametrized calculation of these quantities using 102,400 k-points in the irreducible
Brillouin zone for the “rope” lattice and 800 k-points for the tube.
Results for the electronic structure of an isolated tube and that of the “rope”
are presented in Figure 5.1, where I compare the band structure of an isolated tube
to that of the “rope” in equilibrium. The irreducible Brillouin zone of the triangular
tube lattice, shown in Figure 5.1(a), collapses to the F — A line at large inter-tube
62
.°
0
o
I
pl
0
O
Energy(eV)
Energy(eV)
- 10.00
—15.00
—20.00
PU AF
Figure 5.1: (a) Irreducible part of the hexagonal Brillouin zone of a nanotube crystal,
the “rope”. Band structure of (b) an individual nanotube and (c) the “rope”, along
the tube axis. ((1) Dispersion of the top valence and bottom conduction bands of the
“rope”, in a plane perpendicular to the tube axis, containing the point AF depicted
in (c).
63
separations corresponding to isolated tubes. In this case, depicted in Figure 5.1(b),
the Fermi momentum occurs at the point AF. On the other hand, when nanotubes
are bundled in the “rOpe” lattice, the Fermi point [3,: expands to a Fermi surface
that is normal to the F — A line in the hexagonal Brillouin zone. This Fermi surface
shows a small corrugation, which is induced by the inter-tube interaction, and hence
depends both on the inter-tube separation a and the tube orientation angle (,9, defined
in Figure 3.1(d).
The effect of inter-tube interactions on the electronic structure is easily seen by
comparing the band structure of an isolated tube and the equilibrium “rope” along
the l" — A line in Figures 5.1(b) and (c), respectively. An even more pronounced effect
can be observed in the band dispersion perpendicular to the tube axis in Figure 5.1(d).
In the following, I focus on a precise description of the changing hybridization as a
function of the tube orientation 1;), which describes the coupling of electronic states to
tube librations. It is noted that if a strong electron-libration coupling were present, it
should lead to superconducting behavior that could be described using the Bardeen-
Cooper-Schrieffer (BCS) formalism.
There are two indications that onset of orientational disorder may be signifi-
cantly below room temperature. First, low-frequency infrared modes of the “rope” at
uz15, 22, 40 cm“, some of which may be librations, have been reported to disappear
at Tz30 — 180 K [Holm98].
There is another indication that onset of orientational disorder may be signifi-
cantly below room temperature mentioned in Section 3.1. The transition from non-
metallic to metallic character of the nanotubes occurs near 50 K [F isc97], which is
interpreted to arises from subtle changes of the electronic density of states near E p
in presence of increasing orientational disorder. This is more closely related to the
64
0.4 I ,1 : ’ 0.4 l L] z I
(a) = a (b) 0.021 '1 '1
s: 0.3 1 i - s: 0.3 - ~ - .
I i 0.00 .. .
E : E ’ 3 ' EF 4 i
m (U '
r, 0.21 I 11 J g 02 - I -
E I E i
m 1 .2. 1
co 1 ,1 1 ( (n 1
8 01 r J : ; - 8 0.1 - 1 -
: ’ Tube : Hope
00 I ll '1 l 0.0 l J :i 1
0 2 51:4 E;:6 8 10 0 2 EF4 51:6 8 10
Energy [eV] Energy [eV]
Figure 5.2: Density of states of (a) an isolated nanotube and (b) a nanotube crystal,
the “rope”. The structure of the pseudo-gap in the “rope” is displayed in the inset
on an expanded energy scale. Dashed and dotted lines indicate the position of the
Fermi level E p for the undoped and E’F for the KO; doped systems, respectively.
recently proposed mechanism for the temperature dependence of resistivity due to
inter-tube hopping near defects [Kai398] than temperature-induced changes in the
weak localization of electrons on individual tubes [Lang96].
As discussed above and in Figure 5.1, inter-tube interactions in the “rope” cause
substantial changes in the electronic states which are also reflected in the density
of states. Results, presented in Figure 5.2, indicate that upon bunching of tubes
to a. “rope”, the density of states close to the Fermi level increases by z7% from
1.4x10‘2 states/eV/atom in tubes to 1.5x10‘2 states/eV/atom in “ropes”, nearly
independent of the tube orientation angle cp. Hence I find that states at the Fermi
level do not couple significantly to tube librations, similar to the situation in the
doped C60 solid [Chri92]. Librations are not expected to play an important role in
potentially superconducting behavior induced by electron-phonon coupling in this
marginally metallic system [The396, LeeR97, Fisc97].
An intriguing effect, outlined in the inset of Figure 5.2(b), is the occurrence of a
65
pseudo-gap near E p in the density of states of the “rope” [Kwon98b, Dela98]. This
feature results from breaking the D101, tube symmetry by the triangular lattice, and
should be less significant in the highly symmetric ordered lattice of (6,6) nanotubes
[Char95]. It is expected that in presence of orientational disorder, caused by the
presence of different tubes, or a twisting motion of individual nanotubes in the “rope”
at T > T 0 M, the Brillouin zone should collapse to a point. It is also expected that the
inter-tube spacing increases at T > TOM, in analogy to C60 crystals. The resulting
reduction of the pseudo-gap should mark the onset of metallic behavior of the “ropes”,
as discussed above.
In a situation analogous to graphite intercalation compounds, the electronic struc-
ture of a system doped by potassium to the composition KCg can be described by
a rigid-band model,1 where each K atom transfers its 43 electron to the (otherwise
unmodified) electronic structure of the nanotube crystal. This shifts the Fermi level
from Ep to E’F, as indicated in Figures 5.2(a) and (b). Upon doping to this degree,
the density of states increases by a factor of z12 with respect to the undoped system.
This has been confirmed experimentally by a corresponding increase in conductiv-
ity by one order of magnitude [LeeR97]. As in the undoped solid, characterized by
E p, I do not observe a significant dependence of the density of states at E’F on tube
orientation. This suggests that even in this doped system, librational modes should
not couple significantly to states at the Fermi level. This situation is very similar to
alkali-doped C60 solid, where librational modes were shown not to couple to states at
E p [Chri92] and not to play any significant role in the electron-phonon coupling that
causes superconductivity in these systems [Sch192].
I also calculate electronic structures of ropes consisting other armchair nanotubes
1The electronic structure calculation must, of course, account for the expansion of
the inter-tube spacing a upon intercalation. I expect the electron-libration coupling
to further decrease in such an expanded lattice.
66
T f r Y m I; F ii
\ . 0 I
z : : i
a 0.15 E (313) E E (919) i
8 :: : : -:
é " : 1:
% : i: '
a 0.10 - : 1 ': -
8 :
a 1 1
v: . .
8 0.05 1
1 (a)
0.“, 1 A 1 1 4 I . 1 4
l s s
"g 0.15 (10,10) 1 . 1 (11,11) 5
3 r. , -: ;: :
Q 5 : z: 3.
i: :2 : ' t
E 0.10 " ~ 5‘-
g 1 ..
_, 1
g 0.05 1 ‘
0.00 ~ ‘ ‘
E [eV] E [eV]
Figure 5.3: Electronic density of states of (a) (8,8), (b) (9,9), (c) (10,10), ((1) (11,11)
single-wall carbon nanotubes. Results for the isolated tubes (dotted lines) are com-
pared to those for an infinite ordered lattice or “rope” (solid line).
67
Table 5.1: Calculated energy difference A1 between the first and A2 between the
second pairs of Van Hove singularities in the electronic density of states for isolated
tubes and weakly interacting tubes in a rope. The difference between the tube and
rope values is denoted by (SA. All energy values are in eV units.
(71,71) A1 (tube) A1(rope) 6A1 A2(tube) A2(rope) 6A2
(8,8) 2.03 2.23 0.20 3.69 3.89 0.20
(9,9) 1.83 1.99 0.16 3.39 3.55 0.16
(10,10) 1.64 1.84 0.20 3.12 3.29 0.17
(11,11) 1.49 1.63 0.14 2.87 2.98 0.11
like (8,8), (9, 9), and (11,11). In Figure 5.3, electronic density of states are shown
for each case. Results for the isolated tubes (dotted line) are compared to those for
an infinite ordered lattice or “rope” (solid line). As in the case of (10,10) nanotube
rope, pseudo-gaps of 50.2 eV open near Fermi level due to inter-tube interaction
and symmetry lowering. Calculated energy difference A1 between the first and A2
between the second pairs of Van Hove singularities in the electronic density of states
for isolated tubes and weakly interacting tubes in a rope are summarized in Table 5.1.
The differences, which are in the range of 0.1z0.2, may explain an observation of the
shift in the Raman absorption spectra [RaoA99].
In summary, I used ab initio and parametrized techniques to determine the elec-
tronic properties of an ordered “bundle” of carbon nanotubes. The inter-tube cou-
pling introduces an additional band dispersion of $0.2 eV and opens up a pseudo-gap
of the same magnitude at E p.I do not observe any significant coupling of librational
68
motion to states near E p in the undoped system, nor in the doped KC8 system with a
one-order-of-magnitude higher conductivity. As in the alkali-doped C60 solid, I do not
expect significant influence of tube libration modes on a potentially superconducting
behavior of an ordered lattice of carbon nanotubes.
5.2 Multi-wall nanotubes
An intriguing effect, namely the formation of a pseudo-gap at EF [Dela98, Kwon98b]
has been predicted to occur in ropes of metallic ( 10,10) nanotubes of D10), symmetry
due to the symmetry lowering while forming a triangular lattice, as described in
Section 5.1. It is interesting to investigate whether the weak inter-tube interaction,
which strongly modifies the low-frequency end of the vibrational spectrum and the
density of state near E p in an infinite three-dimensional lattice of identical nanotubes,
may cause even more dramatic effects in a one-dimensional multi-wall tube.
Here I show that the inter-wall coupling, which opens a pseudo-gap in bundled
single-wall nanotubes due to symmetry lowering, may periodically open and close four
such pseudo-gaps near Ep in a “metallic” double-wall nanotube during its librations
around and vibrations normal to the tube axis [Kwon98c]. This type of electron-
phonon coupling has not been observed in librating molecular crystals of €60 fullerenes
or (10,10) nanotubes, that are orientationally frustrated. I study the intriguing inter-
play between geometry and electronic structure during the low-frequency motion of
a double-wall nanotube. Results indicate that the coupling of conduction electrons
to soft tube librations and vibrations is caused by periodic symmetry changes, which
are unique in this system and may lead to its superconducting behavior.
To determine the electronic properties of multi-wall nanotubes, I use the parame-
trized tight-binding technique with parameters determined by ab initio calculations
69
for simpler structures [Toma91], as described in Section 3.2.
The weak inter-tube interaction not only induces new vibrational modes, de-
scribed in Section 3.2, but also modifies significantly the electronic structure near
Ep. Individual (5,5) and (10,10) armchair nanotubes, with their respective D54 and
D10), symmetries, are metallic. The band structure of each tube is characterized by
two crossing linear bands near E 1:, one for “left” and one for “right” moving electrons.
The band structure of a pair of decoupled (5,5) and (10,10) nanotubes, a mere super-
position of the individual band structures, is shown in Figure 5.4(a). The linearity of
the energy bands in-between the van Hove singularities closest to Ep translates into
a constant density of states in that energy region, as shown in Figure 5.5(a).
It has been argued that the symmetry lowering upon bunching identical metallic
(10,10) tubes to a close-packed lattice causes a pseudo-gap to open at Ep [Dela98,
Kwon98b], as described in Section 5.1. As I show in Figure 5.4(b), even the weak
interaction between two adjacent (10,10) tubes gives rise to a band repulsion. In this
lower-symmetry configuration, moreover, a true band gap opens at Ep, as shown in
Figure 5.5(b).
As shown in Figures 5.4(c) and (d), switching on the inter-tube interaction in the
(5,5)@(10,10) double-wall tube removes the near-degeneracy of the bands near E: as
well. In the most stable orientation, characterized by 99m = 0°, defined in Figure 3.2,
the double-wall system is still characterized by the D5,) symmetry of the inner tube.
The four bands cross, with a very small change in slope causing the density of states
to increase by @376, as shown in Figures 5.5(c) and (e). While the same argument
also applies to the least stable orientation 99m = 9°, a markedly different behavior
is obtained at any other tube orientation that lowers the symmetry. As seen in
Figure 5.4(d), the symmetry lowering for «pm 2 3° gives rise to four avoided band
70
99
( ).... (b)...
; 4.00 L f; 4.00 -
(D (D
v v
>. >.
00 OD
3'5 300 - S 300 .
I: . T C .
ca 1::
2.00 ~ - 2.00 -
1.00 1.00
I“ A
O
( ).... (d)...
; 4.00 - 4 f; 4.00 L /
Q) 0 ,A
V v
>. >.
C1) D0
3 300 - 3 300 .
c: ‘ 1: '
m m
2.00 - 2.00 »
1.00 1.00
I‘ A
Figure 5.4: Band structure of aligned nanotube pairs, along the tube axis. Near-
degenerate bands with no gap characterize the (5,5)@(10,10) double-wall nanotube
without inter-tube interaction (a). Inter-tube interaction opens a gap in a pair of
(individually metallic) (10,10) nanotubes (b). In presence of inter-tube interaction,
depending on the mutual tube orientation, the (5,5)@(10,10) system may show zero
gap in the most symmetric, stable configuration at 4pm = 0° (c), or four pseudo-gaps
in a less symmetric and stable configuration at (pin = 3° (d). The Fermi level is shown
by the dashed line.
71
0.4 4 . ,. 1 , 0.4
(a) 2 (b)
50‘ 0.3 - a 0.3 ~
2 2 l E
.9 1 .9
a ; a:
3 0.2 3 4 g 0.2
S ' 5
2. : .2.
a) 3 a)
8 0.1 ~ . 8 0.1
00 1 1 1 :1 L 1 1 0.0 _1 1 1 1 1 1 1
0 2 EF4 6 8 0 2 E54 6 8
Energy [eV] Energy [eV]
0.4 y r . r I 04 r l . I l I 7
(C) 2 (d) s
g 0.3 - g 0.3 - E
E E E S
2 . 9, .
w : w :
g 0.2 3 4 g 0.2 . g 1
5. 3 :3. 1 E
a) f <0 5
8 0.1 L g ’ J 8 0,1 .. 5 1
00 1 1 1 ' 1 1 1 L 00 1 L 1
0 2 EF4 6 8 O 6 8
Energy [eV] Energy [9V]
% I l l I T
(G) g 0.03 - -
9 _ -
on
E 0 02 .-. 3.1 .1 TH: 2'... r.-. 1".-. 1'... .- -. r.-. L". 3'... f... .-.-. “.1 .‘3 3' ......
Iii _ .
.2.
a) 0.01 - .
8 l l l l l
3.4 3.5 6 3.7 3.8 3.9 4.0
Energy [eV]
Figure 5.5: Density of states of aligned nanotube pairs, corresponding to the band
structures in Figure 5.4: The (5,5)@(10,10) double-wall tube with no inter-tube in-
teraction (a); pair of interacting (10,10) nanotubes (b); the (5,5)@(10,10) double-wall
tube as a function of mutual tube orientation, for 90,-" = 0° (c), and cp,,, = 3° ((1). The
Fermi level is shown by the dashed line. The densities of states of (5,5)@(10,10) tubes
near Ep are compared on an expanded energy scale in (e). Appearance of pseudo-
gaps, shown by the solid line for (pin = 3°, is in stark contrast with the flat density
of states for (pin = 0° and (pin = 9°, shown by the long-dashed line. The dotted line
indicates the density of states of non-interacting (5,5) and (10,10) tubes.
72
crossings. This translates into four pseudo-gaps in the density of states near EF, as
shown in Figure 5.5(d) and even better at the expanded energy scale in Figure 5.5(e).
It is expected that the density of states at the Fermi level should be affected not
only by few charged impurities that would shift Ep, but even during the libration
motion of the double-wall tube, since the position of the four pseudo-gaps depends
significantly on the mutual tube orientation. I also would like to emphasize that
the opening and closing of pseudo-gaps is unique to multi-wall nanotubes [Kwon98c]
and does not occur in single-wall nanotube r0pes [Dela98, Kwon98b] as mentioned in
Section 5.1.
The significance of these results extends well beyond the (5,5)@(10,10) system
discussed here. In an n-wall armchair nanotube, I would expect 122 avoided band
crossings to occur, inducing up to n2 pseudo-gaps in the density of states near Ep.
These pseudo—gaps could open and close, depending on the symmetry reduction during
tube librations and vibrations.
(11,0) zig-zag nanotubes are nearly metallic if n is a multiple of three [Mint92,
SaiR92, Hama92] as discussed in Section 1.2. If the curvature-induced gap of few meV
at E p is ignored, the bands describing “left” and “right” moving electrons would cross
at k = I‘. As the inter-tube interaction is switched on in the “metallic” (9,0)@(18,0)
nanotube, the 30 meV gap opens to twice this value at (pin 2 (pout = 0°, and to even
a larger value for lower-symmetry configurations. In the semiconducting (8,0)@(17,0)
system, on the other hand, the inter-tube interaction reduces the 0.6 eV gap of the
non-interacting system by 0.1 eV.
In summary, I studied the effect of inter-tube interaction on electronic structure
of the (5,5)@(10, 10) double-wall nanotube. I found that the weak inter-tube in-
teraction periodically opens and closes four pseudo-gaps near Ep due to symmetry
73
lowering during the low-frequency librational motion about and vibrational motion
normal to the double-tube axis. The strong coupling between the libration mode
and the electronic states at the Fermi level, which is apparently absent in nanotube
and fullerene crystals, may cause superconductivity in systems containing multi-wall
nanotubes.
74
Chapter 6
Transport
The following discussion of the transport behavior of carbon nanotubes follows that
presented in Refs. [Sanv99c, Sanv00].
Electron transport in nanotubes is believed to be ballistic in nature, implying
the absence of inelastic scattering [Fran98]. Recent conductance measurements of
multi-wall carbon nanotubes [Fran98] have raised a significant controversy due to
the observation of unexpected conductance values in apparent disagreement with
theoretical predictions. In these experiments, multi-wall carbon nanotubes, when
brought into contact with liquid mercury, exhibit not only even, but also odd multiples
of the conductance quantum Co = 282/hz(12.9 kQ)‘1, whereas the conductance of
individual tubes has been predicted to be exactly 260 [Chi096, TiaW94, LinM95].
An even bigger surprise was the observation of non-integer quantum conductance
values, such as GmOfiGo, since conductance is believed to be quantized in units of
Go [Land70] .
In this chapter, I demonstrate that the unexpected conductance behavior can
arise from the inter-wall interaction in multi-wall or in bundled nanotubes [Sanv99c].
This interaction may not only block some of the quantum conductance channels, but
also redistribute the current non-uniformly over the individual tubes. I show that
75
ell
mat.
T9391
ram-1
under the experimental conditions described in Ref. [Fran98], this effect may reduce
the conductance of the whole system to well below the expected value of 200.
The electronic band structure, as described in Chapter 5, of single-wall [Mint92,
SaiR92, Hama92] and multi-wall carbon nanotubes [SaiR93, LamP94, Kwon98c], as
well as single-wall nanotube ropes [Dela98, Kwon98b] is now well documented. More
recently, it has been shown that inter-wall coupling leads to the formation of pseudo-
gaps near the Fermi level in multi-wall nanotubes [Kwon98c] and single-wall nanotube
ropes [Dela98, Kwon98b]. These studies have described infinite periodic structures,
the conductance of which is quantized in units of 200. In what follows, I study the
effect of inter-wall coupling on the transport in finite structures.
To determine the conductance of finite multi-wall nanotubes, a linear combination
of atomic orbitals (LCAO) Hamiltonian is combined with a scattering technique de-
veloped recently for magnetic multi-layers [Sanv99a, Sanv98]. The parametrization of
the LCAO matrix elements, based on ab initio results for simpler structures [Toma91],
has been used successfully to describe electronic structure details and total energy
differences in large systems that were untreatable by ab initio techniques. This elec-
tronic Hamiltonian had been used previously to explain the electronic structure and
superconducting properties of the doped C60 solid [Sch192], the opening of a pseudo-
gap near the Fermi level in bundled and multi-wall nanotubes [Kwon98b, Kwon98c].
The scattering technique, which has recently been employed in studies of the giant
magnetoresistance [Sanv99a, Sanv98], determines the quantum-mechanical scattering
matrix S of a phase-coherent “defective” region that is connected to “ideal” external
reservoirs [Sanv99a]. At zero temperature, the energy-dependent electrical conduc-
tance G' (E) is given by the Landauer-Biittiker formula [Biitt85]
C(E) = 2%1113), (6.1)
76
—2 0 2 4 —2 0 2 4
E[eV] E[eV]
Figure 6.1: Electronic density of states (DOS) (a) and conductance G (b) of an
isolated single-wall (10,10) carbon nanotube. The DOS is given in arbitrary units,
and G is given in units of the conductance quantum Go = Qez/hz(12.9 kQ)‘1.
where T(E) is the total transmission coefficient evaluated at the energy E which, in
the case of small bias, is the Fermi energy Ep.1 I summarize a detail description of
this scattering technique in Appendix B.
For a homogeneous system, T(E) assumes integer values corresponding to the
total number of open scattering channels at the energy E. For individual (11, n) “arm-
chair” tubes, this integer is further predicted to be even [Chic96, TiaW94, LinM95],
with a conductance G = 2G0 near the Fermi level. As a reference to previous results
[Chic96, TiaW94, LinM95], the density of states and the calculated conductance of
an isolated (10, 10) nanotube is shown in Figure 6.1.
The corresponding results for the (10,10)@(15,15) double-wall nanotube [Kwon98c]
1When calculating the S-matrix of a doubly—infinite nanotube, comprising a scat-
tering region connected to two semi-infinite single— and multi-wall nanotube leads,
it is noted that the unit cell in each segment consists of two atomic planes along
the nanotube axis. Since the hopping matrix between adjacent unit cells (H1 in
Ref. [Sanv99a]) is singular in this case, I first project out the non-coupled degrees of
freedom before calculating the scattering channels.
77
E
3
§ 0.02
0
E
3. 0.01
U)
C
Q
0.00
6
L3 4
L'J
2
0
Figure 6.2:
(5,5)@(10,10)@(15,15) nanotube [(b) and ((1), respectively].
0.03
1
1 (a) 1 ’ (b) 1
i I U l ‘ h
_ i l 1 1
. (C) 1
-0.2 0.0 0.2 -0.2 0.0 0.2
E [eV] E [eV]
Electronic density of states and conductance of a double-
wall (10,10)@(15,15) nanotube [(a) and (c), respectively], and a triple-wall
78
Ub , iv
1
1165
the
G/60
G/Go
E [eV]
Figure 6.3: Partial conductance of the constituent tubes of (a) a double-wall
(10,10)@(15,15) nanotube and (b) a triple-wall (5,5)@(10,10)@(15,15) nanotube. Val-
ues for the outermost (15,15) tube are given by the solid line, for the (10,10) tube by
the dashed line, and for the innermost (5,5) tube by the dotted line.
79
and the (5,5)@(10,10)@(15,15) triple-wall nanotube, where the inter—wall interaction
significantly modifies the electronic states near the Fermi level, are shown in Fig-
ure 6.2. The density of states of the double- and the triple-wall nanotubes are shown
in Figures 6.2(a) and (b), respectively. The corresponding results for the total con-
ductance are given in Figures 6.2(c) and (d), respectively. The conductance results
suggest that some of the conduction channels have been blocked close to Ep. The
inter-wall interaction, which is responsible for this behavior, also leads to a redistri-
bution of the total conduction current over the individual tube walls. The partial
conductances of the tube walls are defined accordingly as projections of the total
conductance and shown in Figure 6.3. It is noticed that the partial conductance is
strongly non-uniform within the pseudo-gaps, where the effects of inter-tube interac-
tions are stronger.
The experimental set-up of Ref. [Fran98], shown schematically in Figure 6.4(a),
consists of a multi-wall nanotube that is attached to a gold tip of a Scanning Tunneling
Microscope (STM) and used as an electrode. The STM allows the tube to be immersed
at calibrated depth intervals into liquid mercury, acting as a counter-electrode. This
arrangement allows precise conduction measurements to be performed on an isolated
tube. The experimental data of Ref. [Fran98] for the conductance G as a function
of the immersion depth 2 of the tube, reproduced in Figure 6.5, suggest that in a
finite-length multi-wall nanotube, the conductance may achieve values as small as
0.5G0 OI' 1G0.
The key problem in explaining these experimental data is that nothing is known
about the internal structure of the multi-wall nanotube or the nature of the contact
between the tube and the Au and Hg electrodes. A number of different scenarios
have been considered and concluded that the experimental data can only be explained
by assuming that (2') current injection from the gold electrode occurs only into the
80
Au
Figure 6.4: (a) Schematic geometry of a multi-wall nanotube that is being immersed
into mercury up to different depths labeled Hg(#1), Hg(#2), and Hg(#3). Only the
outermost tube is considered to be in contact with the gold STM tip on which it is
suspended. The conductance of this system is given in (b) for the immersion depth
Hg(#1), in (c) for Hg(#2). and in (d) for Hg(#3) as a function of the position of
Ep. The Fermi level may shift with changing immersion depth within a narrow range
Hg(#3)
Hg(#2)
Hg(#1)
indicated by the shaded region.
L:
G
0.5 -
0.0
81
h (b)
l ”’7'“
l
E).
. , Hg(#3)
_J.____
Q
f
l
" ' ‘ H g(#1)
7.1'0"FTF1“5—
E [eV]
G/Go
0.0 .- .
-50 0 50 100 150 200
2 [nm]
Figure 6.5: Conductance G of a multi-wall nanotube as a function of immersion depth
2 in mercury, given in units of the conductance quantum G0 = 282/h%(12.9 kQ)‘1.
Results predicted for the multi-wall nanotube discussed in Figure 6.4, given by the
dashed line, are superimposed on the experimental data of Ref. [Fran98].
outermost tube wall, and that (2'1) the chemical potential equals that of mercury only
within the submersed portion of the tube. In other words, the number of tube walls in
contact with mercury depends on the immersion depth. The first assumption implies
that the electrical contact between the tube and the gold electrode involves only the
outermost wall, as illustrated in Figure 6.4(a). The validity of the second assumption
— in spite of the fact that mercury only wets the outer tubes —— is justified by the
presence of the inter-wall interaction. The main origin of the anomalous conductance
reduction, to be discussed below, is the backscattering of carriers at the interface
of two regions with different numbers of walls due to a discontinuous change of the
conduction current distribution.
The conductance calculation is then reduced to a scattering problem involving
a semi-infinite single-wall nanotube (the one in direct contact with gold) in con-
tact with a scattering region consisting of an inhomogeneous multi-wall tube and the
82
Hg reservoir as the counter-electrode. Depending on the immersion depth, denoted
by Hg(#1), Hg(#2), and Hg(#3) in Figure 6.4(a), portions of the single-wall, the
double-wall, and even the triple-wall sections of the tube are submersed into mercury.
Calculations are performed within the linear-response regime and assume that the
entire submersed portion of the tube is in “direct contact”, i.e. equipotential with
the mercury. Increasing the immersion depth from Hg(#1) to Hg(#2) and Hg(#3),
an increasing number of walls achieve “direct contact” with mercury, thereby chang—
ing the total conductance G(E), as shown in Figures 6.4(b)—(d) and Figure 6.5. It
is also noticed that the conductance of the inhomogeneous multi-wall structure of
Figure 6.4(a) cannot exceed that of a single-wall nanotube, which is the only tube in
electrical contact with the gold electrode.
The calculation underlying Figure 6.4(b) for the submersion depth Hg(#1) con-
siders a scattering region consisting of a finite-length (5,5)@(10,10)@(15,15) nanotube
connected to another finite segment of a (10,10)@(15,15) nanotube. This scattering
region is then connected to external semi-infinite leads consisting of (15,15) nanotubes.
The calculation for the submersion depth Hg(#2), shown in Figure 6.4(c), considers
a scattering region formed of a finite-length (5,5)@(10,10)@( 15,15) nanotube segment
that is attached to a (15,15) nanotube lead on one end and to a (10,10)@(15,15) nan-
otube lead on the other end. Results in Figure 6.4(d) for the submersion depth Hg(#3)
represent the conductance of a (5,5)@(10,10)@(15,15) nanotube lead in contact with
a (15,15) nanotube lead. The calculated conductances depend on the position of the
Fermi level within the tube. Even though Ep may vary with the immersion depth
due to a changing contact potential, these changes are expected to lie within the
narrow energy window of z0.05 eV, indicated by the shaded region. Results shown
in Figures 6.4(b)—(d), suggesting discrete conductance increases from G20.5G0 for
Hg(#1) to GzlGo for Hg(#2) and Hg(#3) are in excellent agreement with the re-
83
Figure 6.6: (a) Schematic geometry of a different three-wall tube, with all nested
tubes in contact with the gold electrode. (b) Conductance for the immersion depths
Hg(#1), Hg(#2), and Hg(#3) as a function of the position of EF. As in Figure 6.4,
the Fermi level may vary with immersion depth within a narrow range indicated by
the shaded region.
cent experimental data of Ref. [Fran98], presented in Figure 6.5.
It is essential to point out that from these calculations a conductance value is
expected to be GmOBGo only when a single tube wall is in direct contact with mercury.
In the case that a single-wall region is long, this small conductance value is expected
to extend over a large range of immersion depths [Fran98]. In absence of such a single-
wall segment, the conductance is expected to be only values of 1G0 and above. It is
believed that the anomalous sample-to-sample variation of the observed conductance
[Fran98] is related to the structural properties of the nanotube and not to defects
which are believed to play only a minor role in transport [And098a, Ando98b].
84
r-
Furthermore, a very difference conductance behavior is expected when more than
on tube is in direct contact with the Au electrode. As a possible scenario, I consider
an inhomogeneous nanotube shown in Figure 6.6(a), where now all of the three tube
walls are in direct contact with the gold electrode. With two conduction channels per
tube wall, the conductance has an upper bound of 6G,. Calculations analogous to
those presented in Figure 6.4 suggest a minimum conductance value GaslGo to oc-
cur for a finite-length (10,10)@(15,15) tube segment sandwiched between (15,15) and
(5,5)@(10,10)@(15,15) nanotube leads, representing the smallest submersion depth
Hg(#1), with mercury in direct contact with only the single-wall portion of the tube.
The conductance value GzZGo is obtained for a (10,10)@(15,15) nanotube lead in
contact with a (5,5)@(10,10)@(15,15) nanotube lead, representing submersion depth
Hg(#2), with mercury in direct contact with a double-wall tube segment. Finally,
depending on the position of EF, the conductance of a triple-wall nanotube sub-
mersed in mercury, modeled by an infinite (5,5)@(10,10)@(15,15) tube, may achieve
conductance values of 4G0 or 6G0, as shown in Figure 6.6(b)-(d). Even though the
inter-wall interaction leads to a significant suppression of the conductance, the pre-
dicted increase in the conductance from 1G, to 2G0 and 4G0 — 6G0 with increasing
submersion depth is much larger than in the scenario of Figure 6.4. Also the predicted
conductance values are very different from the experimental data of Ref. [Fran98],
thus suggesting that only the outermost tube is in electrical contact with the gold
electrode.
In conclusion, it has been shown that the inter-wall interaction in multi-wall nan-
otubes not only blocks certain conduction channels, but also re-distributes the current
non-uniformly across the walls. I have calculated conductance in several structures
and concluded that the puzzling observation of fractional quantum conductance in
multi-wall nanotubes can be explained by assuming that only the outermost tube
85
wall is in electrical contact with the gold electrode. Moreover, I have shown that the
sample-to—sample variations in the conductance are entirely related to the structure
of the nanotube. A very similar behavior is expected for bundled single-wall tubes of
different length submersed into Hg or another metal. These predictions raise impor-
tant questions concerning the nature of the nanotube/ metal interface, which deserve
further investigation both experimentally and theoretically.
86
Chapter 7
Applications
In this chapter, I propose two possible applications of nanotubes and related fullerene
structures. These applications have been filed for US. patents. The following discus-
sion of nanotube—based micro—fastening system and “bucky-shuttle” memory device
follows that presented in Refs. [Pat.1, Pat.2, BerbOO, Kwon99a, Breh99].
7 .1 Nanotube—based micro—fastening system
Fastening systems, albeit on a macro-scale, have generally been in the form of adhesive
bonds or welds occurring between two distinct components brought into relative con-
tact. Numerous potential disadvantages associated with using adhesives are known
such as the irreversible nature of the bonds and the potential for degradation at rel-
atively high temperatures. Further, adhesives require smooth dry interfaces which
are free of impurities to effectuate high quality bonds. Welding results in a physical
deformation of the surfaces being welded and cannot be used effectively for intercon-
necting microscopically small components or large interface areas. Thus, there is a
need for the mechanical micro-fastening system.
The micro—fastening system employs a plurality of mating nanoscale fastening el-
ements which are obtained by structurally modifying, i.e., functionalizing nanotubes.
87
By functionalizing the carbon nanotubes, as described below, the generally cylindrical
shape can be modified to include bent portions. While it has been suggested generally
that carbon nanotubes can be readily functionalized, it has yet to be reported that
carbon nanotubes can be specifically functionalized so as to obtain mating fastening
elements.
Among the various applications for the micro—fastening system are the assembly
of nano—robots useful for micro—surgical procedures, surface coatings, and attachment
of metal contacts to integrated semiconductor devices, etc.
The micro—fastening system comprises a plurality of mating nanoscale fasten-
ing elements manufactured by modifying, i.e., functionalizing nanotubes which are
substantially linear in nature prior to functionalizing. Upon functionalizing the nan-
otubes, fastening elements are obtained in a variety of forms such as hooks, 100ps and
spirals, which facilitates attachment to a corresponding fastening element.
By introducing pentagons and heptagons in a predetermined order, nanotubes
will exhibit a locally positive or negative Gaussian curvature that results in a bend in
the nanotube. By continuing to add pentagons and hexagons in a specific manner, the
bend of the nanotube can be grown until the desired shape is obtained. As previously
noted, among the various fastening elements which can be manufactured are loop
structures, hook structures or spiral shaped structures as shown in Figures 7.1(a), (b)
and (c), respectively.
Upon growing the carbon nanotube to the desired length and shape, one end
of the tubule may be capped or terminated by introducing or forming a fullerene
half dome along the end to be terminated. By providing a fullerene half-dome along
an open end of the carbon nanotube, the formed fastening element end becomes
substantially inert, i.e., non-bonding to other atoms or molecules.
88
(b)
Figure 7.1: The micro—fastening systems can be composed of any functionalized struc-
tures, such as (a) loop, (b) hook, and (c) spiral structures.
89
The opposing end of the fastening element which is Open, i.e., non-terminated,
is bonded to a matrix or substrate which may be in the form of various materials
including metals, carbon (graphite or diamond), silicon, germanium, polymers and
composites. Other materials, provided they are capable of attaining a molten state,
can also be used. Since the open end is highly reactive, it has a natural affinity
for bonding to the desired substrate, by which the fastening element attaches to the
substrate and stands up along the attachment surface. By applying a strong electric
filed in the direction perpendicular to the surface, nanotubes may be assisted in their
alignment in that direction.
Carbon nanotubes having ordered pairs of pentagons and heptagons may occur
spontaneously to a limited extent during synthesis, thus forming hook shaped nan-
otubes [Iiji94]. However, in order to design carbon nanotubes such that they can
be used effectively in micro—fastening systems, atomically dispersed catalysts may be
necessary. For example, transition metals such as Fe and, more preferably, Ni, Co and
Y have been shown to promote formation of single-wall nanotubes or spiral structures
[Ame194].
Spontaneous curving of relatively straight carbon nanotubes may occur by using
a template in proximity to a growing nanotube. In this regard, both on energetic and
entropic grounds, a horizontally growing nanotube, when approaching a vertically
positioned nanotube used as a template, has a higher probability to form C5 and C7
carbon rings, i.e., pentagons and heptagons which would cause the former to wrap
around the latter. As such, specifically functionalized carbon nanotubes useful as
fastening elements can also be prepared without using catalysts.
While the fastening elements can be formed into a number of different configura-
tions, certain configurations are considered to be preferred. For a generic mechanical
90
Figure 7.2: Only a small force Fc is required to deform the nanotube and thus ac-
complish an interconnection between two fastening elements.
91
Figure 7.3: Very large force F0 is required to break the interconnection between two
fastening elements.
92
micro-fastening system, the opening and closing mechanism is shown in Figures 7.2
and 7.3. As shown in Figure 7.2, two fastening elements come into contact as they
advance toward an interlocked position. The fastening elements can easily slide past
each other, because of the angular orientation at the contact surfaces, i.e. only a
small force FC is required to deform two fastening elements and thereby accomplish
an interconnection between them. On the other hand, a much larger force F0 (>> Fe)
is required to open or break this interconnecting bond as demonstrated in Figure 7.3.
The strength of the micro-fastening system relies on the enormous stability of nan-
otubes, their structural rigidity, the strength of nanotube—surface bonds and a large
number of connections possible on a limited surface area. In contrast to mechanical
fasteners which weaken the surfaces to be connected, there is no apparent degrada-
tion of the opposing surfaces to be joined in the micro-fastening system. Adhesives
are typically weaker than most mechanical fasteners and their strength is strongly
diminished at higher temperatures. Welding is not practicable for large interfaces,
whereas the micro—fastening system may be used for both large and microscopically
small interfaces. Bonding technologies excepting the micro—fastening system leave
macroscopically large gaps at the interface. Unlike known bonds between substrates,
the micro—fastening system has an effective thickness of the gap at interface as small
as a few nanometers.
A further advantage of the micro—fastening system is that the surface bonds based
on micro—fastening, while extremely strong, may be re-opened and re-closed, whereas
the surface bonds generated by gluing or welding are permanent. Thus, the micro—
fastening system is selectively reversible which is considered to be highly desirable,
particularly for self-repair.
Still another advantage offered by the micro-fastening system is that the con-
93
ductivity of the fastening elements and their corresponding substrates may be varied
from metallic to insulating, depending largely on the tubule diameter and chirality of
the nanotubes as I discussed in previous chapters.
7 .2 Nanotube-based memory device
Here I present evidence that unusual multi-wall nanotube structures, such as the
“bucky-shuttle” [Yak097], self-assemble from elemental carbon under specific condi-
tions. The molecular dynamics simulations indicate that the bucky-shuttle shows
an unusual dynamical behavior that suggests its use as a nanometer-sized memory
element [Kwon99a, Kwon99a]. I show that a nanotube memory would combine high
switching speed, high packing density and stability with non-volatility of the stored
data.
The system described in this study was produced by thermally annealing diamond
powder of an average diameter of 4 — 6 nm which was prepared by the detonation
method.1 The powder was heated in a graphite crucible in inert argon atmosphere
at 1800°C for 1 hour. This treatment transforms the diamond powder into graphitic
nano-structures presented in transmission electron microsc0pe images shown in Fig-
ure 7.4. A large portion of this material consists of multi-wall capsules with few
layers, the smallest structures being fullerenes with a diameter close to that of C60.
In several cases depicted in Figure 7.4, the enclosed fullerenes may move rather freely
inside the outer capsule, like a bucky—shuttle.
An enlargement of one of such structures in Figure 7.4 is displayed in Fig-
ure 7.5(a). Figure 7.5(b) illustrates a corresponding model, consisting of a C60 en-
capsulated in a C480 capsule. The energetics of the C60@C430 system is shown in
lCluster Diamond, Toron Company Ltd.
94
Figure 7.4: Transmission Electron Microscope images depicting multi-wall carbon
structures that self—assemble during the thermal annealing of nano—diamond powder
under the conditions described in this report. The smallest spherical structures are
C60 molecules that are always found near the end of the capsule, where the attractive
inter—wall interaction is strongest.
95
Figure 7.5: (a) Enlargement of the upper-right section of the Transmission Electron
Microscope image in Figure 7.4. (b) Structural model for an isolated K@CEO@C430
bucky-shuttle, with the K@C5+0 ion in the “bit 0” position.
Energy [eV]
‘
- N ‘
‘
‘ ~ .....
\ ~ -‘,-
u .
n T.\'
. _.- ~ ~
I ,-
. ."I ~ ~
In. ..I \
l l N
.
.
Position [A]
Figure 7.6: Potential energy of K@Cgf0 as a function of its position with respect to the
outer capsule in zero field (solid line) and switching field E, = 0.1 V/ A (dashed lines).
The K@C3L0 ion position, representing the information, can be changed by applying
this switching field between the ends of the capsule. Energy zero corresponds to an
isolated K@Cg*o at infinite separation from the C480 capsule.
96
Figure 7.6. The ends of the outer capsule are halves of the C240 fullerene, the op-
timum structures to hold a C60 molecule at an inter-wall distance of 3.4 A. These
end-caps connect seamlessly to the cylindrical portion of the capsule, a 1.5 nm long
segment of the (10,10) nanotube [The396]. The interaction between the unmodi-
fied C60 molecule and the enclosing capsule is similar to that found in C50 crystals
and nanotube bundles [Thes96]; it is dominated by a Van der Waals and a weak
covalent inter-wall interaction that is proportional to the contact area between the
constituents. An additional image charge interaction, which is nearly independent of
the C50 position, occurs if the C60 molecule carries a net positive charge, as discussed
below. Obviously, the Van der Waals interaction stabilizes the C60 molecule at either
end of the capsule, where the contact area is largest. This is reflected in the potential
energy behavior in Figure 7.6, and results in the likelihood of C60 to be found near
the ends of the capsule, as evidenced in Figures 7.4 and 7.5(a). In the following, I
examine the possibility of information storage in this two-level system.
Usefulness of this nanostructure for data storage implies the possibility to write
and read information fast and reliably. Of equal importance is the capability to
address the stored data efficiently, and the non-volatility of the stored information.
In order to move the encapsulated C50 from one end of the capsule to the other
(the molecular analog of writing) and to determine its position within the capsule (the
molecular analog of reading) most efficiently, the C60 should carry a net charge. In the
K@C60 complex, which is known to form spontaneously under synthesis conditions in
presence of potassium, the valence electron of the encapsulated K atom is completely
transferred to the C60 shell [LiY94]. The C60 is likely to transfer the extra electron to
the graphitic outer capsule, since the ionization potential of K@C60 is smaller than
the work function of graphite. The extra electron will likely be further transferred to
the (graphitic) structure that holds this element in place. Since the enclosed K+ ion
97
does not modify the chemical nature of C60, I model the dynamics of the K@C§’0 ion
in the neutral C480 capsule by uniformly distributing a static charge of +1 e over the
C60 shell.
The writing process corresponds to switching the equilibrium position of the C30
ion between the “bit 0” and the “bit 1” ends of the capsule in an applied electric
field. This is best achieved if the connecting electrodes, supplying the bias voltage,
are integral parts of the end caps, to reduce the field screening by the nanotube
[LouL95]. The energetics of (33;, in the switching field E, = :l:0.1 V/A, generated by
applying a voltage of z1.5 V between the end caps, is displayed in Figure 7.6. One
of the local minima becomes unstable above a critical field strength, causing the C30
ion to move to the only stable position. The switching field E, = 0.1 V/A is small
and will have no effect on the integrity of the carbon bucky-shuttle, since graphitic
structures disintegrate only in fields E 2.3 V/A [Kim897, LeeY97b].
The information, physically stored in the position of the C30 ion within the cap-
sule, can be read non-destructively by detecting the polarity of the capsule. An
alternative destructive read process would involve measuring the current pulse in the
connecting wires, caused by the motion of the C30 ion due to an applied probing
voltage. The total charge transfer associated with the current pulse, which is one
electron in this case, may be increased by connecting several capsules in parallel to
represent one bit, and by using higher charged complexes such as La@C§'{ .
When targeting high storage densities, the addressability of the stored information
becomes important. One possible way to realize a high-density memory board is pre-
sented in Figure 7.7. Maximum density is achieved by packing the nanotube memory
elements like eggs in a carton. Rows of nano—capsules can be connected at the top and
at the bottom by nano—wire electrodes in such a way, that a single memory element is
98
C
"bit1"
\Q "bit 0"
a‘ 6‘; c‘ d a‘ b? c‘ 6‘
TOP View ' ---------- 0 Side view ' --------------- -0
Figure 7.7: Schematic of a high-density memory board in top and side view. When
a switching voltage is applied between conductors b and C, the corresponding bit
information will be stored in the memory element bC at their intersection, shown
shaded.
addressed at their crossing point. Applying a switching voltage between two crossing
electrodes [e.g. the bC pair in Figure 7.7] will generate a nonzero field only in the
memory element labeled bC.‘ As in the ferrite matrix memory, many memory elements
can be addressed in parallel using such an addressing scheme. This arrangement ap-
plies both for the writing and the destructive reading processes described above, and
allows for multiple bits to be written and read in parallel. In the latter case, the sta-
tus of the memory element bC is inspected by applying a switching voltage between
the electrode pair b,C, and monitoring the current in these electrodes.
Unlike in presently used dynamic random access memory (DRAM) elements,
where information has to be sustained by an external power source, the non-volatility
of the stored information results from a non-zero trap potential near the “bit 0”
or “bit 1” end of the capsule. Thermal stability and non-volatility of data depend
on the depth of this trap potential, which in turn can be adjusted by changing the
encapsulated fullerene complex. The calculated trap potential depth of z0.24 eV
for the K@Cgo ion near the ends of the capsule in zero field suggests that stored
99
information should be stable well beyond room temperature and require temperatures
T 23000 K to be destroyed. Further improvement of the thermal stability could be
achieved using higher-charged endohedral complexes containing di- or trivalent donor
atoms, such as La@C82 discussed above.
To study the efficiency of the writing process, I perform a molecular dynamics
simulation of the switching process from “bit 0” to “bit 1” in the microcanonical
ensemble of the C;0@C480 memory element. I use a parametrized linear combination
of atomic orbitals (LCAO) total energy functional [Toma91], augmented by long-
range Van der Waals interactions [Kwon98a]. This computationally efficient 0(N)
approach to determine the forces on individual atoms [Zhon93b] has been previously
used with success to describe the disintegration dynamics of fullerenes [Kim894] and
the growth of multi-wall nanotubes [Kwon97]. This technique is described in detail in
Appendix A. A time step of 5><10‘16 s and a fifth-order Runge-Kutta interpolation
scheme was used to guarantee a total energy conservation of AE / E31040 between
successive time steps.
The results of simulation are shown in Figures 7.8, 7.9, and 7.10. Initially, the C30
ion is equilibrated near the “bit 0” position. At time t = 0, a constant electric field
of 0.1 V/ A is applied along the axis of the outer capsule. The originally stable “bit
0” configuration becomes unstable in the modified total energy surface, depicted in
Figure 7.6. The C30 ion is subject to a constant acceleration to the right, and reaches
the “bit 1” position only 4 ps later, as seen in Figure 7.8. During this switching
process, the potential energy lost by the C30 ion is converted into kinetic energy, as
seen in Figure 7.9. Due to the small (albeit non-negligible) interaction between the
encapsulated ion and the capsule, the kinetic energy gained initially occurs as rigid-
body translational energy of the C30 ion. A nearly negligible energy transfer into the
internal degrees of freedom due to atomic-scale friction, manifested in a very small
100
fig)
raps
pro:
3101‘
oo
«b...»
<2 4 i
X s
C .
.9 0 -'
17; 2
O .
D- :
-4 :
"bit 0"
-8 L 1: 1 I i 1 i i 1 E i
0 2 4 6 8 10 12 14 16
Time [ps]
Figure 7.8: Position of the K@Cg{0 ion with respect to the center of the enclosing C480
capsule as a function of time during a molecular dynamics simulation of switching
process from “bit 0” to “bit 1”, when a constant electric field of 0.1 V/ A is applied
along the axis of the capsule.
6 . . , . . ' . ' . .
I - : . . ‘ “'0‘“ .
4 P : “F";" I” 2 ~. 1 z -
621 '|.’|""\" :
h"- .‘sl i g ' . .
g 2 I. ‘0"... bflf‘l‘hl “01"“ z I d
E 1. .Su ’5"- «than!» N .
H ”Fifty"... i: . ------------ 3:. a: , .
fi 0 ........... f- . _ -°;-:~n-ou-I'.”'—' . _ . 4 any“; fag-o” a“: 6&5“
< '2 F 1 .
-4 _ 3 -
-6 l IE 1 I L. l I i l k
Time [ps]
Figure 7.9: Changes in the potential energy (solid line) and kinetic energy (dashed
line) in the laboratory reference frame as a function of time. The portion of the kinetic
energy, corresponding to the translation of the enclosed K@Cg,*0 ion with respect to
the capsule (dotted line), is seen to decrease as the system temperature rises. The
total energy (dash-dotted line) is conserved. All energies are given per atom.
101
.5
C)
q
:-
..
..
I
..
..
.5
N
Temperature [K]
on
4 r :5 .
0 . 1 1 1: 1 1
0 2 4 6 8 10 12 14 16
Time [ps]
Figure 7.10: Vibrational temperature of the enclosed K@C3L0 ion (dotted line) and the
enclosing capsule (dashed line) as a function of time. The solid lines are backward
convolutions of the vibrational temperature values, using a Gaussian with a full-width
at half maximum of 7.5x10‘13 8.
increase of the vibrational temperature in Figure 7.10, is observed during this initial
stage of the switching process.
4 ps after the switching field is applied, the C30 ion reaches the opposite end
of the capsule, having gained 1.5 eV of net kinetic energy. This kinetic energy is
too small to damage the capsule, as inelastic collisions involving C50 require energies
exceeding 200 eV to occur [Busm91]. Upon impact onto the enclosing capsule from
the inside, a substantial fraction of this energy is converted into heat, thus increasing
the vibrational temperature of the outer capsule by 510 K and that of the C3}, ion
by z2 K. Due to the high heat conductivity and melting temperature TMSA, 000 K
of graphitic nanostructures [Kim894], this modest heat evolution is unlikely to cause
any structural damage even at high access rates.
As seen in Figure 7.9, the net kinetic energy of the encapsulated C30 with respect
102
to the outer capsule is significantly reduced during this collision. The C60 bounces
back towards the middle of the capsule, slowed down by the opposing electric field,
and finally turns again towards the “bit 1” end. Figure 7 .10 indicates that thermal
equilibration in the system after the collision is achieved stepwise. The step period
of 21 ps results from the beats between the low-frequency quadrupolar deformation
modes of the colder encapsulated C30 ion and the hotter enclosing capsule, which
have been excited during the quasi-elastic collision.
One or few oscillations of the cf, ion inside the enclosing capsule, damped by
transferring energy from macroscopic to internal degrees of freedom, are necessary
to stabilize it in the new equilibrium “bit 1” position, with a kinetic energy not
exceeding the depth of the trap potential. As seen in Figure 7.9, this situation occurs
le ps after the initial onset of the switching field, thus resulting in an ideal memory
switching and access rate close to 0.1 THz. In the slower sequential mode, this
translates into a data throughput rate of 10 GB/s, four orders of magnitude faster
than the data throughput rate of 4 — 5 MB / s which is achieved presently in magnetic
mass storage devices.
In order to further reduce the switching time, one may consider increasing the
field to shorten the transfer time between the two states, keeping in mind that the
damping process would be prolonged in such a case. Unlike in this model simulation,
there is no need to apply a constant switching field during the entire bit flip process.
A 0.5 ps pulse of a 0.1 — 0.5 V/ A field is found to suffice to detach the C30 ion from
its stable position and thus to change the memory state. This approach may be of
particular use if an increase of the trap potential, due to a different fullerene complex,
should be desirable.
Mass production of nanotube-based memory devices such as the one discussed
103
here rely on the self-assembly of nanotubes and nano—capsules to ordered close-packed
arrays. There has been encouraging evidence of such a self-assembly mechanism in the
synthesis of free-standing nanotube ropes [Thes96], aligned nanotube columns forming
free-standing membranes [CheG98], multi-wall nanotube columns growing from a
SiC(111) wafer [Kusu98], and most recently many C50 molecules inside long carbon
nanotubes [Smit98]. It is also noted that since any double-wall nano-capsule with
the enclosed structure shorter than the outer capsule behaves as a tunable two-level
system, the functionality of the proposed nanoscale memory is basically independent
of the exact size and shape of the encapsulated ion and the enclosing capsule.
In summary, I have shown that thermal treatment may convert finely dispersed
diamond powder to multi-wall carbon nano—capsules containing fullerenes such as
C50. Using molecular dynamics simulations, I investigated the internal dynamics of a
related model system, consisting of a K@C2,L0 endohedral complex enclosed in a C430
nano—capsule. I showed this to be a tunable two-level system, where transitions be-
tween the two states can be induced by applying an electric field between the C480
end caps. This system, if considered as a memory element, would offer a combina-
tion of high switching speed, high density, non-volatility of data, and relatively easy
read / write access.
104
A
I...-
Chapter 8
Magnetic Iron Clusters
The following discussion of magnetic iron clusters is based on that presented in
Refs. [Serr00].
“Magic numbers”, related to observed abundances in the mass spectra of atomic
clusters, have been explained by the unusual stability gain upon closing atomic shells
in rare gas clusters, or filling electronic shells in alkali clusters. I propose that still a
different sequence of magic numbers is to be expected for systems, where the ground
state structure is determined by competition between the exchange interaction favor-
ing open structures and the covalent bonding that favors compact structures. Tran-
sition metal clusters such as Fe, Co, and Ni, are expected to show this new sequence
of magic numbers. Here, I only focus on iron clusters to demonstrate the interplay
between structure and magnetism in magnetic clusters.
Its interesting properties such as strong directional bonding and ferromagnetism
make iron to be one of the most important metals. Due to its interesting prop-
erties, on the other hand, iron has been one of the toughest materials for predic-
tive calculations. Ab initio local spin—density approximation (LSDA) calculations
[Wang85, Sing91, Hath85, Jans88, Moru86] have not correctly reproduced what ex-
perimentally observed in bulk iron [J one90]. While the body—centered—cubic structure
105
is observed experimentally, the face—centered-cubic phase is suggested to be more sta-
ble in LSDA calculations. Moreover, the calculated cohesive energies, bulk moduli
and lattice constants are far away from those observed in experiments.
These problems has been solved by a technique [Kras89, Zhon93a], in which
electronic total energy calculations are combined with the Stoner theory of itinerant
ferromagnetism [Ston39]. Since ab initio techniques are not suitable for structures
with a low symmetry like clusters due to computational overload, a parametrized one-
electron Hamiltonian is combined with the Stoner model of itinerant ferromagnetism
to investigate electronic, magnetic and structural properties of iron clusters. The
Slater-Koster parametrization, which describes the 3d and 4s electrons of Fe, is based
on Local Density Functional calculations. The total energy functional is based on a
previously published expression [Zhon93a]
-Ecoh : Ebs + Erep + Emag (8'1)
that reproduces accurately structural and magnetic properties of bulk iron in its
different phases. Here, first two terms are corresponding to total energy for usual
paramagnetic system, which can be calculated in the same way as that given in
carbon system, whereas the last term is an essential energy term for ferromagnetic
system which has given the stabilization of the bcc versus fcc phase of Fe [Zhon93a].
The Stoner theory describe the electronic structure of the magnetic system by a rigid
shift of the spin—up and spin down state as
N1(E) = N(E + A511),
(8.2)
Nj(E) = N(E — AEj).
Here, NT and N j are the densities of states for spin-up and spin—down electrons
corresponding to majority and minority subbands, respectively, and N (E) is the
106
density of state for the non-magnetic state. The energy shifts AET and AEL of NT
and N l with respect to N (E) are constrained by the charge conservation
[EF N(E) = [EFME’ N(E) . (8.3)
Ep—AE, Ep
The total magnetic moment is given by p : mug, where m is the number of unpaired
electrons and 113 is the Bohr magneton. m can be obtained by counting the electrons
in the spin—up and spin-down subbands, taking car of the charge neutrality given by
Eq. (8.3), as
EF+AET
m = / N(E)dE
Ep—AE;
rap-meT
= 2 / N(E)dE . (8.4)
E?
The energy difference between a non-magnetic and a ferromagnetic state has two
parts. The first part is the increase in kinetic energy due to the spin flip of electrons
near the Fermi level. The second part is the exchange energy contribution which
depends on the Stoner exchange parameter I . Hence the magnetic energy can be
written as
EF-l-AET
Em... = / (E—EF)N(E>dE
EF
— [BF (E — EF)N(E)dE
Ep—AE,
——m . (8.5)
A self—consistent solution of Eqs. (8.4) and (8.5), combined with the stationary re-
quirement BEmag/am = 0, is now used to determine the magnetic moment 11 and
the magnetic energy Emag. The criterion for the occurrence of a stable ferromagnetic
state is I N (E p) > 1. When calculating Emag using Eq. (8.5), I keep the exchange pa-
rameter I constant, i.e., independent of the structure. The value used is I = 0.632 eV.
107
Table 8.1: Calculated binding energies E, (eV/atom), bond lengths dun (A) and
magnetic moment 11 (pg / atom) of small magnetic iron clusters FeN (N = 2 — 5). (“
all bond lengths are not equal, so average values are given.)
FeN Eb dnn 11
F82 -1.83 2.01 3.00
Fe;, -2.36 2.14 2.67
Fe.; -2.76 225* 2.50
Fe5 -2.36 2.34”“ 2.40
Using this approach and a J-walk Monte Carlo procedure, I determine the equi-
librium geometry, electronic and magnetic structure for aggregates containing up to
tens of iron atoms. Same as bulk iron, I find that many aggregates favor reduced
atomic packing, resulting in narrower “bands” and a higher level density near the
“Fermi level”. This in turn leads to a substantial gain in exchange energy, at the
cost of a reduction in the one-electron “band-structure” energy. Unlike in rare-gas
and alkali clusters, where structural stability can be understood from closed atomic
and electronic shells, respectively, most stable structures and novel magic numbers in
iron clusters result from the non-trivial inter-play between geometry and magnetism
in these systems [Serr00].
To Show a validity of the technique, I start with smaller clusters whose equilibrium
geometries and magnetic moments has been intensively studied [OdaT98, Mor097].
In Figure 8.1, I present atomic binding energy of Fez as a function of distance be-
tween two iron atoms. As shown in the figure, F92 structure is strongly stabilized by
108
I ' I ' I '
Non-Magnetico,..x‘"j
0".
. .. a”. .......................... . .....
u"
O
'0
Magnetic ~
Figure 8.1: Atomic binding energy (a), and magnetic moment (b) of Fez are calculated
as a function of distance between two Fe atoms. For d,,,2.1.8 A, Fez structure is
strongly stabilized by exchange energy gain.
109
Figure 8.2: Optimized geometries of (a) Fe3: triangle, (b) Fe4: tetrahedron, and (c)
Fe5: trigonal bipyramid.
magnetic energy contribution. Optimized equilibrium geometries of Fe3, Fe4, and Fe;
are a triangle, a tetrahedron and trigonal bipyramid, respectively and shown in Fig-
ure 8.2. Our results for these clusters are summarized in Table 8.1. These results are
in excellent agreement in those obtained using other techniques [OdaT98, Mor097].
Results for size dependence of Fe clusters are shown in Figures 8.3 — 8.5. In
Figure 8.3, binding energy and average magnetic moment are shown as a function of
cluster size N up to N = 25. According to the size dependence of binding energy
in Figure 8.3(a), structures at N = 8, 12, and 18 are more stable than others. This
raises a question whether these more stable structures can be “magic” structures
in magnetic iron clusters. To address this question, I calculate average magnetic
moments. As shown in Figure 8.3(b), Feg cluster shows average magnetic moment of
almost zero, whereas Fey), and F813 clusters show relatively high average magnetic
moment. In the case of Feg, structural stability is expected to outweigh magnetic
stability. On the other hand, exchange energy is expected to dominate in the other
two cases.
More detail interpretation is given by calculating size dependence of electronic
energy defined as E8, = Eb, + Emag, magnetic energy, and the contribution of mag-
110
_ -2.0
E
S -2.5
E
E -3.0
Q
m -3.5
5 10 15 20 25
N (Cluster size)
(1))
.—. 3.0 .' """"""""" ' """ j
E . .
e . .
‘5 2.0 - I I -
\m I i
3 . .
/\ 1.0 - -
1 . 4
V I I
0.0 ------- ' ------- ' ‘ ‘ ‘ - '
5 10 15 20 25
N (Cluster size)
Figure 8.3: Cluster size dependence of (a) binding energy E, per atom in eV and (b)
average magnetic moment < ,u > per atom in 113. Arrows indicate unusually stable
Structures corresponding to N = 8, 12, and 18.
111
(
DD
V
1
-3.0
-3.5
-4.0
-4.5
r
I
I
f '7 f r V V V Y r T f v v r V Y 1 V ' v v v v
0.6
0.4: -‘
0.21 I 1 {
0.0 +44
0.20
0.15
0.10
0.05
0.00
A
l Emag I [eV/atom]g Eel [eV/atom]
A
Emag/Eel 3
5 10 15 20 25
N (Cluster size)
Figure 8.4: Cluster size dependence of (a) electronic energy Eel per atom in eV, (b)
absolute value of magnetic energy |Emag| per atom in eV, and (c) the ratio Emag/Ee;
of magnetic energy to electronic energy. Arrows indicate unusually stable structures
corresponding to N = 8, 12, and 18.
112
netic energy from electronic energy. As shown in Figure 8.4(a), these “magic” iron
clusters (N = 8, 12, and 18) show relatively high contribution of electronic energy.
As indicated in Figures 8.4(b) and (c), F93 cluster is non-magnetic structure which
usually obtains energy gain due to tightened covalent bonds, thus forming compact
structure. On the other hand, two other structures have large gain from magnetic
energy due to maximizing exchange contribution resulting in open structure.
To figure out how magnetic behavior is connected to structure quantitatively, I
calculate size dependece of average bond length, and average number of bonds and
show in Figures 8.5(a) and (b). Considering interaction range, I select a cutoff distance
of dc = 3.2 A. These figures seem to contradict to each other. F83, for example, has
large average bond length, which means open structure, but it also has large average
number of bonds, that means compact structure. To resolve this contradiction, I
define a new quantity called structural “compactness” a as
1/2
do
a=%2[; ((1in , (8.6)
which is based on the fact that hopping integral depends on 1/d?
,J, and show in
Figure 8.5(c). In Eq. 8.6, do is a constant value which is independent of clusters and I
use simply do =2 1.0 A. In the figure, dashed line shows a natural trend of compactness
as cluster size increases. Therefore compact structures like F83 are above the dashed
line, whereas open structures such as Fem and F818 are below the line. As discussed
above, open structures are usually magnetic. By applying this formula to bulk system,
iron bcc , which is ferromagnetic and more open than non-magnetic fcc structure, has
a = 0.47 and fcc structure has a = 0.54.
Characterization of magic clusters are shown in Figures 8.6 — 8.8. As shown in
Figure 8.6(a), F e8 is a cube in which nearest and second nearest neighbor distances are
dun = 2.21 Aand (12",, = 3.13 A. Hence its average bond length and number of bonds
113
2.6
E 2.4
A
'5
2.2
2.0
(b) 8.0
.0: ;
4.0: t t I :
2.0 ~ -
J. A. L l A A A A l A A A A l A A A A l A A A A l
per atom
0 0 A A A l A A A A 1 L4 A A l A A A A l A A A A l
O
A
O
V
0.40
0.35
0.30
o, Compactness
ozsc
5 10 15 20 25
N (Cluster size)
Figure 8.5: Cluster size dependence of (a) average bond length < b > per atom in
A, (b) average number of bonds < N, > per atom, and (c) compactness a defined in
Eq. 8.6. Considering interaction range, cutoff distance of dC = 3.2 Ais used to select
bonds. Arrows indicate unusually stable structures corresponding to N = 8, 12, and
18-
114
2.0
5 1.5-
£3
3 1.0 *
8 05
a .
0.0
-l 0 1
E [eV]
Figure 8.6: (a) Optimized geometry of Fes cluster and (b) local density of states of
one atom site. F63 cluster becomes non—magnetic magic cluster because it does not
have spin—exchange energy gain due to very small density of states near Ep.
per atom are < b >2 2.67 Aand < Nb/N >= 6, respectively. Its average magnetic
moment of u/N 2 025MB is very small compared to other clusters resulting from its
small density of states near EF shown in Figure 8.6(b). As a result, F8 cluster, which
is a compact structure, becomes non-magnetic magic cluster.
As shown in Figure 8.7(a), Fen is an icosahedron structure in which nearest
neighbor distance is dun = 2.39 A, and its average number of bonds per atom is
< Nb/N >= 5. Its average magnetic moment of p/N 2 2.67m; is quite large resulting
from its relatively large density of states near Ep shown in Figure 8.7(b). As a result,
F12 cluster, which is an open structure, becomes magnetic magic cluster.
As shown in Figure 8.8(a), F818 is an structure composed of two halfficosahedrons
and a pentagon with a center atom. Neighbor distances are in the range of d,m =
2.39 — 2.70 A, and its average bond length < b >= 2.46 A. Average number of
bonds per atom is < Nb/N >= 6.3. Its average magnetic moment of ,u/N 2 233MB
is quite large resulting from its relatively large density of states near EF shown in
Figure 8.8(b). As a result, F18 cluster, which is an open structure, becomes magnetic
115
(b)
/”\’ if; 4
XX .ng 3:"
.M‘gff'y/v a “i A W 1
E [eV]
Figure 8.7: (3) Optimized geometry of Fen cluster and (b) local density of states
of one atom site. Fen cluster becomes magnetic magic cluster because it has spin—
exchange energy gain due to large density of states near Ep.
(b)
A
93
V
3
fl.\” 5‘ 2.0
XXX/t 21:.
“$9?” 0°0-i\ 0 1
9 E [eV]
Figure 8.8: (a) Optimized geometry of Fem cluster and (b) total density of states.
F913 cluster becomes magnetic magic cluster because it has spin—exchange energy gain
due to large density of states near E p.
116
2.0
1.0
0.0
-l.0
AWN) [eV]
5 10 15 20 25
N (Cluster size)
Figure 8.9: Predicted abundance spectra, based on the second differences in total
energy given in Eq. 8.7. At N = 8, 12, and 18 are shown high peaks which are
indicated by arrows.
magic cluster.
The difference in total energy between adjacent clusters, E (N) — E (N -— 1), is
defined as A(N). The change in this quantity, which is defined as
A‘2)(N) a A(N+1)—A(N)
[E(N+1)—E(N)]—[E(N)—E(N—1)]. (8.7)
This second difference of total energy, shown in Figure 8.9, should be related to
experimentally observable abundance spectra. Therefore, among iron clusters up to
N = 25, Feg, Fem, and Fem are expected to be magic clusters. However, it can be
never sure that those optimized structures are really at global minimum because it is
impossible to scan whole phase space for a large N, which is very complex.
In summary, I discussed the emergence of uncommon structures in the case of
magnetic iron clusters, where the optimum geometry results from a competition be-
tWeen the exchange interaction favoring open structures and the covalent bonding
117
that favors compact structures. Using the total energy calculation combined with
Stoner theory base on itinerant ferromagnetism and J-walk Monte Carlo technique,
I found that iron clusters with N 525 atoms appear especially stable for N = 8,12,
and 18.
118
APPENDICES
119
Appendix A
Total Energy Calculation Based on
Recursion Technique
The ideal formalism to address the dynamics of the large system should at least satisfy
the following three postulates: (2') Reproduction of ab initio or accurate experimen-
tal results for known geometries, and physically sensible interpolation between these
results. (ii) Efficient and parallelizable computation of nonlocal many-body interac-
tions. (iii) Linear scaling with the number of particles.
Parallelization and linear scaling, required by postulates (ii) and (m), are achieved
naturally if the total binding energy Ewhbmd of the system is separated into binding
energies Emnd(z’)’s of the individual atoms i’s as
an... = 2: Emma) , (A1)
and if the computation of these individual binding energies can be performed inde-
pendently on different processors. To really benefit from this partition of the total
energy, I take advantage of the fact that the binding energy of atom 2' depends only
to a negligible degree on the exact position of atoms that are far away.
It is now well established that the total energy of a system, obtained using the
ab initio Density Functional formalism, can be mapped onto a sum of a one-electron
“band structure” or “molecular orbital” energy Embs, and the remaining energy con-
120
tribution Ego”, dominated by the inter-nuclear and closed-shell repulsion, and con-
taining correction terms due to energy double-counting, and an exchange-correlation
energy, as [Cohe94, PapD94]
Etot,b2'nd : Etot,bs + Etot,r - (A'2)
The first term poses the larger challenge, since it describes the non-locality of bonding
due to extended electronic states. I determine this term by first calculating the
electronic density of states N (E) using a parametrized Kohn-Sham operator, and
then integrating over the occupied eigenstates. This scheme describes correctly the
essential physics of nonlocal covalent interactions, as required by postulate (22').
The remaining energy term Eta” is local in nature and can be relatively easily
mapped onto a functional of pairwise nearest-neighbor interactions. It must be chosen
in such a way that the known binding energies of selected structures are correctly
reproduced, hence satisfying the postulate (2').
This framework also allows us to find a meaningful definition of the binding
energy Eb,,,d(i) of atom i, as
Erma”) 1' Eb.(i) + Er(i) - (A3)
Here, Eb,(2') is defined as the local one-electron “band-structure” / “molecular orbital”
energy, which depends mainly on the local environment of site 2' and is obtained using
the site-projected local density of states. Also the local repulsive energy functional
E,(z') depends primarily on the local environment of site 2'.
This energy expression contains the essential physics which governs bonding in
carbon structures. It is superior to semi-empirical “embedded-atom” like potentials
[Ter888], which have been applied to this system [BallQO], since it treats the kinetic
energy quantum mechanically and thus correctly reproduces electronic shell structure
121
effects. Furthermore, it yields J ahn-Teller distortions of symmetric geometries due to
partially occupied degenerate levels at the “Fermi energy”.
In following sections, I present expressions for the local “repulsive” and “band-
structure” energy. I also discuss an efficient parametrization of these expressions for
carbon systems which allows a fast semi-analytical computation of forces acting on
individual atoms. This proves to be very useful in molecular dynamics simulations.
The parametrization introduced here is not only more efficient than previously utilized
schemes, but also corrects some serious deficiencies and inconsistencies of previously
suggested approaches.
A.1 Local repulsive energy
The key quantity in the local repulsive energy functional is the local atomic density
[Cohe94, PapD94] defined as
pi = Z pij- (AA)
#2
Here p.)- is the contribution of pair 2', j and given as
pi]- = [(7%) exp (—%)]pb,~j , (A.5)
where bi, (0 < 1),-,- < 1) determines whether the connection between the sites 2 and j
is to be considered a bond. It is described by a Fermi-Dirac function with a cutoff,
rc which is larger than the nearest-neighbor distance,
bi,- = [1+ exp (Tij _ TON—1. (A.6)
w
In Table Al, I summarize fitting parameters emerging in Eqs. (A.5) and (A.6).
Then the repulsive energy at site 2' is fitted to a finite polynomial of p,- as (see
Table A2)
Ereplpz-l = (190 + 101/J. + P2P?) 6(1).) , (A?)
122
Table A.1: Parametrization of the local atomic density in carbon systems
1.4459 A 3.3610 A 2.4949 2.00 A 0.05 A
Table A2: Parametrization of the repulsive energy in carbon systems
P0 P1 P2
5.7783 12.215 2.3870
pa pb
2.50 5.07
5c £1121 £2122
0.07 0.10 0.05
123
where {(202) is a cutoff function, given by
[1 — exp 07%)] / [I + exp (—P£;uf£)] forp 3 pa
{(P) = [1 + exp (—9£;u§9)]- forpa < p 3 pb (A8)
1 I forpb < p.
A.2 Local band structure energy
The one-electron band-structure energy is given by [Zhon93b]
Ebs = ZEbs(z) (A.9)
i
= Z (I: dEENz-(E) — Emcee.) (A.10)
EF
= 2: f_ dE, (A21)
where 2' represents all atomic sites; 0 atomic orbitals (hereafter I label the Roman
indices as atomic sites and the Greek ones atomic orbitals); E p is the Fermi energy;
N,(E) is the local electronic density of states; and Em,- represents the reference energy
of an isolated atom, which was parametrized in Ref. [Toma91].
In the process of calculating Eb,“ it is necessary to get the local density of states,
N,(E). Of crucial importance for the local electronic density of states is both the
local environment, as defined by the positions and types of neighboring atoms, and a
proper embedding of this cluster environment in a larger entity to avoid the effect of
unsaturated bonds and artificial quantum well states which may be encountered in
any finite—cluster calculations.
The local density of states at the site 2 = 0 is given by
1
N0(E) = 7 11513111100009 + 26) , (A.12)
124
where ImGOO is an imaginary part of 00 element of the Green function matrix, which
is given by the Dyson equation as
I
COME) = (E _ H)00 (A13)
E — (10 —b1 0 —1
—b1 E - a1 —b2 ...
. . . 00
= 1 b, . (A.15)
l
E — a0 — b2
E — a1 — —'2-
To solve Eq. (A.15), it is necessary to consider Hamiltonian describing the system.
Eigenvalues of the Kohn—Sham operator [Hohe64, Kohn64], expressed in a local
basis, are mapped onto a linear combination of atomic orbitals (LCAO) Hamiltonian,
which consequently yields the same eigenstates,
H = 26,,aclacio + Z tiwgclacjg + 12.0. , (A.16)
m 2,3,”;
where cI and c are the creation and annihilation operators for electrons in the re-
spective levels. The on-site energies em and the hopping integrals tin,” can be either
calculated directly from the diagonal and off-diagonal elements of the Kohn—Sham ma-
trix in a local basis, or conveniently parametrized. A simple two-center Slater-Koster
parametrization [Slat54] has been used for four state (.9, p3, pg, 192) nearest-neighbor
tight-binding Hamiltonian [Toma86, Toma87, Toma89]. The parameters have been
obtained from a global fit to density functional calculation within local density approx-
imation [Hohe64, Kohn64] for the electronic structure of C2, a graphite mono-layer
and bulk diamond for different nearest—neighbor distances [Toma88, Toma91].
The continued fraction coefficients an and bn in Eq. (A.15) are obtained by tridi-
125
agonalizing the Hamiltonian matrix
00 b1 0
.. ()1 01 b2 . . .
H : 0 b2 (1.2 . . . ’ (‘A17)
where~ is given to distinguish from the original Hamiltonian matrix. The moments
[Shoh63, Krei59, Wall48] of the density of states of the local orbital is related to the
Hamiltonian, or the continued fraction coefficients as
HI: : chl' (A-18)
Since the Hamiltonian is tridiagonalized, any other elements except for diagonal and
off diagonal ones are zero. (Hereafter, let’s remove” from H and use just H as the
tridiagonalized Hamiltonian.) So, Hf]- can be expressed as the recurrence relation to
obtain the moments from the recursion coefficients an and bn
H}; = ZHfi’lHlj (A.19)
l
= HfJ—erj—IJ + HIE—11911 + Hfflina (A20)
2 Hfifjlbj_1+Hfj_laj_1+H:J-_llbj. (A.2l.)
Since almost all physical information at site 2' are given by lower order of moments
of density of states, which depends only on, at most, the first and second neighbors,
it is sufficient to get the recursion coefficients, (in and bn, up to n = 3 from n =
0. [Zhon93b]. They can be determined by tridiagonalizing not the whole matrix, but
the corresponding small sub-matrix of the Hamiltonian. However, N,(E) requires
not only the local cluster which is built by the neighbor atoms, but also a proper
background (environment) in which this local cluster is embedded. This environment
is needed to avoid the effect of unsaturated bonds and quantum well states which are
126
Table A3: Calculated recursion coefficients an and b3, (492510) for a graphene sheet.
s-orbital p—orbital
72 an b3: an bi
4 4.26726 133.836 1.30797 95.7558
5 -4.22603 52.5404 -0.308042 76.8885
6 3.89913 100.248 1.19842 88.8357
7 -2.12921 98.0581 -0.251197 95.2006
8 0.927010 128.297 0.294314 123.259
9 -0.122119 188.774 0.0152851 179.375
10 0.000000 219.012 0.000000 207.433
127
troublesome in the way of calculating any finite clusters, as discussed above. As a
proper background, previously calculated coefficients, an and bn (4322310) for bulk
graphite structure are used. These values are given in Table A3. To avoid strong
oscillation in the density of states due to this background, the coefficients are mixed
in the similar way to given in Ref. [Zhon93b].
When calculating N0(E), only an and b3, are treated. Therefore it is better to
define new matrix consisting of an and bf, instead of an and bn in the following way,
b b ...b-__
k k 1 2 2 l
.. = H- .___ A.22
'1 Ub1b2...bj_1 ( )
9]- : (5,-,- identity matrix. (A.23)
With k=1, it can be easily shown that
0.0 1 0
b? 0.1 1 . . .
And I can also show that the recurrence relation of H becomes
:3- = £1.11. + Q?,-“a.-_1 + 9:3. 3 (A25)
These relations, Eq. (A25), require much fewer computational time than the direct
matrix multiplication.
The recursion coefficients can be reconstructed from the moments of the density
of states of the local orbital, to combine the recursion coefficients by each local orbital
into those by a single isotropic states. This procedure has been described in several
literatures [Gasp72, Wall48, Gord68], and I summarize it here.
The recursion coefficients are obtained by these relations
an 2 02n+1+02n (A26)
5,2, = 02120121.“, (A27)
128
where
A“) A _
02,, = _A:_1_%_2_ (A28)
Ap-lAp—Q
A A“)
02p+1 #1. (A.29)
Ap—lAp-l
Here, AP and Ag,” are defined by determinants consisting of the moments of the
density of states, pk,
#0 H1 Hp
AP: ”1 ”2 ”’2“ p=0,1,2,... (A.30)
Hp Hp+1 #21;
and
H1 H2 Mp+1
Ag): ”:2 “:3 “’2” p=0,l,2,... (A31)
#1041 Hp+2 - - - #29“
where A_2 = 0, A_1 = 1, and A“) = 0, A97 = 1, as initial conditions.
When calculating these determinants, a floating overflow may often be encoun-
tered. To avoid this, a scaling energy,Eo, which is the order of the band width of the
density of states, is introduced; and new scaled moments are defined by
ilk = #k/Efi- (A32)
Then, the determinants consisting of [1,, can be the order of the unity with proper E0.
(in and I33, obtained from these determinants using the above formulae can be simply
reconverted to an and ()3, with the scaling energy E0,
an 2 (InEo (A33)
23, = 53,193. (4.34)
129
Table A.4: Parametrization of the band-structure energy, given in Eq. (A.37), in
mixing regions.
E1 132 E3 E4 Egg) 7
35.0 eV 38.8 eV 13.0 eV 8.0 eV —36.0 eV ——1.15
It is sufficient to know exact band-structure energy in the region near equilibrium
only in most cases. Therefore, when the atomic distance is either very short or very
long, it will be vain efforts to try to determine exact band-structure energy. Hence,
band-structure energy is replaced by a constant value, Egg), and the scaled repulsive
energy, yErep, at short and long distance, respectively.
To avoid discontinuities in total energy and force, small mixing regions are taken
into account. Since the repulsive energy depends directly on the atomic distances,
mixing regions is determined with mixing ratios defined by
C5 : (E1 - Erep)/(E1 — E2) (A.35)
CI : (E3 _ Erep)/(E3 — E4), (A36)
where subscripts s and l represent short and long distances, respectively. With these
mixing ratios, band-structure energy is determined as
%? final
Ebsl1_ f(Cs)l + Egg)f(Cs) if 0 < Cs < 1
E2. = 7B,“, if g, 2 1 , (A-37)
Ebsll —' f(Cl)] + 7Erepf(Cl) if 0 < Cl < 1
Eb, otherwise
where f (C ) is a mixing function defined by
l l
130
I summarize parameters introduced here in Table A.4.
A.3 Force Calculation
It is indispensable to know the forces acting on each individual atoms in the system
when performing molecular dynamics calculations. The force can be calculated ana-
lytically by taking a negative gradient of the total energy with respect to each atomic
position,
F(r) z —VEtot (A.39)
: _ Z [Vii/lbs“) + VErep(i)l ' (A-40)
131
Appendix B
General Scattering Technique
The scattering technique used in Chapter 6 has been recently employed in stud-
ies of giant magnetoresistance [Sanv99a, Sanv98] and ferromagnetic/ superconductor
structures [Tadd99]. It yields the quantum-mechanical scattering matrix S for a
phase-coherent system attached to external reservoirs. The role of the reservoirs
is to inject and collect incoherent electrons into the scattering region. The energy—
dependent conductance G (E) in the zero—temperature limit is computed by evaluating
the Landauer-—Biittiker formula [Biitt85]
G(E) = 27:40:), (13.1)
where T(E) is the total transmission coefficient evaluated at the energy E (Ep in the
case of zero-bias). The formula of Eq. (8.1) provides an exact relation between the
conductance of a system and its scattering properties.
The transmission coefficient is evaluated using a scattering technique that com-
bines a real space Green function calculation for the incoherent leads and a Gaussian
elimination ( i.e. “decimation”) algorithm for the scattering region. A general scheme
of the technique is presented in Figure 8.1, where I indicate how a transport problem
can be mapped onto a quantum mechanical scattering problem.
Suppose the total Hamiltonian H for the whole system (nanotubes plus external
132
.
c.)
”55650 a -
'u
n a r):
on D an E "
DC HOBO BOD O u
o ODDDDIDO '-
DDDOOOOC -
J a cannon C '
300C 000006 0 .
so ' 2
am: an.
uu -
“D 0 OD BBB 1
u a
no -
o .
unocnnno o o r‘ 4
G O 0 '
UUUUUUU OUODDDDDUUUJUJ‘
DCDCJCDCOCSDDOOCDUJCD‘
Figure 8.1: Scheme of the scattering calculation. The system (a) consists in two
reservoirs with chemical potentials p1 and 122 separated by a scattering region. The
problem is mapped by using the Landauer—Biittiker formalism [Biitt85] onto a quan-
tum mechanical scattering problem (b). The incoming scattering channels in the leads
are calculated through the surface Green function go. The effective coupling matrix
H93 is computed by “decimating” the internal degrees of freedom of the scattering
region. The total transmission T and reflection R coefficients are then calculated by
solving exactly the Dyson’s equation and by using a generalization of the Fisher-Lee
relations [LamC93].
133
leads) can be written
HZHL'l'HL—NT'l'HNT'l‘HNT—R‘l‘HRa (B?)
where HL and H R describe respectively the semi-infinite left-hand side and right-hand
side lead, HL_NT and HNT_R are the coupling matrices between the leads and the
nanotube and H NT is the Hamiltonian of the nanotube. In what follows I will consider
the leads themselves to be carbon nanotubes, whose number of walls depends on the
position of the electrical contacts. This is justified when the transport bottleneck is
formed by the nanotubes and not to the metal-nanotube contacts. As far as I know,
detailed ab-z’m’tz‘o analysis of metal/nanotube interaction is still not available.
The surface Green function 93 of the leads are calculated by numerically evalu-
ating the general semi-analytic formula given in reference 22. One of the key-points
of such a calculation is to compute the scattering channel 3 in the leads. Suppose z
to be the direction of the transport and the Hamiltonian of the leads to be an infinite
matrix of tri-diagonal form with respect to such a direction, with the matrices Ho and
H1, respectively in the diagonal and off-diagonal positions. Therefore, the dispersion
relation for electrons in a Bloch state
1 .
[Ir/)2 Z Wezk‘ZQSk (8.3)
”I:
and moving along 2 with unit flux can be written as
(H0 + H14“c + H-1e-‘k — E)¢,, = 0 , (3.4)
where 2)), is the group velocity corresponding to the state Eq. (8.3) and H_1 2 HI
(H0 2 H3). Note that the matrices H0 and H1 describe respectively the interaction
within a unit cell and the interaction between adjacent cells. If a unit cell possesses
M degrees of freedom, these matrices will be M x M matrices. Moreover d), is a
M dimensional column vector which describes the transverse degrees of freedom of
134
the Bloch-function. The Green function in the leads is constructed by adding up
states of the form of Eq. (8.3) with k both real and imaginary, which means that the
dispersion relation Eq. (8.4) must be solved for real energies in the form k = k(E).
This is the opposite to what is usually computed by ordinary band structure theory
where one is interested in finding all the real energies E = E (k) for a chosen real
k-vector. Moreover in the calculation of k = k(E) instead of solving the equation
det (H0 + Hle“c + H_,e-“c — E) = 0 , (3.5)
which involves the use of a root tracking algorithm in the complex plane, I map the
problem onto an eigenvalue problem by defining the matrix ”H
”H:
( —Hr1(Ho —E) —Hr‘H-1 ) , (B.6)
I 0
where I is the M x A! identity matrix. The eigenvalues of H are the roots 6"" and
the upper half of the eigenvectors of 71 are the corresponding eigenvectors (bk.
The second part of the calculation involves computing an effective coupling matrix
between the surfaces of the scattering region. Note that the purpose of a scattering
technique is to calculate the S matrix between electrons in the leads. Therefore one
is not interested in information regarding the internal degrees of freedom of the scat-
tering region, but only in the resulting coupling between the external interfaces. This
can be achieved by reducing the matrix HL_NT + HNT + HNT_R to an effective cou-
pling matrix Hag. Suppose the total number of degrees of freedom of the Hamiltonian
HL_NT+HNT+HNT_R is N, and the number of degrees of freedom of the lead surfaces
M. One can eliminate the 2' = 1 degree of freedom (not belonging to the external
surfaces) by reducing the N x N total Hamiltonian to an (N — 1) x (N — 1) matrix
with elements
1.92:1)... ”1'le
—— . 8.7
1 1] E _ H11 ( )
135
Repeating this procedure 1 times, I obtain the “decimated” Hamiltonian at l-th order
_ H"‘”H“’”
HS) : Hi“ 1) + I ((13.1) ’ (BS)
and finally after N — AI times, the effective Hamiltonian
_ H:(E) Ha (E)
HE“(E"( Ham) H4713) ) (”‘9’
In the Eq. (B9) the matrices Hf:(E) and H§(E) describe the intra-surface cou-
plings respectively in the left-hand side and right-hand side surfaces, and HfiR(E)
and HfiL(E) describe the effective coupling between these surfaces. From the above
equations it is clear that only matrix elements coupled to the eliminated degree of
freedom are redefined. This exact recursive technique therefore turns out to be very
efficient in the case of short-range interaction like the nearest neighbors tight-binding
model considered here. Two important considerations must be made. First, both the
Green function calculation and the “decimation” require a fixed energy. Once this
has been set the calculation is exact and does not use any approximation. Second,
the calculation of the Green function is completely decoupled by the calculation of
the effective Hamiltonian for the scatterer. This can allow very efficient numerical
optimizations, particularly in the study of disordered systems [Sanv99b].
Once both the surface Green function of the leads 93 and the effective coupling
Hamiltonian Heg(E) are computed then the total Green function GS for the whole
system (leads plus scattering region) are easily calculated by solving the Dyson’s
equation
GS(E) = [(93’(E))“1 - Heffl_l - (310)
Finally the scattering matrix elements are extracted from GS by using a generalization
of the Fisher-Lee relations [LamC93].
136
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