13.»:

.1.
.:.

., n,

 

a. .3.“ .
.unbuwuanr. 1

18“.? .. J .
3. 3.3
h:l.~t.»w..1elmwmwdw

w.

a}: ., 19L... 9

£30314?» .

. E141I11N k
72m 6..

 

1w

0"
'1"

u

swim g

.u ..

 

l

 

 

      

 

 

 

 

12.. Fri. o. 3%
.a . 5. 3m»... :

_kct.

‘II'

 

 

This is to certify that the
dissertation entitled

Self-Assembly and Physical Properties of Atomic Aggregates

presented by

Savas Berber

has been accepted towards fulfillment
of the requirements for the

Ph.D. degree in Physics

 

WWW @mimz

Major Professor’s Signature
Mag 2 .1 2W9

Date

MSU is an Affirmative Action/Equal Opportunity Institution

W;
Michigan State

University

 

 

 

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6/01 cJClFIC/DateDuepGS-p. I 5

SELF-ASSEMBLY AND PHYSICAL PROPERTIES OF ATOMIC AGGREGATES
. By

Savas Berber

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

2004

ABSTRACT
SELF-ASSEMBLY AND PHYSICAL PROPERTIES OF ATOMIC AGGREGATES
By

Savas Berber

In this Thesis, I present a study of nanoscaled atomic aggregates such as ful-
lerenes, nanotubes, diamondoids, and related materials. Nanotubes and fullerenes,
which could be formed of any layered material, show unusual physical properties due
to their lower dimensionality and nanometer size. I have investigated the microscopic
self-assembly mechanisms and the physical properties of these nanoscale atomic ag-
gregates through computer simulations primarily by using molecular dynamics simu-
lations combined with structure optimization and total energy calculations.

First, I determined the stability, optimum geometry and electronic properties
of nanometer-sized capped graphitic cones, called “nanohorns”. My main result,
simulated scanning tunneling microscopy images of the various structures at different
bias voltages, indicate a net electron transfer towards the pentagon vertex sites.

Next, I investigated the absorption of fullerenes in a nanotube during a hierar-
chical self-assembly of “nanopeapods”. I found the absorption through a defect to
be significantly more efficient than the end-on absorption. I also investigated the
energetics and detailed fusion mechanism of fullerenes inside a carbon nanotube. I
found that fullerenes are pulled in by a “capillary” force, which yields an effective
GPa pressure. Fusion of fullerenes continues along the minimum energy path as a
sequence of Stone-Wales transformations.

I have further investigated the microscopic dislocation mechanism leading to struc-
tural transformations in nanostructures. In particular, I studied the relative stability
and the conversion mechanism between multi-wall carbon nanotubes and graphitic

scrolls. I postulated a zipper-like mechanism, which converts a scroll to the more sta—

ble multiwall nanotube. I also found this transformation to proceed very efficiently
due to the unusually low associated activation barrier. A further study related to
defects in nanoscale systems involves the response of defective nanotubes to thermal
and electronic excitations, which are described using ab initio molecular dynamics
calculations, based on the time-dependent density functional formalism. I found that
defective nanotubes, exposed to electronic or thermal excitations, show an unexpected
self-healing ability, which is intimately linked to their nanometer size.

I also explored the suitability of nanotube-based hooks for bonding in nanoscale
electro—mechanical systems (NEMS). Arrays of nanohooks, which are permanently
anchored in solid surfaces, are the nanoscale counterpart of velcro fasteners, forming
tough bonds with a capability of self-repair. I also investigated the possibility of
novel possible “super-hard” crystalline phases of covalently bonded C60 fullerenes
and determined their structural, elastic, and electronic properties.

My studies further involved exploring the physical origin of the unexpected ther-
mal contraction of carbon fullerenes and nanotubes. At low temperatures, these
systems gain structural and vibrational entropy by exploring the configuration space
at little energy cost, which translates into thermal contraction of these systems at
moderate temperatures. I also predicted the thermal conductivity of carbon nano-
tubes. My results suggest an unusually high value Az6, 600 W/m-K for an isolated
(10,10) nanotube at room temperature, which exceeds the thermal conductivity of
the best thermal conductor known, isotopically pure diamond.

Finally, I contributed to a study of magnetism in all-carbon nanostructures using
ab initio spin density functional theory. The main conclusion of that study was that
in this and other non-alternant aromatic systems with negative Gaussian curvature,
unpaired spins can be introduced by sterically protected carbon radicals, so that

particular systems, related to schwarzite, carry a net magnetic moment.

To my Family, Selda, Saadettin and Yunus.

iv

ACKNOWLEDGMENTS

First of all, I thank my Thesis advisor, Professor David Tomanek. I have enjoyed being
in a moderate size research group, and being in close contact with my Thesis advisor
Professor Tomanek. Whenever I read an article about how a good Thesis advisor
should be, I realize that he has done even more than what an excellent mentor does.
He has always encouraged me to present my results by writing papers and giving
talks in conferences. He almost had to carry me on his shoulder to Atlanta, my first
March meeting. He introduced me to almost all the best people in my field in person.
He did everything he could, to support my Ph. D. studies in all aspects. We had
very stimulating discussions on physics, science, technology, science policies, cultures,
religion. He has helped me to get experience on every stage of the academic life. I
always believed he deserves better graduate students than I am. I would not be in
this position without his continuous support and guidance.

I am grateful to the members of my Guidance Committee: Professors Jack Bass,
S.D. Mahanti, Vladimir Zelevinsky, and Richard Enbody. I greatly appreciated their
comments and suggestions.

I was privileged to collaborate with Young-Kyun Kwon, Mina Yoon, Dr. Yoshiyuki
Miyamoto, Dr. Noejung Park, Seunghwu Han, Prof. Eiji Osawa, Prof. Angel Rubio
Secades. Young-Kyun Kwon and Mina Yoon have a special place; I have always
enjoyed group meetings and scientific discussions with them. I also would like to
mention the hospitality of Seoul National University graduate students who made my
visit to Korea an enjoyable and fruitful time. Thank you all!

During my Thesis work, I had the opportunity to discuss scientific issues with the
experts in the field. I thank Sumio Iijima for fruitful discussions on carbon nanohorns
and the synthesis of nanohook structures. I appreciated useful discussions with Ray

Baughman, who drew our attention to thermal expansion properties of carbon nanos-

tructures, and with Alex Soldatov on the experimental aspects of thermal contraction.
I acknowledge useful discussions with Professors Pao—Kuang Kuo and Ctirad Uher on
thermal conductivity of carbon materials. I also feel lucky to happen to know the ex-
perimental work of Andrei Rode on magnetic carbon nanostructured materials before
published. I thank David Luzzi for fruitful discussions on nanopeapods.

I also thank my friends who reminded me that I am supposed graduate some day
by asking me every Friday when I will graduate. They helped me to realize the real
reason why I was at MSU.

I acknowledge financial support from the National Education Ministry of Turkey,
Korean Molecular Logic Devices program, Rosseter Holding Ltd. , Engelhard Cor-
poration, and MSU Graduate School through Dissertation Completion Fellowship. I
appreciated the efforts of Prof. David Tomanek, Prof. S. D. Mahanti, and Ms. Debbie
Simmons during the application for Dissertation Completion Fellowship.

I must also thank my wife Selda for putting up with my late hours, my spoiled

weekends, my bad temper, but above all for putting up with me.

vi

Table of Contents

LIST OF TABLES

LIST OF FIGURES

1 Introduction
1.1 Nomenclature of carbon nanostructures .................

1.2 Outline of the dissertation ........................

2 Electronic and Structural Properties of Carbon Nanohorns
2.1 Introduction ................................
2.2 Carbon nanohorns as a new class of carbon nanostructures ......
2.3 Electronic and structural properties ...................
2.3.1 Structural properties .......................
2.3.2 Electronic structure ........................
2.4 Thermal stability .............................

2.5 Summary .................................

3 Microscopic Formation Mechanism of Carbon Nanopeapods

vii

xii

xiii

14

14

15

16

16

21

24

25

26

3.1 Carbon nanopeapods: A new type of hybrid carbon materials . . . . 26

3.2 Possible formation mechanisms of

nanopeapods ............................... 27
3.3 Role of defects in the formation pathway of nanopeapods ....... 31
3.4 Summary ................................. 37

Energetics of Encapsulation and Fusion of Fullerenes in Nanopea-

pods 38
4.1 Unusual properties of carbon nanopeapods ............... 38
4.2 Energetics of fullerenes in nanopeapods ................. 39
4.3 Chemical reactions inside carbon nanotubes .............. 41

4.3.1 Fusion of fullerenes in carbon nanopeapods ........... 41
4.4 Summary ................................. 46

Defective Structures and Defect-Induced Structural Transformations 48

5.1 Introduction ................................ 48
5.2 Defect engineering in nanostructures .................. 50
5.3 Carbon nanofoam ............................. 50

5.4 Hybrid structures of coexisting scrolls and
multi-wall nanotubes ........................... 52

5.5 Stability differences and conversion

mechanism between nanotubes and scrolls ............... 54
5.6 Continuum energy functional for sp2 bonded carbon systems ..... 56
5.7 Optimum geometries and energies of sp2 carbon nanostructures . . . 59

viii

5.8 Scroll / tube dislocation .......................... 61

5.9 Proposed zipper-like scroll-to-tube

transformation .............................. 64
5.10 Summary ................................. 65
Self Healing in Nanostructured Materials 66
6.1 Introduction ................................ 66
6.2 Method .................................. 67
6.3 Thermal self-healing ........................... 68

6.3.1 Energetics of monatomic vacancies in narrow nanotubes . . . . 71
6.4 Self-healing by photo-excitations ..................... 71
6.5 Conclusions ................................ 76
Bonding and Energy Dissipation in a Nanohook Assembly 78
7.1 Theoretical concept of nanovelcro .................... 78
7.2 Theoretical approach ........................... 79
7.3 Nanohooks based on carbon nanotubes ................. 82
7.4 Dynamics of opening and closing the nanohook assembly ....... 83

7.4.1 Mechanical characteristics of the nanovelcro bond ....... 86
7.5 Summary ................................. 88
Crystalline Solids Based on Polymerized Cso Molecules 89
8.1 Introduction ................................ 89

ix

8.1.1 Polymerized fullerenes ......................

8.2 Inter—fullerene bonding schemes .....................
8.3 Super-hard phases of C60 crystals ....................
8.3.1 Body-centered orthorhombic phases ...............
8.3.2 Body-centered cubic phases ...................
8.3.3 Simple cubic phases ........................
8.3.4 F ace-centered cubic phases ....................

8.4 Conclusions ................................

Thermal Contraction of Carbon Fullerenes and Nanotubes

9.1 Materials exhibiting negative thermal expansion ............

9.2 Computational approach .........................

9.3 Thermal contraction in fullerenes as zero dimensional model systems .

9.4 Thermal contraction in nanotubes as one dimensional model systems
9.4.1 Physical origin of thermal contraction ..............

9.5 Summary .................................

10 Unusually High Thermal Conductivity of Carbon Nanotubes

10.1 Introduction ................................
10.2 Limitation of theoretical approaches describing thermal conductivity .
10.3 Computational results ..........................

10.4 Conclusions ................................

91

95

96

99

102

103

105

105

107

108

110

113

116

116

118

11 Magnetism in All-Carbon Structures with Negative Gaussian Cur-

vature 127
11.1 Magnetism in carbon ........................... 127
11.2 Computational method .......................... 128

11.2.1 The tetrapod as a possible building block of carbon nanofoam 130
11.3 Unpaired spins in a carbon tetrapod .................. 130
11.4 Magnetism due to sterically protected radicals in nanostructured carbon135

11.5 Summary ................................. 137

BIBLIOGRAPHY 139

xi

List of Tables

2.1 Structural data and relative stability of carbon nanohorns ......

8.1 Calculated physical properties of polymerized C60 crystal structures,
as compared to cubic diamond. .....................

xii

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

List of Figures

Unit cell of a nanotube unrolled onto a graphene sheet ........ 6
Relaxed structures of lower diamondoids ................ 9
Relation between the morphology of a nanohorn and a graphene sheet 17

Optimized carbon nanohorn structures ................. 18
Simulated STM images of nanohorns in the tip region ......... 21'
Electronic densities of states of all nanohorn types ........... 23
Perspective views and potential energy surfaces for the encapsulation

of a C60 molecule in a (10, 10) nanotube ................ 29

Probability of C60 encapsulation through the open end of an isolated
(10,10) nanotube ............................. 32

Probability of C60 encapsulation through the open end of bundled
(10,10) nanotubes ............................ 33

C60 encapsulation through a large opening in the wall of a (10,10)
nanotube as the most efficient encapsulation mechanism ....... 34

Fullerene encapsulation energy and the capillary force in nanopeapods 40
Optimum pathway for the 2C60—+C120 fusion reaction ......... 43

Energetics of the fullerene fusion in peapods .............. 44

xiii

5.1

5.2

5.3

5.4

5.5

5.6

6.1

6.2

6.3

7.1

7.2

7.3

8.1

8.2

9.1

9.2

Schematic view of a scroll, a nested multiwall nanotube, and a defect
separating the two systems ........................

Structural description of carbon nanofoam ...............

High-Resolution Transmission Electron Micrograph of a multi wall car-
bon nanotube with a slip-plane defect and irregular layer spacing

Energetics of 3p2 bonded carbon, and structure optimization for multi-
wall structures ...............................

Total energy of multi-wall tubes and scrolls with respect to a reference
graphene strip ...............................

Zipper motion of the scroll / tube dislocation ..............

Thermal self-healing of a defective carbon nanotube ..........

Self-healing process within a (3,3) nanotube with a single vacancy,
induced by illumination ..........................

Time evolution of the energy spectrum during the atomic motion fol-
lowing the initial photo—excitation ....................

Model structures of nanotube-based hooks ...............
Nanovelcro bond based on a two hook assembly ............

Structural deformations and strain energy distribution in nanotube
based nanovelcro during opening process ................

Types of covalent bonds connecting C60 molecules ............

Energy change per atom AE versus the relative volume change AV/ V
for the stiffest BCC and BCO structures .................

Thermal contraction of fullerenes ....................

Thermal contraction of nanotubes ....................

xiv

50

51

53

57

60

62

7O

74

75

81

84

93

97

111

10.1

10.2

10.3

10.4

10.5

11.1

11.2

11.3

Temperature profile during a direct molecular dynamics simulation of
heat conduction in a (10,10) nanotube .................

Velocity autocorrelation function in a heated nanotube ........

Nonequilibrium Molecular Dynamics simulation for heat conduction in
a (10,10) nanotube ............................

Temperature dependence of the thermal conductivity A for a (10,10)
carbon nanotube .............................

Thermal conductivity of various carbon structures as a function of
temperature ................................

Occurrence of trivalent carbon radicals in a tetrapod structure . . . .
Spin-resolved electronic densities of states of tetrapod structures . . .

Charge density associated with the carbon radicals and edge states in
carbon tetrapods and nanotubes .....................

XV

119

120

122

123

125

129

132

133

Chapter 1

Introduction

Science and technology has seen tremendous development in the last century. Our
understanding of the nature and the phenomena happening around us improved by
a great amount. Scientists and engineers now not only understand the nature better,
but have also developed new techniques allowing us to explore unknown phenomena
in relatively little time. Most people agree that we would not be in this position in
science without atomic scale probes, electron microscopes, and powerful computers
to process all the data we collect. The increasing demand of humanity for faster,
better, more reliable, and portable devices and products has pushed the limits of
engineering to sub-micron scales. Micro-scale electromagnetic and mechanical devices
have become the study subjects of mechanical engineers. We face further challenges
when we go to still smaller scales in the emerging field of nanotechnology. The fuel of
scientific development during the last part of the 20th century, including significant
advances in the electronic and computer industry, now faces great challenges. The
demand for higher speed forces us to use smaller building blocks in devices. Within
the next decade, we expect that the electronic industry will reach the quantum limit
for silicon-based device applications, where we can no longer control structural and

functional aspects of future nanoscale devices.

Apart from dealing with small systems, it is a great challenge to manufacture
these devices using conventional methods, such as lithography. One has to employ self-
assembly or directed self-assembly methods to form nanometer-sized building blocks
of future engineering marvels. Nevertheless, we do not have sufficient knowledge of
the phenomena occurring at the nanoscale. Nanostructures constantly surprise with
unexpected properties. Our basic science interest quickly shifts from understanding
those structures to using them in our everyday applications.

In spite of advances in atomic scale probes, there are still many phenomena which
could not be completely understood by using only only experimental techniques. In
particular, it is very hard to interpret experimental results without theoretical or
modeling studies. Computer modeling has proven to be very useful in describing and
understanding phenomena occurring at the nanoscale. Computers also help us to
design new materials, and to predict their physical properties. Recent developments
in computer modeling, and advances in computer technology have made it possible
to attack some of the most important problems we are facing in developing and
understanding nanostructures. Not only physical sciences, but also biological sciences
tend to follow the same direction in recent years. The solution of medical problems
is now being searched at the DNA level, on the nanometer scale.

Some of the most promising building blocks of nanotechnology are based on car-
bon. Elemental carbon exhibits a wide range of properties, which reflects the different
types of bonding between carbon atoms. Research in all-carbon nanostructures was
initiated by the discovery of C60 molecule, which reminded the carbon community
of apparently endless possibilities involving carbon nanostructures. Carbon-based
nanotechnology was subsequently accelerated by the discovery of nanotubes. Nano-
tubes and fullerenes have proven to be promising candidates for future applications.
Current commercial applications include battery applications, and improvements of

mechanical properties in composites.

Nanotube-based transistors are believed to be superior to state of the art sili-
con transistors. Moreover, carbon nanotube transistors are easy to manufacture, and
could be cheaper to produce in the future. Their superior mechanical properties,
thanks to their strong graphitic bonds, make them viable materials for polymer re-
inforcement and nanomanipulation applications. Chemical functionalization makes
nanotubes useful in sensor applications. Moreover, nanotubes appear to be ideal
1-dimensional model systems to study phenomena, such as quantum transport, Lut-
tinger liquid behavior, and Coulomb blockade.

In the meantime, some related materials, such as nanohorns, and some hybrid
structures, such as nanopeapods, have been synthesized.

The fact that these complex structures go beyond the Theorists’ wildest dreams
encouraged the scientific community to suggest new artificial materials and geome—
tries based on sp2 bonded carbon, which could be modeled on a computer. In my
PhD Thesis work, I tried to perform calculations using a variety of techniques to
further enhance general understanding of nanoscale phenomena. I was bold enough
to predict selected physical properties of nanotubes and related nano-materials prior
to experimental observation, including thermal conductivity and thermal contrac-
tion. My studies involved what I considered to be interesting systems or interesting
properties. Scientists initially interested in purely electronic applications of nano-
tubes and related materials quickly realized that knowing only electronic properties
was not enough to construct nanotube-based devices. I tried to fill the gap of knowl-
edge regarding also non-electronic properties of nanotubes and nanostructured carbon
materials. In particular, I focused on electronic, thermal, mechanical, and magnetic
properties of carbon nanostructures.

Following the discovery of nanotubes, the general perception was that these
nanostructures are atomically perfect. Only later on, scientists realized that defects

were present in these structures which could never be absolutely eliminated. Since

there is no simple experimental way of determining the position and morphology
of defects, it is very difficult to estimate how these defects may affect the physical
properties of nanostructures. Computer modeling may be the most efficient way to
determine the effect of defects in nanostructures. Since few atomic vacancies per
nanostructure translate into effective defect densities that are currently considered
unacceptable, industry leaders, such as IBM and NEC, were initially reluctant to
invest in nanotube research before knowing how defective nanotubes would perform
in devices. Since processes associated with device failure are very fast and irrepro—
ducible in most occurrences, little information could be extracted from experimental
data. The most reliable way to determine the microscopic pathway of a failure thus
involves computer simulations under extreme conditions. My simulations have sug-
gested that presence of defects lowers the activation barriers for structural changes
during a directed self-assembly of systems. Even more surprising was my finding that
nanostructures heal themselves.

Recently, diamondoids joined the group of candidates for future nanoscale build-
ing blocks. Successful separation of diamondoid isomers from crude oil opened an
opportunity to use these molecules in nanotechnology. We now expect a possible
synthesis of complex nanostructures based on nanotubes and diamondoids instead of
fullerenes, with potentially interesting applications.

The phenomena we encounter in nanotubes and related materials should not be
limited to graphite-based systems only. Any layered material, including BN, W82, and
others, should form similar nanostructures, which may exhibit similar phenomena.
Even though carbon nanostructures should maintain a special place, I believe other
materials may find even more important applications in the future.

Since this PhD Thesis covers not only different length scales, but also a wide range
of phenomena, I had to use different techniques , which were specific to each study. I

will not discuss the details of computational techniques, including molecular dynamics

simulations, as far as they have been covered in the literature. Each Chapter should
be considered essentially self-contained, including an introduction, a brief description
of methods, summary and conclusions.

In the following, I will briefly summarize the notation which is commonly used
by scientists studying nanotubes and diamondoids. After this brief summary, I will

outline the structure of my Thesis.

1.1 Nomenclature of carbon nanostructures

Most stable allotropes of carbon are 3p2 bonded graphite and sp3 bonded dia-
mond. These two bonding schemes are found also in carbon nanostructures. One of
the most promising building blocks of future nanoscale devices are sp2 bonded ful-
lerenes and nanotubes. Another important building blocks of carbon nanostructures
are diamondoids, which shows 3123 bonding. In the following I will introduce common
notation for carbon nanotubes, and the physical structure of diamondoids.

Carbon nanotubes are classified primarily into achiral and chiral nanotubes [1, 2].
An achiral nanotube exhibits a mirror symmetry with respect to a plane normal to
the tube axis. Chiral nanotubes miss this symmetry, and exhibit a spiral symmetry
instead. There are only two types of achiral nanotubes, namely “armchair” and
“zigzag” nanotubes, discussed below. The structure of a nanotube is specified by the
orientation of the hexagonal carbon rings on the graphene sheet wrapped to a tube
with respect to the tube axis. This orientation is characterized by the chiral index

(n, m) of the nanotube defined by the chiral vector Ch

Ch '33 7131+ 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

 

 

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 OA and OB, respectively. The rectangle OAB’ B defines the unit cell
of the nanotube.

sheet. Its magnitude 0;, represents the circumferential length of the nanotube being
characterized by Ch. The direction perpendicular to Ch becomes the tube axis. Pairs
of integers (n, m) in Eq. (1.1), specifying all possible chiral vectors, defines all different
ways of rolling the graphene sheet to form a nanotube. Zigzag nanotubes, which have
a zigzag shape of the cross-sectional ring are denoted by (n, 0). Armchair nanotubes,
which have an armchair shape of the cross-sectional ring, are denoted by (n, n).

The tube diameter dt is given by

d, = Ch/7r = \/3dCC(n2 + nm + m2)1/2/7r, (1.2)

where doc is the nearest-neighbor distance between two carbon atoms (in graphite
doc = 1.42 A). The chiral angle 6, defined as the angle between the chiral vector Ch

and the lattice vector al, is given by

Ch-a1_ 2n+m
Cha 2\/n2 + nm + m2.

 

 

c050 =

 

(1.3)

The chiral angle 6 is in the range of 0 5 WI 3 30° because of the hexagonal symmetry
of the graphene sheet. Armchair nanotubes, in particular, correspond to 0 = 30° and
zigzag tubes to 0 2 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 of the 1D tube unit cell, can be expressed in terms

of the basis vectors a,- as

1

T:—
(112

—(n + 2m)a1 + (2n + m)a2], (1.4)

where (1;; is the greatest common divisor of (n + 2m) and (2n + m). Furthermore, (1,;

can be expressed in term of the greatest common divisor d of n and m. If (n — m)

is a multiple of 3 so that (n —— m) = 3d, then (in = 3d, and otherwise (1;: = d. Note
that the bigger dR, the smaller is the length of T. For example, T = —a1 + a2 for
any (n,n) armchair nanotube ((1,; = 3d 2 3n) and T = —a1 + 2a2 for any (72,0)
zigzag nanotubes (d3 = n). A (6,3) nanotube (d = (1;; = 3), shown in Figure 1.1,
hasT=—4 a1+5a2.

Unlike nanotubes, diamondoids show 3133 bonding between carbon atoms, and
could be described as hydrogen terminated nanoscale diamond fragments. These
nanoscale building blocks are very different from fullerenes and nanotubes, but com-
pare well in stability and light weight with fullerenes and nanotubes. Being es-
sentially hydrogen-terminated nanometer sized diamond fragments, they may occur
in a large variety of shapes, as shown in Fig. 1.2. The smallest possible diamon-
doid is adamantane[3], consisting of ten carbon atoms arranged as a single diamond
cage[4], surrounded by sixteen hydrogen atoms, as shown in Fig. 1.2(a). Larger
diamondoids[5, 6, 7] are created by connecting more diamond cages and are cate-
gorized according to the number of diamond cages they contain. The diamondoids
shown in Fig. 1.2 contain up to tens of carbon atoms and are only a few nanometers

in diameter.

1.2 Outline of the dissertation

This dissertation contains 11 Chapters, including the introduction chapter. The
computational method of choice throughout the dissertation involves molecular dy-
namics simulations employing an electronic Hamiltonian based on the linear combina-
tion of atomic orbitals (LCAO). Where appropriate, this electronic Hamiltonian was
substituted by the computationally less demanding Tersoff bond-order potential. For
electronic structure calculations I either used ab initio techniques or a parametrized

LCAO formalism. Magnetic properties were investigated using spin resolved density

(a) Adamantane (b) Diamantane

 

 

 

 

Figure 1.2: Relaxed structures of lower diamondoids. Hydrogen atoms, terminating
the carbon skeleton, are omitted from the diagrams for clarity. (a) The smallest
diamondoid, adamantane, consisting of a single diamond cage. (b) Diamantane with
two diamond cages. (c) Tetramantane with four diamond cages. (d) Decamantane
with ten diamond cages.

functional theory, and time dependent phenomena were explored by time-dependent
density functional theory. The recursion technique for calculating energies and forces
in molecular dynamics simulations has been described previously [8, 9].

The dissertation starts in Chapter 2. There, I use parameterized linear combi-
nation of atomic orbitals calculations to determine the stability, optimum geometry
and electronic properties of nanometer-sized capped graphitic cones, called “nano-
horns”. Different nanohorn morphologies are considered, which differ in the relative
location of the five terminating pentagons. Simulated scanning tunneling microscopy
images of the various structures at different bias voltages reflect a net electron trans-
fer towards the pentagon vertex sites. I find that the density of states at the tip,
observable by scanning tunneling spectroscOpy, can be used to discriminate between
different tip structures. My molecular dynamics simulations indicate that disintegra-
tion of nanohorns at high temperatures starts in the highest-strain region near the
tip.

Chapters 3 and 4 deals with a completely different, although related, nanostruc-
ture, the nanopeapod, consisting of fullerenes encapsulated in a nanotube. Using
molecular dynamics calculations, I investigate the absorption of a C60 molecule in a
(10, 10) nanotube either through the open end or a large defect in the tube wall as pos-
sible scenarios for the hierarchical self-assembly of (C60)n@(10, 10) “nanopeapods”.
In view of the complex potential energy surface of this system, the encapsulation
probability depends in a non-trivial manner on the local topology, as well as the
launch angle and velocity of the fullerene. I find the absorption through a defect to
be significantly more efficient than the end-on absorption. This process occurs most
likely within a narrow launch velocity range for the fullerene, which agrees well with
the observed optimum temperature window for peapod formation.

After shedding light on the microscopic formation mechanism of nanopeapods

in Chapter 3, I investigate the equilibrium packing and detailed fusion mechanism

10

of fullerenes inside a carbon nanotube using total energy calculations in Chapter 4.
Fullerenes are pulled in by a “capillary” force, which yields an effective GPa pressure
and improves packing. The (2 + 2) cycloaddition reaction, a necessary precursor for
fusion, may be accelerated inside the nanotube template. Fusion continues along the
minimum energy path as a sequence of Stone-Wales transformations. Based on my
total energy calculations, I found that diamondoids should encapsulate in nanotubes
through self-assembly, and later fuse into one dimensional diamond nanowires.

In Chapters 5 and 6, I switch gears, and deal with the adversity and advantages
of defects in nanotubes. Chapter 5 describes a unique transformation mechanism,
reminiscent of a zipper, which converts scrolls into nanotubes. Using total energy
and structure optimization calculations, I investigate the relative stability and the
conversion mechanism between multi-wall carbon nanotubes and graphitic scrolls. I
suggest that axial segments of a nanotube and a scroll may coexist within the same
tubular nanostructure, separated by a dislocation region. A scroll may convert to the
more stable nanotube by concerted bond-rearrangement rather than bond breaking.
I postulate the zipper-like conversion to proceed very efficiently due to the unusually
low associated activation barrier.

Chapter 6 covers my computer simulations to study the defect tolerance of car-
bon nanotubes under extreme conditions. The response of the system to electronic
excitations is investigated using ab initio molecular dynamics calculations, based on
the time-dependent density functional. N osé—Hoover molecular dynamics simulations
are used to gain insight into the stability of defective nanotubes at high temperatures.
I find that defective nanotubes, exposed to electronic or thermal excitations, show an
unexpected self-healing ability, which is intimately linked to their nanometer size.

In Chapter 7, I explore the feasibility of a new scheme to permanently bond
nanostructures. Combining total energy and molecular dynamics calculations, I ex-

plore the suitability of nanotube-based hooks for bonding. My results indicate that

11

a large force of 3.0 nN is required to disengage two hooks, which are formed by the
insertion of pentagon-heptagon pairs in a (7, 0) carbon nanotube. N anohooks based
on various nanotubes are resilient and keep their structural integrity during the open-
ing process. Arrays of hooks, which are permanently anchored in solid surfaces, are
a nanoscale counterpart of velcro fasteners, forming tough bonds with a capability of
self-repair.

Chapter 8 presents another unusual self-assembled system, 3D polymerized fuller-
enes, that shows unusual mechanical properties. Combining total energy and struc-
ture optimization calculations, I explored new possible crystalline phases of cova-
lently bonded C60 fullerenes and determined their structural, elastic, and electronic
properties. Motivated by reported observations that bulk structures of polymerized
fullerenes may be stiffer than diamond, I have explored possible ways of fullerene
polymerization and have identified 12 stable crystal structures as potential candi-
dates for ”super-hard” carbon. Even though all these phases are very stiff, none of
them exceeds the bulk modulus of diamond. The electronic structure of 3D crystals
of polymerized C60 depends mainly on the packing structure of the system, with only
minor modifications due to the specific inter-fullerene bonding.

Chapter 9 is devoted to the thermal contraction of fullerenes and nanotubes. I
perform molecular dynamics simulations to study shape changes of carbon fullerenes
and nanotubes with increasing temperature. At low temperatures, these systems gain
structural and vibrational entropy by exploring the configuration space at little energy
cost. I find that the soft phonon modes, which couple most strongly to the shape,
maintain the surface area of these hollow nanostructures. In nanotubes, the gain
in entropy translates into a longitudinal contraction, which reaches a maximum at

@800 K. Only at much higher temperatures do the anharmonicities in the vibration
modes cause an overall expansion.

In Chapter 10, I report a quantitative calculation of the thermal conductivity of

12

carbon nanotubes, which was done before any reliable experimental data were avail-
able. Combining equilibrium and non-equilibrium molecular dynamics simulations
with accurate carbon interaction potentials, I determine the thermal conductivity
A of carbon nanotubes and its dependence on temperature. My results suggest an
unusually high value Az6, 600 W/m-K for an isolated (10, 10) nanotube at room tem-
perature, comparable to the thermal conductivity of a hypothetical isolated graphene
monolayer, and exceeding that of isotopically pure diamond. I find that these high
values of A are associated with the large phonon mean free path in these systems;
substantially lower values are predicted and observed for the basal plane of bulk
graphite.

Finally, Chapter 11 deals with the occurrence of a permanent magnetic moment
in carbon nanostructures. Magnetic behavior in selected all-carbon nanostructures
is investigated by ab initio spin density functional theory. Calculational results indi-
cate that particular systems, which are related to schwarzite and contain no under-
coordinated carbon atoms, carry a net magnetic moment in the ground state. I pos-
tulate that in this and other non-alternant aromatic systems with negative Gaussian
curvature, unpaired spins can be introduced by sterically protected carbon radicals.

Some of the figures presented in this dissertation were originally prepared in
color. Only one of those color figures may lose some of the information when they
are printed in greyscale. The figure that requires color reproduction is denoted in the
first line of the caption as ”(Color)”. Other color figures should not lose any essential

information when reproduced in greyscale.

13

Chapter 2

Electronic and Structural

Pr0perties of Carbon Nanohorns

The following discussion of the structural and electronic prOperties of carbon

nanohorns follows that presented in Reference [10].

2. 1 Introduction

Since their first discovery [11], carbon nanotubes have drawn the attention of
both scientists and engineers due to the large number of interesting new phenomena
they exhibit [1, 12, 13, 2], and due to their potential use in nanoscale devices: quan-
tum wires [14], nonlinear electronic elements [15], transistors [16], molecular memory
devices [17], and electron field emitters [18, 19, 20, 21]. Even though nanotubes have
not yet found commercially viable applications, projections indicate that this should
occur in the very near future, with the advent of molecular electronics and further
miniaturization of micro-electromechanical devices (MEMS). Among the most unique
features of nanotubes are their electronic properties. It has been predicted that single-
wall carbon nanotubes [22, 23] can be either metallic or semiconducting, depending

on their diameter and chirality [24, 25, 26]. Recently, the correlation between the chi-

14

rality and conducting behavior of nanotubes has been confirmed by high resolution
scanning tunneling microscopy (STM) studies [27, 28].

Even though these studies have demonstrated that atomic resolution can be
achieved [27, 28, 29], the precise determination of the atomic configuration, charac-
terized by the chiral vector, diameter, distortion, and position of atomic defects, is
still a very difficult task to achieve in nanotubes. Much of the difficulty arises from
the fact that the electronic states at the Fermi level are only indirectly related to the
atomic positions. Theoretical modeling of STM images has been found crucial to cor-
rectly interpret experimental data for graphite [30, 31], and has been recently applied
to carbon nanotubes [32, 33, 34, 35]. As an alternative technique, scanning tunneling
spectroscopy combined with modeling has been used to investigate the effect of the

terminating cap on the electronic structure of nanotubes [36, 37].

2.2 Carbon nanohorns as a new class of carbon
nanostructures

Among the more unusual systems that have been synthesized in the past few
years are cone-shaped graphitic carbon structures [38, 39]. Whereas similar structures
have been observed previously near the end of multi-wall nanotubes [40], it is only
recently that an unusually high production rate of up to 10 g/ h has been achieved
for single-walled cone-shaped structures, called “nanohorns”, using the C02 laser
ablation technique at room temperature in absence of a metal catalyst [41]. These
conical nanohorns have the unique opening angle of 2:20?

I consider a microscopic understanding of the electronic and structural properties
of nanohorns a crucial prerequisite for understanding the role of terminating caps in
the physical behavior of contacts between nanotube-based nano-devices. So far, nei-

ther nanohorns nor other cone-shaped structures have been investigated theoretically.

15

In the following, I study the structural stability of the various tip morphologies, and
the inter-relationship between the atomic arrangement and the electronic structure
at the terminating cap, as well as the disintegration behavior of nanohorns at high

temperatures.

2.3 Electronic and structural properties

2.3.1 Structural properties

Cones can be formed by cutting a wedge from planar graphite and connecting
the exposed edges in a seamless manner, as depicted in Figure 2.1. The Opening angle
of the wedge, called the disclination angle, is n(7r/3), with OSnSG. This disclination
angle is related to the opening angle of the cone by 0 = 23in"1(1 — n/6). Two—
dimensional planar structures (e.g. a graphene sheet) are associated with n = 0, and
one-dimensional cylindrical structures, such as the nanotubes, are described by n = 6.
All other possible graphitic cone structures with 0 < n < 6 have been observed in a
sample generated by pyrolysis of hydrocarbons [39]. According to Euler’s rule, the
terminating cap of a cone with the disclination angle n(7r / 3) contains 72 pentagon(s)
that substitute for the hexagonal rings of planar graphite.

The observed cone opening angle of z20°, corresponding to a 57r / 3 disclination,
implies that all nanohorns contain exactly five pentagons near the tip. I classify the
structure of nanohorns by distinguishing the relative positions of the carbon pentagons
at the apex which determine the morphology of the terminating cap. My study will
focus on the influence of the relative position of these five pentagons on the properties
of nanohorns.

The cap morphologies investigated in this study are presented in Figure 2.2.
Nanohorns with all five pentagons at the “shoulder” of the cone, yielding a blunt tip,

are shown in Figures 2.2(a)—(c). Nanohorns with a pentagon at the apex of the tip,

16

o 0.0mmoooooo
' ‘3“. ‘00" '0
a... 4 c «.0; 9
in 9...‘.‘ .
‘9 39‘

b

.900

o 0
one...»
2.:.:.*.:~2.: . .
.g ”0:." 430:}
13080! 15‘ 309. 4
.0 .0“0;~O:O_“ 03.6.30:
to 0'. c #‘o'o'c—ma‘o'o

 

Figure 2.1: Relation between the morphology of a nanohorn and a graphene sheet.
Nanohorns can be obtained by cutting off the green (light) parts, folding, and con-
necting the exposed edges in a seamless manner. The red (dark) area would form the
tip of the carbon nanohorn. The labels show the carbon nanohorn types A-F, which

are the all possible types taking into account the typical diameter of the nanohorns
with the disclination angle 57r/ 3.

17

 

Figure 2.2: Optimized carbon nanohorn structures with a total disclination angle of
5(7r/3), containing five isolated pentagons at the terminating cap. Structures (a) — (c)
contain all pentagons at the conical “shoulder”, whereas structures (d) — (f) contain
a pentagon at the apex. The pentagons are highlighted by a darker color.

18

surrounded by the other four pentagons at the shoulder, are shown in Figures 2.2(d)-
(f). Note that the cone angle of each nanohorn is z20°, even though the size of the
terminating cap varies with the relative position of the pentagons.

To determine the structural and electronic properties of carbon nanohorns, I
used the parameterized linear combination of atomic orbitals (LCAO) technique with
parameters determined by ab initio calculations for simpler structures [42]. 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 techniquesl. Some of the problems tackled successfully by this technique are
the electronic structure and superconducting properties of the doped C60 solid [43], the
opening of pseudo-gaps near the Fermi level in a (10,10) nanotubes rope [44, 45] and
a (5,5)@(10,10) double-wall nanotube [46], as well as fractional quantum conductance
in nanotubes [47]. This technique, combined with the recursion technique to achieve
an 0(N) scaling, can determine very efficiently the forces on individual atoms [8],
and had previously been used with success to describe the disintegration dynamics
of fullerenes [48], the growth of multi-wall nanotubes [49] and the dynamics of a
“bucky-shuttle” [17].

To investigate the structural stability and electronic properties of carbon nano-
horns, I first optimized the structures with various cap morphologies, shown in Fig-
ure 2.2. For the sake of an easier interpretation of my results, I distinguish the
Ncapz40 — 50 atoms at the terminating cap from those within the cone-shaped man-
tle, that is terminated by Nedge atoms at the other end. I associate the tip region of
a hypothetically infinite nanohorn with all the sites excluding the edge. Structural

details and the results of my stability calculations are presented in Table 2.1. These

 

1 During structure optimization, I keep all sites charge neutral by modifying the diagonal elements
of the Hamiltonian, to suppress structural artifacts near the edge. Once the geometry is optimized,
the on-site energies are kept at the unperturbed values. Due to the absence of a Hubbard term,
this procedure tends to exaggerate the net charge transfer when compared to truly self-consistent
calculations.

19

Quantity (a) (b) (C) (d) (e) (f)

 

Ntot 205 272 296 290 308 217
Ntip 172 233 257 251 270 180
Nedge 33 39 39 39 38 37

<Ecoh,,o,>(eV) —7.28 —7.29 —7.30 -7.30 —7.31 —7.28
<Ecoh,,,,,> (eV) —7.36 —7.36 —7.37 —7.36 —7.37 —7.36
<Ecoh,.dge>(eV) —6.88 —6.88 —6.88 —6.88 —6.87 —6.89
<Ecoh,p.n,> (eV) —7.28 —7.28 —7.28 —7.28 -7.28 —7.28

 

Table 2.1: Structural data and stability results for carbon nanohorn structures (a)—
(f), presented in Figure 2.2. NM = Nap + Nedge is the total number of atoms, which
are subdivided into tip and edge atoms. < Emma” > is the average binding energy,
taken over the entire structure, and < Emma], > the corresponding value excluding the
edge region. < Ecohfidge > is the binding energy of the edge atoms, and < Ecohmem >
is the average over the pentagon sites in each system.
results indicate that atoms in nanohorns are only z0.1 eV less stable than in graphite.
The relative differences in < Emma, > reflect the strain energy changes induced by
the different pentagon arrangements. To minimize the effect of under-coordinated
atoms at the edge on the relative stabilities, I excluded the edge atoms from the aver-
age when calculating < Ecomip >. Since my results for < Ecoh,tip > and < Emma >
follow the same trends, I believe that the effect of edge atoms on the physical prop-
erties can be neglected for structures containing hundreds of atoms. Even though
the total energy differences may appear minute on a per-atom basis, they translate
into few electron-volts when related to the entire structure. My results suggest that
the under-coordinated edge atoms are all less stable than the cone mantle atoms by
z0.5 eV. Also atoms in pentagons are less stable than those in hexagons by z0.1 eV,
resulting in an energy penalty of z0.5 eV to create a pentagon if the strain energy
induced by bending the lattice could be ignored.

When comparing the stabilities of the tip regions, described by < Emmi], >, I
found no large difference between blunt tips that have all the pentagons distributed

along the cylinder mantle and pointed tips containing a pentagon at the apex. I found

the structure shown in Figure 2.2(c) to be more stable than the other blunt structures

20

 

Figure 2.3: Simulated STM images in the tip region for (a) the nanohorn shown in
Figure 2.2(c) and (b) the nanohorn shown in Figure 2.2(d). These results for the
occupied electronic states near the Fermi level, corresponding to the bias voltage
of Vb = 0.2 V, are suggestive of a net electron transfer from the hexagonal to the
pentagonal sites. The charge density contour displayed corresponds to the value of
p = 1.35x10‘3 electrons/A3. Dark lines depict the atomic bonds to guide the eye.

with no pentagon at the apex. Similarly, the structure shown in Figure 2.2(e) is most
stable among the pointed tips containing a pentagon at the apex. Equilibrium carbon-
carbon bond lengths in the cap region are doc = 1.43 — 1.44 A at the pentagonal sites
and doc = 1.39 A at the hexagonal sites, as compared to doc = 1.41 — 1.42 A in the
mantle. This implies that the “single bonds” found in pentagons should be weaker

than the “double bonds” connecting hexagonal sites, thus confirming my results in

Table 2.1 and the analogous behavior in the C60 molecule.

2.3.2 Electronic structure

Since pentagon sites are defects in an all-hexagon structure, they may carry a net
charge [50, 51]. To characterize the nature of the defect states associated with these
sites, I calculated the electronic structure at the tip of the nanohorns. The charge
density associated with states near E p, corresponding to the local density of states at

that particular position and energy, is proportional to the current observed in STM

21

experiments. To compute the local charge density associated with a given eigenstate,
I projected that state onto a local atomic basis. The projection coefficients were
used in conjunction with real-space atomic wave functions from density functional
calculations [31] to determine the charge density corresponding to a particular level
or the total charge density. To mimic a large structure, I convoluted the discrete level
spectrum by a Gaussian with a full-width at half-maximum of 0.3 eV. Using this
convoluted spectrum, I also determined the charge density associated with particular
energy intervals corresponding to STM data for a given bias voltage.

In Figure 2.3, I present such simulated STM images for the nanohorns represented
in Figures 2.2(c) and 2.2(d). I show the charge density associated with occupied states
within a narrow energy interval of 0.2 eV below the Fermi level2 as three-dimensional
charge density contours, for the density value of p = 1.35 x 10"3 electrons/A3. Very
similar results to those presented in Figure 2.3 were obtained at a higher bias voltage
of 0.4 eV. As seen in Figure 2.3, ppvr interactions dominate the spectrum near Ep.
These images also show a net excess of electrons on the pentagonal sites as compared
to the hexagonal sites. This extra negative charge at the apex should make pointed
nanohorn structures with a pentagon at the apex better candidates for field emitters
[18, 19, 20, 21] than structures with no pentagon at the apex and a relatively blunt
tip.

It has been shown previously that theoretical modeling of STM images is essential
for the correct interpretation of experimental data. Atomically resolved STM images,
however, are very hard to obtain especially near the terminating caps of tubes [36,
37] and cones, due to the large surface curvature that can not be probed efficiently

using current cone-shaped STM tips. A better way to identify the tip structure may

 

2The nanohorn structures considered in my calculation are clusters containing edge atoms. I
found that due to the electron transfer towards the under-coordinated atoms at the edge, the “Fermi
level” of these clusters may drop by as much as z1.0 - 1.2 eV with respect to that of a graphene
monolayer. To compensate for this artifact, I excluded the edge region from the calculated density
of states and found the Ep value to lie very close to that of a graphene monolayer.

22

 

 

 

 

DOS [a.u.]

 

 

 

 

 

 

 

 

 

E [eV] E [6V]

Figure 2.4: Electronic densities of states, normalized per atom and shown in consistent
arbitrary units, for the terminating cap of the tip (solid line) and for the pentagon
sites only (dashed line) of nanohorn structures shown in Figure 2.2. The similarity
between the results for the entire cap and the pentagon sites necessitated a vertical
oflset of one set of data for an easy distinction, thus indicating that variations in the
arrangement of pentagons affect the densities of states not only at the pentagonal
sites, but within the entire cap region. The most marked differences are noted near
the Fermi level, at E30 eV.

23

consist of scanning tunneling spectroscopy (STS) measurements in the vicinity of the
tip. This approach is based on the fact that in STS experiments, the normalized
conductance (V/I)(dI/dV) is proportional to the density of states which, in turn,
is structure sensitive. I have calculated the density of states at the terminating cap
for the different nanohorn structures shown in Figure 2.2. My results are shown in
Figure 2.4, convoluted using a Gaussian with a full-width at half-maximum of 0.3 eV.

To investigate the efl'ect of pentagonal sites on the electronic structure at the
tip, I first calculated the density of states only at the 25 atoms contained in the five
terminating pentagons. The corresponding densities of states, shown by the dashed
curve in Figure 2.4, are found to vary significantly from structure to structure near
the Fermi level. Thus, a comparison between the densities of states at EzEF should
offer a new way to discriminate between the various tip morphologies. For an easy
comparison with experiments, I also calculated the density of states in the entire
terminating cap, including all five pentagons and consisting of Neapz40 — 50 atoms,
depending on the structure. The corresponding density of states, given by the solid
line in Figure 2.4, is vertically displaced for easier comparison. My results show that
the densities of states, both normalized per atom, are very similar. Thus, I conclude
that the pentagonal sites determine all essential features of the electronic structure
near the Fermi level at the tip. I note that a perturbation of the electronic structure

far from defects has also been predicted in nanotubes [52].

2.4 Thermal stability

Next, I have studied the heat resilience [41] as well as the decay mechanism of
nanohorns at extremely high temperatures using molecular dynamics simulations [53].
In my canonical molecular dynamics simulations, I keep the structure at a constant

temperature using a Nose-Hoover thermostat [54, 55], and use a fifth—order Runge—

24

Kutta interpolation scheme to integrate the equations of motion, with a time step of
At = 5 x 10‘16 s. I found the system to remain structurally intact within the tem-
perature range from T = 2, 000 — 4, 000 K. Then, I heated up the system gradually
from T = 4, 000 K to 5, 000 K within 4,000 time steps, corresponding to a time inter-
val of 2 ps. My molecular simulations show that nanohorn structures are extremely
heat resilient up to Tz4, 500 K. At higher temperatures, I find these structures to
disintegrate preferentially in the vicinity of the pentagon sites. A simultaneous dis-
integration of the nanohorn structures at the exposed edge, which also occurs in my
simulations, is ignored as an artifact of finite-size systems. The preferential disinte-
gration in the higher strain region near the pentagon sites, associated with a large
local curvature, is one reason for the observation that nanohorn tips are opened easily

at high temperatures, in presence of oxygen [41].

2.5 Summary

In summary, I used parameterized linear combination of atomic orbitals calcu-
lations to determine the stability, optimum geometry and electronic properties of
nanometer-sized capped graphitic cones, called nanohorns. I considered different
nanohorn morphologies that differ in the relative location of the five terminating
pentagons. I found a net electron transfer to the pentagonal sites of the cap. This
negative excess charge is seen in simulated scanning tunneling microscopy images of
the various structures at different bias voltages. I found that the density of states at
the tip, observable by scanning tunneling spectroscopy, can be used to discriminate
between different tip structures. My molecular dynamics simulations indicate that
disintegration of nanohorns at high temperatures starts in the highest-strain region

near the tip.

25

Chapter 3

Microscopic Formation Mechanism

of Carbon Nanopeapods

The following discussion of possible microscopic formation mechanisms of carbon

nanopeapods follows that of Reference [56].

3.1 Carbon nanopeapods: A new type of hybrid
carbon materials

A new type of self-assembled hybrid structures called “nanopeapods,” consisting
of fullerene arrays inside single-wall carbon nanotubes, have recently been reported
[57, 58, 59, 60, 61, 62]. Potential uses of nanopeapods range from nanometer-sized con-
tainers for chemical reactions [60] to data storage [17], and possibly high-temperature
superconductors [63]. Even though the encapsulation of fullerenes (such as C60) in
nanotubes is favorable on energetic grounds [17 , 64, 65] and occurs rapidly by expos-
ing nanotubes to sublimed fullerenes [57 , 58, 59, 60, 61], virtually nothing is known

about how the fullerenes enter the nanotubes when forming the nanopeapods.

26

3.2 Possible formation mechanisms of
nanopeapods

A statistical mechanics approach, which considers only the initial and final state,
describes only the equilibrium state after an infinite time [65]. It can say nothing
about the likelihood of fullerene encapsulation in a finite time, as it ignores any
activation barriers along the reaction path. Only trajectory calculations based on the
complex potential energy surface of the system can provide detailed answers about
the dynamics and intermediate steps involved in the formation of nanopeapods.

An important current controversy involves the question whether fullerenes enter
through the open ends of perfect nanotubes or whether defects in the wall play an
important role as entry channels. The extremely high filling ratio of up to 100% [61]
appears hard to achieve even within few days if the only entry channels are the two
open ends of each tube. In such a case, the fullerene kinetics is limited and even
a single impurity may block an entry channel. On the other hand, only few open-
ings in the tube walls have been observed in High-Resolution Transmission Electron
Microscopy (HRTEM) images [66]. Another important question, which a statistical
approach can not answer, is whether the encapsulation process, be it through an open
end or through a defect, occurs directly from the gas phase, or involves a diffusive
motion along the wall.

Here, I present a large-scale simulation for the entry of a C60 molecule into a
(10,10) carbon nanotube via different scenarios, illustrated in Fig. 3.1. I consider
the encapsulation through the open end of a free-standing nanotube [Fig. 3.1(a)]
and through the open end of a nanotube bundle [Fig. 3.1(b)]. These scenarios are
compared to the C60 entry through a defect, in my model a large Opening in the
nanotube wall [Fig. 3.1(c)]. I propose that what has been interpreted as harmless

“amorphous overcoating” of perfect single-wall nanotubes in HRTEM images may

27

indeed represent defective nanotube segments with openings large enough to let a
fullerene pass through. Formation of such large defects may occur, I believe, during
the harsh purification process that is a necessary prerequisite for peapod formation
[60]. By assuming that the C60 adsorbs on the wall prior to encapsulation, I take
advantage of the enhanced probability of the fullerene to find an entry channel into
a nanotube within the lower-dimensional phase space probed by the C60 diffusing
along the wall, as compared to a fullerene in the gas phase. My calculations elucidate
not only which of these processes is most likely, but also the optimum temperature
window for the peapod formation.

I calculate the potential energy of the fullerene-nanotube system using an elec-
tronic Hamiltonian that had been used successfully to describe the formation of multi-
wall nanotubes [49] and the dynamics of a “bucky-shuttle” device [17]. This approach
describes accurately not only the covalent bonding within the graphitic substructures,
but also the modification of the fullerene-nanotube interaction due to the inter-wall
interaction and possibly the presence of unsaturated bonds at defect sites or tube
edges. I find the use of an electronic Hamiltonian to be required in this system, as
analytical bond-order potentials do not describe the rehybridization at defect sites
and tube ends with a sufficient precision. Once the quantum nature of the interatomic
interactions has been taken into account, the trajectories of the heavy carbon atoms
are described reliably in classical manner.

In Fig. 3.1 I present schematic views of the encapsulation scenarios together with
the calculated potential energy surfaces in the 1: — y plane that contains the tube axes,
the center of the C60 molecule and the defect. The tube structures underlying the
potential energy surfaces have been globally optimized. They reflect the relaxation at
the Open end of the tube and the edges of a large defect in the tube wall, representing
an extended vacancy formed during the harsh purification process. I consider a defect

formed by removing 76 atoms from the wall, with a diameter of @136 A, just large

28

 

 

-15 -10 -5 0 5 10 15

 

 

y [A]

 

 

 

 

.15 l
.15 -10 -5 o 5 10 15
xIA]

Figure 3.1: Perspective views (left panels) and potential energy surfaces (right pan-
els) for the encapsulation of a C60 molecule in a (10,10) nanotube under different
scenarios. Dotted lines indicate possible C60 trajectories, characterized by the launch
velocity v and launch angle 0 with respect to the tube axis. The potential energy
surfaces are shown in the z — y plane containing the center of the fullerene and the
tube axis 2:. Adjacent energy contours are separated by 0.01 eV; the position of the
tube walls is indicated by the solid gray areas. The scenarios involve Ceo encapsu—
lation through (a) the open end of an isolated (10,10) nanotube, (b) the open end
of bundled (10,10) nanotubes, and (c) a large opening in the (10,10) nanotube wall,
centered in the x — y plane.

29

enough to permit an unhindered entry of a C60 molecule.

When following an “optimum” trajectory starting outside the tube, a C50 mole-
cule may first physisorb on the outer wall, thus gaining @007 eV. My energy contour
plots in Fig. 3.1 suggest that the fullerene diffuses virtually freely along the tube wall.
Since the adsorption potential is relatively shallow, the fullerene’s angular velocity
along the perimeter is limited by requiring the centrifugal force not to exceed a critical
value for radial detachment. Consequently, the fullerene should follow a near-axial
helical trajectory within a narrow 1D channel in the a: — y plane during its diffusive
motion along the tube surface. Due to the large inertia of the C50 molecule and the
absence of corrugations along the potential energy surface, virtually no rotations are
excited as the fullerene slides along the tube wall. Upon entering the nanotube, the
C60 gains an additional 0.36 eV [67].

The absence of activation barriers for all fullerene encapsulation scenarios de-
picted in Fig. 3.1 is surprising for two reasons. First, it suggests that the strong
attraction to the undercoordinated atoms at the tube or defect edges does not trap
the fullerene, but only modifies the entry channel without hindering the encapsula-
tionl. Second, it leaves the origin of the observed optimum temperature of z400°C
for C50 encapsulation an open question.

The likelihood of C50 encapsulation depends on its initial conditions and the
topology of the entire potential energy surface, as suggested by the sample trajec-
tories in Fig. 3.1. In my molecular dynamics simulations, I initially placed the C60
molecule at equilibrium radial distance from the tube. I then calculated C60 trajec-
tories in the a; — y plane using time steps of 5x10‘16 3, by varying the launch speed 2)
from 0 - 300 m/s in 1 m/s increments and the launch angle 0 with respect to the tube

axis from —90° to +90° in 1° increments. The calculation of all 54, 000 trajectories

 

1Due to their enhanced reactivity, unsaturated edges form a stronger bond with the fullerenes
than edges saturated with hydrogen or other elements. It is conceivable that fullerenes could get
trapped near vacancies that are smaller or comparable in size with the fullerene.

30

for each encapsulation pathway would not be computationally feasible without ap-
proximations. I described the fullerene-nanotube interaction by the potential energy
surfaces presented in Fig. 3.1, thus implying that both the C60 and the nanotube are
structurally rigid.

For each set of initial conditions, I found that the ultimate outcome of each
trajectory regarding encapsulation is decided within less than 106 time steps. I also
found that all physically interesting phenomena occur within a velocity range between
50 — 200 m / s, the lower bound marking the onset of C50 mobility along the tube. The
upper end of this range corresponds to high temperatures, at which fragmentation
and tube fusion are to be expected. Consequently, I will discuss the encapsulation

under the specific scenarios within this velocity range.

3.3 Role of defects in the formation pathway of
nanopeapods

The outcome of the encapsulation process through the end of an isolated nano-
tube, depicted in Fig. 3.1(a), is shown in Fig. 3.2. The scatter diagram of Fig. 3.2(a)
condenses the information of all trajectories regarding encapsulation. A black mark
denotes a trajectory with a launch velocity and angle that leads to encapsulation,
such as that shown in Fig. 3.2(b). Similarly, a white field marks failed encapsulation,
such as depicted in the trajectory of Fig. 3.2(c). Consequently, the fraction of the
dark area is proportional to the probability of encapsulation. The scatter diagram,
dominated by white color, suggests that encapsulation through the open end of an
isolated nanotube should be very rare. The main reason is the relatively narrow and
shallow channel in the potential energy surface of Fig. 3.1(a), which the fullerene
must follow to reach the inside of the tube. At very low velocities, the fullerene gets

trapped in one of the shallow potential energy wells outside the tube wall. At too

31

 

 

O
(a) 80 I '7 V V 7 I T—" V Y I I V I V '

(b)

 

 

 

40° - = ._ .
. '3‘ . .%.:.- 1
,, . ‘4‘, .
i
CD 0° - .
x, ' l
‘E .
.. ‘ 1|" l
.40" L ' 4

 

 

 

-30°’ RUN-..
50 100 150 200

v [m/s]

 

 

Iii I

 

Figure 3.2: (a) Scatter diagram showing the probable outcome of a C50 encapsulation
through the open end of an isolated (10,10) nanotube, as a function of the fullerene
launch velocity v and launch angle 0. Initial conditions leading to a successful encap-
sulation are marked in black, those which do not lead to encapsulation are marked in
white in the scatter diagram. A sample successful encapsulation trajectory is depicted
in (b), an unsuccessful trajectory in (c).

high velocities, on the other hand, the C50 is incapable of following the tight “U-turn”
in the potential energy surface, as illustrated in Fig. 3.2(c).

In view of the low estimated probability of encapsulation through the end of
an isolated tube, I considered next the analogous process at the end of a nanotube
bundle, consisting of (10,10) tubes with an inter-wall separation of 3.4 A, depicted
in Fig. 3.1(b). As suggested by the two successful trajectories in the left panel of
Fig. 3.1(b), the more complex potential energy surface for this system increases the
number of entry channels, thus enhancing the probability of encapsulation. The
results of my trajectory calculations, summarized in the scatter diagram of Fig. 3.3(a),

indeed show a somewhat higher encapsulation probability as compared to the single-

32

 

4;
=2

 

 

 

 

50 100 150 200
v [m/s]

 

Figure 3.3: (a) Scatter diagram showing the probable outcome of a C60 encapsulation
through the open end of bundled (10,10) nanotubes, as a function of the fullerene
launch velocity v and launch angle 9 . Initial conditions leading to a successful
encapsulation are marked in black, those which do not lead to encapsulation are
marked in white in the scatter diagram. Sample trajectories, describing successful
encapsulation in two different tubes, are depicted in (b) and (c).

tube scenario. In comparison to Fig. 3.2(a), I observe two solid dark regions for
velocities beyond z150 m/s and 0 % i40°. These regions correspond to trajectories
that first follow the wall of one tube and continue inside the neighboring tube, as
shown in Fig. 3.2(b). This new channel is relatively narrow due to the “S”—shape
of the tube-fullerene interaction potential well, shown in the right-hand panel of
Fig. 3.1(b). As in the case of an isolated tube, trajectories involving a “U-turn”,
depicted in Fig. 3.3(c), form an additional entry channel, responsible for the rare
encapsulation events in the remaining region of the scatter diagram. I conclude that
even in the case of a bundle, the probability of encapsulation through the end is rather

low.

33

 

 

3|

 

 

 

I
I

A
O
V

 

 

 

 

 

 

|
J

 

50 100 150 200
v [m/s]

F

 

 

 

Figure 3.4: (Color) (a) Scatter diagram showing the probable outcome of a C60 en-
capsulation through a large opening in the wall of a (10, 10) nanotube, as a function
of the fullerene launch velocity v and launch angle 0. Events are color coded ac-
cording to the final state. Conditions that do not lead to encapsulation are marked
in white. Conditions leading to encapsulation, with a sample trajectory shown in
(b), are marked in red. Initial conditions and corresponding trajectories involving a
temporary entry of the fullerene into the tube, followed by an escape after multiple
scattering events, are shown in green if the final velocity has a negative ar-component
(c) and in blue if the final velocity has a positive z-component (d).

The outcome of the encapsulation process through a large opening in the side
wall of the (10,10) nanotube, depicted in Fig. 3.1(c), is shown as a scatter diagram
in Fig. 3.4(a). In comparison to Figs. 3.2(a) and 3.3(a), the dark area in the scatter
diagram of Fig. 3.4(a) is substantially larger, yielding a 30-fold increase of the cross-
section for the C60 entry. In the following, I focus on the encapsulation through the
large wall opening as the most likely process leading to peapod formation.

The scatter diagram of Fig. 3.4(a) shows a complex pattern consisting of large

areas corresponding to the C60 entering the tube. Among these events, I have marked

34

those trajectories leading to a definitive encapsulation, as depicted in Fig. 3.4(b), in
red. The “green” and “blue” trajectories, shown in Figs. 3.4(c) and (d), involve
a temporary entry of the C60 molecule into the tube. In that case, after multiple
scattering events, the fullerene may eventually escape in the backward (“green event”)
or the forward direction (“blue event”). Since such multiple scattering processes are
chaotic, the “blue” and “green” events are homogeneously intermixed rather than
separated into specific regions in Fig. 3.4(a).

To interpret the complex pattern in the scatter diagram of Fig. 3.4(a), I took a
closer look at the individual trajectories. For launch velocities below 50 m/s, the C50
becomes trapped in one of the local minima and never reaches the defect. The highest
probability of a successful encapsulation with an axial launch (0 = 0°) is achieved at
velocities ranging from 50 m/ 35122120 m/s. Axial launches with velocities exceeding
@120 m/s overshoot the hole. C60 encapsulation is still possible for fast launches,
but requires large launch angles and involves one or several radial reflections off the
wall, as shown schematically in Fig. 3.1(c). For a given nonzero launch angle, the
largest launch velocity allowing the fullerene to enter the tube involves a straight
trajectory with no radial reflections. This is the case for the trajectories depicted in
Figs. 3.4(b)-(d), for launch angles 0%10° and v§200 m/s. At lower launch velocities,
a radial reflection from the tube wall causes the fullerene to overshoot the hole. As
the velocity is reduced further, the point of first radial reflection is shifted closer to
the launch site, allowing the fullerene to successfully enter the nanotube again. At a
constant launch angle such as 0 = 40°, I can distinguish up to seven alternating regions
associated with “successful” and “unsuccessful” trajectories. Each of these contiguous
regions in Fig. 3.4(a) can be characterized by the number of radial reflections between
the launch site and the hole edge.

Maybe my most important result is the finding that, in absence of inelastic

scattering, the probability of successful encapsulation (“red” events) is largest in the

35

velocity range between 80 -— 120 m/s and almost independent of the launch angle 0.
This suggests that not the launch direction, but rather the kinetic energy of the Can
molecule decides the outcome. In view of the restricted motion of the C50 molecule
along the tube surface, which reduces the number of degrees of freedom, I find that
this velocity window translates into a kinetic temperature of the C60 close to 400°C.

It is remarkable that the optimum temperature for peapod formation, as pre-
dicted by my calculation, falls into the observed temperature range from 350 - 450°C
[60]. Previous explanations of this optimum temperature window were only based
on plausibility arguments suggesting a low C60 mobility at low temperatures and a
closure of the wall defects or fullerene fusion [68] under annealing conditions. I be-
lieve that the quantitative agreement between theory and observation also provides
an indirect evidence that the microscopic encapsulation mechanism involves defects
in the tube wall.

My calculations discussed so far have been based on the assumption that the
microscopic degrees of freedom are decoupled from the macroscopic motion of the
C60 molecule. Effectively, the possibility of energy transfer between the macroscopic
and microscopic degrees of freedom introduces a damping mechanism. This new
inelastic channel could convert some of the “green” and “blue” trajectories, depicted
in Fig. 3.4, into successful encapsulation events, thus further increasing the cross-
section for C60 encapsulation.

To investigate additional phenomena that may occur when none of the micro-
scopic degrees of freedom are frozen, I performed a molecular dynamics simulation
of the fullerene-tube interaction in a canonical simulation at synthesis temperature.
Considering the motion of all the atoms carries a large computational overhead, since
calculating the electronic structure self-consistently at each time step slows down the
simulation by five orders of magnitude. An important structural change I observed

at elevated temperatures was the occurrence of quadrupolar tube deformations near

36

the hole due to the reduced structural rigidity at that site. This, together with a
strong tube bending mode that reduces the size of the entry channel, decreases the

probability of C60 capture.

3.4 Summary

In summary, I investigated the microscopic mechanism of (C60)n@(10, 10) peapod
formation by considering the entry of the fullerene into the tube either through an
extended vacancy in the wall or the open end. My total energy calculations indicate
that presence of dangling bonds at the unsaturated edge of the uncapped end or of a
large enough vacancy increases the fullerene-nanotube interaction, thus promoting —
rather than hindering — the fullerene entry into the tube. Using these potential energy
surfaces, the likelihood of a C60 encapsulation was investigated for scenarios involving
end-on encapsulation in isolated and bundled tubes, and for encapsulation through
large defects in the wall. Extensive molecular dynamics simulations, performed under
these scenarios as a function of the C60 launch velocity and angle with respect to
the tube axis, indicate that the most efficient peapod formation mechanism involves
diffusive motion along the wall and encapsulation through a large opening in tube
wall. The C60 encapsulation occurs most likely within a narrow range of launch
velocities that agrees well with the experimentally observed temperature window.
The efficiency of the filling process through openings in the tube wall is expected
to improve with increasing concentration of defects up to the point where, due to
a significant fraction of atoms missing or displaced, the tubes lose their structural

integrity.

37

Chapter 4

Energetics of Encapsulation and
Fusion of Fullerenes in

Nanopeapods

The following discussion of the encapsulation and fusion of fullerenes inside car-
bon nanotubes, namely fusion of fullerenes inside carbon nanopeapods, follows that
presented in References [69, 70]. In analogy to fullerene fusion, I also comment on
the possibility of diamondoid fusion inside nanotubes based on ab initio total energy

calculations reported in Reference [71].

4.1 Unusual properties of carbon nanopeapods

The first observation of the hierarchical self-assembly of C60 molecules and single-
wall carbon nanotubes (SWNTs) to peapods by high-resolution transmission electron
microscopy (HRTEM) [57] was followed by reports of encapsulating other fullerenes
and metallofullerenes in single-walled carbon nanotubes [72, 61, 73, 74, 75, 76, 77].

Several intriguing observations regarding peapods await explanation. For one,

high-resolution transmission electron microscopy images [61, 73, 74, 75, 76, 78], elec-

38

tron diffraction [77] and Raman spectra [79] suggest the observed equilibrium spacing
between fullerenes in peapods to be smaller by 3 — 4% than in three-dimensional
crystals, but larger than in polymerized solids. Also, in contrast to 2-D and 3—D
crystals [80, 81], where polymerization of fullerenes does not easily occur even under
high pressures and temperatures, fullerenes fuse rather readily to long capsules when
encapsulated in nanotubes [57, 68, 60]. For fusion to be enabled at 1100°C, signifi-
cantly below the decomposition temperature of fullerenes [48] close to 4, 000°C, the
enclosing nanotube must play a crucial role. Here I investigate the packing of fuller-
enes in various nanotubes and determine the optimum energy path for thermal fusion
of fullerenes, including the identification of activation barriers and transition-state

geometries.

4.2 Energetics of fullerenes in nanopeapods

Figure 4.1 (a) shows schematically the static force associated with the energy gain
during the encapsulation of a fullerene in a nanotube. To decide which fullerene may
enter into which nanotube, in Figure 4.1 (b) I plot the encapsulation energy of C60
and C34 in single—wall nanotubes with radii RNT in the range 0.6 nmSRNTSOB run.
As expected, encapsulation is favored only in nanotubes that are sufficiently wide.
On the other hand, the encapsulation energy in very wide tubes should be smaller
and comparable to the adsorption energy on graphite. These results are in general
agreement with reported results based on an alternative potential [65, 82, 83]. My
results suggest that each fullerene of radius RF shows a preference for nanotubes
with RNTzRF + 0.3 nm. For this snug fit, encapsulation results in an energy gain of
zOA eV for C60@(10, 10) and Cg4@(11, 11) due to the attractive interaction between
the tube wall and the fullerene. This value agrees with ab initio calculations [67],

but is lower than empirical fits to experimental data [65, 82, 83, 84]. I found that

39

(b) (C)

 

 

 

 

 

 

 

“fl.é A2 15 0'4
_1.0 92':
-—-0.5 .2. '
<1 00 L502
_0.5 _ 4 . . . -0.4
0.65 0.70 0.75 0.80 1.0 0.0 -1.0
RNTlnm] 2 [nm]

Figure 4.1: Energetics of fullerene encapsulation. (a) During the insertion into a nano-
tube, a fullerene is pulled in by a “capillary” force F, which is linked to the energy gain
upon axial displacement Az. (b) The encapsulation energy AE of C50 (0) and C84 (0)
in single-wall carbon nanotubes with radii RNT ranging from 0.6 nngNTSOB nm.
Best fitting fullerene/nanotube pairs show an encapsulation energy of z0.4 eV. (c)
Energy change during the fullerene insertion process along the tube axis 2, with z = 0
at the tube end. C60 and C84 are pulled into the best fitting (10, 10) and (11, 11) nano-
tubes by Fz0.1 nN, reflected in a constant constant slope of AE/Az near the tube
end. The high energy cost prevents the entry of the C84 fullerene into the narrow
(10,10) nanotube.
the encapsulation energy of fullerenes is represented well in a continuum approxima-
tion [85], shown by the solid lines in Figure 4.1 (b), by modeling fullerenes as spheres
inside atomically smooth nanotubes. No special preference regarding tube chirality
was observed, suggesting the possibility to use peapod formation as a tool to separate
tubes by diameter.

In Figure 4.1 (c), I display the energy associated with the insertion of C60 and
C34 in the (10,10) and (11,11) nanotubes. For the energetically most favorable pairs,
open-end insertion is associated with a “capillary” force F = AE/AzzOJ nN near
the tube end at 2:0. As shown in Figure 4.1 (c), insertion of the large C84 fullerene
in the narrower (10,10) nanotube is energetically unfavorable.

The static capillary force F z0.1 nN, when divided by the cross-section area of

the enclosing nanotube, yields a static pressure in the sub-GPa range. At nonzero

temperature, fullerenes packed inside a peapod are also subject to a dynamical pres-

40

sure, resulting from collisions with neighboring fullerenes. In the temperature range
close to 1,000 K, when fusion occurs [68], collisions between thermally equilibrated
fullerenes lead to forces close to 1 nN, representing accordingly higher nominal pres-
sures in the GPa range. This nominal pressure is found to be a very important factor
in determining the equilibrium structure of nanopeapods [70], and could account for
the distance reduction between fullerenes when they form one dimensional crystals

inside nanotubes [61, 77].

4.3 Chemical reactions inside carbon nanotubes

The postulated high nominal pressure inside nanotubes suggests a possible use
of nanotubes as nanoscale autoclaves to promote chemical reactions. For example,
HRTEM observations indicate that under electron irradiation [57] or at elevated tem-
peratures [68, 60], fullerenes fuse more readily inside a nanotube than in 2 D and
3 D crystals [80, 81]. In addition to fullerenes, carbon nanotubes could also encapsu-
late other species such as diamondoid molecules [5] to form one dimensional diamond
wires inside carbon nanotubes by taking advantage of the physical confinement pro-
vided by nanotube template, and a high nominal pressure [71]. Ab initio calculations
indicate that diamondoids gain energy after encapsulation inside a nanotube. Similar
to fullerenes, each diamondoid molecule has a preferred diameter range for efficient
encapsulation. I found that doped diamondoid molecules should fuse into one dimen-
sional nanometer sized diamond whiskers more easily inside a nanotube than in free

space.

4.3.1 Fusion of fullerenes in carbon nanopeapods

Large-scale structural changes, including fusion, can be achieved by a finite se-

quence of generalized Stone-Wales transformations, which involve only bond rotations

41

and avoid bond breaking. Using a graphical search program, I determine the opti-
mum reaction pathway for thermal fusion of fullerenes. Search of the phase space
by the ‘string method’ provides detailed information about the optimum pathway,
including the identification of activation barriers and transition-state geometries. I
find the fusion process to be exothermic. The fusion dynamics is fast in spite of the
formidable total activation barrier close to 5 eV, associated with each Stone-Wales
transformation, since bond rotations turn out to be multi-step processes with lower
individual activation barriers.

The fusion of two C60 molecules to 3 C120 capsule, which has been observed in
peapods [60, 68], is driven by the energy gain associated with reducing the local cur-
vature in the system. Still, this reaction involves a large-scale morphological change
and will only occur if the required activation barrier is small.

It appears that the most likely fusion path may involve a sequence of bond rota-
tions, called generalized Stone-Wales (GSW) transformations. GSW transformations
are known to require much lower activation energies than processes involving bond
breaking, and have been studied extensively in 3p2 bonded carbon structures [86].
Optimum pathway for the 2C60—)C120 fusion reaction, involving the smallest number
of generalized Stone-Wales bond rotations is depicted in Fig. 4.2.

Close inspection of the reaction energy along the contiguous optimum fusion
path in Fig. 4.3 indicates a sequence of 23 activated processes connecting the 24
metastable states. I find the activation energy barriers AEGSWz5 eV of these GSW
transformations to be significantly lower than in graphite, as expected for Stone-
Wales processes in non-planar structures due to the deviation from spz-bonding. In
presence of extra carbon atoms, the activation barriers for GSW transformations
may be lowered further to below 4 eV by autocatalytic reactions [87, 88]. Also, under
electron irradiation, this process can proceed relatively fast in view of the high rate

of sub-threshold energy transfer to the structure [89]. In extended fullerene systems,

42

 

@4233 ”firm ”5% 9

iii ““3 “5‘3 “1%? 4%

20

Figure 4.2: Optimum pathway for the 2C60—+C120 fusion reaction, involving the small-
est number of generalized Stone-Wales bond rotations, determined by a graphical
search of all possible bond rotation sequences. Polygons other than hexagons are
emphasized by color and shading.

moreover, the energy release during the fusion process should heat up the structure
locally, thus further promoting activated processes in the local vicinity.

In spite of its lower activation barrier in comparison to the GSW steps, the
initial (2 + 2) cycloaddition reaction between the structures “0” and “1” may play
an important role, and possibly even limit the rate of the fusion process. Fusion
can only be initiated in the optimum geometry, where two double bonds in adjacent
fullerenes face each other at the contact point. The probability of this configuration
will multiply the attempt frequency V of the 0—->1 reaction in the Arrhenius formula1
and thus reduce the reaction rate, since the low activation barrier of m0] eV, only
applies to attempts with the optimum fullerene orientation.

To estimate the probability of the configuration required for the (2 + 2) cycload-

dition to occur, I first consider the phase space describing the motion of two rigid

 

1 The Arrhenius formula determines the reaction rate as v exp (-AE/k3T), where u is the attempt
frequency, AE the activation barrier, k3 the Boltzmann constant, and T the temperature.

43

AE[eV]

Figure 4.3: Energetics of the fullerene fusion in peapods. (a) Energy change along the
optimum reaction path, given by the solid line. Energy results for the 25 intermediate
structures, shown in Fig. 4.2, based on my total energy functional (0), are compared
to ab initio Density Functional results (+). The contiguous minimum energy path in

 

10 Illl

lllllllllrllllll

 

 

 

1O 15
reaction coordinate

configurational space was identified using a ‘string’ technique.

44

fullerenes at constant equilibrium distance (structure “0” in Fig. 4.2), which are freely
rotating in space. The 8-dimensional configurational space, spanned by the three Eu-
ler angles defining the orientation of each fullerene and the two-dimensional vector
defining the orientation of the inter-fullerene connection, is explored uniformly by the
rotating fullerenes. Next, I assume that the difference between a “correct” and an “in-
correct” fullerene alignment corresponds to a misorientation exceeding A902,? in any
dimension, which naturally introduces a grain size for the discretized configurational
space.

In view of the fact that each fullerene has thirty double bonds, each of which
can have two orientations, 3,600 out of 3x1019 cells in this space represent favorable
configurations. Assuming that the configurational space exploration by the freely ro-
tating fullerenes occurs at random in-between two cycloaddition attempts, separated
by the period of the inter-fullerene vibration, the probability of finding an optimum
configuration is @10‘14. Using 11 = 7x1012 Hz for the inter-fullerene vibration fre-
quency [85], the (2 + 2) cycloaddition step with an activation barrier AE = 0.725 eV
should occur on the time scale of one week at 1,100°C, significantly longer than the
time frame of a GSW transformation. Thus, this step should be rate limiting in a
close-packed three-dimensional C60 system, which — while molten at this temperature
— could be prevented from evaporation by external pressure.

Restricting the configurational space to one dimension, which occurs when chains
of fullerenes are packed in peapods, increases the fusion probability substantially. The
crucial role played by the enclosing nanotube is to keep adjacent fullerenes in place
long enough for them to probe the configurational space at close range. Since the
vector connecting adjacent fullerenes coincides with the nanotube axis, the possibility
of non-central collisions is eliminated, the dimensionality of the configurational space
is reduced to six, and the number of discrete cells to only 5x10”. This increases

the probability of the optimum fullerene orientation by five orders of magnitude, and

45

reduces the reaction time of the (2 + 2) cycloaddition step to only 7 s at 1,100°C. I
conclude that fusion should occur more easily, when fullerenes are packed in peapods,
than in three-dimensional bulk C60.

In a three-dimensional C60 system, the fusion rate should further be reduced
due to the fact that more than one GSW transformation involving the same fuller-
ene may occur simultaneously. Each C60 molecule has initially the ability to form
at least four initial connections with neighboring fullerenes by the (2 + 2) cycloaddi-
tion reaction [81]. Considering the finite size of the C60 molecule, GSW transforma-
tions associated with one fusion reaction are likely to interfere with transformation
necessary for a separate fusion reaction, occurring concurrently. Due to resulting
frustration, the activation barriers of individual GSW transformations could increase
significantly, possibly even stopping the fusion. Since this effect is less severe in lower
dimensions, the reduction of the overall fusion rate associated with concurring binary

fusion reactions should be much less important in one-dimensional peapods than in

bulk C60.

4.4 Summary

In summary, I studied the energetics and the fusion mechanism of fullerenes en-
capsulated in nanotubes. I found that insertion of a fullerene inside an optimum
nanotube host is associated with an energy gain of zOA eV. The “capillary” force
produced by the entering fullerene gives rise to a static pressure inside the nanotube.
This pressure is expected to increase due to collisions between fullerenes and reach
GPa magnitude under experimental conditions. I investigated the energetically most
favorable pathway for fusion of two isolated C60 molecules inside a (10, 10) nanotube.
Using a graphical search program, I identified the shortest fusion pathway as a se-

quence of only 23 generalized Stone-Wales transformations, which can be viewed as

46

bond-rotations, involving relatively low activation barriers.

47

Chapter 5

Defective Structures and
Defect-Induced Structural

Transformations

The following discussion of defective carbon nanostructures and defect-induced

transformations follows that presented in References [90, 91, 92, 93].

5. 1 Introduction

When structurally perfect, carbon nanostructures are very stable even under ex-
treme conditions. Specific structural defects offer an unusual, energetically accessible
path to an inter-conversion between different isomers. More complex systems follow
a hierarchical self-assembly formation process.

In recent years, following the discovery of the C60 “buckyball” [94], a plethora of
nanostructures attracted the attention of carbon scientists. Examples of these intrigu-
ing systems are other fullerenes, onions [95], nanotubes [11], and hybrid structures
such as peapods [57]. Whereas most of these sp2 bonded structures are known to

grow from carbon vapor under extreme conditions, systems like multi-wall nanocap—

48

sules may form by activated bond rearrangement from ultra-disperse diamond [17].
All fullerene-based nanostructures show an amazing mechanical strength, chemical
inertness, and often an unusually low content of atomic defects.

What appears to be common to all these structures are synthesis conditions that
involve extremely high temperatures of close to (or beyond) 1, 000°C. While showing
no signs of spontaneous decay at room temperature, all fullerene-derived structures
are less stable than bulk graphite. Small variations in synthesis conditions, such as
change of temperature or partial pressure of the cooling inert gas, appear to have
an unusually large effect on the morphology of the product. Obviously, all these
structures are formed under non-equilibrium conditions, which are characterized by
a competition between the gain in free energy (which contains a significant amount
of vibrational entrOpy) and the fast synthesis kinetics. Due to the fast dynamics of
bond formation and fragmentation under the extreme synthesis conditions, only lim-
ited information can be extracted experimentally about the nature of the microscopic
processes which occur while these structures are formed. Simulations and theoretical
modeling of the self-assembly process may play a key role not only in the understand-
ing of nanostructure formation, but also in optimizing synthesis conditions to form
particular nanostructures.

This Chapter addresses the physical properties of some of these nanostructures,
in particular their stability. It will be shown how macroscopic concepts from elas-
ticity theory translate down to the nanometer scale, where they provide quantitative
predictions for the relative stability of nanotubes, scrolls, fullerenes, and peapods.
With the help of ab initio total energy and molecular dynamics calculations, it is now
possible to study the formation and inter-conversion mechanism of unusual systems,

such as carbon foam [91] or peapods [56].

49

 

Figure 5.1: Schematic view of (a) a nested multi-wall nanotube, (b) a scroll , and (c)
a defect separating the two morphologies within one tube.

5.2 Defect engineering in nanostructures

In fullerenes, the Stone-Wales transformation has been shown to assist efficiently
in the ejection of C2 molecules, thus shrink-wrapping the fullerenes. In nanotubes,
Stone—Wales defects are associated with deformations. As described in the following,
still other defects may be involved in transforming defective nanotubes, which may
contain scroll segments, into defect-free nanostructures. It is conceivable that inser-
tion of particular defects may even be used in the deterministic construction process
of carbon nanostructures, such as nanotubes with a particular chirality. A particular
type of defect, illustrated in Figure 5.1(c), may be involved in transforming a scroll
[Figure 5.1(b)] into a more stable multi-wall system of nested cylinders [Figure 5.1(a)]
by a “zipper-like” bond rearrangement transformation, which will be discussed later

in this Chapter.

5.3 Carbon nanofoam

A three-dimensional periodic arrangement of the type of defects depicted in Fig-
ure 5.1(c) may result in a new crystalline structure, which has been discussed exten-

sively in Reference [91]. Initially a hybrid connecting 3p2 and sp3 carbon atoms, the

50

sp2 spz

sp2

(a)

 

 

 

(b) SP3 8:03

 

 

 

 

 

Figure 5.2: Structural description of carbon foam. (a) Schematic illustrating the
structural relationship between carbon foam and layered graphite. (b) Conventional
orthorhombic unit cell of carbon foam, holding 64 atoms. Unit cell variations are
possible by changing the number of carbon atoms along the a and b axes. Initially
fourfold coordinated “sp3” atoms are grey shaded, and the initial bond direction to
their fourth neighbors is indicated by the dashed lines. (c) Perspective view of the
3D foam lattice structure, based on the above unit cell.

51

unusual “foam” structure shown in Figure 5.2 combines a large active surface area,
associated with graphite layers, with the structural rigidity of the diamond lattice.
The “origami-style” folding process, depicted in Figure 5.2(a), illustrates the close
relationship between this structure and graphite. Quite intriguing is the fact that by
changing the terrace size a, the foam covers the phase space between layered graphite
and cubic diamond, the most stable carbon allotropes. While changing the size of

the pores, the system maintains a metallic character [91].

5.4 Hybrid structures of coexisting scrolls and
multi-wall nanotubes

Visual inspection of a large number of high-resolution electron microscopy trans-
mission (HRTEM) images of multi-wall carbon nanotubes (MWCNTs)clearly illus-
trates that these materials have an unexpected intriguing internal structure fraught
with numerous defects. [90] It is believed that the chaotic environment associated
with evaporation of graphite electrodes in an arc leads to these complex structures.
Here I discuss the structure of a hybrid MWCNT composed of scrolled and nested
segments, with the respective features typically separated from each other by defects
such as edge dislocations, depicted in Figure 5.3. Within the nested tubes, helicity
changes little from layer to layer and depends only on layer diameter. Experimental
findings [90] suggest the formation of MWCNTS that involves the formation of a scroll
of a given helicity which converts, assisted by defects, into the thermodynamically

more stable multi-wall structure composed of nested cylinders.

52

 

Figure 5.3: High-Resolution Transmission Electron Micrograph of a multi wall carbon
nanotube with a slip-plane defect and irregular layer spacing. (from Reference [90])

53

5.5 Stability differences and conversion
mechanism between nanotubes and scrolls

Combining atomic-scale perfection with a nanometer-size diameter and millime-
ter-size length, carbon nanotubes [11, 1] are considered important pioneers of nan-
otechnology. These narrow graphitic cylinders with tunable electronic pr0perties[24,
25, 26] are being considered as functional building blocks of nanoscale devices. An
important challenge in nanotechnology will be to connect these substructures to more
complex systems by directed hierarchical self-assembly, using bond rearrangements
as an atomic-scale counterpart of welding.

When observed in the High Resolution Transmission Electron Microscopy (HR-
TEM) images of carbon arc-discharge deposits [11], the symmetric, evenly spaced
line patterns have been interpreted as images of coaxial, nested graphitic tubules.
Alternatively, the same images have been attributed to graphitic scroll structures
[96]. Since a distinction between the two appears to be impossible based on electron
micrographs alone [97], the scroll-tube controversy needs to be settled by different
means. The unusual high stability and oxidation resistance of nanotubes seems to
exclude scrolls as candidate structures due to their long exposed edge [11]. Presence
of scrolls, on the other hand, was conjectured from the unusually large thermal ex-
pansion in the radial direction [98] and the relative ease to intercalate metals in these
multi-walled structures[99, 100, 101]. Both scrolls and tubes have been observed in
HRTEM images of the much larger tubules of vanadium oxide [102], which show a
layered structure similar to graphite. Also, dark-field transmission electron diffraction
contrast images [103, 104, 105] and x—ray diffraction data [106] of tubular graphitic
structures have been taken as indications for scrolls radially embedded within nested
cylinders. More recent HRTEM observations even suggest the coexistence of axial

scroll and multi-wall segments within the same tubular structure [90]. This observa-

54

tion introduces two intriguing problems, namely the detailed morphology within the
coexistence region, and the possibility of a scroll-to-tube conversion.

Here I present the first study of the relative stability and the microscopic inter-
conversion mechanism between nested graphitic cylinders and scrolls. I show that
scroll segments, consisting of rolled-up graphene sheets, may coexist with nested
tube segments within a contiguous tubular nanostructure. I propose a concerted
bond rearrangement mechanism that avoids bond breaking and results in an axial
zipper-like motion of the scroll/tube dislocation. The energy cost associated with
creating this dislocation is low enough to form scroll/nanotube junctions at tempera-
tures occurring during nanotube synthesis. The activation barrier for the subsequent
scroll-to-nanotube transformation is sufficiently low to enable conversion even at room
temperature.

Available experimental data suggest that nested tubes and scrolls only coexist in
structures with very many walls [90]. Due to their large size and lack of periodicity,
these systems pose an awesome computational challenge. The most suitable formalism
to describe accurately the relative energies of large, defect-free tube or scroll segments
is based on continuum elasticity theory [107, 108, 109]. This approach proved to be
accurate in comparison to ab initio total energy calculations for selected single-wall
nanotubes [108, 109], and also offers intuitive insight into the origin of structural
changes. The major limitation of the continuum approach is its inability to deal
with atomic defects. For accurate structural and energy information about defective
structures, including coexisting scrolls and tubes, I combine the continuum approach

with atomistic calculations.

55

5.6 Continuum energy functional for sp2 bonded
carbon systems

It has been shown that deformation energies of sp2 bonded carbon nanostruc—
tures are described more accurately by Continuum Elasticity Theory, which ignores
discrete atomic positions, than by parametrized bond-order potentials.[107, 108] The
continuum energy functional, which describes defect-free single— and multi-wall tubes
as well as scrolls, contains the strain energy AEs, the interwall interaction AEiw, and
an energy term AEe associated with the creation of an exposed edge in scrolls. In
order to avoid energy terms associated with tube ends, I only consider infinitely long
structures. I only compare structures with the same total area A = WL, where W
is the nominal width and L the length of a rectangular graphene layer taken as a
reference. This reference system has no unsaturated edges and the same number of
atoms as the sp2 bonded tubular structures.

The strain energy, caused by deforming a graphene monolayer into a cylinder of
local radius R, depicted in Figure 5.4(a), is given by AE, = ecyzL/ R, where L is the
length of the tubular structure. Ab initio calculations for single-wall tubes suggest
a value of 66y; = 4.43 eV, related to the flexural rigidity of a graphene monolayer
[108]. Since strain energy favors geometries with minimum curvature, it must be
compensated by other energy terms to stabilize tubular structures. In multi-walled
systems, this compensating energy is provided by the attractive inter-layer interaction
ABM = eiAc, where Ac is the contact area between sp2 bonded layersl. Since the
inter-wall separation of AR = 0.34 nm is common to multi-wall tubes, scrolls and
graphite, I use 6.- = 2.48 eV/nmz, based on the interlayer interaction in bulk graphite.

In scrolls, I also need to consider the energy penalty associated with creating an

 

1I note that the contact area A; = A — 7r(R,~n + Rout)L is smaller than the total area A, since the
innermost and outermost walls with the respective radii Rm and Rout have only one neighboring
layer.

56

 

 

 

(b)50\"""'E'
~ ...... A
H 0 -..-.-...‘T? ...... S

E

S -50 W=100nm

‘2' "100 'Rin,opt

§ -150 - AES+AEiw

-200 K ........... AEiw

 

 

 

12345678910
Rinlnml

 

 

 

 

 

l': , "' ' ' I Scroll.

 

 

4

50 100 150 200
W [nm]

 

 

Figure 5.4: (a) Schematic of the transformation from a planar graphene strip of width
W to a scroll and a multi—wall nanotube. (b) Dependence of the strain energy AE, and
the inter-wall interaction energy AE,-w on the interior radius R," of a scroll, formed
from a W = 100 nm wide reference strip. The optimum value of Ramp; results from a
compromise between reducing strain and maximizing the inter-layer attraction. The
dashed zero line indicates the energy of the reference structure. (c) Dependence of the
optimum interior radius of a scroll (solid line) and a multi-wall nanotube (dashed line)
on W. Abrupt changes of 12,-mop, occur in multi—wall structures, when the optimum
number of walls changes.

57

exposed edge, given by ABC 2 (CL. I use 68 = 21 eV/nm as an average value for
graphite [110], thus ignoring the small difference between a zigzag and an armchair
edge.

In tubular systems, all these energy terms are proportional to the tube length L.

For a multi-wall nanotube, the energy per unit length is given by

 

E—L‘b‘ = an: (R... +(n1—1)AR) “ “[W " «(12... + 12...». (5'1)

with the summation extending over the nested walls.

The corresponding expression for a scroll is obtained by replacing the summation
by an integral over the deformed layer, augmented by the energy penalty associated
with the exposed edge. I use the angle 0 in cylindrical coordinates as a convenient

integration variable, yielding

AE, ecy, 9...... 1
—— = — —— 0 .2
L 27r 0 R(0)d ’ (5 )

where 12(0) = Rm + BAR/2w is the local radius of curvature, and (9me is determined
by the initial width W of the reference layer. Combining this strain energy with the

other remaining terms leads to a scroll energy per unit length of

 

Escroll 6cyl Rout
— - —— ln

L — AR Rm) — €i[W — ”(Rm + Rout” + 25c - (5.3)

The last term is the exposed edge penalty, which does not depend on the width W of
the reference layer and becomes negligibly small in comparison to the other terms in
multi-wall systems. I note that providing W and Rm defines uniquely the structure
of the system, including the number of walls.

Since the edge energy does not depend on Rm, I only need to consider the strain

and the inter-wall energy when optimizing the scroll structure. The dependence of

58

AEs / L and AEW / L on the interior radius is shown in Figure 5.4(b) for a fixed width
W = 100 nm of the reference layer. I find that the strain energy AE, decreases with
increasing interior radius and asymptotically approaches the zero value of the planar
reference layer. Increasing R," for a fixed value of W, on the other hand, reduces
the number of walls and the attractive inter-wall interaction AEiw. The sum of these
energies, given by the dash-dotted line in Figure 5.4(b), shows a shallow minimum
at the optimum value 12,-Mp, of the interior radius. This fact suggests that small
deviations from the equilibrium structure cost very little energy and are within the

reach of thermal processes.

5.7 Optimum geometries and energies of sp2 car-
bon nanostructures

In Figure 5.4(c), I display the dependence of the optimum interior radius on the
reference layer width. In scrolls, represented by the solid line, I find that Rm increases
monotonically with increasing W. The optimum value of Rm tends to minimize strain
while maximizing the inter-wall attraction, and slowly approaches Rin,optz0.9 nm.
The saturation behavior of Rgn,opt(W), seen in Figure 5.4(c), results from the fact
that an increase of Rm beyond a certain value should have little effect on the strain
energy, but would significantly reduce the inter-wall attraction. Thus, an important
confirmation of my picture is the observation that most multiwall structures have large
interior diameters close to 1.8 nm. Multi-wall tubes follow a similar general trend as
scrolls with the exception that the optimum number of walls changes abruptly. Such
changes occur whenever the width exceeds certain threshold values Wn, where the
strain energy sacrifice associated with more walls of smaller radius is compensated by
the increased interwall attraction.

My energy results for optimized multi-wall tubes and scrolls are summarized

59

10

 

 

EI ' 1 fi'i l l l WT f
:I
':|
E‘
:1
3|
:I
a O
3: Single Wall
'1!
3
I
1‘
NF. - i -
E 5 ‘
I;
\'.
S i“:
a) \'-.
ha \°.
3
<

 

 

 

 

 

100
W [nm]

Figure 5.5: Total energy of multi-wall tubes and scrolls with respect to a reference
graphene strip of the same length and the width W. For sufficiently large values of
W, tubular structures are preferred to the planar graphene strip and approach the

energy of bulk graphite. The higher stability of multi-wall tubes over scrolls results
from the absence of exposed edges.

60

in Figure 5.5. All energies are given per area and referred to an infinite graphene
layer, shown as the dashed line at AE/A = 0. Bulk graphite, the most stable carbon
allotrope, is stabilized by the inter-layer interaction with respect to the monolayer and
represented by the horizontal dash-dotted line. The energy of a graphene strip, given
by the dotted line, is dominated by the exposed edge penalty, when the width W is
small. At very small values of W, cylindrical deformation of the strip is energetically
prohibitive. Only for wide enough strips, the inter-wall attraction may outweigh the
strain energy, stabilizing the scroll with respect to a planar structure. According to
Figure 5.5, the narrowest scroll may form when Wz8.4 nm. For W228 nm, the
stabilizing inter-wall interaction outweighs even the exposed edge energy and the
scroll becomes more stable than the infinite reference layer.

Due to the absence of the exposed edge energy penalty, I always can find tubes
with one or multiple walls that are more stable than a scroll, as seen in Figure 5.5.
Since the relative importance of the exposed edge decreases with increasing W, I
find the energy difference between multi-wall tubes and scrolls to decrease rapidly
with increasing number of walls. In very large multi-wall structures, I postulate that
scroll and nanotube segments may even coexist. I also identify a stepwise conversion

mechanism of scroll segments into more stable multi-wall nanotube segments.

5.8 Scroll / tube dislocation

A possible way to connect a scroll with a nanotube within a contiguous struc-
ture is shown schematically in Figure 5.6(a). In this origami-style counterpart of the
system, the tube segment is separated from the scroll by a dislocation region, formed
by axially cutting and reconnecting adjacent layers. Obviously, the local morphology,
depicted in Figure 5.6(b), can be extended to any number of walls. By axially dis-

placing the dislocation in the direction of the arrow in Figure 5.6(b), a scroll can be

61

 

 

 

 

 

 

 

5
z [A]
Figure 5.6: (a) A hybrid multi-wall structure with scroll and multi-wall tube segments,
shown from the tube end. (b) The detailed structure of two adjacent layers close to
the dislocation core near the nanotube/ scroll junction. The axial dislocation motion
is indicated by the arrow. (c) Equilibrium structure of three innermost walls within
an optimized multi-wall system. (d) Atomic structure within the unit cell of an
infinite periodic stack of dislocations. The zipper-like propagation of the dislocation
along the tube axis 2 is indicated by the arrow. (e) Total energy change AEW
per graphitic layer during the zipper-like scroll-to—tube conversion within the hybrid
structure. 2 > 0 denotes the axial position of the dislocation, and z < 0 corresponds
to a dislocation-free structure.

62

gradually converted to a multi-wall nanotube.

To globally optimize the infinite structure containing a nanotube / scroll junction,
depicted in Figure 5.6(c), I separate the total energy into two parts. Far away from
the junction, the energy and equilibrium structure of the infinite scroll and multi-
wall nanotube segments is determined by continuum elasticity theory, as described
above. As seen in Figure 5.4(c), optimized structures with very many walls have
nearly identical innermost and outermost radii and can be joined smoothly with
the exception of the dislocation region. To optimize the atomic structure in the finite
region near the junction, shown in Figure 5.6(d), I make use of an electronic structure
calculation, based on the parametrized linear combination of atomic orbitals (LCAO)
formalism [42], which has been used successfully to describe rebonding and growth
at the edge of multi-wall nanotubes [49].

Junctions of multi-wall nanotubes with scrolls, depicted in Figures 5.6(a) and
(c), contain the type of dislocations, represented in Figures 5.6(b) and (d), in each
layer. The total strain is minimized when the dislocation line is shortest, correspond-
ing to radially stacked dislocations in a tubular structure. In systems with very many
walls, where scroll/nanotube coexistence has been observed [90], the average local
curvature is small. The optimum geometry and strain energy in a radial stack of tube
dislocations is represented well by a periodic stack of defects connecting graphitic
layers, illustrated in Figure 5.6(b), with the atomic structure of the unit cell depicted
in Figure 5.6(d). Such an infinite periodic stack of dislocations is formed by rigidly
connecting layers without introducing dangling bonds, in analogy to graphitic ‘foam’
[91]. Prior to relaxation, each layer in the defect region contains two fourfold coordi—
nated, sp3 bonded atoms. My results suggest that formation of this defect requires
an energy investment of only z0.2 eV per layer in a structure consisting of infinitely

many infinitely large layers2. Thus, nanotube / scroll hybrids may form under synthesis

 

2To avoid edge effects, my calculation has been performed using periodic boundary conditions
and finite unit cells. Constraints, imposed by the boundary conditions, increase the value of the

63

conditions, which typically involve high temperatures close to 1000 K.

5.9 PrOposed zipper-like scroll-to-tube
transformation

The zipper-like scroll-to—nanotube transformation is achieved by displacing the
dislocation region in the direction of the arrows in Figs. 5.6(b) and (d). Displacing the
dislocation core by one unit cell of the 3102 structure yields a morphologically identical
system with the same energy, provided the structure is periodic in all directions.
Only when the number of layers N L is finite and the energy of the exposed edges in
the scroll segment can not be neglected, does the total energy per layer change by
8(Etot(z)/NL)/Bz = 2ce/NL = (42 eV/nm)/NL due to the axial dislocation motion.
The associated energy gain is the driving force for the scroll-to—nanotube conversion.

In order to determine possible activation barriers that would hinder the zipper
motion, 1 calculated the energy associated with the concerted bond rearrangement,
involving an 3132/3103 rehybridization, using the atomistic approach. The total energy
per layer in an infinite stack of graphitic layers is displayed in Fig. 5.6(e) as a function
of the axial position z of the dislocation, where z < 0 corresponds to a defect-free
graphitic structure. In structures with a finite number of layers, the z dependence of
BEtot/NL at z > 0 acquires an additional gradient due to the average energy gain per
layer associated with the elimination of exposed edges.

My results in Fig. 5.6(e) indicate that the activation barrier associated with
axially displacing the dislocation region by one unit cell is surprisingly small, namely

only $0.02 eV. In scrolls with N L layers and exposed edges, this activation barrier is

 

apparent dislocation energy for small unit cell sizes. In my calculations, I enlarged the unit cell
size beyond that depicted in Fig. 5.6(d), until reaching convergence for the structure within the
dislocation region and the value of the dislocation energy. I found my results to agree with ab initio
results for moderate unit cell sizes, which are accessible to first-principles calculations.

64

further reduced due to the gradient of 8(Etot(z)/NL)/Bz. In systems with less than
hundred walls, the gradient is sufficiently large to make the scroll-to—tube conversion
activation free. Since the activation barrier is so small in any multi-wall system, I
expect the zipper-like scroll-to-tube transformation to occur efficiently even at room
temperature. Only much higher activation barriers, possibly associated with defects

or impurities, could stop the scroll-to—tube conversion in a transition state [90].

5.10 Summary

In summary, using total energy calculations, I investigated the relative stability
of various sp2 bonded structures, including graphitic scrolls and multi-wall nanotubes.
My structure optimization studies have shown that a tubular shape with many walls
is favored for very large systems. Such structures should share a similar interior
diameter approaching 1.8 nm, in agreement with electron microscopy observations. I
found that segments of nested multi-wall nanotubes may coexist with scrolls within
the same tubular nanostructure, separated by a dislocation region. The low energy
cost to create this defect suggests that hybrid multi-wall structures should likely form
under common synthesis conditions. I proposed a concerted bond-rearrangement
mechanism that would displace the dislocation axially, like a zipper, thus converting
the less stable scroll into a multi-wall nanotube. Due to the low energy cost of the
bond rearrangement, I find that the scroll-to—nanotube conversion may proceed even

at room temperature.

65

Chapter 6

Self Healing in Nanostructured

Materials

The following discussion of the defect tolerance of carbon nanotubes under ex-
treme thermal and electronic excitations follows that presented in References [111,

112].

6. 1 Introduction

The continuing trend towards miniaturization, rapidly approaching the atomic
scale [113], raises serious concerns about the required degree of perfection and de-
fect tolerance of nanoscale devices. Carbon nanotubes [11], considered as func-
tional building blocks [114] of such devices, can sustain very large current densi-
ties [114, 115] of 109 A/cmz, and show an unusually high thermal and mechanical
stability [1]. Nanotube—based transistors out-perform state-of-the—art silicon-based
elements in terms of speed, on/off ratio, and the maximum current [115, 116].

However, substantial defect densities in currently available nanotubes raise a
concern about the performance and reliability of nanotube-based devices. Theoretical

studies showed that even individual vacancies change transport properties of these

66

quasi one-dimensional systems significantly [117, 118]. At particular bias voltages,
drastic conductance reduction in nanotubes results from perturbing the 7r electron
system and from the presence of dangling bonds [118]. Another point of concern is,
whether defects may trigger failure earlier in nanoscale-devices than in current Si-
based devices. This concern is based on the fact that a ‘ten-nine’ purity requirement
is currently being placed on silicon wafers in order to avoid device failure, which
appears to be initiated by processes occurring at defect sites [119, 120]. So far, little
is known about possible microscopic processes that could lead to failure of defective
nanotubes.

Here I study the defect tolerance of nanotube components by monitoring the
behavior of carbon nanotubes with atomic vacancies under electronic excitations and
at elevated temperatures. Since nanostructures appear to be more susceptible to
defects than bulk systems, I intuitively expect them to fail earlier. The major result of
my theoretical study attests to the contrary. Rather than showing signs of premature
failure, I find that nanotubes exhibit a unique self-healing mechanism, associated with
a new bond formation near the vacancy. Such a self-healing process does not occur
in submicron—scale Si-based devices, and is intimately linked to the nanometer size of
my system.

The atomic-scale processes associated with failure, which occur in the vicinity of
defects on the sub-picosecond to picosecond time scale, are very hard to track experi-
mentally. As a viable alternative, I performed molecular dynamics (MD) simulations

of defective carbon nanotubes subject to extreme thermal and electronic excitations.

6.2 Method

To investigate the processes occurring at elevated temperatures, I coupled the

defective nanotube to a Nose-Hoover thermostat [54, 55] and determined the atomic

67

trajectories using the Linear Combination of Atomic Orbitals (LCAO) total energy
functional [42], with parameters based on ab initio calculations. This approach has
been used successfully to describe bond breaking and forming during the melting
process in fullerenes [48].

Describing the response of defective nanotubes to electronic excitations turned
out to be particularly challenging, since the study precludes the solution of the time-
dependent Schréidinger equation for the time evolution of the electronic wave functions
in presence of the ionic motion, caused by the changing charge distribution. This
computer intensive calculation has been performed using the time-dependent density
functional theory coupled to molecular dynamics (TDDFT-MD) [121, 122].

Since I suspect intrinsically strained narrow nanotubes to be most susceptible to
failure, I focus in my study on the (3, 3) nanotube with a monovacancy as a test case.
With only 0.4 nm in diameter, this nanotube is one of the narrowest observed [123,
124]. To distinguish nanotube-specific intrinsic behavior from effects caused by the
nanotube environment in the experimental situation [124], I focus on an isolated
nanotube in vacuum. I use periodic boundary conditions throughout my simulations,
also implying a periodic arrangement of vacancies with an axial separation of four to
six primitive unit cell sizes of the defect-free nanotube. Consequently, the unit cells

used in my calculation hold between 47 and 71 atoms.

6.3 Thermal self-healing

In the first study, addressing a possible device failure due to overheating, I mon-
itor atomic-level processes at elevated temperatures. In my molecular dynamics sim-
ulations, I used time steps of 0.5 fs to guarantee energy conservation. I used the
relaxed monovacancy structure, depicted in the left panel of Figure 6.1(c), as the

starting point. The temperature of the Nose-Hoover thermostat coupled to the (3, 3)

68

nanotube was increased at 0.75 ps intervals in discrete steps of 400 K. The results of
my simulations are discussed in Figure 6.1.

Figure 6.1(a) displays the average total energy per atom as a function of temper-
ature, imposed by the thermostat. To verify the reliability of Nosé—Hoover molecular
dynamics results, presented by the data points marked by (0), I compared these
data to those obtained using microcanonical molecular dynamics, shown by the data
points marked by (x) in Figure 6.1(a). The blue dotted line connecting the data
points, provided to guide the untrained eye, suggests the occurrence of a phase tran-
sition at a temperature TM just above 4, 000 K, close to the melting temperature of
graphite [125] or fullerenes [48]. This relatively high value of TM comes as surprise,
since this narrow nanotube was intuitively expected to melt prematurely, starting
at the vacancy site. The slope of the E(T) curve in Figure 6.1(a), representing the
specific heat c, has the classical value of 3kg below TM, indicative of a single phase.
Above 4, 000 K, I observe a distinct step in E(T), suggesting a phase transition. The
transition appears to be gradual in my simulation, which is a consequence of using
finite-size unit cells. At temperatures beyond TM, the specific heat appears to recover
its classical value c 2 3kg.

In order to understand the microscopic origin of the transition at TM and the
absence of pre-melting near the defect site, I have monitored the distance between
the edge atoms adjacent to the vacancy during my simulation. The temperature
dependence of the three corresponding interatomic bond lengths d.- is shown in Fig-
ure 6.1(b). At selected temperatures, I display the bond length fluctuations by the
standard deviations of di. Contrary to my anticipation, I observed significant struc-
tural changes and re-bonding at the vacancy site, as illustrated in Figure 6.1(c). The
final structure, achieved at 4, 800 K and depicted in the last panel of Figure 6.1(c),
contained two new bonds, emphasized by color and shading. These new bonds formed

a bridge across the initial gap and saturated all dangling bonds at the vacancy site,

69

 

 

 

 

 

 

 

 

 

(a) -55 (b) 0.35
:53. ---4

A ,z 0.30
E
0
§ E 0.25-
% 1':- 0.20
LT!

0.15

0.10 "- -

o 2000 4000 6000

A
O
V

 

   

T=3600 K T=4200 K T=4800 K

Figure 6.1: Thermal self-healing of a carbon nanotube. (a) Total energy per atom
during a molecular dynamics (MD) simulation of the narrow (3,3) nanotube with
a single vacancy as a function of temperature. Results based on Nosé-Hoover MD
are shown by (0) in red, those based on microcanonical molecular dynamics by (x)
in green. The blue dashed line, a guide to the eye, indicates a phase transition
near 4, 000 K. The standard deviations of energy and temperature of the system are
shown by error bars for selected data points. (b) Interatomic distances (1,- near the
vacancy site as a function of temperature, as defined in the inset. all is shown by
the red solid line, d2 by the blue dashed line, and d3 by the green dotted line. (c)
Snapshots of structures encountered during Nose-Hoover MD simulations, as the heat
bath temperature increases. The three atoms adjacent to the vacancy and the newly
forming C-C bonds are emphasized in the figure.

70

restoring part of the initial structural rigidity. Even more important, this recon-
struction is expected to remove the earlier mentioned reduction in the conductance,
associated with dangling bonds [118]. My total energy calculations suggested that
this transition requires an initial energy investment of 0.3 eV, followed by a net energy
gain of up to 1.76 eV per vacancy. Consequently, this reconstruction can be viewed

as a thermally activated self-healing process.

6.3.1 Energetics of monatomic vacancies in narrow nano-

tubes

To understand this energy gain in a broader context, I performed structure op-
timization on monovacancies in wider (5, 5), (7,7), and (10,10) nanotubes. I found
the energy gain due to re-bonding near the vacancy to be the largest in the (3, 3)
nanotube and to decrease with increasing tube diameter. In the more flexible narrow
nanotubes, the strain induced by the reconstruction could be accommodated locally
at little energy cost. With increasing nanotube diameter, the strain extended over an
increasing region, making this reconstruction energetically less favorable. Ultimately,
in a graphene monolayer, the energy gain due to the formation of new bonds was
not sufficient to compensate for the associated large strain energy due to re—bonding.
Thus, the thermally activated self-healing process was found to be intimately linked

to the larger flexibility of nanostructures.

6.4 Self-healing by photo-excitations

As mentioned earlier, structural changes may be achieved also by electronic ex-
citations, with no need for thermal activation. I studied the atomic-scale processes
following a photo-excitation near a monovacancy in a (3, 3) carbon nanotube using ab

initio TDDFT-MD calculations, based on the local density approximation (LDA) to

71

the density functional theory (DFT) [126, 127]. Valence electron states were described
using Troullier-Martins type pseudopotentials [128] and a plane wave basis with the
kinetic energy cutoff of 40 Ry. Due to the large size of the unit cell representing the
defective tube, I limited my sampling of the momentum space to the I‘ point.

Among the many possible excitations, I selected the particle—hole state with the
highest likelihood to destroy the nanotube, by comparing the outward components of
Hellmann-Feynman forces near the vacancy. I found this to be the case by promoting
an electron from the highest occupied state to the second lowest unoccupied state.
The energy of 0.9 eV, associated with this excitation, was determined from the total
energy difference between the excited and the ground state in a (3, 3) nanotube with
a 47-atom unit cell. The subsequent time evolution of the electronic and atomic
structure in this Franck-Condon like process was computed within the TDDFT-MD
formalism, by maintaining self-consistency between the wave function and potential,
which both evolve in time [121, 122].

In the beginning of the simulation, I froze all the atoms in the relaxed ground
state geometry and obtained the static solution of the electronically excited state.
The ion dynamics induced by this excitation is depicted as a time sequence of ge-
ometries in Figure 6.2(a). I found that the vacancy initially started to open in the
direction of the Hellmann-Feynman forces. 200 fs after the excitation, two of the three
atoms adjacent to the vacancy approached each other sufliciently close to form a new
bond. I interpret this new bond formation as a self-healing process, in analogy to the
previously discussed behavior at elevated temperatures. The difference with respect
to thermal self-healing is the absence of an activation barrier and the extremely short
time scale of the photo- induced process. The time evolution of the electronic states
during the ionic motion is presented in Figure 6.2(b) as a series of contour maps
depicting the norm of the excited electron wavefunction. As time progresses, I find

the excited electron distribution to change from an anti-bonding to a dangling bond

72

characteristics.

I should emphasize that in contrast to conventional ab initio MD simulations, I
monitor the time evolution of each populated and empty single-electron state during
the ionic motion. This approach provides information not only about the charge
distribution, but also the lifetime of electronic excitations in presence of ionic motion.
An important indication for the onset of a non-adiabatic decay is a sudden increase
in the off-diagonal matrix elements of the Kohn-Sham Hamiltonian in the basis of the
time-evolving eigenstates. Following a non-adiabatic decay, rapid Rabi oscillations
appear in the electronic levels, even if the atomic motion is smooth.

With the charge distribution determined by the time-evolving populated states,
forces acting on atoms are determined using the Hellmann—Feynman theorem [121,
122]. This approach offers a significant advantage over the conventional ab initio
MD simulations, where forces acting on atoms are calculated from static solutions
of the Kohn-Sham equations at each time step, presuming the population of each
Kohn—Sham state is known. Predicting the correct sequence of populated levels is
a challenging undertaking especially in case of level alternation, which occurs in the
single-electron spectrum following the photo-excitation of my system, as shown in
Figure 6.3. In my system, I found the character of the hole wave function to differ
significantly from the valence band character, and the character of the excited electron
to also differ from the conduction band states. Thus, the level alternation seen in
Figure 6.3 did not cause a non-adiabatic decay of the excitation, and the system
remained on the adiabatic potential energy surface during my simulation.

Unlike in bulk systems, where electron-hole excitations typically decay on a fem-
tosecond time scale, I did not find any indication for such a non-adiabatic decay of the
excitation in the defective nanotube during my MD simulation, extending over 200 fs.
Due to the apparent long lifetime of the electronic excitation, a significant fraction of

the ion dynamics occurs on the excited state potential energy surface, which is very

73

 

 

 

 

 

 

 

 

 

 

 

 

 

0 fs 100 fs 200 fs

Figure 6.2: Self-healing process within a (3,3) nanotube with a single vacancy, in-
duced by illumination. (a) Time evolution of the geometry. The three atoms adjacent
to the vacancy and the new C-C bond, which forms 200 fs after the photo-excitation,
are emphasized by color and shading. (b) Time evolution of the excited electron state.
The lowest contour lines are common to the three panels.

74

 

wuL:J-’E21-Ju:tr

l 'L...l-r"|“'l'--l'“l-J——-I~J"l-~+-'-l--I-,_

‘

p—a

o v

     
  

 

Illllllll

I
[\J
Il|ll

llllJlllllllllL Illl

 

 

One—electron—expectation value (eV)
I

5%}

 

 

 

—3

150 200

100
Tlme (fs)

Figure 6.3: Time evolution of the energy spectrum during the atomic motion following
the initial photo-excitation. Fully occupied states are denoted by solid black lines,
completely empty states by dashed black lines. Time evolution of the electron-hole
pair, created by the photo-excitation, is shown by the heavier red solid lines.

75

different from the ground state potential energy surface. One such difference concerns
the activation barrier, which is absent in the photo-excited state, but has a value of
0.3 eV in the electronic ground state. In contrast to the slow, thermally activated
self-healing process, occurring in the electronic ground state, I find the photo-induced
self-healing process to occur very fast even at low temperatures due to the absence

of the activation barrier in the excited state.

6.5 Conclusions

In summary, I performed molecular dynamics simulations of defective nanotubes
subject to elevated temperatures and electronic excitations. I have identified a self—
healing process of defects, which is intimately linked to the nanometer size of these
systems and absent in their bulk counterparts, and bears promise for future nanotube
applications. I found that the stability and conductivity of narrow nanotubes with
monatomic vacancies should improve following the formation of new bonds among the
atoms at the vacancy edge. Self-healing was observed under both thermal and elec-
tronic excitations. This process was energetically favorable due to the extra flexibility
of nanostructures in response to strain.

I should point out that my simulations correspond to vacuum conditions and
that defect-gas interactions cannot be ignored in more realistic situations. I also
explored the possibility of healing other imperfections, such as oxygen contamination
or Stone-Wales type defects, which may exist in nanotubes. My preliminary results
suggest that electronic excitations do not eliminate Stone-Wales defects, but rather
induce localized vibrations in their vicinity.

Another type of common defects in carbon nanotubes are Stone-Wales ( SW )
defects. SW defects in nanotubes can be healed through electromagnetic excitations.

A specific wavelength of the electromagnetic spectrum seems to excite resonance

76

vibrations at SW defect sites. This could be used to determine if there are any
SW defects in a sample. Alternatively, it should be possible to determine defect
densities simply by doing vibration spectroscopy after exciting the sample with a

specific wavelength. This could be a useful alternative to STM imaging in search of

SW defects.

77

Chapter 7

Bonding and Energy Dissipation in

a Nanohook Assembly

The following discussion of the nanovelcro assembly of nanoscale hooks based on

carbon nanotubes follows that presented in Reference [129].

7 .1 Theoretical concept of nanovelcro

Carbon nanotubes [11], consisting of graphite layers rolled up to seamless, nano-
meter-wide cylinders, are now considered important building blocks for nanotechnol-
ogy [2]. Their extraordinary mechanical properties, including high stiffness [130, 131,
132] and axial strength [133, 134], are related to the unparalleled tensile strength
of graphite [135]. In single-wall nanotubes [22, 23], substitution of hexagons by
pentagon-heptagon pairs is known to cause a permanent bend in the tube and to
change its chirality [136]. The morphology of a nanotube deformed to a hook is il-
lustrated schematically in Figure 7.1(a). A High-Resolution Transmission Electron
Micrograph (HRTEM) of this system [137] is reproduced in Figure 7.1(b), and a Scan-
ning Electron Micrograph (SEM) of nanohooks [138] is shown in Figure 7.1(c). So

far, studies of nanotubes containing pentagon-heptagon pairs have concentrated on

78

their intriguing electronic properties [139, 140].

Here I explore the suitability of nanotubes, permanently deformed to hooks or
other non-cylindrical structures, to efl'ectuate bonding between solid surfaces,1 as
a nanometer-scale counterpart of velcro. I study the physical properties, including
mechanical strength and resilience, of a micro-fastening system consisting of solid
surfaces covered with nano—hooks, illustrated in Figure 7.2(a) and Figure 7 .1(c), which
is called ’nanovelcro’. By studying the atomic-scale processes during closure and
opening, I show that a nanovelcro junction should be ductile rather than brittle,
and exhibit large toughness. I find that nanovelcro offers significant advantages over
conventional adhesives and welding, including thermal stability to 4,000 K, and a self-
repair mechanism under local shear. With a large density of hooks per area, strong

bonding can be achieved in parallel to mechanically decoupling the connected parts.

7 .2 Theoretical approach

To determine the physical behavior of the nanovelcro micro—fastening system, I
combine total energy and structure optimization calculations with molecular dynam-
ics simulations. In order to describe realistically a possible sp2—>sp3 rehybridization
during the opening and closing of nanovelcro bonds, I use an electronic Hamiltonian
that had been applied successfully to describe the formation of peapods [56], multi-
wall nanotubes [49], the dynamics of the “bucky-shuttle” [17], and the melting of
fullerenes [48]. Total energies and forces are compared to those based on the Tersoff
potential [141] for strained structures that maintain sp2 bonding. When modeling
the dynamical processes during opening and closure of the hook assembly in Fig-
ure 7.2(a), I subject the grey-shaded rigid anchor sections to either a constant force

or a constant velocity in the desired direction.

 

1 “Micro-Fastening System and Method of Manufacture”. U.S. Patent Application of D. Tomanek,
Richard J. Enbody, and Young-Kyun Kwon, filed February 12, 1998.

79

. - . - .I . -
19 .;2 .
v' I ‘ . l
u . - . , ~ ' . .
i..-s_n_,.n..- q

1:5? _. 32-»
‘m’ ‘

33%;
i,

.l I
. ’» "I y~ ; ’
{w W »‘ "
PW“... "WW
4 ’9 1.1"»; ' I
'. 'v‘ 15‘“? , 5 ¥
* q " ' . 7' I: ' .7.
.~ “.1 ,K ;.V .141.-

 

Figure 7.1: Structure of a nanotube-based hook. (a) Schematic view of a hook,
formed by inserting pentagon-heptagon pairs in an all-hexagon tubular structure,
and the equilibrium structure of hooks based on a (7,0) and a (12,0) nanotube. (b)
Transmission Electron Micrograph of a nanotube—based hook, published in Ref. [137].
(c) Scanning Electron Micrograph of an array of nanohooks grown on a surface [138].
(d) Atomic binding energies in (7,0) and (12,0) nanohooks. The grey scale coding
reflects the energy scale on the right.

80

     
      

 

E 2.0
“- 1.0
0.0

   

           

.‘a,
-
.1-
a

 

‘v' 9‘
‘. 2:309

.- ';'
k |

 

- o'peni'ng —'>-
closing --<--

 

 

 

Figure 7.2: (a) Schematic drawing of a two-hook assembly, defining the anchor dis-
tance :r, the direction of the opening force F0 and the closing force Fe. Snap shots
of the (7, 0) nanohooks during (b) closure and (0) opening of the nanohook assembly.
(d) Force acting on the nanohooks during the opening and closure of the assembly as a
function of the relative anchor displacement A11). The labels correspond to structures
depicted in (b) and (c). The grey-shaded area, depicting the hysteresis, represents
the energy dissipated during an opening-closing cycle.

81

7 .3 Nanohooks based on carbon nanotubes

As illustrated in Figure 7.1(a), a set of six pentagon-heptagon pairs causes a
permanent deformation of a (7, 0) single-wall nanotube to a nanohook [137]. Euler’s
theorem suggests that a hook deformation due to pentagon-heptagon insertion does
not depend on the chiral index of the tube. This is illustrated by comparing the
relaxed structures of the (7, 0) and the wider (12,0) nanotube in the right panel of
Figure 7.1(a). Even though substitution of hexagons by pentagon-heptagon pairs in a
nanotube is energetically unfavorable, the system is sufficiently flexible to redistribute
the strain in the vicinity of the pentagons and heptagons.

To visualize this strain redistribution in the structure, I grey-shaded the spheres
representing individual atoms according to their binding energy in Figure 7.1(d). I
found all the atomic binding energies to be lower than the 7.4 eV value of graphite.
The least stable atoms, indicated by the darkest shading in Figure 7.1(d), are lo-
cated in the caps. The atomic arrangement at the hemisphere terminating the (7, 0)
nanotube is similar to the strained C24 fullerene, with atomic binding energies of
only z6.4 eV. The cap structure at the end of the (12, 0) nanotube resembles that of
the more stable C34 fullerene, with atomic binding energies close to 7.1 eV. In gen-
eral, I expect the occurrence of pentagon-heptagon defects, causing permanently bent
structures shown in Figures 7.1(b) and (c), primarily at lower synthesis temperatures,
where such defects cannot be annealed easily [137, 138].

A pair of mating nanohooks is illustrated schematically in Figure 7 .2(a), together
with the direction of the Opening force F0 and the closing force Fc. The nanohooks are
to be considered permanently anchored in the surfaces to be connected. The anchor
regions are emphasized by the dark color and separated by the distance :22. The forces
are given by the gradients for the total energy of the nanohook structure with the
exception of the rigid edge regions, emphasized by the dark color in Figures 7.1(a)

and 7.2(a)-(c). Snap shots of the (7,0) nanohook engagement process are shown in

82

Figure 7.2(b). In spite of significant structural deformations during this transition, I
found no signs of irreversibility associated with a possible local sp2—>sp3 transition or a
permanent structural change, reflecting the resilience of the nanohooks to mechanical

deformations.

7 .4 Dynamics of opening and closing the nanohook
assembly

In my molecular dynamics simulation, I subject the anchor region of the hooks,
to a constant velocity 0c 2 25 m/s, and monitored the force Fe during the closure
process. Comparing results for diflerent velocities, I found the force Fc(:r,vc) to
depend only on the relative distance a: between the anchor regions at low displacement
velocities. My results indicate that the value of Fc(a:), based on molecular dynamics
simulations, agrees with static results based on static structure Optimization with
constrained anchor regions. At velocities 06275 m/s I observed an increase in Fc(:r, vc)
due to the inertia of the nanostructure. For the sake of convenience, I defined a: =
$0 + A33, where 270 is the shortest distance between the anchor regions, at which the
substructures started interacting. Numerical results for FC(A:r), displayed by the
dashed line in Figure 7.2(d), indicate that closure of the (7 , 0) nanohook assembly
requires an average force of < Fe > z0.9 nN.

The dynamics of the opening process is illustrated by snap shots in Figure 7.2(c).
As during the closing process, I subjected the anchor regions of the hooks to a low
constant velocity 00 = 25 m/s and monitored the force F0 during the opening process.
The results, given by the solid line in Figure 7.2(d), indicate an average opening force
of < F0 > zl.7 nN, about twice the value of the closing force. The opening force
increases as the hook becomes stiffer while stretched, and reaches the maximum value

of Foz3.0 nN. In spite of this considerable force, I have not observed any irreversible

83

 

  

 

 

 

 

 

 

 

 

 

(a) g ' V . ' 'stfucture'®'—-L
“:23 2 _ structure®------
.D
9
Q r D
E "-’ ~.--~~~~~--~
0 1 '-
1.40 o 1.50
dcclA]
(b) g 4 _—'strucfure® I 1
B structure®
g .
a 2
E5
O

 

Figure 7.3: Distribution of (a) bond lengths and (b) atomic binding energies in the
(7, 0) nanohook assembly at different stages of the disengagement process. Results
for the unstrained structure 6) of Figure 7 .2(c), given by the solid line, are compared
to those for the strained structure @, given by the dotted line.

84

structural changes in the nanohook assembly, including the least stable terminating
cap. Even in the most strained structure @ in Figure 7.2(c), the closest inter-wall
distance between the hook substructures was in excess of 2.1 A, thus preventing a
local sp2—>sp3 rebonding [142].

When exploring the suitability of nanohooks for bonding, I assume that the
nanohooks are permanently anchored in the substrate by covalent bonds. These
bonds are particularly strong for nanohooks grown on diamond, metals and carbides.
Uprooting the hook would require a very high force, possibly comparable to the
average force of z50 nN associated with cleaving the tube axially. The forces needed
to open and close the nanovelcro bond are much smaller and will not detach the hook
assembly from the anchor points.

To determine which parts of the nanohook assembly are most prone to damage,
I plotted the distribution of bond lengths and atomic binding energies during the
opening process in Figure 7 .3. As seen in Figure 7.3(a), most bond lengths are close
to the graphite value doc = 1.42 A in the initial structure (5). The corresponding
binding energy distribution in Figure 7.3(b) shows a large peak near 7.3 eV, reflecting
the small strain in the nanotube as compared to a graphene monolayer. Only the cap
atoms show a much lower binding energy Ecohas6.4 eV, depicted in Figure 7.1(d). In
the strained hook structure Q), the distortion is accommodated by a large portion
of the system, reflecting the ductility of the bond. In the vicinity of the hook, the
average bond length increases by ’£0.05 A and the distribution broadens significantly.
This behavior is reflected in the binding energy distribution, which shifts to smaller
binding energy values and also broadens considerably. I do not observe new peaks
at significantly larger bond lengths or smaller binding energies, which would indicate
the onset of a crack. These findings, together with the structural snap shots shown
in Figures 7.2(b) and (c), confirm that nanovelcro maintains its structural integrity

during repeated opening and closing.

85

7 .4.1 Mechanical characteristics of the nanovelcro bond

My results in Figure 7.2(d) also provide quantitative information about the
toughness of the (7,0) nanovelcro bond, defined as the energy needed tO open the
nanohook assembly. The calculated toughness Of z30 eV is very high, almost twice
the energy investment of 15.4 eV to cleave a perfect (7,0) nanotube. The simple
reason for this unexpected result is that an average Opening force < F0 > zl.7 nN,
acting over a large distance Of 30 A, performs more work than the much higher force
of z50 nN, which cleaves axial bonds across a distance of 0.5 A. Upon opening, the
energy stored in the strained hook structure is deposited into the internal degrees
of freedom, heating up the nanostructure locally up to near 1,000 K. This energy
is efficiently carried away due to the excellent thermal conductivity Of carbon nano-
tubes [143], thus preventing irreversible structural changes.

Next, I define the stability of the nanovelcro bond as the energy to close and
reopen the nanohook assembly, given by AE;J = [:0 (Fo(:r) — FC($)) dz: . I find a large
value Of AEbz24 eV for the stability Of the (7, 0) nanohook system, corresponding
to the shaded area in Figure 7.2(d). The relatively small difference between the
toughness and the stability Of the bond is due tO the low amount of energy required
tO close the hook, given by the area under the dotted line.

The usefulness of nanovelcro for permanent bonding becomes obvious especially
when considering two flat solid surfaces covered by an array of nanohooks. In view
of the small nanohook cross-section, We may find up to one nanohook per nm2,
corresponding to an ideal coverage Of 1018 nanohooks per m2. Thus, detachment of
nanovelcro bonded areas should require an energy investment of 35 J /m2. This is
significantly more than the energy to cleave most crystals, which is twice their surface
energy, and is responsible for the unusual toughness of the nanovelcro bond. In view
Of the large force required to Open a nanohook assembly, the ultimate strength Of

nanovelcro should approach 3 GPa, more than in most solids. Under tensile load, I

86

expect the solids to break first, while the nanovelcro bonds remain engaged.

I found that the bonding ability deteriorates with increasing nanotube diameter.
In the wider nanohook based on the (12,0) nanotube, I found that the resilience
Of the system is reduced by its tendency to collapse upon bending. This particular
deficit can be compensated by using multi-wall nanotubes or peapods [57] instead of
single-wall systems.

As I did not Observe any sp2—>sp3 rebonding during my simulation, I compared
my total energies to calculations based on the Tersoff potential [141]. Since this bond-
Order potential considers nearest-neighbor bonds only, the Optimized hook structures
were slightly wider and the work associated with their Opening and closing turned
out to be about 10% lower than in the results presented above.

It is essential to notice that the crucial feature Of the nanovelcro bond is not the
shape of the deformed nanotubes, but rather the area under the force-displacement
hysteresis curve in Figure 7 .2(d). Other structures, including coils [144], can be com-
bined with hooks and other deformed tubes for efficient bonding. I expect the bonded
area tO show good electrical and thermal conductivity, reflecting the intrinsic prop-
erties of nanotubes. Conductivity measurements can also be used to monitor the
local bonding in real time. A uniform surface coverage by nanohooks can be achieved
using Chemical Vapor Deposition Of hydrocarbons on catalyst-covered surfaces [145],
as seen in Figure 7.1(c). The requirement of a low growth temperature for the forma-
tion of book structures also extends the range Of substrates, on which nanovelcro can
be grown. The ability Of hooks to open and close reversibly results in a unique self-
repairing capability. This is of particular interest when bonding solids with different
thermal expansion, such as applying a diamond coating to metals, since self-repair

should prevent delamination in case Of large temperature fluctuations.

87

7 .5 Summary

In conclusion, I combined total energy and molecular dynamics calculations to
explore the suitability of nanotube-based hooks for bonding. Hooks, coils, and similar
structures form upon inserting pentagon-heptagon pairs in the honeycomb structure Of
straight carbon nanotubes. I postulated that surfaces covered with an array Of hooks,
which are covalently anchored in the substrate, can be pressed together and form
permanent bonds as a nanoscale counterpart of velcro fasteners. My results indicate
that a large force Of 3.0 nN is required to disengage two books based on a (7, 0) carbon
nanotube. Performing simulations for (7,0) and (12, 0) systems, I found nanohooks
to be generally resilient and tO keep their structural integrity during the Opening
and closing process. In view of the high tensile strength of individual nanotubes and
the stability of nanotube-substrate bonds, arrays of hooks may connect solids ranging
from metals to carbides and diamond with a tough, heat resistant bond. This bonding
scheme shows a capability for self-repair that may prevent delamination caused by
difl'erential thermal expansion. Nanovelcro bears promise as a micro-fastening system
for the next generation of nano-robots and nanometer-scale mechanical and electronic

components.

88

Chapter 8

Crystalline Solids Based on

Polymerized C50 Molecules

The following discussion of physical properties, in particular rigidity of self-

assembled fullerene crystals, follows that presented in Reference [146].

8. 1 Introduction

Owing to the fact that the binding energy Of carbon atoms in diamond and in
graphite is roughly the same, whereas the atomic coordination number in graphite
is lower than in diamond, the interatomic sp2 bonds in graphite are intrinsically
stronger than the sp3 bonds in diamond. Still, diamond is known as the solid with
the highest bulk modulus[147] Of 443 GPa. Obviously, substantial effort has been
invested to harness the toughness Of the 3p2 bond in a 3D atomic arrangement with a
bulk modulus superior to that of diamond. Only on the nanometer scale, non-planar
sp2 bonded carbon structures including fullerenes[94] and nanotubes[11, 22] show an
extraordinary stability and stiffness[148, 130, 131, 132, 133, 134]. The bulk modulus
of a single C50 molecule is predicted to reach the value Of[148] 717 GPa. Nevertheless,

this superior stiffness of individual fullerenes is not reflected in the elastic properties

89

of molecular crystals formed of fullerenes[85], which are soft due to the weak inter-
fullerene interaction.

Recent Observations suggest that C60 fullerenes, which have polymerized into
3D crystals under high pressure and high temperature conditions, surpass diamond
in hardness[81, 149, 150]. Some of the polymerized C60 crystals have been studied
theoretically[151, 152, 153]. Optimized packing within selected finite-size C60 aggre-
gates, but not infinite structures, has been suggested based on the comparison between
simulated X-ray patterns and experimental data[154, 155]. Theoretical calculations
so far have failed to identify any structure with a bulk modulus comparable to the
reported experimental data, which would exceed the value Of bulk diamond. It has
also been pointed out that the experimental methods used to identify the structure
Of polymerized fullerenes may not be conclusive[156]. The uncertainty regarding the
atomic arrangement within super-hard polymerized C60 structures requires extensive
additional studies.

Previous computational attempts to explore candidate systems for hard poly-
merized fullerenes were limited to few structures[151, 152, 153], since the rehybridiza-
tion occurring in fullerene polymers can not be described reliably by analytical bond
order potentials, and thus requires more sophisticated, computationally demanding
total energy functionals. Due to the large number of degrees Of freedom, uncon-
strained Optimization Of the atomic coordinates as a function Of volume, and thus a
reliable determination Of the total energy and the bulk modulus is computationally
prohibitive with ab initio total energy functionals.

Here I calculate the total energy Of 12 stable polymerized fullerene phases using
an electronic Hamiltonian that had been applied successfully to describe the formation
Of peapods [56], multi-wall nanotubes [49], the dynamics of the “bucky-shuttle” [17],
and the melting Of fullerenes [48]. The underlying parameterized total energy func-

tional [42, 8] is efficient enough to explore many structures, providing an adequate

90

description of total energy changes associated with different bonding geometries. My
approach reproduces correctly the Observed bulk modulus Of cubic diamond. Also,
as I discuss below, my results compare favorably with ab initio results[151, 152, 153],
which have been reported for a limited number Of candidate systems. Besides the
Optimized geometry, I also determine the physical properties Of candidate structures

for super-hard crystalline materials.

8 . 1 . 1 Polymerized fullerenes

Fullerenes such as C60 are known to form stable 1D and 2D polymers by the
‘cycloaddition’ reaction, with ‘double bonds’ facing each other in adjacent fullerenes
[157, 80, 158, 159]. Selected 1D and 2D polymer structures are shown in Fig. 8.1(a),
with the nature Of the bond depicted in Fig. 8.1(b). The interaction between such
low-dimensional polymers in 3D assemblies is generally weak, similar to the inter-layer
interaction in graphite. The prerequisite for making bulk structures incompressible
is the formation Of strong, covalent bonds between these low-dimensional polymers.
Such a rehybridization is indeed expected to occur spontaneously under high temper-
atures and pressures, similar to the conversion Of rhombohedral graphite tO hexagonal
diamond[142].

Due to structural constraints, maximizing the inter-fullerene bonding is not as
easy in 3D crystals as it is in one and two dimensions. Polymerization by cycloaddi-
tion, shown in Fig. 8.1(b), occurs at moderate temperatures and pressures. Under less
favorable conditions, other covalent inter-fullerene bonds may be established, which
would increase the number Of favorable local bonding geometries and thus improve
the degree of covalent bonding in bulk structures. Fullerene polymerization by reac-
tions other than cycloaddition may require high temperature and pressure conditions,
similar to those reported in Ref. [81, 149, 150]. In my studies Of bulk fullerene poly-

mers I considered alternative inter-fullerene bonds, depicted by the different bonding

91

schemes in Figs. 8.1(c)-(i), which have been suggested previously[160, 161, 162]. Such
strong inter-fullerene bonds have been recently Observed to connect fullerenes dur-
ing their fusion inside carbon nanotubes[57, 60, 68], while subjected to effective high
pressure conditions[69].

The bonding schemes between C60 molecules, which I consider in the following,
are depicted in Figs. 8.1(b)-(i). The most common polymerization involves the 66/66
(2+2) cycloaddition, depicted in Fig. 8.1(b), involving ‘double bonds’ at common
hexagon-hexagon edges in adjacent fullerenes, which face each other. This bonding
scheme connects fullerenes tO form chain polymers, labeled by C in Fig. 8.1(a), by
converting pairs Of ‘double bonds’, facing each other in adjacent fullerenes, to single
bonds, and leads to the formation Of two new ’single bonds’ connecting the fullerenes.
Due to their partial sp3 nature, the new bonds cause a corresponding structural re-
laxation within the fullerenes. Structures depicted in Figs. 8.1(c)-(e) may be derived
from the structure in Fig. 8.1(b) by subsequent bond breaking and bond rotation.
Disrupting the two intra-fullerene bonds, involved in the (2+2) cycloaddition, sta-
bilizes the inter-fullerene bonds and partly relieves structural strain, as depicted in
Fig. 8.1(c). Due to the smaller number Of structural constraints in this double chain
(DC) configuration, this structure is less rigid than that in Fig. 8.1(b). In this sys-
tem, all atoms have 3 neighbors, and all interatomic bonds are spz-like. Rotating the
inter-fullerene bonds by 90° yields an Open hinge (OH) structure with sp2 bonded
bridges, depicted in Fig. 8.1(d). I also note that these structures occur during the
stepwise conversion Of two C50 fullerenes to a C120 capsule by generalized Stone-Wales
transformations[69, 86].

The Open-hinge structure in Fig. 8.1(d) bears promise as a building block in
materials with a high bulk modulus, since even its narrowest structural elements
are graphene strips with an unusually high tensile strength. Among all the possible

lattice geometries based on this bonding type, only one turned out to remain stable

92

 

Figure 8.1: Types of covalent bonding between C60 molecules. (a) Arrangement of
polymerized Ceo chains (C), a square (S) and a triangular (T) 2D lattice of poly-
merized Cso molecules. Polymerization in these low—dimensional structures occurs

93

by the ‘cycloaddition’ reaction, depicted in (b). The different inter-fullerene bonding
schemes considered here are shown in (b)-(i). (b) C60 dimerization by the 66/ 66 (2+2)
cycloaddition reaction, which converts pairs Of ‘double bonds’, facing each other in ad-
jacent fullerenes, to single bonds, and leads to the formation of two new ’single bonds’
connecting the fullerenes. (0) Starting with the structure (b), disruption Of the two
intra—fullerene bonds, affected by the cycloaddition, strengthens the inter-fullerene
bonds and partly relieves structural strain. (d) Starting with the structure (0), rota-
tion Of the inter-fullerene bonds normal to the plane Of the figure leads to the ‘Open
hinge’ structure. (e) Compressing structure (d), the hinges may approach each other
to form a four membered common ring. (f) 56/65 (2+2) cycloaddition, related to
structure (b), but involving a pair Of ‘single bonds’ at the common pentagon-hexagon
edge, rather than ‘double bonds’ at the common hexagon-hexagon edge. (g) Starting
with the structure (c), rotation Of the inter-fullerene bonds normal to the plane of
the figure leads to a new bonding scheme, which I call the 56/65 four membered
common ring. (h) Occurring mainly in body-centered orthorhombic fullerene lattices,
the (3+3) cycloaddition establishes a covalent bond along the cell diagonal between
the closest atoms in adjacent fullerenes. (i) Occurring mainly in body-centered cu-
bic fullerene lattices, (6+6) cycloaddition connects two facing hexagons in adjacent
fullerenes along the cell diagonal.

under compression. Compressing the Open-hinge structure in Fig. 8.1(d) leads to
a spontaneous formation Of a bond connecting the bridges and the formation Of a
four membered common ring (FCR) between fullerene pairs. This compact covalent
bonding arrangement results in a more stable and rigid bonding scheme, as depicted in
Fig. 8.1(e), and appears to be another promising candidate for an ultra-hard fullerene
based material. It is conceivable that the rigidity Of the bulk material will scale with
the number Of four membered common rings it could accommodate.

With the particular symmetry of fullerene molecules and the geometrical con-
straints imposed by a lattice structure, it is impossible to simultaneously connect
adjacent fullerenes by a particular type of bond. Design Of an ultimately hard crys-
tal may involve inter-fullerene bonding structures not considered before. One Of the
bonding schemes, which has not been considered before when constructing candi-
date super-hard C60 structures polymerized in 3D, is the 56/ 65 (2+2) cycloaddition,
very similar to 66/66 (2+2) cycloaddition. The structure Obtained by the 56/65

(2+2) cycloaddition, shown in Fig. 8.1(f), is closely related to the structure shown

94

in Fig. 8.1(b), but involves a pair Of ‘single bonds’ at the common pentagon-hexagon
edge, rather than ‘double bonds’ at the common hexagon-hexagon edge. The equi-
librium structure and electronic properties of 2D polymerized C60 using this bonding
type have been reported recently [159]. A similar bond rotation, which led to the
FCR bonded structure in Fig. 8.1(e), could be carried out in the structure shown in

Fig. 8.1(f), to yield a different bonding scheme, depicted in Fig. 8.1(g).

8.2 Inter-fullerene bonding schemes

The bonding schemes described so far can form strongly connected 2D lattices,
including those in Fig. 8.1(a), which can be stacked to form a 3D lattice. In close
packed 3D structures, not all bonds between adjacent planes Of polymerized fuller-
enes are normal to these planes. The bonding scheme is dictated by the size Of the
fullerene and the lattice type. There are two straightforward ways to connect C60
molecules along a diagonal in a unit cell, namely the (3+3) cycloaddition and the
(6+6) cycloaddition, depicted in Figs. 8.1(b) and (i), respectively. In some lattices
the inter-fullerene nearest neighbor distance along a particular direction may be too
large for a covalent bond to form.

Using the different bonding schemes mentioned above, I have arranged C60
molecules in 16 different crystalline lattices, such as the simple cubic (SC), body-
centered orthorhombic (BCO), face-centered cubic (FCC), and body-centered cubic
(BCC) lattice. For the sake of convenience, I consider all these lattices as orthorhom-
bic lattices with a basis, spanned by the orthogonal lattice vectors a, b, and c. I
oriented the fullerenes in a way to form one Of the bonds depicted in Figs. 8.1(b)-(g)
along the lattice vector directions. Among these structures, the BCC phase formed
by (2+2) cycloaddition, and some Of the BCO phases have been discussed in the

literature previously. My results agree very well with those Of ab initio calculations

95

for structural parameters and bulk modulus Of the the published BCC and BCO
phases [151, 152, 153], and also with other structure Optimization calculations for
BCO phases [154, 155]. Due to the constraints imposed by the unit cell geometry, the
bond along the space diagonal in the BCO lattice is formed by the (3+3) cycload-
dition. The corresponding bond between the closest carbon atoms on neighboring
fullerenes is shown in Fig. 8.1(h). In the SC lattice, all bonds occur along a, b,
and c. In the BCC lattice, new bonds along the space diagonal are formed by the
(6+6) cycloaddition. In the F CC lattice, all inter-fullerene bonds, regardless of the
direction, are formed by the 56/ 65 (2+2) cycloaddition, as depicted in Fig. 8.1(f), or

bonds involving a 56/ 65 four membered common ring, depicted in Fig. 8.1(g).

8.3 Super-hard phases of C60 crystals

To determine the physical properties of these systems, I first Optimized the unit
cell size for each structure using a conjugate gradient energy minimization, starting
from the initial structures discussed above. I have used conventional orthorhombic
unit cells in my total energy and electronic structure calculations. TO determine the
bulk modulus, I have calculated the total energy Of the system as a function Of unit
cell size. All the atoms have been fully relaxed for each volume, and total energies
have been determined for the relaxed structures. In Fig. 8.2 I show the energy per
atom versus the relative volume change for the stiffest BCC and BCO structures.
Unlike the BCC structure, the BCO phase maintains its stiffness under both tensile
and compressive stress. I determine the bulk modulus, which is related to the second
derivative Of the energy around the equilibrium, from the fitted polynomial functions.

I summarize the calculated structural, mechanical and electronic prOperties of
the stable systems considered here in Table 8.1. I denote the structures according

to the lattice and bonding type. Since the type of diagonal inter-fullerene bonds is

96

 

0 BCC (2+2)

A \

g o BCO FCR

“c3 “

S 0.2 . \ ,

a) 'b .\\ I”

V b a \\ ’I‘gd

L” I \G \ ’I. u
a - i * I” I.

q 0 1 l- G\ \\ ’g .

[- G.&\‘ ,0” .

O f. l r l G 111‘ . ‘ . l . . '

 

 

 

 

Figure 8.2: Energy change per atom AE versus the relative volume change AV/ V for
the stiffest BCC and BCO structures. Solid circles are data points for BCC crystals,
with bonds along the sides of the conventional unit cell formed by (2+2) cycloaddition.
A polynomial fit to the BCC data points, representing relaxed structures, is given
by the dashed line. Open circles are the data points for the BCO lattice, containing
four membered common ring (FCR) bonds between fullerenes. A polynomial fit to
the BCO data points is given by the dash-dotted line.

97

.23: Mom 98V wmmv 9% H m mm «3.? Racofiflmaxm 63.9.. 333030

 

533838 G .e .e 3 55% 5298.:le 28 .la. .la on...
mac? :8 fin: 383558 mg .8 ONE 2: was 62 £8 fin: $838280 05838:qu on... E was? 5980 mo .3585: 2: ohm
02w H .>o Sm 98586 E QMVZ 6%: “88m 2: as 83% mo meadow 23 one am $.85 new v53 2: i 36:28 augment/3w
on... .N 31608 5:53 2: £803 pmpflofl 3 “6on8 fits 88w SQ zoom 355 $628 Omaha; 2: am: H 688338 8303?.
283m :28 Sm .vnofiip Base 8 839:8 mm £83058 #3960 80 UONMHOEEOQ .3 85.8553 EUREQ «33.2530 "mm 2an.

 

 

 

 

3m 3m 3.” w od 3.. 3m as. 83: Bases
3.2 3.2 3.2 as am 85 was 3.: was: due ooa
3.2 3.2 32 9a 3 2:. am.“ as: 23: Am+3 oom
5w Es 3w 8 as was :2 32 33.. mo Om
new new new 8 as 2: s: is: 5:: mom Om
was was was co 3 a; m3 3}. $3: a+mv Om
was was was ca 3 86 EN 32.. £3: mom oom
Es Ra Ra o2 as SN SN 3% was: on oom
Ra Ra Es as as 8N as 3% name: 3+3 oom
m2 8.x 8s as a: cod was 3.3 $3: mom com
”.2 was as OS 3. 8a was $3 was: 55% com
2: Es 9.; OS 3: 86 as was 33: 35% com
52 8s 2% ea 3 85 as $3 23' 3+3 com
:3... 3% Eu .2. €sz 93$ races 3on A>e§m esteem

 

98

determined by the lattice type, I do not include this information in the structure
notation. For each structure I list the cohesive energy Econ, the bulk modulus B, the
gravimetric density p, the fundamental band gap energy E9, the density of states at
the Fermi level N(Ep), the number Of atoms in the orthorhombic conventional unit
cell NC, and the dimensions a, b, c Of the conventional unit cell. Four Of the BCC
structures, which I initially considered and which turned out to be unstable, are not
listed in the table. Table 8.1 contains only one stable Open hinge (OH) structure,
since the OH bond in other structures is metastable and transforms into an FCR

bond under compression.

8.3.1 Body-centered orthorhombic phases

My results include physical properties Of 4 different BCO phases. All BCO phases
are rather close-packed and have a gravimetric density around 2.4 g/cm3, similar to
the density reported in the experiment[81, 149, 150], and show a metallic character.
In the BCO (2+2) structure, all inter-fullerene bonds along a and b are formed by
(2+2) cycloaddition, and no covalent bonds occur along the c direction. The bulk
modulus Bz166.7 GPa Of this structure is comparable to or higher than that of
any elemental metal, yet its binding energy is still comparable to that Of isolated
C60 molecules. The rigidity Of this BCO crystal increases, as inter-fullerene bonds
along the a-axis are converted into four membered common rings, resulting in the
BCO SFCR—I structure. Since the FCR bonding scheme is stiffer than that resulting
from the (2+2) cycloaddition, the bulk modulus in this system increases by 30%. I
also find that the unit cell shrinks in the a and b directions, but expands along c
direction. The energy gained by bringing the fullerenes closer to each other in the
a — 0 plane outweighs the energy loss, associated with elongating the bonds along the
cell diagonal.

The energetically more stable BCO SFCR—II structure with a single FCR is Ob-

99

tained by breaking the bonds along the b axis, formed by (2+2) cycloaddition, which
relieves some Of the strain along the diagonal (3+3) bonds, increasing the cell depth
0, as listed in Table 8.1. This phase is formed Of arrays Of 1D fullerene chains, poly-
merized with FCR bonds, inter-connected by (3+3) bonds to a 3D structure. Due to
the lower number of covalent inter-fullerene bonds, the bulk modulus is lower in this
system, close to the value Of the BCO (2+2) phase. I find that for the same number
of covalent inter-fullerene bonds, increasing the fraction of FCR bonds leads to higher
bulk modulus values. Indeed, according to Table 8.1, the stiffest BCO structure is the
BCO FCR phase, with all inter-fullerene bonds along a and b formed by four mem-
bered common rings. The bulk modulus Of this structure, B = 254.1 GPa, compares
favorably with the value found in cubic diamond[147], Bdia = 443 GPa, while the
gravimetric density Of the fullerene structure is lower than that Of diamond. All the
BCO phases I have investigated are very stable, with cohesive energies comparable
to that of diamond. Also, based on comparing simulated x—ray patterns[154, 155]
with Observed x-ray data[81, 149, 150], BCO structures appear as the most likely
components of the reported super-hard carbon phases. Since the highest bulk mod-
ulus of any BCO phase amounts to barely more than half the diamond value, these
structures cannot account for the reported high bulk modulus values[81, 149, 150] of

@500 — 900 GPa.

8.3.2 Body-centered cubic phases

Since even a strong covalent connection between fullerenes along the a and b
direction is not sufficient to yield a very high bulk modulus, I considered alternative
bonding schemes with strong bonds also along the c direction. One realization Of such
a bonding scheme is the BCC lattice, with fullerenes in adjacent a — b planes close
enough to form covalent bonds along the c direction. According to Table 8.1, all BCC

phases have higher bulk modulus values than BCO or any other lattice types. As a

100

matter Of fact, the bulk modulus of the stiffest BCC phase is only 20% lower than
that of cubic diamond, while the gravimetric density of this new phase is still 25%
smaller than in diamond. In view Of the Optimized cell dimensions, which contain
relatively large inter-fullerene distances that are incompatible with strong covalent
bonds, I would expect the the BCC phase to be barely stable. Under compressions,
however, the high stiffness Of this phase benefits from the rather incompressible (6+6)
bonds along the cell diagonal, resulting in the highest bulk modulus value identified
in this study.

The anticipated low stability Of the elongated inter-fullerene bonds in the BCC
phase is reflected in its lower stability with respect to the BCO phase according
to Table 8.1. As a matter Of fact, in absence Of steric constraints protecting the
cubic symmetry, I should expect a spontaneous symmetry breaking, leading to a
transformation Of the BCC to the more stable BCO phase. This indeed occurs in the
metastable BCC phase of C60 molecules, polymerized with four membered common
rings in the a—b plane, with no covalent bonds along the c axis, and (6+6) connections
along the unit cell diagonal.

As suggested above, the stiffness Of the BCC structures under compression is
in stark contrast to their much lower strength under tension. This is seen from the
asymmetry of the elastic response Of the BCC (2+2) lattice, shown in Fig. 8.2, which
is very different from the almost symmetric elastic response Of the BCO lattice subject
to relative volume changes. Moreover, beyond a critical tensile strain, all BCC phases
undergo structural changes, reflected in the lack Of data points for the BCC (2+2)
phase at large positive relative volume changes AV/ V in Fig. 8.2.

All stable BCC phases listed in Table 8.1 exhibit identical bonding along the a, b,
and c axes. I found all these structures to be insulators with a l — 2 eV fundamental
band gap, and a gravimetric density close to 2.6 g/cm3. The stiffest BCC crystal
is BCC (2 + 2) with the bulk modulus Bz370 GPa. Since the double chain (DC)

101

inter-fullerene connection is weaker, the BCC DC structure has a lower bulk modulus
than BCC (2 + 2), but is still stiffer than other lattice types due to the presence of
the incompressible (6 + 6) inter-fullerene connections along the cell diagonal. The
dominant role Of (6 + 6) bonding in achieving a high stiffness is reflected in the
similar bulk moduli Of the BCC FCR and the BCC (2 + 2) structures. Apparently,
the intrinsic higher stiffness Of FCR connections, which lead tO high bulk modulus

values in BCO lattices, is Of secondary importance in this case.

8.3.3 Simple cubic phases

As mentioned above, the marginal stability Of the BCC lattice was caused by the
large size Of the C60 molecule in the center Of the cubic unit cell. I may expect that
removal of the central fullerene will stabilize the cubic lattice by reducing the strain
along the a, b, and c directions. Even though the resulting simple cubic phase should
be energetically more favorable, the lower packing fraction Of fullerenes may lower the
bulk modulus. I have studied 4 different SC phases. Out of these, SC with a double
chain bond transforms into the SC (2+2) structure even under small pressure, thus
eliminating it as a candidate for a super hard structure. Since the compressibility Of
the SC lattice directly reflects the rigidity Of the inter-fullerene connections along the
cube edges, I can easily rationalize that the bulk modulus Of SC FCR is the highest
among the SC lattices, followed SC OH and finally SC (2+2), the softest SC lattice.
As seen in Table 8.1, the gravimetric densities Of all SC phases are approximately half
the value Of diamond. All SC fullerene crystals are insulators with a m1 — 2 eV band
gap. The SC OH phase with Open hinge connections is unique in containing only
3-fold coordinated carbon atoms, implying sp2 inter-atomic bonding throughout the
structure. This results in a high bulk modulus Bz166 GPa at a relatively low density
Of 1.81 g/cm3. As a logical consequence, schwarzite or Mackay structures consisting

Of a simple cubic lattice formed by nanotubes with perfect sp2 interconnects, should

102

be extremely rigid. Due to the constraints associated with the cubic symmetry of
such a structure, I find the number of atoms in C50 tO be insufficient tO form such

perfect 3102 interconnects at the vertices.

8.3.4 Face-centered cubic phases

The last lattice type I have investigated is FCC, where all inter-fullerene bonds
are identical. I have considered 56/65 connections, displayed in Fig. 8.1(f), and
56/65 four membered common rings, shown in Fig. 8.1(g). In spite of the different
bonding, the two structures share similar values Of the bulk modulus, unit cell size and
gravimetric density. The unexpected insensitivity Of the bulk modulus Bz180 GPa
tO the bonding scheme is associated with extreme structural deformations Of the
fullerenes, which degrade substantially their initial structural rigidity. Consequently,
the presence Of strong covalent inter-fullerene bonds is only beneficial for an overall
high structural stiffness, if fullerene deformations do not reduce the intrinsic rigidity
Of the molecule. I find both FCC phases to be metallic with a similar density Of states
at the Fermi level. As seen in Table 8.1, the metallic behavior of FCC structures is

unique among the cubic lattices.

8.4 Conclusions

A major advantage Of fullerene based super-hard materials is their formation
mechanism by self-assembly from sub-nanometer sized fullerenes. This is particularly
beneficial, when micrometer-sized voids are to be filled with a rigid structure to
enhance the overall stiffness. The high structural rigidity of polymerized fullerenes
is coupled with a low gravimetric density, which lies below the diamond value in all
the compounds studied here, mainly due to the empty space inside the fullerenes.

Nevertheless, I must notice that the stiffest phases tend to have the highest mass

103

densities. As a matter Of fact, among all crystalline materials, the system with the
highest bulk modulus, diamond, distinguishes itself also by the largest number of
atoms per unit volume.

In conclusion, I combined total energy and structure Optimization calculations
to explore the physical properties of new hard phases consisting Of fullerenes poly-
merized in 3D. I have identified 12 different stable crystal structures that are very
stiff. The bulk moduli I have found in 3D fullerenes crystals are as high as 80% Of
the diamond value, in contradiction to experiments reporting materials stiffer than
diamond[81, 149, 150]. The BCC lattice structure, which shows the highest bulk
modulus value among those addressed here, turned out to be the least stable among
the structures investigated. The bulk modulus of other structures, which should form
more readily under moderate conditions, lies below half the diamond value, but still
exceeds that of most metals. The possibility Of self-assembling these rigid structures
from sub-nanometer C60 fullerenes, which may be filled easily into nano—cavities in
bulk structures, may lead to new materials with a superior stiffness and stability at
high temperatures. The electronic properties Of 3D structures formed Of polymerized
fullerenes depend on the lattice type, with BCO systems exhibiting metallic behavior,
whereas the majority Of cubic systems are insulators with a #1 — 2 eV fundamental

band gap.

104

Chapter 9

Thermal Contraction of Carbon

Fullerenes and Nanotubes

The following discussion of thermal expansion properties of fullerenes and nano-

tubes follows that presented in Reference [163].

9.1 Materials exhibiting negative thermal expan-
sion

There has been increasing demand for high performance composites with near-
zero thermal expansion [164, 165]. In such composites, thermal expansion Of one
component may be compensated by a thermal contraction Of other components [166].
Since most materials expand thermally, there has been extensive search for mate-
rials with a negative thermal expansion coefficient. Thermal contraction has been
Observed in quasi one-dimensional polymers such as rubber, polyethylene, polyacety-
lene, polydiacetylenes, and selenium [167 , 168, 169, 170, 171], layered systems such as
graphite and boron nitride [172, 173, 174], and three-dimensional covalent networks

found in oxides [175, 176] such as NaTi2P3012 and ZI'WgOg. Since carbon fullerenes

105

and nanotubes consist of sp2 bonded carbon networks related to graphite, I expect
them to also contract at low temperatures.

Thermally induced contraction and expansion result from a competition be-
tween internal energy and entropy. In the harmonic regime at low temperatures,
low-dimensional systems gain structural and vibrational entropy by exploring the
voids in configurational space at relatively little energy cost. The dominating effect
of entropy in this regime results in a contraction. Only at higher temperatures do
the anharmonicities in the interatomic interaction play a significant role, causing an
overall expansion. I expect the transition from contraction at low temperatures to
expansion at high temperatures to be a universal phenomenon in low-dimensional ma-
terials, and only the crossover temperature to change from system to system. As I will
discuss in the following, the interplay between energy and entropy has a particularly
interesting effect on the thermal behavior Of nanostructures.

Following the discovery of fullerenes [94] such as C60, carbon nanotubes [11, 22],
perfect cylinders formed Of rolled graphite monolayers, are now being considered as
important building blocks for nanotechnology [177, 115]. Thermal expansion prop-
erties Of carbon nanotubes are Of great importance, since the electrical characteristic
Of complex nanotube-based circuitry will change significantly due tO difl'erential con-
traction imposing internal stress [178]. Another important aspect Of the thermal
expansion of nanotubes is the effect on the mechanical prOperties Of nanotube-based
high-performance composites [179]. Even though the same sp2 bonding is shared
by 2D graphite, 1D nanotubes and 0D fullerenes, I find it important to determine
the effect Of dimensionality on the thermal contraction at low temperatures and the

crossover point to thermal expansion.

106

9.2 Computational approach

In this Chapter, I use N osé-Hoover molecular dynamics simulations [54, 55] to de-
termine how temperature affects the volume Of C50 molecules and carbon nanotubes,
and also to determine how their length changes with increasing temperature. I show
that at low temperatures, both fullerene molecules and carbon nanotubes contract
in length and volume. Only at temperatures beyond several hundred degrees Kelvin,
the initial contraction changes to an overall expansion, driven by exciting anharmonic
phonon modes. I determine temperature-induced structural changes by investigating
the coupling between vibrational modes and the shape of fullerenes and nanotubes.
I identify those vibration modes that play a dominant role in how the volume and
length of fullerenes and nanotubes change with temperature.

TO correctly reproduce the change in length and volume of nanostructures in
response to temperature, I need to describe precisely those energy changes associated
with shape deformations in the anharmonic regime. I found that the popular Ter-
soff potential [141] introduces a spurious soft “harmonica twist” mode that modifies
the length Of a straight tube. TO avoid this problem, I use a parameterized linear
combination Of atomic orbitals formalism based on a universal parametrization Of
ab initio density functional results for structures as different as C2, carbon chains,
graphite, and diamond [42]. This energy functional is particularly suited for my pur-
pose, as it is based on an electronic Hamiltonian that goes well beyond the nearest
neighbor approximation, thus also describing anharmonic effects due to the sp2—>sp3
rehybridization during deformation. I make use Of the computationally efficient ap-
proach [8] that has been previously used with success to describe the formation Of
multiwall nanotubes [49], nanopeapods [56], the dynamics Of the “bucky—shuttle” [17],

and the disintegration of fullerenes [48].

107

 

(a ) 0.004 -

 

 

 

 

 

—0.002

(b) 2x105 -

1x105-

 

   

.r i”’lh\‘\"

“as

  

200 400 600
T [K]

    

 

BIK'I]

 

.1x10'S -

 

 

 

 

0 100 200 300 400
T [K]

Figure 9.1: (a) Relative volume change AV/Vo in a C60 molecule as a function of
temperature T. (b) Temperature dependence Of the thermal volumetric expansion
coefficient [3. (c) Schematic snap shots Of the ellipsoidal deformation mode, responsi-
ble for volumetric contraction at low temperatures.

9.3 Thermal contraction in fullerenes as zero di-
mensional model systems

As suggested earlier, the most pronounced deviation from the thermal expansion,
which is typically associated with three-dimensional objects, can be expected in the
zero-dimensional C50 molecule. To study volumetric changes with temperature, I cou—
pled the C60 molecule to a thermal heat bath at temperatures ranging from T = 0 K

to 1, 000 K. My molecular dynamics simulations indicate that the C60 molecule shows

108

an ellipsoidal deformation, one of the softest vibration modes that is depicted in Fig-
ure 9.1(c), even at low temperatures. In view Of this fact, I diagonalized the gyration
tensor Of a deformed C60 molecule to Obtain the principal axes and to estimate the
volume at any moment in time. In my simulations, I changed the heat bath tem-
perature stepwise, and discarded data Obtained during the first 0.2 ps following a
temperature change to allow for equilibration with the heat bath. My data for the
volume change AV(T), shown in Figure 9.1(a), represent the average over 104 - 105
time steps of At = 5x10"16 3. These results suggest that a C60 molecule shows a
volumetric contraction, rather than an expansion, up to Tz260 K, and expands only
at higher temperatures. Figure 9.1(b) depicts the corresponding thermal volumetric

expansion coefficient, which is defined by

_1dV

s _ Val—T“ (9.1)

I find that the thermal volumetric expansion coefficient starts at zero at T = 0 K,
and initially decreases linearly to 5% —1x10“5 K‘l. The minimum in the 6 (T) curve,
corresponding to the maximum thermal volumetric contraction, occurs at Tz70 K.
Above this temperature, the volumetric contraction Of C60 slows down. At Tz150 K,
C60 achieves its smallest volume, and its volumetric contraction turns tO an expansion.
Further analysis Of the changes in the principal axes with temperature showed that
among the soft vibration modes, it is the quadrupolar deformation, which keeps the
surface area constant, that dominates the thermal contraction at low temperatures.
Thermal expansion at higher temperatures results from a bond expansion, which is

reflected in a radial breathing mode that changes the surface area Of the fullerene.

109

9.4 Thermal contraction in nanotubes as one di-
mensional model systems

Next, I investigate the effect Of temperature on the volume and length of car-
bon nanotubes as model one-dimensional Objects. Due to their natural abundance
among single-wall carbon nanotubes grown by pulsed-laser vaporization [180], I focus
on (10, 10) nanotubes in my simulations. Also, nanotubes with a similar diameter are
the most likely constituents Of “bucky paper”, for which initial experimental measure-
ments Of thermal expansion have been performed [181]. I considered 50— 1, 500 A long
Open-ended segments Of an isolated (10, 10) nanotube in order tO calculate the effect Of
temperature on its volume and total length. As in the C60 simulations, the nanotube
was coupled to a Nose-Hoover thermostat in my canonical molecular dynamics sim-
ulations. Considering an Open-ended segment rather than an infinite nanotube with
periodic boundary conditions, I avoided introduction of spurious phonon modes and
problems due to a fluctuating unit cell size. Since the vibrations of an Open-ended
segment are altered by the presence Of under-coordinated carbon atoms, I ignored
the five terminating layersl when evaluating the total length and volume of the tube
segment.

The estimation of the volume of a deformed nanotube is less trivial than for
the C60 molecule due to the large number of relevant phonon modes, illustrated in
Figure 9.2(e)—(h). I determine the volume Of a deformed nanotube as a sum Of partial
volumes associated with each deformed layer, given by a product Of its cross section
area and the distance between adjacent layers. In a deformed nanotube segment, I
first determine the center Of mass Of the relevant layers and use the distance between

them as the length. I approximate the cross-section area Of each layer by that of a

 

lIn achiral nanotubes aligned with the z axis, I define the ensemble of atoms with the same 2
coordinate as a ‘layer’. In the (10, 10) nanotube, a layer consists Of a 20 atom ring along the tube
circumference.

110

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-5

A . A -8x10 . L A A

0 400 800 1200 0 200 400 600 800 1000
T[K] T[K]

e ’ .m _~.4‘0‘IF-"?l
( ) m'iefffifi‘.‘*:"‘ -.-. -.*.-x\~:-.~.anew-.qu--:1—,~9-.--4‘jv.§vt>'~-‘»~fsyt~.~y~~fsr.taxi-rew-fl‘“ g I ' ‘ -, , ,,,,-‘r"'“
, , . . . ...
. ' _, i
v; v w.:i
v."
. . ‘ 7.. . . . . . , -v.
.th‘ . \,y,- ~ 1' 1.. 4.-|--‘.|]\ .~s......,., ‘ ,_ ‘ ' , fi‘; ' « ar>\~.~m. A
, 1 . . . . , . . . . l ‘. I - . . '. . ' ’ '

 

Figure 9.2: (a) Relative length change AL/Lo in a (10, 10) carbon nanotube as a func-
tion Of temperature T. (b) Temperature dependence of the thermal linear expansion
coefficient 6. (c) Relative volume change AV/Vo and (d) thermal volumetric expan-
sion coefficient 6 Of the (10,10) nanotube as a function of temperature. Schematic
snap shots of selected soft phonon modes that couple strongly to the length and vol-
ume Of the nanotube: (e) reference structure Of the unperturbed tube, (f) Optical
pinch mode, (g) transverse acoustic bending mode, (h) twist mode. The relative con-
tribution Of the pinch mode to the volume change is shown by the dashed line in (c).

111

circle with radius r. This radius is given by the average distance Of each atom from
the center Of mass Of that layer. Similar to my treatment Of the C60 molecule, I
first determine the three principal axes for each nanotube layer by diagonalizing its
gyration tensor. Then, I define the volume as the product Of the elliptical cross-section
area normal tO the tube axis and the distance between adjacent layers.

My results for the response of the nanotube to changing temperature are dis-
played in Figure 9.2. Similar to my results for the C50 molecule, data points rep-
resenting the volume and length at a particular temperature have been Obtained by
taking a time average over 10" — 105 time steps of At = 5x10’16 8 after equilibration
with the heat bath. I have filtered out high-frequency noise using a simplified Fourier
analysis of the total volume and length as a function of time at 0 K, 200 K, 400 K,
800 K, 1, 000 K, and 1, 100 K. The error bars reflect also the additional uncertainty
associated with this fitting process.

My results for the temperature dependence of the tube length, shown in Fig-
ure 9.2(a), suggest that carbon nanotubes should contract up to very high tempera-
tures close to 1, 300 K. These findings are in agreement with the Observed contraction
in “bucky paper” below room temperature [181]. The solid line is a fit to the same
functional form as used in the thermal expansion analysis of graphite [135], which
also shows an in-plane contraction at low temperatures. In analogy to Eq. (9.1), the

linear thermal expansion coefficient a is defined as

_1dL

Similar to the volumetric expansion coefficient Of C60 depicted in Figure 9.1(b), the
linear expansion coefficient a of a nanotube decreases from its initial zero value at
T = 0 K, and becomes negative. The minimum value in the a(T) curve, corresponding

to the maximum thermal contraction, is a z —1.2x10“5 K‘1 and occurs at Tz400 K.

112

Contraction continues up tO Tz900 K, where a vanishes and the tube length is
minimum. Above 900 K, 0 becomes positive, indicating onset Of thermal expansion,
as the anharmonic part Of the inter-atomic potential gains in importance. Above
1, 300 K, the tube should exceed its initial length at T = 0 K.

My results reveal that not only do nanotubes contract axially, but they also
Show volumetric contraction up tO very high temperatures, due to radial deforma-
tions discussed below. As shown in Figure 9.2(c), the volume decreases gradually,
as the temperature increases from 0 K to Tz800 K. At Tz800 K, the volume starts
expanding again, and exceeds its original value at Tz1200 K. The larger error bars
are a consequence Of a combined uncertainty in the average tube radius and length.
The volumetric thermal expansion coefficient, presented in Figure 9.2(d), decreases
from zero to its minimum value 6% — 7x10‘5 at Tz400 K, and then increases to

reach zero value at Tz800 K, corresponding tO the smallest tube volume.

9.4.1 Physical origin of thermal contraction

In order to understand the origin Of thermal contraction, I analyzed my molec-
ular dynamics simulations in terms of normal vibration modes that couple strongly
to the length and volume. As for Cso, I focused on the soft modes that do not
change the surface area of the reference tube, shown in Figure 9.2(e). The signifi-
cant modes are the pinch mode, the bending mode, and the twist mode, depicted in
Figures 9.2(f)—(h), respectively. Whereas short-wavelength modes, such as the pinch
mode, can be described in my atomistic molecular dynamics simulations, description
Of long-wavelength modes requires long tube segments exceeding my computational
resources. To describe the long-wavelength bending and twisting phonon modes, I
performed molecular dynamics simulations based on a model assumption Of rigid
atomic layers that interact according to the energy functional described above.

My long-wavelength calculations suggest the transverse acoustic bending mode,

113

depicted in Figure 9.2(g), to be very soft, as it involves a shear motion that requires
a minimum Of bond stretching and bending. My simulations indicate that length
contraction due to this mode saturates in (10, 10) tube segments with a length close
to 100 A. I find the bending mode tO contribute almost one third towards the length
contraction Of a (10,10) nanotube. Also the soft twisting mode, depicted in Fig-
ure 9.2(h), can be viewed as a shear mode. I found this mode to be active in the
orientational melting within a nanotube mm at low temperatures [182], and to con-
tribute up to z 10% towards the tube length contraction. Since none Of these modes
affects the tube cross-section, their effect on the volumetric contraction is very small.

My calculations suggest that an important contribution towards the longitudi-
nal contraction, and the dominant contribution towards the volumetric contraction,
comes from the Optical “pinch” mode, shown in Figure 9.2(f). The dashed line in Fig-
ure 9.2(c) indicates the analytically calculated contribution Of a pure pinch phonon
mode to the total volume change. In the pinch mode, each tube layer in the pinch
mode is subject to an elliptical deformation. TO calculate the volumetric contraction
in this mode, I assumed the circumference C’ of the elliptical cross-section to be con-
stant. In this case, the area S Of the ellipse decreases monotonically with increasing
eccentricity tEb/a, defined as the ratio Of its major and minor axes a and b. Making
use of the complete elliptic integral Of the second kind E (as), the ellipse area is given by
S (t) = (7rC2 / 16) (t/[E(1 — 132)]2). Assuming that t changes linearly between tmm and
1 along the axial direction, the total volume V Of the deformed nanotube segment Of
length L changes as V = fol” dlS(t) = L < S >, where < S >= ft?“ dtS(t)/(1 -— tmin).
I find that the pinch mode accounts for nearly one half Of the volume change, but
only one third Of the total length change.

The relative contribution of the changing cross-section area S and length L to-
wards the volume change is determined by the decomposition AV/l/b = AL/Lo +
AS/So + (AL/L0)(AS/So). This expression allows us to extract additional infor-

114

mation about the thermal behavior of nanotubes, in particular the radial thermal
expansion coefficient (1,, from the known length and volume changes, presented in
Figures 9.2(a)—(d). I find that in the temperature range Of interest, the average tube
radius should contract. This result agrees with experimental Observations [183], but
contradicts simulations based on Tersoff potential [184]. My numerical value for the
radial contraction at low temperatures are: — 2x 10"5 K’1 is about one order Of mag-
nitude larger than X-ray results reported for single-wall nanotube ropes [183], where
the dominating pinch mode is likely to be suppressed by inter-tube interactions. In
agreement with the experimental data [183], I expect a volumetric thermal expansion
Of nanotube ropes due to an increasing inter-tube distance, in analogy to the volu-
metric expansion Of bulk graphite, which is dominated by the expansion along the

c—axis.

9.5 Summary

In summary, I investigated thermal expansion behavior Of fullerenes and carbon
nanotubes using molecular dynamics simulations. I showed that C60 molecule exhibits
volumetric thermal contraction up to T3260 K, whereas carbon nanotubes contract
in length and volume for Tz800 K. Among the different active vibration modes, I
found the ellipsoidal deformation mode to dominate the volumetric contraction of
C60. In nanotubes, length contraction is shared by bending, twist and pinch modes,
whereas the volumetric contraction is dominated by the pinch mode. Combining
thermal contraction Of nanotubes with thermal expansion Of a host material may
yield high performance composites showing negligible thermal expansion up to high

temperatures.

115

Chapter 10

Unusually High Thermal

Conductivity of Carbon Nanotubes

The following discussion of thermal conductivity in carbon nanotubes follows

that presented in Reference [143].

10.1 Introduction

With the continually decreasing size of electronic and micromechanical devices,
there is an increasing interest in materials that conduct heat efficiently, thus pre-
venting structural damage. The stiff sp3 bonds, resulting in a high speed of sound,
make monocrystalline isotopically pure diamond one of the best thermal conduc-
tors [185]. An unusually high thermal conductance should also be expected in carbon
nanotubes[ll, 1], which are held together by even stronger sp2 bonds. These systems,
consisting of seamless and atomically perfect graphitic cylinders few nanometers in
diameter, are self-supporting. The rigidity of these systems, combined with virtual
absence of atomic defects or coupling to soft phonon modes of the embedding medium,
should make isolated nanotubes very good candidates for efficient thermal conduc-

tors. This conjecture has been confirmed by experimental data that are consistent

116

with a very high thermal conductivity for nanotubes [186, 187].

In the following, I will present results of molecular dynamics simulations using
the Tersoff potential [141], augmented by Van der Waals interactions in graphite,
for the temperature dependence of the thermal conductivity of nanotubes and other
carbon allotropes. I will show that isolated nanotubes are at least as good heat
conductors as high-purity diamond. My comparison with graphitic carbon shows
that inter-layer coupling reduces thermal conductivity of graphite within the basal
plane by one order of magnitude with respect to the nanotube value which lies close
to that for a hypothetical isolated graphene monolayer.

The thermal conductivity /\ of a solid along a particular direction, taken here as

the z axis, is related to the heat flowing down a long rod with a temperature gradient

dT/ dz by

1 d dT

25? = —/\d;’ (10.1)
where dQ is the energy transmitted across the area A in the time interval dt. In solids
where the phonon contribution to the heat conductance dominates, /\ is proportional
to 0121, the product of the heat capacity per unit volume C, the speed of sound 1), and
the phonon mean free path I. The latter quantity is limited by scattering from sample
boundaries (related to grain sizes), point defects, and by umklapp processes. In an
experiment, the strong dependence of the thermal conductivity A on I translates into
an unusual sensitivity to isotopic and other atomic defects. This is best illustrated by
the reported thermal conductivity values in the basal plane of graphite [188] which
scatter by nearly two orders of magnitude. As similar uncertainties may be associated

with thermal conductivity measurements in “mats” of nanotubes [186, 187], I decided

to determine this quantity using molecular dynamics simulations.

117

10.2 Limitation of theoretical approaches describ-
ing thermal conductivity

The first approach used to calculate /\ was based on a direct molecular dynamics
simulation. Heat exchange with a periodic array of hot and cold regions along the
nanotube has been achieved by velocity rescaling, following a method that had been
successfully applied to the thermal conductivity of glasses [189]. Unlike glasses, how-
ever, nanotubes exhibit an unusually high degree of long-range order over hundreds
of nanometers. The perturbations imposed by the heat transfer reduce the effective
phonon mean free path to below the unit cell size. I found it hard to achieve con-
vergence, since the phonon mean free path in nanotubes is significantly larger than
unit cell sizes tractable in molecular dynamics simulations. This method requires
long nanotubes to converge in order to separate regions coupled to the thermostat
from “unperturbed” tube sections. This can not be guaranteed in short nanotubes,
resulting in a nonlinear temperature profile, shown in Figure 10.1, in contrary to the
expected linear temperature profile at 220.

As an alternate approach to determine the thermal conductivity, I used equilib-
rium molecular dynamics simulations [190, 191] based on the Green-Kubo expression
that relates this quantity to the integral over time t of the heat flux autocorrelation

function by [192]

1 00
= . t. 1 .2
A 3VchT2/o <J(t) J(0)>d (0 )

Here, k3 is the Boltzmann constant, V is the volume, T the temperature of the

sample, and the angled brackets denote an ensemble average. The heat flux vector

J (t) is defined by

J(t) = %ZI§A€,‘ 2‘ ZviAe, — Z Z r,j(f,-j ' Vi) , (10.3)

i j(¢i)

118

 

 

1610-...-.

 

 

. O

O
r . .-
158.0 » - ' ' -
H _ . . ‘
E .. 0.0... *
154.0 -- ' ' .
. . .
150.0411A1A1-1-- 4

 

-15 -10 -5 0 5 10 15
z [A]

Figure 10.1: Temperature profile in a direct molecular dynamics simulation of heat
flow in a (10,10) nanotube. Temperature is controlled by thermostats located close
to z = —15A and z = 15A. Since the nanotube is not long enough, the temperature
profile failed to converge, reaching a linear temperature profile near 2 = 0.

where A6,- = e,— < e > is the excess energy of atom z" with respect to the average
energy per atom < e >. r,- is the position and v,- the velocity of atom z', and rij =
r, — r,-. Assuming that the total potential energy U = 2,21,- can be expressed as a
sum of binding energies 11.,- of individual atoms, then fij = —V,uj, where V,- is the
gradient with respect to the position of atom 2'.

In low-dimensional systems, such as nanotubes or graphene monolayers, I infer
the volume from the way that these systems pack in space (nanotubes form bun-
dles and graphite a layered structure, both with an inter-wall separation of z3.4 A)
in order to convert thermal conductance of a system to thermal conductivity of a
material.

Once J (t) is known, the thermal conductivity can be calculated using Eq. (10.2).

I found, however, that these results depend sensitively on the initial conditions of each

simulation, thus necessitating a large ensemble of simulations. This high computa-

119

1.00~-~

 

 

 

 

"a?

... 0.75

A

§

7N 0.50

:2 0.25

V I
0 5 10 15 20 25

t[pS]

Figure 10.2: Velocity autocorrelation function, averaged over a large number of en-
sembles. The timescale shown is not long enough to obtain a converged value of the
velocity autocorrelation. The sharp drop at two corresponds to soft phonon modes,
which contribute little to the thermal conductivity.
tional demand was further increased by the slow convergence of the autocorrelation
function, requiring long integration time periods. In Figure 10.2 I show the veloc-
ity autocorrelation function averaged over a large number of ensembles, obtained by
randomizing the initial coordinates and velocities. The fact that the autocorrelation
function < J (t) - J (0) > does not converge to 0 indicates that time periods much
longer than 25 ps are required to correctly describe thermal conduction. This is a
clear indication of a long phonon lifetime, and long phonon mean free path.

These disadvantages have been shown to be strongly reduced in an alternate
approach [193] that combines the Green-Kubo formula with nonequilibrium thermo-

dynamics [194, 195] in a computationally efficient manner [196]. In this approach,

the thermal conductivity along the z axis is given by

_, , <Jz(Fe,t)>
“113303.132. perv ’

 

(10.4)

120

where T is the temperature of the sample, regulated by a Nose-Hoover thermostat
[54, 55], and V is the volume of the sample. Jz(Fe, t) is the z component of the heat
flux vector for a particular time t. Fe is a small fictitious “thermal force” (with a
dimension of inverse length) that is applied to individual atoms. This fictitious force
Fe and the Nosé-Hoover thermostat impose an additional force AF,- on each atom 2'.

This additional force modifies the gradient of the potential energy and is given by

AF,- -_- AeiFe — Z ram-re) + 1%: Z {Mg-VF.) — ap, . (10.5)

1(751') j k(#:i)
Here, a is the N osé—Hoover thermostat multiplier acting on the momentum p, of atom
2'. a is calculated using the time integral of the difference between the instantaneous
kinetic temperature T of the system and the heat bath temperature Teq, from ('1 =
(T — Teq)/ Q, where Q is the thermal inertia. The third term in Eq. (10.5) guarantees

that the net force acting on the entire N -atom system vanishes.

10.3 Computational results

In Figure 10.3 I present the results of my nonequilibrium molecular dynamics
simulations for the thermal conductance of an isolated (10,10) nanotube aligned
along the z axis. In my calculation, I consider 400 atoms per unit cell, and use
periodic boundary conditions. Each molecular dynamics simulation run consists of
50,000 time steps of 5.0x10"16 3. My results for the time dependence of the heat
current for the particular value Fe = 0.2 A‘l, shown in Figure 10.3(a), suggest that
Jz(t) converges within the first few picoseconds to its limiting value for t——>oo in the
temperatures range below 400 K. The same is true for the quantity Jz(t)/T, shown
in Figure 10.3(b), the average of which is pr0portional to the thermal conductivity
/\ according to Eq. (10.4). My molecular dynamics simulations have been performed

for a total time length of 25 ps to represent well the long-time behavior.

121

(

(

Figure 10.3: (a) Time dependence of the axial heat flux Jz(t) in a (10,10) carbon
nanotube. Results of nonequilibrium molecular dynamics simulation at a fixed applied
thermal force Fe = 0.2 A", are shown at temperatures T = 50 K (dashed line),
100 K (solid line), and 300 K (dotted line). (b) Time dependence of Jz(t)/T, a key
quantity for the calculation of the thermal conductivity, for Fe = 0.2 A‘1 and the
same temperature values. (c) Dependence of the heat transport on the applied heat
force Fe in the simulations for T = 100 K. The dashed line represents an analytical
expression that is used to determine the thermal conductivity A by extrapolating the

A

99

 

80.0

v

60.0

1,0) [eV—A/s]

 

40.0 :5’

v v rT vvvvvvvvvvvvvvv

 

 

 

)-.

 

 

(t)/T [ev-A/s-K] D"

 

Jz

 

 

 

 

 

vvvvv

 

 

300 _
C) . \
I \
E 200: \
g : QR
100 '-
lr< .
: GM‘ ~e..._9
_£_€_0
0 - I ...............
0.0 0.1 0.2 0.3 0.4
Fem'l]

simulation data points :\ for Fe—>0.

122

 

4x104 + -
3x104 E -

2x104 [ 1

XINNWHrEfl

1x104 ’ i

 

 

 

0x104 "‘ -" -- -“ ‘---‘ --- -"
0 100 200 300 400

T [K]

Figure 10.4: Temperature dependence of the thermal conductivity /\ for a (10,10)
carbon nanotube for temperatures below 400 K.

In Figure 10.3(c) I show the dependence of the quantity

~__ , <Jz(Fe,t) >
A2530 FeTV

 

mom

on Fe. I have found that direct calculations of :\ for very small thermal forces carry a
substantial error, as they require a division of two very small numbers in Eq. (10.6). I
base my calculations of the thermal conductivity at each temperature on 16 simulation
runs, with F8 values ranging from 0.4 — 0.05 A‘l. As shown in Figure 10.3(c), data
for :\ can be extrapolated analytically for Fe—>0 to yield the thermal conductivity A,
shown in Figure 10.4.

My results for the temperature dependence of the thermal conductivity of an
isolated (10,10) carbon nanotube, shown in Figure 10.4, reflect the fact that A is

proportional to the heat capacity C' and the phonon mean free path I. At low tem-

123

peratures, l is nearly constant, and the temperature dependence of A follows that of
the specific heat. At high temperatures, where the specific heat is constant, A de-
creases as the phonon mean free path becomes smaller due to umklapp processes. My
calculations suggest that at T = 100 K, carbon nanotubes show an unusually high
thermal conductivity value of 37, 000 W/m-K. This value lies very close to the highest
value observed in any solid, A = 41, 000 W/m.K, that has been reported [185] for a
99.9% pure 12C crystal at 104 K. In spite of the decrease of A above 100 K, the room
temperature value of 6, 600 W/m-K is still very high, exceeding the reported thermal
conductivity value of 3, 320 W/m-K for nearly isotopically pure diamond [197].

I found it useful to compare the thermal conductivity of a (10,10) nanotube to
that of an isolated graphene monolayer as well as bulk graphite. For the graphene
monolayer, I unrolled the 400—atom large unit cell of the (10, 10) nanotube into a plane.
The periodically repeated unit cell used in the bulk graphite calculation contained
720 atoms, arranged in three layers. The results of my calculations, presented in
Figure 10.5, suggest that an isolated nanotube shows a very similar thermal transport
behavior as a hypothetical isolated graphene monolayer, in general agreement with
available experimental data [198, 199, 200]. Whereas even larger thermal conductivity
should be expected for a monolayer than for a nanotube, I must consider that unlike
the nanotube, a graphene monolayer is not self-supporting in vacuum. For all carbon
allotropes considered here, I also find that the thermal conductivity decreases with
increasing temperature in the range depicted in Figure 10.5.

Very interesting is the fact that once graphene layers are stacked in graphite,
the inter-layer interactions quench the thermal conductivity of this system by nearly
one order of magnitude. For the latter case of crystalline graphite, I also found my
calculated thermal conductivity values to be confirmed by corresponding observations
in the basal plane of highest-purity synthetic graphite [198, 199, 200] which are also

reproduced in the figure. I would like to note that experimental data suggest that

124

 

 

4x104 _....g..,.. _-

4 . 1 3x103
3x10 ' \

093-8 -

' 71 1 I I

 

 

 

 
    

g \~\ 2X103 T ‘1‘- :c. '

E 4 '. ' : a: ‘
3 2X10 : \‘\.1X103 '. - _ - . ‘f:£: ']
f: i . 00 300 400 ;

1x104 L

~n§.~

 

 

4 - '... . f; 1': '-'$'.' '.' .' “34': 2' : '-°
0X10

iT[Efl

Figure 10.5: Thermal conductivity A for a (10,10) carbon nanotube (solid line), in
comparison to a constrained graphite monolayer (dash-dotted line), and the basal
plane of AA graphite (dotted line) at temperatures between 200 K and 400 K. The
inset reproduces the graphite data on an expanded scale. The calculated values (solid
triangles) are compared to the experimental data of Refs. [198] (open circles), [199]
(open diamonds), and [200] (open squares) for graphite.

125

the thermal conductivity in the basal plane of graphite peaks near 100 K, similar to

my nanotube results.

10.4 Conclusions

Based on the above described difference in the conductivity between a graphene
monolayer and graphite, I should expect a similar reduction of the thermal conduc-
tivity when a nanotube is brought into contact with other systems. This should
occur when nanotubes form a bundle or rope, become nested in multi-wall nano-
tubes, or interact with other nanotubes in the “nanotube mat” of “bucky-paper” and
could be verified experimentally. Consistent with my conjecture is the low value of
A2107 W/m-K reported for the bulk nanotube mat at room temperature [186, 187].

In summary, I combined results of equilibrium and non-equilibrium molecular
dynamics simulations with accurate carbon potentials to determine the thermal con-
ductivity A of carbon nanotubes and its dependence on temperature. My results sug-
gest an unusually high value Az6,600 W/m-K for an isolated (10, 10) nanotube at
room temperature, comparable to the thermal conductivity of a hypothetical isolated
graphene monolayer or graphite. I believe that these high values of A are associated
with the large phonon mean free paths in these systems. My numerical data indicate
that in presence of inter-layer coupling in graphite and related systems, the thermal
conductivity is reduced significantly to fall into the experimentally observed value

range.

126

Chapter 11

Magnetism in All-Carbon
Structures with Negative Gaussian

Curvature

The following discussion of magnetism in nanostructured carbon materials follows

that presented in Reference [201].

11.1 Magnetism in carbon

The discovery of new nanostructured carbon phases, including fullerenes [94] and
nanotubes [11], has opened a new era in materials science. Minute changes in the
spatial arrangement of carbon atoms can profoundly alter the electronic properties of
these systems from a semiconductor to a metal or superconductor [24, 25, 26, 43]. One
could expect that structural rearrangements might also significantly change the mag-
netic properties of all-carbon allotropes from their known diamagnetic behaviour.
Only recently, a weakly magnetic all-carbon structure has been reported, formed
under high-pressure and high-temperature conditions, consisting of rhombohedrally

polymerized C60 molecules [202]. Other experimental observations suggest the occur-

127

rence of ferromagnetism in a semiconducting nanostructured carbon foam with a low
mass density [203]. As I show in the following, the unexpected magnetic behaviour
of all-carbon systems can be quantitatively interpreted using spin-polarized ab initio
calculations. These results suggest that unpaired spins are introduced by sterically
protected carbon radicals, which are immobilised in the non-alternant aromatic sys-
tem of 3192 bonded carbon with negative Gaussian curvature. This new mechanism
to generate unpaired spins in semiconductors may find a useful application in the

emerging field of spintronics.

11.2 Computational method

Here I present results for electronic and magnetic properties of an sp2 bonded all-
carbon nanostructure, consisting of a nanotube junction that is structurally related
to schwarzite [204, 205, 206, 207]. These results have been obtained in collaboration
with theory team members, who are listed as co-authors in Reference [201]. The
tetrapod system, depicted in Figure 11.1 (a), is a possible building block of the recently
synthesized nanostructured carbon foam [208, 209, 203]. Our most striking finding is
that of a ground state with a net magnetic moment. We find the origin of magnetism
in this system to be different from previous reports [210, 211, 212], which related
the spin polarization at a graphite edge to the presence of undercoordinated carbon
atoms. Our calculations suggest that unpaired spins may also be introduced by carbon
radicals, not only by undercoordinated carbon atoms. In the structure we propose,
the radicals are sterically protected in a non-alternant aromatic system of carbons

with negative Gaussian curvature.

128

 

Figure 11.1: (a) The structure of a carbon tetrapod, the building block of schwarzite.
The four extremities consist of (9,0) nanotubes with a zigzag edge. (b) Introduction
of trivalent carbon radicals, emphasized by the gray spheres, in the aromatic system
of the otherwise tetravalent carbon atoms in the tetrapod core. The radicals are asso-
ciated with heptagons, indicated by gray shading. The structure of an (c) 3192- and
(d) sp3—terminated (9,0) nanotube, with the smaller spheres indicating the hydrogen
atoms.

129

11.2.1 The tetrapod as a possible building block of carbon

nanofoam

Our electronic and magnetic structure calculations are based on the ‘tetrapod’
geometry shown in Figure 11.1 (a). The core, depicted in Figure 11.1 (b), consists
of the warped sp2 bonded building block of the Plumber’s Nightmare structure [205].
It is formed by a network of hexagons and heptagons only, and connected to four
(9,0) nanotube extremities. Depending on the degree of hydrogen passivation, the
exposed nanotube edges are terminated by 3p2 or 3123 carbon atoms [213]. In this type
of schwarzite-related structures, tubular segments of zero Gaussian curvature coexist
with saddle-like regions of negative Gaussian curvature. We expect that a range of
schwarzite-like building blocks similar to ours could be cross-linked to form a foam

structure with a very low mass density, and carry a magnetic moment.

11.3 Unpaired spins in a carbon tetrapod

We calculate the electronic structure of different tetrapods using the density
functional theory within the local spin density approximation (LSDA), employing the
Ceperley-Alder exchange-correlation functional for the local spin density [126, 127]
and ab initio pseudopotentials [128, 214]. To facilitate the numerical treatment of
very large aggregates containing hundreds of atoms within LSDA, we make use of our
recently developed approach based on an atom-centered numerical basis set [215],
proven to describe correctly similar carbon systems [216]. We use Pulay correc-
tions [217] to the Hellman-Feynman forces during structure optimization.

To discriminate between the effect of the sp2 core region and the edge region in
our magnetic structure calculations, we have considered hydrogen terminated C264
and C335 clusters with the same core structure, but a different length of the nanotube

extremities. We assume that the edge of these extremities may be terminated in three

130

different ways. Besides the bare, unterminated edge, we consider an sp2 termination
by one hydrogen atom per edge atom, shown in Figure 11.1(c). In the sp3 terminated
structure shown in Figure 11.1(d), each carbon edge atom is connected to two hy-
drogen atoms. The interface between the sp3 edge atoms and the sp2 atoms of the
nanotube is analogous to that found in diamond-graphite hybrid structures [218]. In
the following, we disregard structures with a bare edge, since they are likely to be
passivated fast under experimental conditions due to their high reactivity.

The electronic density of states for the sp2 terminated tetrapod is presented in
Figure 11.2(a). The projection of the total density of states onto the 192 carbon
atoms in the core region, including twelve heptagon rings, is shown by the solid line,
and that of the 72 atoms at the edges by the dashed line. We focus on a 1 eV wide
energy window around the Fermi level, which determines the magnetic behavior, since
electrons with higher binding energies are all paired. Artificial level broadening was
used only in the figure, for the sake of visual clarity; no broadening has been applied
during the self-consistent calculations. The density of states indicates the presence
of twelve unpaired spins in this spZ-terminated geometry. The decomposition of the
density of states suggests that eight of the twelve spins are strongly localized at the
zigzag edges, with two unpaired spins at each edge.

Presence of the remaining four unpaired spin states, which are unrelated to the
edge states, is independent of the length of the nanotube extremities. A detailed
analysis revealed that the electronic states associated with the majority spin are not
localized, but rather spread out across the entire cluster, as shown in Figure 11.3(a).
We found that the exchange splitting of the four states decreases with increasing nano-
tube length. This can be understood in terms of the exchange interaction between
two electrons, which occupy the same region. The origin of the exchange splitting is
the Coulomb repulsion between these electrons, which decreases as the region grows.

To clarify the nature of the unpaired electrons near the tetrapod edges, we per-

131

(a ) 20 I. T I V ( b) 20 h I f
.12: f: #:12913 E A
= 10 - :: r S 10 ~
5. “1" ill S
E 0 W .«E 0
g V V 5]] 'u' g
g-m- :l i g-IO’ , ‘

'l
l
O -20 Q -20 L A _
-0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4
Energy (eV) Energy (eV)

(C)A 20 - I ‘I l (d) 20 '- l r I l 4
:2 9:414]; .5
E 10 - 5 10
s A 3
3‘3 0 :‘E'. 0
E l E ]
m '10 ' " a) '10
8 -20 - 8 -20 - ‘

-0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4
Energy (eV) Energy (eV)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 11.2: Spin-resolved electronic densities of states of carbon nanostructures re-
lated to Figure 11.1, with the majority spins represented in the upper panels and
the minority spins in the lower panels. (a) Density of states of the C264 tetrapod of
Figure 11.1(a). The solid line represents the projection of the density of states onto
the 192 core carbon atoms. The dashed line represents that of the 72 carbon atoms
at the zigzag edges. (b) Electronic density of states of a 126—atom segment of an sp2
terminated (9,0) nanotube, shown in Figure 11.1(c). The solid line represents the
density of states of all the carbon atoms, and the dashed line its projection onto the
36 carbon atoms at the zigzag edge. The states denoted by “A” are doubly degen-
erate and localized at the edge. The counterparts of (a) and (b) for 3173 terminated
structures are shown in (c) for the C264 tetrapod and in (d) for the (9,0) nanotube
segment. The discrete level spectrum has been convoluted with a Gaussian with a
full width at half maximum of 0.012 eV. The Fermi level lies at E = 0.

132

 

Figure 11.3: (a) Charge density associated with the four unpaired spins, which are
unrelated to the edge states in an sp2 terminated tetrapod. (b) Wave function of a
spin-polarized edge state in an 3102 terminated (9, 0) nanotube. We use color shading
to represent the phase of the wave function.

formed an independent calculation for the (9,0) nanotube with the same edge ge-
ometry. Similar to our results for the tetrapod in Figure 11.2(a), the spin-resolved
density of states of the nanotube, shown in Figure 11.2(b), indicates the presence
of two unpaired spins per zigzag edge. Character analysis suggests that the states
in the narrow energy window used are mainly carbon-based, with only a negligi-
ble contribution from hydrogen. We show the total density of states of all carbon
atoms by the solid line, and that of the carbon edge atoms by the dashed line in
Figure 11.2(b). The spin-polarized electrons occupy doubly degenerate 7r‘ states at
each edge, labeled by A. We find one of the edge state wave functions, displayed in
Figure 11.3(b), to be symmetric and the other one antisymmetric with respect to a
mirror plane of the nanotube. The exchange splitting of about 0.5 eV, separating
the spin-up from the spin-down edge states, is comparable to that found by ab initio

plane-wave calculations for doped C-BN composite nanotubes [219].

133

To study the effect of edge termination on the net magnetic moment of the
tetrapod, we compared our above electronic structure calculations for 3p2 terminated
structures to those showing 3123 termination. In Figure 11.2(c) we present the elec-
tronic density of states for the sp3 terminated tetrapod. To model an sp3 termination,
we attached two hydrogen atoms instead of one to each carbon edge atom, as shown
in Figure 11.1(d). As we noted earlier, hydrogen-induced states within this bonding
geometry occur far away from the Fermi level, in analogy to the a-states of .9193-
bonded carbon, and do not interfere with carbon-related states within the narrow
energy window in Figure 11.2(c). Also, the localized 7r states, which we found at the
edge of the sp2 terminated structure, vanish in the sp3 bonding geometry. In this
case, we do not expect any unpaired spins associated with the sp3 edge atoms. The
projection of the density of states onto the inner 192 carbon atoms of the tetrapod
is represented by the solid line, and that of the 72 edge atoms by the dashed line in
Figure 11.2(c). In the sp3 terminated tetrapod, we find four unpaired spins, similar
to the 3p2 terminated tetrapod, where we identified four unpaired spins, which were
unrelated to the edge states.

To confirm the absence of spin polarization at the sp3 terminated edges, we show
in Figure 11.2(d) the density of states of an 3103 terminated (9,0) nanotube, as a
counterpart to results shown in Figure 11.2(b). Our spin-resolved density of states
confirms that the sp3 edge should not be spin polarized, in stark contrast to our
results for the .3122 terminated edge. These results agree with those reported for 3192
and 3193 terminated graphene strips [213].

Our findings raise several intriguing questions. The first important question is,
what distinguishes the magnetic structure, shown in Figure 11.1(a), from other carbon
structures, which are nonmagnetic, or systems, where magnetism arises due to under-
coordinated edge atoms. Equally important is to find the reason for the occurrence

of the spin polarization and its apparent robustness with respect to selected struc-

134

tural changes. To answer these questions, we calculated the magnetic ground state of
numerous other 3192 bonded carbon structures, including tetrapods terminated by C60-
derived caps, fullerenes in the size range from C20 to C240, nanotube Y—junctions, and
schwarzite—related structures with negative Gaussian curvature, containing octagons.

In our calculations, we found the number of unpaired spins not to be affected by
changing size and boundary shape of the tetrapod, but to depend sensitively on the
bonding topology. Setting aside the role of unsaturated graphitic edges in magnetism,
which has been discussed before [210, 211, 212, 213], we found the presence of higher
polygons, introducing a negative Gaussian curvature, to be a necessary condition for
a magnetic ground state. To investigate a possible conjecture that negative curvature
is also a sufficient condition to establish spin polarization, we calculated the electronic
ground state of a tetrapod, where all four extremities were terminated by Cso—derived
caps. Our finding of an energy gap of 0.34 eV between the highest occupied and the
lowest unoccupied state in this structure, which excludes the possibility of a Fermi

instability, proved this conjecture incorrect.

11.4 Magnetism due to sterically protected radi-
cals in nanostructured carbon

The only viable explanation for the occurrence of magnetism in an all-carbon
structure is the introduction and stabilization of carbon radicals. The occurrence of
radicals, which can introduce an unpaired spin, is directly related to the possibility
of pairing all valence electrons in covalent bonds. No radicals should be expected to
occur if bonds in the system can be assigned in a way that would keep all carbon
atoms tetravalent. In a graphitic structure, each tetravalent carbon atom is connected
to three neighbors by one double and two single bonds. In such a case, all electrons

should be paired and the structure should be diamagnetic. This is the case in C50

135

and other fullerenes, where all carbon atoms are kept tetravalent by associating single
bonds with the edges of pentagons. Also in a planar graphene sheet or an infinite
nanotube, all carbon atoms are equivalent and tetravalent.

Failure to find a global bonding scheme, where each carbon atom is associated
with one double and two single bonds, leads to the introduction of radicals due to
a frustration in the electronic system. As we expand on below, this is the case in
the tetrapod structure of Figure 11.1(a), where twelve trivalent carbon radicals are
introduced as a consequence of inserting twelve heptagons in the otherwise alternant
aromatic system of the graphitic honeycomb lattice. These trivalent carbons, empha-
sized in Figure 11.1(b) by the gray spheres, are the source of spin polarization. In
the structure discussed here, these radicals are sterically protected within the system
of ’single’ and ’double’ bonds imposed by the tetrapod t0pology, and occur in groups
of three.

This conclusion can be reached by analyzing the bonding scheme in the sp3
terminated tetrapod. The system of single and double bonds, shown in Figure 11.1(b),
follows the bonding at the edge of the terminating nanotube. Since two valence
electrons of the terminating carbon atoms are paired with those of hydrogen, all
bonds along the zigzag edge must be single bonds. To keep carbon atoms at the
edge tetravalent, adjacent bonds in the axial direction must be double bonds, as
shown in Figure 11.1(b). Following this scheme into the core region of the structure
necessarily leads to the above mentioned frustration, which leaves several sites, namely
the trivalent carbon radicals with a lone electron, connected to their neighbors by only
three single bonds. Following this reasoning, we can thus correlate the occurrence of
a magnetic moment in a carbon nanostructure with the bonding topology.

The lone electrons need not be localized at the radical sites, but rather gain
energy by delocalizing across the entire structure, as shown in Figure 11.3(a). We

find the radicals to form groups of three in the tetrapods. Assuming that two of the

136

three spins are paired, each group of three radicals should contribute one unpaired spin
to the structure. With four groups of radicals per tetrapod, we would expect four
unpaired spins in total, in agreement with our spin density functional calculation.
With increasing size of the nanostructure, as the unpaired spins delocalize over a
larger area, the exchange splitting should gradually decrease to zero, thus eventually
eliminating magnetism at nonzero temperatures.

Stabilization of carbon radicals by steric protection has been known since the
synthesis of triphenylmethyl in 1900 by Gomberg [220]. Radicals such as perinaph-
thenyl (or phenalenyl) have even been investigated as potential molecular conductors
by Haddon [221]. Whereas sterically unshielded carbon-based radicals has been re-
ported recently in organic systems [222], our study is the first report to our knowledge
that identifies carbon radicals in an undoped all-carbon structure with no apparent
exposed edges. We should emphasize that the magnetism found in this study is
caused by the presence of sterically protected radicals within nanometer-sized con-
ducting segments of sp2 carbon. It could also occur in other nanostructured solids,
which may be different from the tetrapod structure discussed here. We believe this to
be a general phenomenon, the discovery of which may strongly influence the emerging

field of spintronics.

11.5 Summary

In summary, we studied several sp2-bonded carbon nanostructures and identi-
fied a system of unpaired electrons that is distinct from the spin polarized states at
the zigzag edge or dangling-bond states. Our spin density functional calculations for
the tetrapod motif, which contains negatively curved graphitic surfaces, indicate the
presence of four unpaired spins in the electronic ground state. This magnetic behav-

ior originates in the presence of trivalent carbon radicals, which are introduced and

137

sterically stabilized within the aromatic system of the otherwise tetravalent carbon
atoms in the core of the tetrapod structure. We expect a similar type of magnetism

to occur in other graphitic structures with a negative Gaussian curvature.

138

BIBLIOGRAPHY

[1] M. Dresselhaus, G. Dresselhaus, and P. Eklund, Science of Fullerenes and
Carbon Nanotubes, Academic Press, San Diego, 1996.

[2] R. Saito, G. Dresselhaus, and M. Dresselhaus, Physical Properties of Carbon
Nanotubes, Imperial College Press, 1998.

[3] P. R. von Schleyer, A simple preparation of adamantane, J. Am. Chem. Soc.
79, 3292 (1957).

[4] W. H. Bragg and W. L. Bragg, Nature 91, 554 (1913).

[5] C. Cupas, P. R. von Schleyer, and D. J. Trecker, Congressane, J. Am. Chem.
Soc. 87, 917 (1965).

[6] V. Z. W. Jr., P. R. von Schleyer, G. J. Gleicher, and L. B. Rodewald, Triaman-
tane, J. Am. Chem. Soc. 88, 3862 (1966).

[7] D. Farcasiu, H. Bohm, and P. R. von Schleyer, Stepwise Elaboration of Diamon-
doid Hydrocarbons - Synthesis of Diamantane From Adamantane, J. Organic
Chem. 42, 96 (1977).

[8] W. Zhong, D. Tomanek, and G. Bertsch, Total Energy Calculations for E2:-
tremely Large Clusters: The Recursive Approach, Solid State Communications
86, 607 (1993).

[9] Y.-K. Kwon, Equilibrium Geometry and Electronic Properties of Nanostruc-
tures, PhD thesis, Michigan State University, 1999.

[10] S. Berber, Y.-K. Kwon, and D. Tomanek, Electronic and Structural Propeties
of Carbon Nanohorns, Physical Review B 62, R2291 (2000).

[11] S. Iijima, Helical microtubules of graphitic carbon, Nature 354, 56 (1991).

[12] T. Ebbesen, Carbon Nanotubes — Review of nanotubes, Physics Today 49, 26
(1996).

[13] B. Yakobson and R. Smalley, Fullerene nanotubes: 01,000,000 and beyond, Amer-
ican Scientific 85, 324 (1997).

139

[14] S. Tans, M. H. Devoret, H. Dai, A. Thess, R. E. Smalley, L. Geerligs, and
C. Dekker, Individual single-wall carbon nanotubes as quantum wires, Nature
386, 474 (1997).

[15] M. Bockrath, D. H. Cobden, P. L. Mchen, N. G. Chopra, A. Zettl, A. Thess,
and R. Smalley, Single-electron transport in ropes of carbon nanotubes, Science
275, 1922 (1997).

[16] S. Tans, A. Verschueren, and C. Dekker, Room temperature transistor based on
a single carbon nanotube, Nature 393, 49 (1998).

[17] Y.-K. Kwon, D. Tomanek, and S. Iijima, “Bucky Shuttle” Memory Device:
Synthetic Approach and Molecular Dynamics Simulations, Physical Review
Letters 82, 1470 (1999).

[18] A. Rinzler, J. Hafner, P. Nikolaev, L. Lou, S. Kim, D. Tomanek, P. Nordlander,
D. Colbert, and R. Smalley, Unraveling nanotubes - field-emission from an
atomic wire, Science 269, 1550 (1995).

[19] W. de Heer, A. Chatelain, and D. Ugarte, A carbon nanotube field-emission
electron source, Science 270, 1179 (1995).

[20] L. Chernozatonskii, Y. Gulyaev, Z. Kosakovskaja, N. Sinitsyn, G. Torgashov,
Y. Zakharchenko, E. Fedorov, and V. Valchuk, Electron field-emission from
nano-filament carbon-films, Chemical Physics Letters 233, 63 (1995).

[21] W. Choi, D. Chung, J. Kang, H. Kim, Y. Jin, I. Han, Y. Lee, J. Jung,
N. Lee, G. Park, and J. Kim, Fully sealed, high-brightness carbon-nanotube
field-emission display, Applied Physics Letters 7 5, 3129 (1999).

[22] S. Iijima and T. Ichihashi, Single-shell carbon nanotubes of I-nm diameter
S WnT growth, catalytic (Fe-Co), Nature 363, 603 (1993).

[23] D. Bethune, C. Kiang, M. de Vries, G. Gorman, R. Savoy, J. Vazquez, and
R. Beyers, Cobalt-catalysed growth of carbon nanotubes with single-atomic-layer
walls, Nature 363, 605 (1993).

[24] J. Mintmire, B. Dunlap, and C. White, Are Fullerene tubulues metallic, Physical
Review Letters 68, 631 (1992).

[25] R. Saito, M. Fujita, G. Dresselhaus, and M. Dresselhaus, Electronic-structure
of chiral graphene tubulues, Applied Physics Letters 60, 2204 (1992).

[26] N. Hamada, S. Sawada, and A. Oshiyama, New one-dimensional conductor -
graphitic microtubules, Physical Review Letters 68, 1579 (1992).

[27] J. Wild6er, L. Venema, A. Rinzler, R. Smalley, and C. Dekker, Electronic
structure of atomically resolved carbon nanotubes, Nature 391, 59 (1998).

140

[28] T. Odom, J .-L. Huang, P. Kim, and C. Lieber, Atomic structure and electronic
properties of single-walled carbon nanotubes, Nature 391, 62 (1998).

[29] W. Clauss, D. Bergeron, and A. Johnson, Atomic resolution STM imaging of
a twisted single-wall carbon nanotube, Physical Review B 58, R4266 (1998).

[30] D. Tomanek, S. G. Louie, H. J. Mamin, D. W. Abraham, R. E. Thomson,
E. Ganz, and J. Clarke, Theory and observation of highly asymmetric atomic

structure in scanning-tunneling-microscopy images of graphite, Physical Review
B 35, 7790 (1987).

[31] D. Tomanek and S. G. Louie, First-principles calculation of highly asymmet—
ric structure in scanning-tunneling-microscopy images of graphite. Uses orbital

wave functions of carbon atoms in real space, Physical Review B 37, 8327
(1988).

[32] V. Meunier and P. Lambin, Tight-Binding Computation of the S TM Image of
Bent Carbon Nanotubes, Physical Review Letters 81, 5588 (1998).

[33] A. Rubio, Spectroscopic properties and S TM images of carbon nanotubes, Ap-
plied Physics A 68, 275 (1999).

[34] V. Meunier, P. Senet, and P. Lambin, Scanning tunneling spectroscopy signature
of finite—size and connected nanotubes: A tight-binding study, Physical Review
B 60, 7792 (1999).

[35] C. Kane and E. Mele, Broken symmetries in scanning tunneling images of
carbon nanotubes, Physical Review B 59, R12759 (1999).

[36] D. Carroll, P. Redlich, P. Ajayan, J. Charlier, X. Blase, A. DeVita, and R. Car,
Electronic structure and localized states at carbon nanotubes, Physical Review
Letters 78, 2811 (1997).

[37] P. Kim, T. W. Odom, J.-L. Huang, and C. M. Lieber, Electronic Density
of States of Atomically Resolved Single- Walled Carbon Nanotubes: Van Hove
Singularities and End States, Physical Review Letters 82, 1225 (1999).

[38] M. Ge and K. Sattler, Observation of fullerene cones, Chemical Physics Letters
220, 192 (1994).

[39] A. Krishnan, E. Dujardin, M. Treacy, J. Hugdahl, S. Lynum, and T. Ebbesen,
Graphitic cones and the nucleation of curved carbon surfaces, Nature 388, 451
(1997).

[40] S. Iijima, T. Ichihashi, and Y. Ando, Pentagons, heptagons and negative cur-
vature in graphite microtubule growth - multi-wall cones terminating CNTs,
Nature 356, 776 (1992).

141

[41] S. Iijima, M. Yudasaka, R. Yamada, S. Bandow, K. Suenaga, F. Kokai, and
K. Takahashi, Nana-aggregates of single-walled graphitic carbon nano-horns,
Chemical Physics Letters 309, 165 (1999).

[42] D. Tomanek and M. A. Schluter, Growth regimes of carbon clusters, Physical
Review Letters 67, 2331 (1991).

[43] M. Schluter, M. Lannoo, M. Needels, G. Baraff, and D. Tomanek, Electro-
Phonon Coupling and Superconductivity in Alkali-Intercalated Geo Solid, Phys-
ical Review Letters 68, 526 (1992).

[44] Y.-K. Kwon, S. Saito, and D. Tomanek, Effect of inter-tube coupling on the
electronic structure of carbon nanotube ropes, Physical Review B 58, R13314
(1998).

[45] Y.-K. Kwon, D. Tomanek, Y. H. Lee, K. H. Lee, and S. Saito, Do carbon
nanotubes spin when bundled?, J. Mater. Res. 13, 2363 (1998).

[46] Y.-K. Kwon and D. Tomanek, Electronic and structural properties of multiwall
carbon nanotubes, Physical Review B 58, R16001 (1998).

[47] S. Sanvito, Y.-K. Kwon, D. Tomanek, and C. J. Lambert, Fractional quantum
conductance in carbon nanotubes, Physical Review Letters 84, 1974 (2000).

[48] S. Kim and D. Tomanek, Melting the fullerenes: A molecular dynamics study,
Physical Review Letters 72, 2418 (1994).

[49] Y.—K. Kwon, Y. H. Lee, S. G. Kim, P. Jund, D. Tomanek, and R. E. Smalley,
Morphology and Stability of Growing Multi- Wall Carbon Nanotubes, Physical
Review Letters 79, 2065 (1997).

[50] P. Lambin, A. Fonseca, J. Vigneron, J. Nagy, and A. Lucas, Structural and elec-
tronic properties of bent nanotubes, Chemical Physics Letters 245, 85 (1995).

[51] G. Treboux, P. Lapstun, and K. Silverbrook, An intrinsic carbon nanotube
heterojunction diode, Journal of Physical Chemistry B 103, 1871 (1999).

[52] V. Meunier, L. Henrard, and P. Lambin, Energetics of bent carbon nanotubes,
Physical Review B 57, 2586 (1998).

[53] M. Allen and D. Tildesley, Computer Simulation of Liquids, Clarendon Press,
Oxford, 1987.

[54] S. Nosé, A Molecular-Dynamics method for simulations in The Canonical En-
semble, Molecular Physics 52, 255 (1984).

[55] W. Hoover, Canonical Dynamics - Equilibrium Phase-Space Distributions,
Physical Review A 31, 1695 (1985).

142

[56] S. Berber, Y.—K. Kwon, and D. Tomanek, Microscopic Formation Mechanism
of Nanotube Peapods, Physical Review Letters 88, 185502 (2002).

[57] B. Smith, M. Monthioux, and D. Luzzi, Encapsulated 060 in carbon nanotubes,
Nature 396, 323 (1998).

[58] B. Burteaux, A. Claye, B. Smith, M. Monthioux, D. Luzzi, and J. Fischer, Abun-
dance of encapsulated 060 in single-wall carbon nanotubes, Chemical Physics
Letters 310, 21 (1999).

[59] B. Smith, M. Monthioux, and D. Luzzi, Carbon nanotube encapsulated fuller-
enes: a unique class of hybrid materials, Chemical Physics Letters 315, 31
(1999).

[60] B. W. Smith and D. E. Luzzi, Formation mechanism of fullerene peapods and
coaxial tubes: a path to large scale synthesis, Chemical Physics Letters 321,
169 (2000).

[61] K. Hirahara, K. Suenaga, S. Bandow, H. Kato, T. Okazaki, H. Shinohara, and
S. Iijima, One-Dimensional Metallofullerene Crystal Generated Inside Single-
Walled Carbon Nanotubes, Physical Review Letters 85, 5384 (2000).

[62] R. Dunin-Borkowski, J. Hutchinson, K. Coleman, V. Williams, J. Claridge,
A. York, C. Xu, S. Bailey, G. Brown, S. Friedrichs, and M. Green, The size
distribution, imaging and obstructing properties of C-60 and higher fullerenes

formed within arc-grown single walled carbon nanotubes, Chemical Physics Let-
ters 316, 191 (2000).

[63] R. F. Service, Nanotube ’Peapods’ Show Electrifying Promise, Science 292, 45
(2001).

[64] L. A. Girifalco, M. Hodak, and R. S. Lee, Carbon nanotubes, buckyballs, ropes,
and a universal graphitic potential, Physical Review B 62, 13104 (2000).

[65] M. Hodak and L. A. Girifalco, Quasi-one-dimensional system of molecules
inside carbon nanotubes. Exact solution for the lattice gas model and its appli-
cation to fullerene-filled nanotube, Physical Review B 64, 35407 (2001).

[66] M. Monthioux, B. W. Smith, B. Burteaux, A. Claye, J. E. Fischer, and D. E.
Luzzi, Sensitivity of single-wall carbon nanotubes to chemical processing: an
electron microscopy investigation, Carbon 39, 1251 (2001).

[67] S. Okada, S. Saito, and A. Oshiyama, Energetics and Electronic Structures of
Encapsulated C60 in a Carbon Nanotube, Physical Review Letters 86, 3835
(2001).

[68] S. Bandow, M. Takizawa, K. Hirahara, M. Yudasaka, and S. Iijima, Raman
scattering study of double-wall carbon nanotubes derived from the chains of

fullerenes in single-wall carbon nanotubes, Chemical Physics Letters 337, 48
(2001).

143

[69]

[70]

[71]

[72]

[73]

[74]

[75]

[76]

[77]

[78]

[79]

[80]

[81]

S. Han, M. Yoon, S. Berber, N. Park, E. Osawa, J. Ihm, and D. Tomanek,
Microscopic Mechanism of Fullerene Fusion, Physical Review Letters .

M. Yoon, S. Berber, and D. Tomanek, Energetics and Packing of Fullerenes in
Nanotube Peapods, Submitted to Physical Review Letters .

G. C. McIntosh, M. Yoon, S. Berber, and D. Tomanek, Diamondoids as Building
Blocks of Functional Nanostructures, Physical Review B (2004).

B. Smith, D. Luzzi, and Y. Achiba, Tumbling atoms and evidence for charge
transfer in La‘2 @030 @S WN T, Chemical Physics Letters 331, 137 (2000).

D. L. Dorset and J. R. Fryer, Electron Crystallographic Analysis of Sublimed
C70 Buckminsterfullerene, Journal of Physical Chemistry B 101, 3968 (1997).

H. Kawada, Y. Fujii, H. Nakao, Y. Murakami, T. Watanuki, H. Suematsu,
K. Kikuchi, Y. Achiba, and I. Ikemoto, Structural aspects of 032 and C76
crystals studied by z—ray diffraction, Physical Review B 51, 8723 (1995).

Y. Saito, T. Yoshikawa, N. Fujimoto, and H. Shinohara, 034 thin films grown
epitaxially on mica, Physical Review B 48, 9182 (1993).

R. Beyers, C. H. Kiang, R. D. Johnson, J. R. Salem, M. S. Devries, C. S.
Yannoni, D. S. Bethune, H. C. Dorn, P. Burbank, K. Harich, and S. Stevenson,

Preparation and structure of crystals of the metallofullerenes S 02 @034, Nature
370, 196 (1994).

K. Hirahara, S. Bandow, K. Suenaga, H. Kato, T. Okazaki, H. Shinohara, and
S. Ijima, Electron diflraction study of one-dimensional crystals of fullerenes,
Physical Review B 64, 115420 (2001).

B. Smith, R. M. Russo, S. B. Chikkannanavar, and D. E. Luzzi, High-yield syn-
thesis and one-dimensional structure of 060 encapsulated in single-wall carbon
nanotubes, Journal of Applied Physics 91, 9333 (2002).

H. Kuzmany, R. Pfeiffer, C. Kramberger, T. Pichler, X. Liu, M. Knupfer,
J. Fink, H. Kataura, Y. Achiba, B. Smith, and D. Luzzi, Analysis of the con-

centration of C60 fullerenes in single wall carbon nanotubes, Applied Physics
A 76, 449 (2003).

K. P. Meletov, S. Assimopoulos, I. Tsilija, G. A. Kourouklis, J. Arvanitidis,
S. Ves, B. Sundqvist, and T. Wagberg, High-pressure-induced metastable phase
in tetragonal 2D polymeric Geo, Chemical Physics Letters 341, 435 (2001).

V. D. Blank, S. G. Buga, N. R. Serebryanaya, V. N. Denisov, G. A. Dubitsky,
A. N. Ivlev, B. N. Mavrin, and M. Y. Popov, Ultrahard and superhard car-
bon phases produced from C60 by heating at high-pressure-structural and Raman
studies, Physical Letters A 205, 208 (1995).

144

[82] M. Hodak and L. Girifalco, Cohesive properties of fullerene-filled nanotube
ropes, Chemical Physics Letters 363, 93 (2002).

[83] M. Hodak and L. Girifalco, Ordered phases of fullerene molecules formed inside
carbon nanotubes, Physical Review B 67, 075419 (2003).

[84] H. Ulbricht, G. Moos, and T. Hertel, Interaction of C60 with carbon nanotubes
and graphite, Physical Review Letters 90, 095501 (2003).

[85] Y. Wang, D. Tomanek, and G. F. Bertsch, Stiffness of a solid composed of C60
clusters, Physical Review B 44, R6562 (1991).

[86] A. J. Stone and D. J. Wales, Theoretical studies of icosahedral C60 and some
related species, Chemical Physics Letters 128, 501 (1986).

[87] B. R. Eggen, M. I. Heggie, G. Jungnickel, C. D. Latham, R. Jones, and P. R.
Briddon, Autocatalysis during fullerene growth, Science 272, 87 (1996).

[88] Z. Slanina, X. Zhao, F. Uhlik, M. Ozawa, and E. Osawa, Computational Mod-
elling of the Elemental Catalysis in the Stone- Wales Fullerene Rearrangements,
Journal of Organometal. Chemistry 599, 57 (2000).

[89] F. Banhart, Irradiation effects in carbon nanostructures, Rep. Prog. Phys. 62,
1181 (1999).

[90] J. G. Lavin, S. Subramoney, S. B. R. S. Ruoff, and D. Tomanek, Scrolls and
nested tubes in multiwall carbon nanotubes, Carbon 40, 1123 (2002).

[91] K. Umemoto, S. Saito, S. Berber, and D. Tomanek, Carbon foam: Spanning
the phase space between graphite and diamond, Physical Review B 64, 193409
(2001).

[92] D. Tomanek, K. Umemoto, S. Saito, and S. Berber, Hierarchical Assembly of
Nanostructured Carbon Foam, Molecular Crystals and Liquid Crystals 386, 189
(2002).

[93] S. Berber and D. Tomanek, Stability differences and conversion mechanism
between nanotubes and scrolls, Physical Review B , BR (2004).

[94] H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl, and R. E. Smalley, C60:
Buckminsterfullerene, Nature 318, 162 (1985).

[95] D. Ugarte, Curling and closure of graphitic networks under electron-beam irra—
diation, Nature 359, 707 (1992).

[96] X. F. Zhang, X. B. Zhang, G. V. Tendeloo, S. Amelinckx, M. O. de Beeck, and

J. V. Landuyt, Carbon nanotubes - their formation process and observation by
electron-microscopy, Journal of Crystal Growth 130, 368 (1993).

145

[97] O. Zhou, R. M. Fleming, D. W. Murphy, C. H. Chen, R. C. Haddon, A. P.
Ramirez, and S. H. Glarum, Defects in Carbon Nanostructures, Science 263,
1744 (1994).

[98] S. Bandow, Radial thermal expansion of purified multiwall carbon nanotubes
measured by X-ray diffraction, Japanese Journal of Applied Physics: Part 2-
Letters 36, L1403 (1997).

[99] V. Z. Mordkovich, M. Baxendale, S. Yoshimura, and R. P. H. Chang, Intercu-
lation into carbon nanotubes, Carbon 34, 1301 (1996).

[100] F. Leroux, K. Metenier, S. Gautier, E. Frackowiak, S. Bonnamy, and F. Beguin,
Electrochemical insertion of lithium in catalytic multi-walled carbon nanotubes,
Journal of Power Sources 82, 317 (1999).

[101] E. Frackowiak, K. Metenier, V. Bertagna, and F. Beguin, Supercapacitor elec-
trodes from multiwalled carbon nanotubes, Applied Physics Letters 77, 2421
(2000).

[102] F. Krumeich, H. J. Muhr, M. Niederberger, F. Bieri, and R. Nesper, The cross-
sectional structure of vanadium oxide nanotubes studied by transmission electron
microscopy and electron spectroscopic imaging, Zeitschrift Fur Anorganische
und Allgemeine Chemie 626, 2208 (2000).

[103] S. Amelinckx, D. Bernearts, X. B. Zhang, G. V. Tendeloo, and J. V. Landuyt, A
structure model and growth-mechanism for multishell carbon nanotubes, Science
267, 1334 (1995).

[104] D. Bernaerts, A. A. Lucas, and S. Amelinckx, On a peculiar contrast effect
associated with carbon nanotubes, Philosophical Magazine A 76, 267 (1997).

[105] D. Bernaerts, S. Amelinckx, G. V. Tendeloo, and J. V. Landuyt, Microstructure
and formation mechanism of cylindrical and conical scrolls of the misfit layer
compounds PbanS2n+1, Journal of Crystal Growth 172, 433 (1997).

[106] Y. Maniwa, R. Fujiwara, H. Kira, E. N. H. Tou, M. Takata, M. Sakata, A. Fu-
jiwara, X. L. Z. S. Iijima, and Y. Ando, Multiwalled carbon nanotubes grown
in hydrogen atmosphere: An x-ray diffraction study, Physical Review B 64,
073105 (2001).

[107] A. Love, A Treatise on the Mathematical Theory of Elasticity, Dover Publica-
tions, New York, 1944.

[108] D. Tomanek, W. Zhong, and E. Krastev, Stability of multishell fullerenes,
Physical Review B 48, 15461 (1993).

[109] D. Robertson, D. Brenner, and J. Mintmire, Energetics of nanoscale graphitic
tubules, Physical Review B 45, 12592 (1992).

146

[110] V. P. Dravid, X. Lin, Y. Wang, X. K. Wang, A. Yee, J. B. Ketterson, and
R. H. P. Chang, Buckytubes and derivatives - their growth and implications for
buckyball formation, Science 259, 1601 (1993).

[111] Y. Miyamoto, S. Berber, M. Yoon, A. Rubio, and D. Tomanek, Onset of
nanotube decay under extreme thermal and electronic excitations, Physica B
323, 78 (2002).

[112] Y. Miyamoto, S. Berber, M. Yoon, A. Rubio, and D. Tomanek, Reconstruction
of Defective Carbon Nanotubes Induced by Electronic Excitations, Submitted
to Physical Review B (2004).

[113] P. Ball, Designing the molecular world, Princeton University Press, Princeton,
1994.

[114] P. C. Collins, M. S. Arnold, and P. Avouris, Engineering carbon nanotubes and
nanotube circuits using electrical breakdown, Science 292, 706 (2001).

[115] J. Appenzeller, R. Martel, V. Derycke, M. Radosavjevic, S. Wind, D. Neu-
mayer, and P. Avouris, Carbon nanotubes as potential building blocks for future
nanoelectronics, Microelectronic Engineering 64, 391 (2002).

[116] R. Martel, V. Derycke, C. Lavoie, J. Appenzeller, K. K. Chan, J. Tersoff, and
P. Avouris, Ambipolar electrical transport in semiconducting single-wall carbon
nanotubes, Physical Review Letters 87 , 256805 (2001).

[117] M. Igami, T. Nakanishi, and T. Ando, Conductance of carbon nanotubes with
a vacancy, Journal of Physical Society of Japan 68, 716 (1999).

[118] H. J. Choi, J. Ihm, S. G. Louie, and M. L. Cohen, Defects, quasibound states,
and quantum conductance in metallic carbon nanotubes, Physical Review Let-
ters 84, 2917 (2000).

[119] P. E. Bléichl and J. H. Stathis, Hydrogen Electrochemistry and Stress-Induced
Leakage Current in Silica, Physical Review Letters 83, 372 (1999).

[120] A. Yokozawa and Y. Miyamoto, Hydrogen dynamics in SiOg triggered by elec-
tronic excitations, Journal of Applied Physics 88, 4542 (2000).

[121] O. Sugino and Y. Miyamoto, Density functional approach to electron dynamics:
Stable simulation under a self-consistent field, Physical Review B 59, 2579
(1999).

[122] O. Sugino and Y. Miyamoto, Density-functional approach to electron dynamics:
Stable simulation under a self-consistent field (vol 59, pg 2579, 19.99), Physical
Review B 66, 89901 (2002).

[123] L. C. Qin, X. L. Zhao, K. Hirahara, Y. Miyamoto, Y. Ando, and S. Iijima,
Materials science - The smallest carbon nanotube, Nature 408, 50 (2000).

147

[124] N. Wang, Z. K. Tang, G. D. Li, and J. S. Chen, Materials science - Single-walled
4 angstrom carbon nanotube arrays, Nature 408, 50 (2000).

[125] C. Ronchi, R. Beukers, H. Heinz, J. P. Hiernaut, and R. Selfslag, Graphite
Melting under Laser-Pulse Heating, International Journal of ThermOphysics
13, 107 (1992).

[126] D. M. Ceperley and B. J. Alder, Ground-state of the electron-gas by a stochastic
method, Physical Review Letters 45, 566 (1980).

[127] J. P. Perdew and A. Zunger, Self—interaction correction to density-functional
approximations for many-electron systems, Physical Review B 23, 5048 (1981).

[128] N. Troullier and J. L. Martins, Efficient pseudopotentials for plane-wave calcu-
lations, Physical Review B 43, 1993 (1991).

[129] S. Berber, Y.-K. Kwon, and D. Tomanek, Bonding and Energy Dissipation in
a Nanohook Assembly, Physical Review Letters 91, 165503 (2003).

[130] G. Overney, W. Zhong, and D. Tomanek, Structural rigidity and low frequency
vibrational modes of long carbon tubules, Z. Phys. D 27, 93 (1993).

[131] M. Treacy, T. Ebbesen, and J. Gibson, Exceptionally high Young’s modulus
observed for individual carbon nanotubes, Nature 381, 678 (1996).

[132] M. R. Falvo, G. J. Clary, R. M. T. II, V. Chi, F. P. B. Jr, S. Washburn, and
R. Superfine, Bending and buckling of carbon nanotubes under large strain,
Nature 389, 582 (1997).

[133] B. I. Yakobson, C. Brabec, and J. Bernholc, Nanomechanics of Carbon Tubes:
Instabilities beyond Linear Response, Physical Review Letters 76, 2511 (1996).

[134] J. P. Lu, Elastic properties of carbon nanotubes and nanoropes, Physical Review
Letters 79, 1297 (1997).

[135] W. N. Reynolds, Physical Poperties of Graphite, Elsevier, Amsterdam, 1968.

[136] R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Tunneling conductance of
connected carbon nanotubes, Physical Review B 53, 2044 (1996).

[137] S. Iijima, Carbon Nanotubes, MRS Bulletin 19, 43 (1994).
[138] J .-C. Gabriel, (private communication).
[139] B. Dunlap, Connecting carbon tubules, Physical Review B 46, 1933 (1992).

[140] J. C. Charlier, T. W. Ebbesen, and P. Lambin, Structural and electronic prop-
erties of pentagon-heptagon pair defects in carbon nanotubes, Physical Review
B 53, 11108 (1996).

148

[141] J. Tersoff, Empirical interatomic potential for carbon, with applications to
amorphous-carbon, Physical Review Letters 61, 2879 (1988).

[142] S. Fahy, S. Louie, and M. Cohen, Pseudopotential total-energy study of the
transition from rhombohedral graphite to diamond, Physical Review B 34, 1191
(1986).

[143] S. Berber, Y.-K. Kwon, and D. Tomanek, Unusually High Thermal Conductivity
of Carbon Nanotubes, Physical Review Letters 84, 4613 (2000).

[144] P. Simonis, A. Volodin, E. Seynaeve, P. Lambin, and C. V. Haesendonck, Com-
parative study of a coiled carbon nanotube by atomic force misroscopy and scan-
ning electron misroscopy, in Science and Application of Nanotubes, edited by
D. Tomanek and R. J. Enbody (Kluwer Academic, New York, 2000), p. 83 .

[145] Z. F. Ren, Z. P. Huang, J. W. Xu, J. H. Wang, P. Bush, M. P. Siegal, and P. N.
Provencio, Synthesis of large arrays of well-aligned carbon nanotubes on glass,
Science 282, 1105 (1999).

[146] S. Berber, E. Osawa, and D. Tomanek, Rigid Crystalline Phases of Polymerized
Fullerenes, Submitted to Physcial Review B (2004).

[147] C. Kittel, Introduction to Solid State Physics, Wiley, New York, 1996.

[148] S. J. Woo, E. Kim, S. H. Lee, Y. H. Lee, I. C. Jeon, and S. Y. Hwang, Bulk
modulus of C60 molecule via tight-binding method, Phys. Lett. A 162, 501
(1992).

[149] N. R. Serebryanaya, V. D. Blank, V. A. Ivdenko, and L. A. Chernozatonskii,
Pressure-induced superhard phase of 060, Solid State Commun. 118, 183 (2001).

[150] A. V. Talyzin, L. S. Dubrovinsky, M. Oden, T. L. Bihan, and U. Jansson, In situ
x-ray diffraction study of 060 polymerization at high pressure and temperature,
Phys. Rev. B 66, 165409 (2002).

[151] S. Okada, S. Saito, and A. Oshiyama, New Metallic Crystalline Carbon: Three
Dimensionally Polymerized 060 Fulllerite, Phys. Rev. Lett. 83, 1986 (1999).

[152] E. Burgos, E. Halac, R. Weht, H. Bonadeo, E. Artacho, and P. Ordejon, New
superhard phases for three-dimensional C—60-based fullerites, Phys. Rev. Lett.
85, 2328 (2000).

[153] C. A. Perottoni and J. A. H. da Jornada, First-principles calculation of the

structure and elastic properties of a 3D-polymerized fullerite, Phys. Rev. B 65,
224208 (2002).

[154] N. R. Serebryanaya and L. A. Chernozatonskii, Modelling and interpretation
of the experimental data on the 3D polymerized C60 fullerites, Solid State
Commun. 114, 537 (2000).

149

[155] L. Chernozatonskii, N. Serebryanaya, and B. Mavrin, The superhard crystalline
three-dimensional polymerized C—60 phase, Chem. Phys. Lett. 316, 199 (2000).

[156] S. Okada, S. Saito, and A. Oshiyama, Comment on ”New metallic crystalline
carbon: Three dimensionally polymerized C-60 fullerite” - Okada, Saito, and
Oshiyama reply, Phys. Rev. Lett. 85, 5672 (2000).

[157] A. M. Rao, P. C. Eklund, U. D. Venkateswaran, J. Tucker, M. A. Duncan, G. M.
Bendele, P. W. Stephens, J .-L. Hodeau, L. Marques, M. Nunez-Regueiro, I. O.
Bashkin, E. G. Ponyatovsky, and A. P. Morovsky, Properties of C60 polymerized
under high pressure and temperature, Applied Physics A 44, R6562 (1991).

[158] K. P. Meletov, J. Arvanitidis, S. Assimopoulos, G. A. Kourouklis, and
B. Sundqvist, Pressure-induced transformations and optical properties of the

two-dimensional tetragonal polymer of C60 at pressures up to 30 GPa, J. Exp.
Theor. Phys. 95, 736 (2002).

[159] A. V. Okotrub, V. V. Belavin, L. G. Bulusheva, V. A. Davydov, T. L. Makarova,
and D. Tomanek, Electronic structure and properties of rhombohedrally poly-
merized C60, J. Chem. Phys. 115, 5637 (2001).

[160] K. Honda, E. Osawa, Z. Slanina, and T. Matsumoto, Thermal transformation
of ring-opened [2+2/Cso dimer into a wide-bridged Cm isomer. A computa-
tional evaluation of fulvalene-naphthalene rearrangements, Fullerene Science
and Technology 4, 819 (1996).

[161] S. Osawa, M. Sakai, and E. Osawa, Nature of Cyclobutane Bonds in the Neutral
[2+2] Dimer of 060, Journal of Physical Chemistry A 101, 1378 (1997).

[162] H. Ueno, S. Osawa, E. Osawa, and K. Takeuchi, Stone- Wales rearrangement
pathways from the hinge-opened [2+2] Cso dimer to IPR 0120 fullerenes: vi-
brational analysis of intermediates, Fullerene Science and Technology 6, 319
(1998).

[163] Y.-K. Kwon, S. Berber, and D. Tomanek, Thermal Contraction of Carbon
Fullerenes and Nanotubes, Physical Review Letters 92, 015901 (2004).

[164] H. Mavoori and S. Jin, Low thermal-expansion copper composites via negative
CTE metallic elements, JOM-Journal of The Minerals Metals & Materials
Society 50, 70 (1998).

[165] X. C. Li, J. Stampfl, and F. B. Prinz, Mechanical and Thermal Expansion Be-
havior of Laser Deposited Metal Matrix Composites of Invar and TiC, Materials
Science and Engineering A 282, 86 (2000).

[166] T. Ito, T. Suganuma, and K. Wakashima, A micromechanics-based analysis
for tailoring glass-fiber—reinforced thermoplastic laminates with near-zero coef-
ficients of thermal expansion, Composites Science and Technology 60, 1851
(2000).

150

[167] R. Kubo, Rubber Elasticity, Kawade Shobo Shinsha, Tokyo, 1946.

[168] D. Lacks and G. Rutledge, Mechanisms for axial thermal contraction in polymer
crystals - Polyethylene vs isotactic Polypropylene, Chemical Engineering Science
49, 2881 (1994).

[169] P. Papanek and J. Fischer, Molecular-Dynamics simulation of crystalline Trans-
polyacetylene, Physical Review B 48, 12566 (1993).

[170] R. Baughman and E. Turi, Negative thermal-expansion of a Polydiacetylene
single-crystal, Journal of Poylmer Science Part B-Polymer Physics 11, 2453
(1973).

[171] R. Munn, Role of elastic-constants in negative thermal-expansion of axial solids,
Journal of Physics C: Solid State Physics 5, 535 (1972).

[172] D. Riley, The thermal expansion of graphite: Part II theoretical, Proceedings
of The Physical Society of London 57 , 486 (1945).

[173] W. Morgan, T hermal-Expansion coefficients of graphite crystals, Carbon 10,
73 (1972).

[174] G. Belenkii, E. Salaev, R. Suleimanov, N. Abdullaev, and V. Shteinshraiber,
The nature of negative linear expansion in layer crystals C, BN, GaS, GaSe
AND InSe, Solid State Communications 53, 967 (1985).

[175] R. Roy and D. Agrawal, Thermal-expansion materials not so new, Nature 388,
433 (1997).

[176] T. Mary, J. Evans, T. Vogt, , and A. Sleight, Negative thermal expansion from
0.3 to 1050 Kelvin in ZrW2 08, Science 272, 90 (1996).

[177] G. Tseng and J. Ellenbogen, Nanotechnology - Toward nanocomputers, Science
294, 1293 (2001).

[178] E. Minot, Y. Yaish, V. Sazonova, J .-Y. Park, M. Brink, and P. Mchen, Tuning
Carbon Nanotube Band Gaps with Strain, Physical Review Letters 90, 156401
(2003).

[179] M. Cadek, J. Coleman, V. Barron, K. Hedicke, and W. Blau, Morphological and
mechanical properties of carbon-nanotube-reinforced semicrystalline and amor-
phous polymer composites, Applied Physics Letters 81, 5123 (2002).

[180] A. Thess, R. Lee, P. Nikolaev, H. Dai, P. Petit, J. Robert, C. Xu, Y. Lee, S. Kim,
D. Colbert, G. Scuseria, D. Tomanek, J. Fischer, and R. Smalley, Crystalline
ropes of metallic carbon nanotubes, Science 273, 483 (1996).

[181] R. H. Baughman and A. Soldatov, (private communication).

151

[182] Y.-K. Kwon and D. Tomanek, Orientational melting in carbon nanotube ropes,
Physical Review Letters 84, 1483 (2000).

[183] Y. Maniwa, R. Fujiwara, H. Kira, H. Tou, H. Kataura, S. Suzuki, Y. Achiba,
E. Nishibori, M. Takata, M. Sakata, A. Fujiwara, , and H. Suematsu, Thermal
expansion of single-walled carbon nanotube (SWNT) bundles: X-ray difiraction
studies, Physical Review B 64, 241402 (2001).

[184] N. Raravikar, P. Keblinski, A. Rao, M. Dresselhaus, L. Schadler, and P. Ajayan,
Temperature dependence of radial breathing mode Raman frequency of single-
walled carbon nanotubes, Physical Review B 66, 235424 (2002).

[185] L. Wei, P. Kuo, R. Thomas, T. Anthony, and W. Banholzer, Thermal conduc-
tivity of isotopically modified single crystal diamond, Physical Review Letters
70, 3764 (1993).

[186] J. Hone, M. Whitney, C. Piskoti, and A. Zettl, Thermal conductivity of single-
walled carbon nanotubes, Physical Review B 59, R2514 (1999).

[187] J. Hone, M. Whitney, and A. Zettl, Thermal conductivity of single-walled carbon
nanotubes, Synthetic Metals 103, 2498 (1999).

[188] C. Uher, Thermal conductivity of graphite, in Landolt—Bb'rnstein: Numeri-
cal Data and Functional Relationships in Science and Technology, edited by
O. Madelung and GK. White, New Series, Group III, Vol. 15c (Springer-Verlag,
Berlin, 1991), pp. 426-448 .

[189] P. Jund and R. Jullien, Molecular—dynamics calculation of the thermal conduc-
tivity of vitreous silica, Physical Review B 59, 13707 (1999).

[190] M. Schoen and C. Hoheisel, The shear viscosity of a Lennard-Jones fluid cal-
culated by equlibrium molecular dynamics, Molecular Physics 56, 653 (1985).

[191] D. Levesque and L. Verlet, Molecular-Dynamics calculations of transport-
coefl‘iecients, Molecular Physics 61, 143 (1987).

[192] D. McQuarrie, Statistical Mechanics, Harper and Row, London, 1976.

[193] A. Maeda and T. Munakata, Thermal Conductivity of ID lattice by NEMD,
Physical Review E 52, 234 (1995).

[194] D. Evans, Homogeneous NEMD algorithm for thermal-conductivity - Applica-
tion of non-canonical linear response theory, Physics Letters A 91, 457 (1982).

[195] D. Hansen and D. Evans, A generalized heat-flow algorithm, Molecular Physics
81, 767 (1994).

[196] D. Rapaport, The Art of Molecular Dynamics Simulation, Cambridge Univer-
sity Press, Cambridge, 1998.

152

[197] T. Anthony, W. Banholzer, J.F.Fleischer, L. Wei, P. Kuo, R. Thomas, and
R. Pryor, Thermal diffusivity of isotopically enriched 120 diamond, Physical
Review B 42, 1104 (1990).

[198] T. Nihira and T. Iwata, Thermal resistivity changes in electron-irradiated
pyrolytic-graphite, Japanese Journal of Applied Physics 14, 1099 (1975).

[199] M. Holland, C. Klein, and W. Straub, The lorenz number of graphite at very
low temperatures, Journal of Physics and Chemistry of Solids 27 , 903 (1966).

[200] A. de Combarieu, Conductibilite’ Thermique de Graphite Quais Monocristallin
et Effets D’irradiation Aux Neutrons I. Mesures, Journal de Physique (Paris)
28, 951 (1967).

[201] N. Park, M. Yoon, S. Berber, J. Ihm, E. Osawa, and D. Tomanek, Magnetism
in all-carbon structures with negative Gaussian Curvature, Physical Review
Letters 91, 237204 (2003).

[202] T. L. Makarova, B. Sundqvist, R. Héhne, P. Esquinazi, Y. Kopelevich,
P. Scharff, V. A. Davydov, L. S. Kashevarova, and A. V. Rakhmanina, Magnetic
carbon, Nature 413, 716 (2001).

[203] A. Rode, (private communication).
[204] A. L. Mackay and H. Terrones, Diamond from graphite, Nature 352, 762 (1991).

[205] D. Vanderbilt and J. Tersoff, Negative-curvature fullerene analog of C60, Phys-
ical Review Letters 68, 511 (1992).

[206] L. A. Bursill and L. Bourgeois, Image-analysis of a negatively curved graphitic
sheet model for amorphous-carbon, Physics Letters B 9, 1461 (1995).

[207] S. T. Hyde and M. O’Keeffe, Elastic warping of graphitic carbon sheets: Relative
energies of some fullerenes, schwarzites and buckytubes, Phil. Trans. Roy. Soc.
Lond. A 354, 1999 (1996).

[208] A. V. Rode, S. T. Hyde, E. G. Gamaly, R. G. Elliman, D. R. McKenzie, and
S. Bulcock, Structural analysis of a carbon foam formed by high pulse-rate laser
ablation, Applied Physics A-Materials Science & Processing 69, S755 (1999).

[209] A. V. Rode, E. G. Gamaly, and B. Luther-Davies, Formation of cluster-
assembled carbon nano-foam by high—repetition-rate laser ablation, Applied
Physics A-Materials Science & Processing 70, 135 (2000).

[210] M. Fujita, K. Wakabayashi, K. Nakada, and K. Kusakabe, Peculiar localized
state at zigzag graphite edge, Journal of The Physical Society of Japan 65, 1920
(1996).

153

[211] K. Wakabayashi, M. Sigrist, and M. Fujita, Spin wave mode of edge-localized
magnetic states in nanographite zigzag ribbons, Journal of The Physical Society
of Japan 67, 2089 (1998).

[212] K. Nakada, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus, Edge state in
graphene ribbons: Nanometer size efiect and edge shape dependence, Physical
Review B 54, 17954 (1996).

[213] K. Kusakabe and M. Maruyama, Magnetic nanographite, Physical Review B
67, 092406 (2003).

[214] L. Kleinman and D. M. Bylander, Eflicacious form for model pseudopotentials,
Physical Review Letters 48, 1425 (1982).

[215] O. F. Sankey and D. J. Niklewski, Ab initio multicenter tight-binding model
for molecular-dynamics simulations and other applications in covalent system,
Physical Review B 40, 3979 (1989).

[216] N. Park, K. Lee, S. Han, J. Yu, and J. Ihm, Energetics of large carbon clus-
ters:Crossover from fullerenes to nanotube, Physical Review B 65, 121405
(2002).

[217] M. Scheffier, J. P. Vigneron, and G. B. Bachelet, Energetics of large carbon
clusters: Crossover from fullerenes to nanotubes, Physical Review B 31, 6541
(1985).

[218] A. T. Balaban, D. J. Klein, and C. A. Folden, Diamond-graphite hybrids,
Chemical Physics Letters 217, 266 (1994).

[219] J. Choi, Y.-H. Kim, K. J. Chang, and D. Tomanek, Itinerant ferromagnetism
in heterostructured C/BN nanotubes, Physical Review B 67 , 125421 (2003).

[220] M. Gomberg, An Instance of trivalent carbon: Triphenylmethyl, Journal of
American Chemical Society 22, 757 (1900).

[221] R. C. Haddon, Design of Organic Metals and Superconductors, Nature (London)
256, 394 (1975).

[222] M. E. Itkis, X. Chi, A. W. Cordes, and R. C. Haddon, Magneto-Opto-Electronic
Bistability in a Phenalenyl-Based Neutral Radical, Science 296, 1443 (2002).

154

IIIIIIIIIIIIIIIIIIIIIIIIIIII

[Illl]lll][[l]]l]]l]l][]]]ll]]]]l]lll