Carbon Nanotubes - DiVA portal722664/FULLTEXT01.pdf · an insulator. These ranges of properties of course make carbon nanotubes highly interesting for many applications. Carbon nanotubes

Carbon Nanotubes

A Theoretical study of Young's modulus

Kolnanorör

En teoretisk studie av Youngs modul

Tore Fredriksson

Health, Science and Technology

Physics

30

Thijs Holleboom

Lars Johansson

2014-06

Faculty of Health, Science and Technology

Department of Engineering and Physics

Carbon Nanotubes

A Theoretical study of Young’s modulus

Author:Tore Fredriksson

Supervisor:Thijs Holleboom

Examiner:Lars Johansson

January 29, 2014

Abstract

Carbon nanotubes have extraordinary mechanical, electrical, thermal andoptical properties. They are harder than diamond yet flexible, have betterelectrical conductor than copper, but can also be a semiconductor or evenan insulator. These ranges of properties of course make carbon nanotubeshighly interesting for many applications. Carbon nanotubes are already usedin products as hockey sticks and tennis rackets for improving strength andflexibility. Soon there are mobile phones with flexible screens made fromcarbon nanotubes. Also, car- and airplane bodies will probably be mademuch lighter and stronger, if carbon nanotubes are included in the construc-tion. However, the real game changers are; nanoelectromechanical systems(NEMS) and computer processors based on graphene and carbon nanotubes.

In this work, we study Young’s modulus in the axial direction of carbonnanotubes. This has been done by performing density functional theorycalculations. The unit cell has been chosen as to accommodate for tubes ofdifferent radii. This allows for modelling the effect of bending of the bondsbetween the carbon atoms in the carbon nanotubes of different radii. Theresults show that Young’s modulus decreases as the radius decreases. Ineffect, the Young’s modulus declines from 1 to 0.8 TPa. This effect can beunderstood because the bending diminishes the pure sp2 character of thebonds.

These results are important and useful in construction, not only when usingcarbon nanotubes but also when using graphene. Our results point towardsa Young’s modulus that is a material constant and, above a certain crit-ical value, only weakly dependent on the radius of the carbon nanotube.Graphene can be seen as a carbon nanotube with infinite radius.

I

Acknowledge

First I want to thank my supervisor Thijs Holleboom for the many hoursof discussions on how to make this the best thesis it can be. Also, KristerSvensson for his many insights into carbon nanotubes that he has sharedwith me. My fiancee deserves one huge thank you for her support duringthis whole ordeal. Finally to the rest of my family and my friends for makingthe hard times durable, and the good times even better.

II

Contents

1 Introduction 5

2 Carbon 72.1 Diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.3 Lonsdaleite . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Graphite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Fullerenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.1 Buckminsterfullerenes . . . . . . . . . . . . . . . . . . 102.3.2 Carbon nanotubes . . . . . . . . . . . . . . . . . . . . 102.3.3 Nanobuds . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.4 Nanofoam . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.5 Carbyne . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4.1 Electronic Properties . . . . . . . . . . . . . . . . . . . 122.4.2 Mechanical Properties . . . . . . . . . . . . . . . . . . 132.4.3 Optical Properties . . . . . . . . . . . . . . . . . . . . 14

3 Carbon nanotube 153.1 Discovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Categories of carbon nanotubes . . . . . . . . . . . . . . . . . 17

3.2.1 Chirality . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.2 Single walled . . . . . . . . . . . . . . . . . . . . . . . 183.2.3 Multi walled . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1

CONTENTS 2

3.3.1 Strength . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3.2 Electrical properties . . . . . . . . . . . . . . . . . . . 213.3.3 Thermal properties . . . . . . . . . . . . . . . . . . . . 22

3.4 Extreme carbon nanotubes . . . . . . . . . . . . . . . . . . . . 223.5 Future uses for graphene and carbon nanotubes . . . . . . . . 24

3.5.1 Material additives . . . . . . . . . . . . . . . . . . . . . 243.5.2 Nanomechanics . . . . . . . . . . . . . . . . . . . . . . 243.5.3 Nanoelectromechanical systems . . . . . . . . . . . . . 253.5.4 Solar cells . . . . . . . . . . . . . . . . . . . . . . . . . 263.5.5 Energy storage . . . . . . . . . . . . . . . . . . . . . . 263.5.6 Biomedicine . . . . . . . . . . . . . . . . . . . . . . . . 27

3.6 Toxicity and biocompatibility graphene and carbon nanotubes 29

4 Orbital Hybridization 304.1 spn Hybridization . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.1.1 sp Hybridization (linear) . . . . . . . . . . . . . . . . . 314.1.2 sp2 Hybridization (trigonal) . . . . . . . . . . . . . . . 324.1.3 sp3 Hybridization (tetrahedral) . . . . . . . . . . . . . 32

4.2 π bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Mechanical properties 345.1 Tensile strength . . . . . . . . . . . . . . . . . . . . . . . . . . 345.2 Young’s modulus . . . . . . . . . . . . . . . . . . . . . . . . . 34

6 Density Functional Theory 386.1 Born Oppenheimer Approximation . . . . . . . . . . . . . . . 396.2 The theorems of Hohenberg Kohn . . . . . . . . . . . . . . . . 39

6.2.1 The first Hohenberg Kohn theorem . . . . . . . . . . . 406.2.2 The second Hohenberg Kohn theorem . . . . . . . . . . 40

6.3 Kohn Sham equations . . . . . . . . . . . . . . . . . . . . . . 416.4 Local Density Approximation . . . . . . . . . . . . . . . . . . 42

7 Augmented Plane Wave Method 447.1 Linearized APW . . . . . . . . . . . . . . . . . . . . . . . . . 457.2 Full Potential Linearized Augmented Plane Wave . . . . . . . 46

8 Results 488.1 ELK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

CONTENTS 3

8.1.1 Periodic Structures and unit cells . . . . . . . . . . . . 498.2 Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518.3 Carbon nanotubes . . . . . . . . . . . . . . . . . . . . . . . . 51

8.3.1 Zigzag . . . . . . . . . . . . . . . . . . . . . . . . . . . 518.3.2 Armchair . . . . . . . . . . . . . . . . . . . . . . . . . 53

8.4 Y(r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548.5 In situ Si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

9 Discussion and Conclusions 56

A Tables 58

List of Tables

3.1 Mechanical properties . . . . . . . . . . . . . . . . . . . . . . . 21

8.1 E(ε) = a+ bx+ cx2 . . . . . . . . . . . . . . . . . . . . . . . . 538.2 Y (r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

A.1 E(ε) - Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . 58A.2 E(ε) - Zigzag r = 3.21301 . . . . . . . . . . . . . . . . . . . . 59A.3 E(ε) - Armchair r = 2.74452 . . . . . . . . . . . . . . . . . . . 60A.4 E(ε) - Armchair r = 3.41653 . . . . . . . . . . . . . . . . . . . 60A.5 E(ε) - Armchair r = 4.09065 . . . . . . . . . . . . . . . . . . . 61A.6 E(ε) - Armchair r = 4.76598 . . . . . . . . . . . . . . . . . . . 61A.7 E(ε) - Armchair r = 5.44211 . . . . . . . . . . . . . . . . . . . 62A.8 E(ε) - Armchair r = 6.79564 . . . . . . . . . . . . . . . . . . . 62A.9 E(ε) - Armchair r = 6.79564 . . . . . . . . . . . . . . . . . . . 63

4

Chapter 1

Introduction

“Graphene - a single layer of carbon atoms - may be the mostamazing and versatile substance available to mankind”

- Graphene Flagship

When Iijima 1991 [16] made carbon nanotubes known to the broad scientificcommunity, he set a revolution in motion. Albeit slow at first but from2004, when Geim and Novoselov [28] successfully uncovered one free layerof graphene, there has been a boom seldom seen. This was accentuated bygraphene being the first of EU’s flagships1, granting e1 B over a ten yearperiod, the biggest research initiative ever.

Graphene is one monolayer of carbon atoms packed into a honeycomb lattice.It was the first two dimensional material to be discovered and is the motherof all graphitic allotropes. It can be wrapped up into zero-dimensional struc-tures known as fullerenes. Stacked layer upon layer to three-dimensionalgraphite. Rolled up into one dimensional carbon nanotubes, the structurewe have chosen to investigate further.

In mid 1930s Peierls and Landau [29, 21] taught us that two dimensionalmaterials can not exist as they are thermodynamically unstable. Howeverthese strange two dimensional materials where still theoretically interesting

1http://graphene-flagship.eu/

5

http://graphene-flagship.eu/

CHAPTER 1. INTRODUCTION 6

thus they were discussed and researched for purely academic reasons. In 1991Iijima published a high impact article [16]. Here he claimed to have seencarbon structures shaped like needles, this was not the first evidence thatcarbon nanotubes existed. Despite not being first, Iijimas carbon needles,triggered a first boom in the interest around low dimensional materials. Thisboom was enough for some to disregard Peierls and Landau’s results. In2004 Konstantin Novoselov and Andre Geim [28], successfully extracted asingle layer of graphene from the graphite in a pencil, using only scotch tape.The ensuing second boom this created was immense, making carbon andits allotropes one of the biggest fields of research. The discovery yieldedNovoselov and Geim the Nobel prize in physics 2010.

Despite many tries, no one is yet successful in unambiguously determining theYoung’s modulus of carbon nanotubes. Reported values are ranging between0.6-5.5 TPa [38]. However, they seem to be converging to 1 TPa [26]. Thisis our try to cast some light on the subject, and hopefully some new insightscan emerge from our work.

Chapter 2

Carbon

Carbon only exists as single atoms in extreme temperature environments. Inambient environment carbon reacts with other atoms to stabilize, to formmulti atomic compounds. The most important of these compounds must beattributed to organic chemistry and carbons role in makeup the nucleic acidsin DNA. Also, the proteins which are the building blocks for all life, thusmaking carbon the basis of life.

Now, a short description of how carbon bonds to carbon forming what isknown as allotropes1, especially graphene and carbon nanotubes.

The most fascinating about the carbon allotropes is the multitude of dif-ferent properties they display, most notably electrical, range from stronglyinsulating to nearly perfect conductors. Thermal, most thermally conduct-ing, and mechanical, diamond is the hardest naturally occurring material andgraphite one of the softest, carbyne has the highest Young’s modulus evermeasured.

A short description of the most common and interesting carbon allotropes,together with some of their extreme properties follows below.

1Carbon allotropes include, but is not limited to; amorphous carbon, graphite, dia-mond, fullerenes (buckyballs, carbon nanotubes, carbon nanobuds and nanofibers), lons-daleite, glassy carbon, carbon nanofoam, graphene and linear acetylenic carbon (carbyne)

7

CHAPTER 2. CARBON 8

2.1 Diamond

In an environment with extreme pressure (4.5-6.0 GPa) and temperatures(900-1300 C), carbon forms the compact allotrope diamond.

2.1.1 Structure

The Diamond lattice is a face centered cubic crystal structure, where eachatom is bonded tetrahedrally to four other carbon atoms in a sp3 bonding,see chapter 4, thus making a three dimensional network of puckered six mem-bered rings of atoms; it is the same structure as silicon and germanium, butdue to the strength of the carbon-carbon bonds, it is the hardest naturallyoccurring material in terms of resistance to scratching.

2.1.2 Properties

Hardness 10 000 kg mm−2, scratch resistance 160 GPa and a Young’s modulusof 1.22 TPa.

Very high electric resistivity 1013 − 1016 Ω cm, and a bandgap of 5.45 eV.

2.1.3 Lonsdaleite

Under some conditions, carbon crystallizes as Lonsdaleite. This form has ahexagonal crystal lattice where all atoms are covalently bonded. Therefore,all properties of Lonsdaleite are close to those of diamond.

Imperfections in natural Lonsdaleite reduce hardness, while artificial materialhas been tested harder than diamond with indentation pressures of 152 GPain the same direction diamonds break at 97 GPa.

CHAPTER 2. CARBON 9

2.2 Graphite

In ambient conditions carbon takes the form of graphite, in which each atomis sp2 bonded to three others in a plane composed of fused hexagonal rings,just like those in aromatic hydrocarbons. The resulting network is two dimen-sional, and the resulting flat sheets are stacked and loosely bonded throughweak van der Waals forces. This gives graphite its softness and its cleavingproperties, the sheets slip easily past one another. Because of the delocaliza-tion of one of the outer electrons of each atom to form a π-cloud, graphiteconducts electricity. However only in the plane of each covalently bondedsheet, resulting in a lower bulk electrical conductivity for carbon than formost metals.

2.2.1 Structure

Graphite has a layered, planar structure. In each layer, the carbon atomsare arranged in a honeycomb lattice with atomic distance of 1.42 A, andinterplanar spacing 3.35 - 3.45 A. The two known forms of graphite, hexag-onal (α) and rhombohedral (β), have very similar physical properties. Theonly difference is stacking, either with the atoms inline or a slight shift ofthe layers. The hexagonal graphite may be either flat or buckled. The αform can be converted to the β form through mechanical treatment and theβ form reverts to the α form when it is heated above 1300 C.

2.2.2 Properties

In plane, graphite is a good conductor of both thermal energy and electricity,but between the planes there is almost no conduction at all. This is due tothat both phonons and electrons move in plane but not between them.

CHAPTER 2. CARBON 10

2.3 Fullerenes

Fullerenes have a graphite like structure, but instead of purely hexagonalpacking, they also contain pentagons and heptagons of carbon atoms, bend-ing it to spheres, ellipses or cylinders.

2.3.1 Buckminsterfullerenes

Buckminsterfullerene or C60 is a spheroid fullerene molecule. It is a trun-cated icosahedron with same structure as a soccer ball, i.e. made of twentyhexagons and twelve pentagons, with a carbon atom at each vertex of eachpolygon and a bond along each polygon edge. The buckyballs are fairly large

Figure 2.1: C60 molecule 60 carbon atoms in 12 pentagons and 20 hexagons

molecules formed completely of carbon bonded trigonally, forming spheroids.

2.3.2 Carbon nanotubes

Carbon nanotubes are structurally similar to C60, except that each atom isbonded with sp2 hybridization, in a curved sheet that forms a hollow cylinder,see chapter 3.


2.3.3 Nanobuds

Nanobuds were first reported in 2007 [24]. They are a hybrid of a carbonnanotube and a Buckministerfullerene, combining the properties of both ina single structure.

2.3.4 Nanofoam

Carbon nanofoam is a ferromagnetic allotrope discovered in 1997 [34]. Itconsists of a low-density cluster assembly of carbon atoms strung together ina loose three-dimensional web, in which the atoms are bonded trigonally insix- and seven membered rings. It is among the lightest known solids, witha density of about 2 kg m−3.

2.3.5 Carbyne

Carbyne, or linear acetylenic carbon, is an infinite chain of sp hybridized,see chapter 4, carbon atoms [22] with alternating single and triple bonds,fig. 2.2. This type of carbyne is of considerable interest to nanotechnology

Figure 2.2: Alternating single and triple bonds structure of carbybne

as its Young’s modulus of 32.7 TPa is forty times that of diamond [22]. Atensile stiffness, see chapter 5.1 of C = 95.56 eV/A ∼ 109 Nm kg−1, which isdouble the stiffness of graphene, and three times stiffer than diamond. Thereare many other interesting physical applications of carbyne that have beenproposed theoretically, including nanoelectronic- and spintronic devices, alsohydrogen storage. The carbyne ring structure is the ground state for small(up to about 20 atoms) carbon clusters.

Indications of naturally formed carbyne were observed in such environmentsas shock compressed graphite, interstellar dust, and meteorites.


2.4 Graphene

“It would take an elephant, balanced on a pencil, to break througha sheet of graphene the thickness of Saran Wrap”

- Prof. James Hone Columbia University

Graphene is a flat monolayer of carbon atoms tightly packed (Atomic distance1.42 A) into a two-dimensional honeycomb lattice, fig. 2.3, and is a basicbuilding block for graphitic materials of all other dimensionalities. It is partof the zero dimensional fullerenes, rolled into one dimension nanotubes orstacked into three dimension graphite [9].

Figure 2.3: Honeycomb structure of Graphene

Graphene is the material of superlatives; conducts electricity 1 000 timesbetter than copper, is 300 times stronger than steel or even more impressive;it is harder than diamond.

2.4.1 Electronic Properties

Graphene is a zero overlap semi-metal or zero gap semiconductor.

In the vicinity of the six corners of the Brillouin zone the dispersion relationfor low energies becomes conical, which leads to that the electrons (and holes)has a zero effective mass, fig. 2.4, [26].


Figure 2.4: Graphene Brillouin Zone and Electronic Energy Dispersion [32, 25]

Due to this conical dispersion relation, electrons and holes near these sixpoints behave like relativistic particles, which are described by the Diracequation for spin 1/2 particles. Hence, the electrons and holes are called Diracfermions, or graphinos, and the six corners of the Brillouin zone are calledthe Dirac points. The equation describing the electrons’ conical dispersionrelation is

E = hvF

√k2x + k2y

where the Fermi velocity vF ∼ 106 m/s, and the wavevector k is measuredfrom the Dirac points.

Theoretically the upper limit of electron mobility in graphene is 2 · 105 cm2

V−1 s−1 with a carrier density of 1012 cm2, limited almost only by graphene’sacoustic phonons. Experimentally of course the quality of the graphene andsubstrate plays a role for the electron mobility, which is why it is reportedto values in the range of 2 · 104 cm2 V−1 s−1. Despite the significant gap be-tween theory and experiments, these velocities mean that the charge carrierscan travel several µm before scattering, leading to a phenomenon known asballistic transport. The resistivity of the graphene sheet is 10−6 Ω cm, whichis less than the resistivity of silver, the lowest resistivity substance known atroom temperature.

2.4.2 Mechanical Properties

“In our 1 m2 hammock tied between two trees you could place aweight of approximately 4 kg before it would break. It should thusbe possible to make an almost invisible hammock out of graphene


that could hold a cat without breaking. The hammock would weighless than one mg (0.77 mg), corresponding to the weight of oneof the cats whiskers.”

- Nobel Committee, 2010

Measurements have shown that graphene has a breaking strength over 100times greater than a hypothetical one atom steel film, with its Young’s mod-ulus of about 1 TPa.

Using an atomic force microscope (AFM), the spring constant of suspendedgraphene sheets has been measured. Graphene sheets, held together by vander Waals forces, were suspended over SiO2 cavities where an AFM tip wasused to test its mechanical properties [8]. Its spring constant was in therange 1−5 N m−1 and the Young’s modulus was 0.5 TPa, which differs fromthat of the bulk graphite. These high values make graphene very strong andrigid.

2.4.3 Optical Properties

Graphene’s unique optical properties produce an unexpectedly high opacityfor an atomic monolayer, absorbing πα ≈ 2.3% of white light, where α is thefine structure constant [7].

The bandgap of graphene can be tuned from 0 to 0.25 eV by applying avoltage to a dual gate bilayer graphene field effect transistor (FET) at roomtemperature. The optical response of graphene nanoribbons has also beenshown to be tunable into the THz regime by an applied magnetic field. It hasbeen shown that graphene/graphene oxide system exhibits electrochromicbehavior, allowing tuning of both linear and ultrafast optical properties.

Chapter 3

Carbon nanotube

Carbon nanotubes are another allotrope of carbon. They can be thoughtof as rolled up graphene with the edges fused together. Just as graphene,carbon nanotubes have some extreme properties. If they can be utilized theywill revolutionize great many fields, including but not limited to materials,electronic and optical there will probably even arise new fields in the waketoo. The first areas where carbon nanotubes are being used is in new andbetter materials foremost as a small amount additive into carbon fibers ini.e. clubs, bats and car parts.

The chemical bonding of nanotubes is composed entirely of sp2 hybridizedbonds, similar to those of graphene. These bonds, which are stronger than thesp3 bonds found in diamond, provide nanotubes with their unique strength,and the associated π bonds are the reason for their electrical properties.

3.1 Discovery

Carbon nanotubes are actually relatively easy to produce. Using a piece ofgraphite as anode and have a high voltage current go through will createall kinds of fullerenes, among other carbon nanotubes. The main problemhas been in detection, before 1931 when Knoll & Ruska invented the firsttransmission electron microscope (TEM) it was not possible to see such a

15

CHAPTER 3. CARBON NANOTUBE 16

small object. Also using this method there is no way of knowing the diameter,length or chirality of the carbon nanotubes createed. Thus it is not usablewhen any of these properties are relevant.

Once the TEM was invented there has been different groups detecting nanoscale carbon structures, but when Iijima in 1991 [16] made the broad scientificcommunity aware of the potentials of carbon nanotubes a rush to understandthe new material ensued. This has yielded Iijima a somewhat unearnedreputation as the sole discoverer of carbon nanotubes.

Below follows a short description of the small steps towards a discovery ofcarbon nanotubes.

1952 L. V. Radushkevich and V. M. Lukyanovich [33] published clear imagesof 50 nanometer diameter tubes made of carbon. This was publishedin an Soviet magazine, and never reached western scientists

1979 John Abrahamson presented evidence of carbon nanotubes at the 14thBiennial Conference of Carbon at Pennsylvania State University. Theconference paper described carbon nanotubes as carbon fibers that wereproduced on carbon anodes during arc discharge. A characterizationof these fibers was given as well as hypotheses for their growth in anitrogen atmosphere at low pressures

1981 a group of Soviet scientists published the results of chemical and struc-tural characterization of carbon nanoparticles produced by a thermo-catalytical disproportionation of carbon monoxide. Using TEM imagesand XRD patterns, the authors suggested that their carbon multi-layertubular crystals were formed by rolling graphene layers into cylinders.They speculated that by rolling graphene layers into a cylinder, manydifferent arrangements of graphene hexagonal nets are possible. Theysuggested two possibilities of such arrangements: circular arrangement(armchair nanotube) and a spiral, helical arrangement (chiral tube)

1987 Howard G. Tennett of Hyperion Catalysis was issued a U.S. patent forthe production of “cylindrical discrete carbon fibrils”

1991 Iijima [16], despite not being the first to show carbon nanotubes, Iijimacategorized and described them in much greater detail than anyone


before him. Iijima also concluded that carbon nanotubes are not ofthe “scroll” type, as he could only detect “russian doll” tubes. Thisgave rise to an unprecedented interest in carbon nanotubes, and startedthe intense research in the field of nano technology that has since onlyincreased.

3.2 Categories of carbon nanotubes

Carbon nanotubes are categorized first on how many walls they have, single-or multi walled (also double-, triple- and many walled are sometimes used)

Also, the directionality of the rolled up graphene is important. This phe-nomenon is known as chirality.

3.2.1 Chirality

The way the graphene sheet is wrapped is represented by a pair of indices(m,n) (fig. 3.1). The integers n and m denote the number of vectors ~a1,~a2in the honeycomb crystal lattice of graphene. Where

~a1 =

(3

2,

√3

2

)ac−c, ~a2 =

(3

2,−√

3

2

)ac−c (3.1)

with ac−c = 1.421 A, the distance between two carbon atoms in graphite.

The chirality vector (fig. 3.1) ~Ch = m~a1 + n~a2, is a vector in the graphenehexagonal lattice. Rolling up the graphene such that the start and end ofthe vector join to form the circumstantial circle of the carbon nanotube. ~Ch

have the lengthCh = a

√n2 + nm+m2 (3.2)

where a =√

3ac−c = 2.461 A. The radius of an nanotube is thus just

d =Ch

π=a

π

√n2 + nm+m2 (3.3)


Armchair

Zigzag

a1

a2

Ch

Figure 3.1: How to name carbon nanotubes by their chirality

Naming convention for carbon nanotubes using (fig. 3.1) is

• If m = 0, the nanotubes are called zigzag nanotubes

• If n = m, the nanotubes are called armchair nanotubes

• Else they are called chiral

3.2.2 Single walled

Single walled carbon nanotubes (SWNT) have a diameter in the range 3 −1000 A, and a lengths normally in the micrometer range but known to beranging up to centimeters. The structure of a single walled carbon nanotubecan be conceptualized by wrapping one sheet of graphene into a seamlesscylinder.

Single walled carbon nanotubes are an important variety of carbon nanotubebecause most of their properties change significantly with the chirality, andthis dependence is non monotonic. In particular, their band gap can varyfrom 0 to about 2 eV, and their electrical conductivity can show metallic orsemiconducting behavior.


(a) Singel walled carbon nanotube,zigzag chirality (m, 0)

(b) Singel walled carbon nanotube,armchair chirality (m,n)

Figure 3.2: Singel walled carbon nanotubes with the two extreme chiralities(zigzag and armchair)

3.2.3 Multi walled

Multi walled carbon nanotubes (MWNT) consist of multiple layers of graphene.There are two different ways to create multi walled nanotubes.

• Russian Doll model. Single walled nanotube within a larger singlewalled nanotube, repeated many times.

• Parchment model. A single sheet of graphite is rolled in around itself,resembling a scroll of parchment.

The interlayer distance in multi walled nanotubes is close to the distancebetween graphene layers in graphite, 3.35−3.45 A. Individual layers are singlewalled carbon nanotubes. Since different single walled carbon nanotubes havedifferent electronic properties, multi walled nanotubes are almost always zerogap metals.


Figure 3.3: A double walled carbon nanotube

Double walled carbon nanotubes

Double walled carbon nanotubes (DWNT) form a special class of nanotubes.Their morphology and properties are similar to those of single walled carbonnanotubes, but their resistance to chemicals are significantly improved. Thisis especially important when adding atoms to the surface of the tube, i.e.nanobuds. In the case of single walled carbon nanotube, this will break somedouble bonds, destroying the structure on the nanotube thus, modifying bothits mechanical and electrical properties. In the case of double walled carbonnanotube, only the outer wall is modified.

3.3 Properties

3.3.1 Strength

Carbon nanotubes are among the strongest and stiffest materials discoveredin terms of tensile strength and Young’s modulus respectively. This strengthresults from the covalent sp2 bonds formed between the individual carbonatoms.


Carbon nanotube have been tested to have a tensile strength of up to ∼ 100GPa. This corresponds to a human hair, diameter 100 µm, lifting 100 kg.Since carbon nanotubes have a density of 1.3 g cm−3, its specific strength is48 000 kN m kg−1 is the best of known materials, compared to high carbonsteel’s 154 kN m kg−1.

Under excessive tensile strain, the tubes will undergo plastic deformation,which means the deformation is permanent. This deformation begins atstrains of approximately 5% and increases the maximum strain the tubesundergo before fracture by releasing strain energy. Due to the hollow struc-

Table 3.1: Mechanical properties, a comparison between carbon nanotubes andother selected materials. εmax is the maximum extension before breaking the ma-terial. Values within brackets are theoretical maximums

MaterialYoung’s modulus

TPaTensile strength

GPaεmax

%SWNT 0.65-5.5 126 16-23MWNT 0.2-0.95 >63 (300)

Stainless steel 0.186-0.214 0.38-1.55 15-50Kevlar 0.06-0.18 3.6-3.8 2

Diamond 1.22 >60 (225)

ture, carbon nanotubes are rather weak in its radial direction. This forcesthem to undergo buckling when placed under compressive, torsional, or bend-ing stress. Single walled carbon nanotubes can withstand a radial pressureup to 25 GPa without permanent deformation. After that they undergo atransformation to superhard phase nanotubes [31]. The bulk modulus ofsuperhard phase nanotubes is 462 to 546 GPa, even higher than that of adiamond’s 420 GPa.

3.3.2 Electrical properties

Because of the symmetry and unique electronic structure of graphene, thestructure of a nanotube strongly affects its electrical properties. For a given(n,m) nanotube there are some well defined types,


• n = m metallic carbon nanotube

• n−m = 3k semiconducting with a very small band gap

This however is not true for the smallest carbon nanotubes.

In theory, metallic nanotubes can carry an electric current density of 4 · 109

A cm−2, which is more than 1 000 times greater than copper.

Because of their nanoscale cross section, electrons propagate only along thetube’s axis, and electron transport involves quantum effects. As a result,carbon nanotubes are frequently referred to as one-dimensional conductors.The maximum electrical conductance of a single-walled carbon nanotube is2G0, where G0 = 2e2/h is the conductance of a single ballistic quantumchannel.

3.3.3 Thermal properties

All nanotubes are very good thermal conductors along the tube, exhibitinga property known as “ballistic conduction”, but good insulators laterally tothe tube axis. Measurements show that a single walled carbon nanotubeshas a room-temperature thermal conductivity along its axis of about 3500W m−1 K−1; compare this to copper, a metal well known for its goodthermal conductivity, which transmits 385 W m−1 K−1. A single walledcarbon nanotube has a room temperature thermal conductivity across itsaxis (in the radial direction) of about 1.52 W m−1 K−1, which is about asthermally conductive as soil.

3.4 Extreme carbon nanotubes

• Length;

– Longest carbon nanotube is 18.5 cm long [39]

– Shortest carbon nanotube is the organic compound cyclopara-phenylene [36]


• Width;

– Thinnest carbon nanotube is a (2, 2) with the diameter of 3 A [44].Carbon nanotubes this small can only exist as the innermost tubein a MWNT.

Thinnest single walled carbon nanotube is 4.3 A in diameter


3.5 Future uses for graphene and carbon nan-

otubes

In the near future there are several areas that will be greatly influenced bygraphene and carbon nanotubes. Below are some areas discussed but ofcourse there are several more and new are discovered every day [4, 19].

3.5.1 Material additives

Carbon nanotubes are strong and light. If there is no requirement on theirexact properties like chirality and number of walls they are easy and cheap toproduce. This makes them perfect for material additives, yielding strongerand lighter materials at a very low cost. In principle this is easy, just addcarbon nanotubes to the mixture and the mixture has new and improvedproperties, reality however is not always that easy.

This is today used in many sporting equipments like; bats, hockey sticks andtennis rackets. However there will soon be car- and airplane bodies done inthis way.

With lighter and stronger cars and airplanes, less fuel is needed for transportand it becomes a lot safer.

3.5.2 Nanomechanics

The field of nanomechanics applies the properties of nanomaterials to createmachines on a nanometer scale. The problem, and benefit, here is of coursethat on the nano scale it is required to not only follow the normal mechanicalrules but also those of the nanoworld.


Bearings

Multi walled carbon nanotubes move almost frictionless inside each other,creating an almost perfect atomic linear- or rotational bearing.

3.5.3 Nanoelectromechanical systems

Nanoelectromechanical systems (NEMS), as multi walled carbon nanotubeshas superior properties, together with them being almost frictionless bearingsyields applications in the NEMS field that previously only could be dreamtof, i.e. electrical wires, nanomotors, switches and high frequency oscillators.

Wires and connectors

Due to single walled carbon nanotubes’ low scattering, high carrier capacityand almost zero electro migration, they are suggested to replace copper asconnectors and wires in NEMS.

Nanomotors

Nanomotors made of carbon nanotubes generate a rotating motion via asuspended multi walled carbon nanotube with a rotor attached between twostator electrodes. This contraption is free to rotate with virtually no friction.

Switches

Switches are the primary concept for carbon nanotube computer memory.By laying out spin coated carbon nanotubes in a square pattern, the state ofthe intersections can read with a weak current and the state changed with astronger current.


High frequency Oscillators

Using carbon nanotubes high mechanical stiffness yields possibilities of os-cillations in the ≈ 50 GHz range.

3.5.4 Solar cells

In addition to the for mentioned electronic properties graphene have a lowrate of carrier recombination, this makes them perfect candidates for photo-voltaic or solar cells.

Until recently carbon nanotubes mostly have been used as a replacement ofindium tin oxide as electrodes in organic solar cells. However, recent studieshave used graphene flakes in a “polymer blend bulk heterojunction” solarcell [42]. This increases the donor to acceptor ratio, and with only a smallamount of graphene the efficiency is greatly improved.

3.5.5 Energy storage

For energy storage, the electronic properties of graphene is the most obvi-ous usage. However the physical properties of carbon nanotubes is anotherpossibility.

Multi walled carbon nanotubes are widely used in lithium ion batteries, thisgives the batteries faster recharge time and higher storage rate. The fastrecharge and discharge rate of graphene is the primary use in the superca-pacitor.

Supercapacitor

For real time high power applications, it is critical to have high specificcapacitance with fast charging time at high current density. This makesgraphene ideal for these kind of applications. [18].


The main reason for not having electric cars yet is the recharge time of thebatteries. With a fully loaded battery a car can be driven about 400 km,but the recharge time to get back is 43 h. A capacitor can be charges in afraction of that time, but wont hold nearly as much energy. This is where thegraphene based supercapacitor comes in play, they can be recharged fasterthan normal capacitors and hold almost as much energy as a battery. Thus,the recharging of a car is a matter of minutes instead of hours, maintainingits mileage.

Gravimetric energy storage

Using carbon nanotubes mechanical properties as a way to store energy isdone much as the same way a steel spring stores energy in a mechanical clock[14]. Here the Young’s modulus of 1 TPa and a theoretical strains of about20%, will yield an energy storage capacity three orders of magnitude higherthan ordinary steel springs.

3.5.6 Biomedicine

Since carbon nanotubes have a high surface area, chemical stability, andrich electronic polyaromatic structure, they are able to adsorb or conjugatewith a wide variety of molecules. This makes them perfect candidates forbiomedicine applications [12].

Drug delivery

Carbon nanotubes have been proven to be an excellent vehicle for drug deliv-ery by penetrating into the cells directly and keeping the drug intact withoutmetabolism during transport in the body.

More importantly carbon nanotubes can be set to target specific cells, e.g.by making the carbon nanotubes magnetic and pull them to the target area.Thus carbon nanotubes are perfect for transporting medicine directly intothe cells and destroying them, instead of treating the entire body.


Biosensor

A biosensor is used to measure chemicals in the body, most prominent isthe research to couple carbon nanotubes with glucose-oxidase biosensors tocontrol the blood sugar level in diabetic patients.


3.6 Toxicity and biocompatibility graphene

and carbon nanotubes

The big bump in the road for graphene and carbon nanotubes is the lackof research on its toxicity and biocompatibility. Also, the few results thatexists are often inconclusive or contradictory. Until there are some thoroughstudies, most applications will not reach the market. Despite this, rater largeproblem, there are a plethora of applications proposed in the biomedical fieldalone [12].

One positive side note is that previous use of carbon based biomaterials showhigh biocompatibility. However, it is suggested that due to the presence oftransition metal catalysts carbon nanotubes are some what toxic, and if theyreach the organs they can induce inflammation or fibrotic reactions. Though,chemically functionalized carbon nanotubes for drug delivery have so far notdemonstrated any toxicity.

Obviously, it is urged to use caution when handling carbon nanotubes andgraphene, and the introduction of safety measures especially in larger scalemanufacturing facilities must be considered. Most importantly, the success ofcarbon nanotubes technology is dependent upon the continuation of researchinto the toxicology of carbon nanotubes and graphene.

Chapter 4

Orbital Hybridization

The solution to the schrodinger equation yields a region where the electrondensity is expected to be highest. This region is called an orbital. WhenLinus Pauling in the 1950s tried to calculate the orbits of atoms heavierthan hydrogen, he failed; as everyone else. He, therefore, set out to create amathematical model that could explain the different orbitals.

Orbital hybridization is the mathematical concept of mixing atomic wavefunctions into completely new hybrid orbitals, with a new shape and (lower)energy. These new orbitals better explains electron’s bonding together inmolecules. The concept of hybridization explains how atoms forms hy-bridized bonds but not why it does so [23]. Also, orbital hybridization isonly an explanatory model.

Consider the element whose bonding is of most interest to us: carbon. Itselectronic configuration is 1s2 2s2 2p2. 2s and 2p differ very little in energy;therefore, the wave functions can mix when carbon is bonded.

4.1 spn Hybridization

spn is a notation of how the orbitals are hybridized, i.e. a sp2 has one s (33%)and two p (66%) orbitals. Even though several orbitals comes together and

30

CHAPTER 4. ORBITAL HYBRIDIZATION 31

(a) s orbital (b) p orbital

Figure 4.1: s and p orbitals

form new, there are the same number of orbitals available for bonding.

How to determine which hybridization a carbon atoms has sp3, sp2 or sp

• 1 triple bond and 1 single bond (or 2 double bonds, in the case of CO2),then it is sp hybridized (linear)

• 2 single bonds and 1 double bond, it is sp2 hybridized (trigonal planar)

• 4 single bonds, it is sp3 hybridized (tetrahedral)

4.1.1 sp Hybridization (linear)

Mixing one 2s and one 2p wave function of carbon, two new hybrids areobtained, called sp orbitals, made up of 50% s and 50% p character. Themajor parts of the orbitals point away from each other at an angle of 180.There are two additional minor back lobes (one for each sp hybrid) withopposite sign. The remaining two p orbitals are left unchanged, these twoforms two π bonds.

The 180 angle that results from this hybridization scheme minimizes electronrepulsion. The oversized front lobes of the hybrid orbitals also overlap betterthan lobes of unhybridized orbitals; the result is energy reduction due toimproved bonding. This is the hybridization that gives rise to Carbyne, oneof the materials harder than diamond.


4.1.2 sp2 Hybridization (trigonal)

The sp2 hybridization is the mixing of one 2s and two 2p atomic orbital,which involves the promotion of one electron in the s orbital to one of the 2patomic orbital. The combination of these atomic orbitals creates three newhybrid orbital equal in energy-level. The hybrid orbital is higher in energythan the s orbital but lower in energy than the p orbital, but they are closerin energy to the p orbital. The new set of formed hybrid orbital createstrigonal structures, creating a molecular geometry of 120.

The combination of an s orbital, and two p orbital from the same valenceshell provides a set of three equivalent sp2 hybridized orbital that point indirections separated by 120. The directions of these new, hybridized orbitalare the dictators of the spatial arrangement for bonding. The sp2 hybridizedorbital are the same in size, energy shape but different in the spatial orienta-tion. This unique orientation is imperative and is what characterizes an sp2

hybridized orbital from other hybridized orbital.

Figure 4.2: sp2 Hybridization orbitals

4.1.3 sp3 Hybridization (tetrahedral)

To achieve sp3 bonding, first promotion of one electron from 2s to 2p resultsin four singly filled orbitals. Then, the 2s orbital is hybridized with allthree 2p orbitals. This makes four equivalent sp3 orbitals with tetrahedralsymmetry, for electron repulsion minimization. Each has 75% p and 25% scharacter and occupied by one electron. The bond angles of a tetrahedron is109.58.

Any combination of atomic and hybrid orbitals may overlap to form bonds.C-C bonds are generated by overlap of hybrid orbitals. The diamond cu-


bic crystal lattice is two tetrahedrally bonded atoms in each primitive cell.Separated by 1/4 of the width of the unit cell in each dimension.

4.2 π bond

A π bond is a covalent bond formed between two neighboring atom’s un-bonded p orbitals (fig. 4.3). Since all carbon allotropes with sp and sp2

Figure 4.3: Two p orbitals forming a π bond.

hybridization have a free p orbital, they will form π bonds.

π bonds are usually weaker than σ bonds1. This bond’s weakness is explainedby; a greater extension from the positive charge of the atomic nucleus. Alsoby a significantly less overlap between the p orbitals due to their parallel ori-entation. This is contrasted by σ bonds which form bonding orbitals directlybetween the nucleus of the bonding atoms, resulting in greater overlap. Aπ bond by itself is weaker than a σ bond, but π bonds are only found incombination with σ bonds, so the combination of the two bonds is strongerthan either bond would be by itself.

Electrons in π bonds are sometimes referred to as π electrons. These πelectrons are closest to the Fermi level and thus crucial for the electronicproperties in graphene and carbon nanotubes.

1σ bonds are two overlapping s orbitals

Chapter 5

Mechanical properties

5.1 Tensile strength

Tensile strength is the force required to pull something to its breaking point.There are two different definitions of tensile strength

• Yield strength, the stress required to permanently deform the objectbeing pulled

• Ultimate strength, maximum stress before material breaks

5.2 Young’s modulus

Here a short derivation of how a classical spring can be used to calculate theYoung’s modulus of a carbon nanotube is done.

First the definition of young’s modulus

Y =σ

ε(5.1)

where, σ = F/A with F applied force on the area A. ε = ∆z/z with, z thelength of an unperturbed carbon nanotube and ∆z the change of the length

34

CHAPTER 5. MECHANICAL PROPERTIES 35

of the stretched carbon nanotube. Thus,

F = Y Aε (5.2)

Secondly Hooke’s lawF = k∆z (5.3)

with k as the spring constant. Rewriting and using eq. (5.2) yields,

k =F

∆z=Y Aε

∆z. (5.4)

Also from Hooke’s law the potential energy stored in the spring is obtainable,

E =

∫Fdz =

1

2k(∆z)2 (5.5)

now combining eqs. (5.4, 5.5), yields

E =1

2

Y Aε

∆z(∆z)2

=1

2Y Aε∆z.

(5.6)

Thus, the formula for young’s modulus can be written as

Y =2E

A∆zε(5.7)

now using

∆z =∆z

z· z = εz (5.8)

yields

Y =2E

Azε2. (5.9)

A can be chosen in several different ways, which probably is the reason forsuch a widespread range of the Young’s modulus.

If the area is chosen to be entire cross sectional area of the carbon nanotube;

A = 2πr2. (5.10)


This will not take into account the empty part of the tube. The area willgrow with the square of the radius but the number of atoms will only growlinearly, thus a large radius carbon nanotube will have an very small Young’smodulus.

Taking into account the hole in the cylinder the problem will instead be; howthick the cylinder walls are.

• At first glance the radius of the carbon atom rC = 70 pm, might be agood choice. But this will yield a Young’s modulus much larger thanany measurements have ever been close to.

• Remaining is, the distance between two layers of graphite, t = 3.35 A[5].

Here is set with the carbon atoms centered at r and a thickness, t spreadequally around inside and out, thus the area is described as

A = π((r + t/2)2 − (r − t/2)2

)= 2πrt. (5.11)

Thus the final equation that is needed in calculating the Young’s modulusof a carbon nanotube is

Y =E

πrtzε2. (5.12)

One important note here is that this is the energy between two carbon atoms.For carbon nanotubes, the calculations have 4 or 8 carbon atoms in their cellsand the formula must thus be properly edited for that purpose.

Now only the energy is missing, and is therefore what will be required to findout, how this is done is described below.


r

t

Figure 5.1: Description of how r and t is chosen, the gray dots represents carbonatoms

Chapter 6

Density Functional Theory

“The underlying laws necessary for the mathematical theory oflarge parts of physics and the whole of chemistry are thus com-pletely known, and the difficulty is only that the exact applicationof these laws leads to equations much too complicated to be solu-ble.”

- Dirac, P. A. M.

Density Functional Theory (DFT) is a Quantum mechanical method to solvemany body problems. The central part of DFT is that it determines theelectron density, with respect to spatial dependency, ρ(~r), and not the wavefunction, φ. The total energy of the molecule, E [ρ], is a functional1 of theelectron density, ρ(~r). Thus Density Functional Theory. It is also assumedthat the density is slowly varying.

The electron density is calculated using

ρ(~r) =∑i

φ∗iφi (6.1)

Any Quantum mechanical system is described by a Hamiltonian, its “full”

1in mathematics a function of a function is known as a functional

38

CHAPTER 6. DENSITY FUNCTIONAL THEORY 39

and exact form the Hamiltonian2 looks like;

H =− h2

2

∑i

∇2~Ri

Mi

− h2

2me

∑i

∇2~ri

− e2

4πε0

∑i,j

Zi

|~Ri − ~rj|+

e2

8πε0

∑i 6=j

1

|~ri − ~rj|+

e2

8πε0

∑i 6=j

ZiZj

|~Ri − ~Rj|

(6.2)

where me, ~ri and Mi, ~Ri is the mass and position for the electron and nucleoidparticles, respectively. This is not doable for any atomic system larger thanHydrogen atom, thus some approximations are needed.

6.1 Born Oppenheimer Approximation

The first approximation is the Born Oppenheimer Approximation proposedin 1927 [2]. In an atom, the core contains almost all the mass, in the hydrogencase the core (one proton) is 1836.15 times heavier than the electron. Thusthe nuclear core movements are insignificant and much slower compared tothe electrons. This yields an separable wave function or,

φ = φelectrons · φnuclear (6.3)

6.2 The theorems of Hohenberg Kohn

Early DFT was discussed using the Thomas Fermi model3, but not untilHohenberg Kohn put forward their theorems in 1964 [15] had there been anytheoretical evidence for the theory.

2only position and charge is here considered; most notably lacking is the spin3a quantum mechanical theory for the electronic structure of many body systems de-

veloped semiclassically shortly after the introduction of the Schrodinger equation.


6.2.1 The first Hohenberg Kohn theorem

The first Hohenberg Kohn theorem demonstrates that the ground state prop-erties of a many electron system are uniquely determined by an electron den-sity that depends on only 3 spatial coordinates. It lays the groundwork forreducing the many body problem of N electrons with 3N spatial coordinatesto 3 spatial coordinates, through the use of functionals of the electron den-sity. This theorem can be extended to the time dependent domain to developtime dependent density functional theory (TDDFT), which can be used todescribe excited states [3].

Theorem 1 There is a one-to-one correspondence between the ground statedensity ρ(~r) of a many electron system and the external potential Vext(~r).

An immediate consequence is that the ground state expectation value of anyobservable O is a unique functional of the exact ground-state electron density:⟨

φ|O|φ⟩

= O[ρ] (6.4)

6.2.2 The second Hohenberg Kohn theorem

The second Hohenberg Kohn theorem defines an energy functional for thesystem and proves that the correct ground state electron density minimizesthis energy functional.

Theorem 2 if N interacting electrons move in an external potential U(~r),the ground-state electron density ρ0(~r) minimises the functional

E [ρ] = F [ρ] + ρ(~r)U(~r)d~r (6.5)

and the minimum value of the functional E is E0 the exact ground state elec-tronic energy.

Putting together Born Oppenheimer Approximation and The theorems ofHohenberg Kohn yields; the nuclear core can be said to be stationary, and


the electrons instantaneously in equilibrium with the nuclear core. This yieldsa highly simplified version of eq. 6.2, where only N interacting electrons inan external potential, Vext(~r), remains, thus the kinetic energy of the nucleusis zero, and only the kinetic energy of the electrons, their potential energyand the electron-electron interactions remains, yielding

H[ρ] = T [ρ] + V [ρ] + Vext(~r), (6.6)

where the kinetic part, T [ρ], and the electron-electron interaction part, V [ρ],are universal parts free from the dependence on protons, even which kind ofmany electron system is rendered irrelevant, for brevity F [ρ] = T [ρ] + V [ρ]is used. The system specific (non universal) parts are given by the externalpotential Vext.

6.3 Kohn Sham equations

The Kohn Sham equations are used to find the ground state density ρ(~r),until they where published in 1965 there were no practical methods for usingDFT [20, 3].

The Kohn Sham equations are based on the ansatz;

Ansatz 1 The exact ground-state density can be represented by the ground-state density of an auxiliary system of noninteracting particles, called “non-interacting V-representability.”

From Hohenberg Kohn [15] it is known that the ground-state density func-tional can be written as

E [ρ] =

∫Vext(~r)ρ(~r)d~r + F [ρ] (6.7)

where F [ρ] can be redivided into it’s three original parts

F [ρ] = T [ρ] +1

2

∫ρ(~r)ρ(~r ′)

|~r − ~r ′|d~rd~r ′ + Exc[ρ]. (6.8)


Here Exc[ρ] is the exchange and correlation of a system with density ρ(~r)given by,

Exc[ρ] =

∫ρ(~r)εxc[ρ]d~r, (6.9)

where εxc[ρ] is the exchange and correlation energy per electron of a uniformelectron gas.

Applying the condition of slowly varying density∫δρ(~r)d~r = 0 (6.10)

yields, ∫δρ(~r)

[Vext(~r) +

∫ρ(~r ′)

|~r − ~r ′|+δTs[ρ]

δρ(v~r)+ µxc[ρ]

]d~r = 0 (6.11)

here

µxc =d

dnnεxc[ρ] (6.12)

6.4 Local Density Approximation

Local Density Approximation (LDA) is a class of more sophisticated ap-proximations to the exchange correlation energy functional [30]. They onlydepend on the value of the electronic density at each point in space, ρ(~r).When approximating the exchange correlation energy, there are several ap-proches. However by far the most successful is the homogeneous electrongas model. Since the homogeneous electron gas model yields extremely goodresults, LDA is often synonymous with the homogeneous electron gas func-tionals.

The LDA exchange correlation energy is found by

ELDAxc [ρ] =

∫ρ(~r)εxc[ρ]d~r

where εxc is the exchange correlation energy density. The exchange correla-tion energy can be separated into two pieces;

Exc = Ex + Ec ,


so that separate expressions for Ex and Ec are sought. The exchange termoften takes on a simple analytic form for the homogeneous electron gas. Onlylimiting expressions for the correlation density are known exactly, leading tonumerous different approximations for εc.

The exchange correlation potential corresponding to the exchange correlationenergy for a local density approximation is given by

V LDAxc (~r) =

δELDAxc

δρ= εxc[ρ] + ρ

∂εxc[ρ]

∂ρ. (6.13)

Despite being a sophisticated approximation there are one large fall back.The LDA potential decays much too fast, especially in finite systems. Thismanifests itself especially in electron rich atoms, falsely stating they areunable to bind some electrons, yielding an prediction of the atom beingunstable.

Chapter 7

Augmented Plane WaveMethod

The Augmented Plane Wave (APW) method is a method that uses the muf-fin tin approximation to approximate the electrons energy states in a crystallattice. For the potential a spherical symmetry is assumed centered at theatomic nucleus, the electron can thus be considered as they where in a freeatom. In the region between the spheres it is constant, in most cases; zero,thus the electron can be considered to be completely free. By solving theSchrodinger equation and matching the solutions to the spheres and intersti-tial region, then optimizing by using the variational method the augmentedplane waves are constructed [3, 37].

It is now clear that the space can be separated into two distinct regions, callthe region occupied by the spheres, S, and the interstitial region, I, yielding

φ~k~K

(~r, E) =

ei(

~k+ ~K)·~r ~r ∈ I∑∞l=0

∑lm=−lAlmRl(r)Y

lm(θ, ψ) ~r ∈ S

(7.1)

where ~k is a Bloch vector in the first Brillouin zone, ~K the reciprocal latticevector. Rl(r) is the solutions to the radial Schrodinger equations:

− 1

2r2d

dr

[r2dRl(r)

dr

]+

[l(l + 1)

2r2+ V (r)

]Rl(r) = ERl(r) (7.2)

44

CHAPTER 7. AUGMENTED PLANE WAVE METHOD 45

which are solvable. Y lm(θ, ψ) is the spherical harmonic [41] and Alm are pa-

rameters for matching solutions between S and I. The problem with match-ing the different regions is that in I there is a plane-wave and in S sphericalharmonic wave and the wave must be matched over the entire surface of S.Thus, the expanded plane-wave becomes;

ei~q·~r = 4π∞∑l=0

l∑m=−l

iljl(qr)Yl∗m (θ~q, ψ~q)Y

lm(θ, ψ) (7.3)

where ~q = ~k + ~K, (r, θ, ψ) and (q, θ~q, ψ~q) are the spherical coordinates of ~rand ~q respectively. jl is the Bessel function of order l. It is highly impracticalto have a infinite summation, thus a maximum is chosen for l, called lmax.The coefficients Alm are determined by requiring continuity over the entiresphere, thus equaling the expanded plane-wave and the spherical harmonicwave from eq. 7.1, fixes Alm as;

Alm =4πilei~q·~r

R(R)jl(qR)Y l∗

m (θ~q, ψ~q) (7.4)

yielding,

φ~q(~r) = 4π∑lm

il[jl(qR)

Rl(R)

]Rl(r)Y

l∗m (θ~q, ψ~q)Y

lm(θ, ψ) (7.5)

as wave function inside the sphere.

The problem here is that to solve this and find the energy solutions Ei,requires the parameter E to be calculated. Thus the only way to solve thisis to guess a solution, calculate and guess a new solution until the guess andsolution line up.

7.1 Linearized APW

As in APW method, the Linearized APW (LAPW) has a interstitial regionwith a free electron wave function, and inside the spheres linear combinationsof ψlm(~r), matching not only in value but also its derivative [1, 3].


This matching is done by first calculating R(r, E) using the ground stateenergy, E0, this can be Taylor expanded to find nearby energies.

R(r, E) = R(r, E0) + (E − E0)R(r, E0) (7.6)

where the dot represents derivation with respect to energy,

f(r, E) =∂

∂Ef(r, E) and

f ′(r, E) =∂

∂rf(r, E).

(7.7)

Like in the APW method the in- and outside of the sphere needs to matched,but now there is twice as many radial functions to fit, not just Rl but alsoRl yielding,

φ~q(~r) =

ei~q·~r ~r ∈ I∑

l,m

[AlmRl(r, E0) +BlmRl(r, E0)

]Y lm(θ, ψ) ~r ∈ S (7.8)

and the constants Alm, Blm are fixed by matching conditions. Now the energydependence in the wave function has been eliminated, and they are perfectlysmooth around the entire boundary. Alas, the trade off is the exactness ofthe energies inside the sphere is not as good.

7.2 Full Potential Linearized Augmented Plane

Wave

This method become possible only with the development of techniques forobtaining the coulomb potential with a charge density is general, periodicand shape approximation free. This is as it sounds very complicated and anyattempt in a derivation is material for a complete thesis. Thus, herein willonly a brief discussion about the method and its usability be included. Areader that wants a fuller explanation should read [17] or go directly to thesource papers [11, 40].

Full Potential Linearized Augmented Plane Wave (FPLAPW), combines thechoice of the LAPW basis set with the treatment of the full potential and


charge density without any shape approximations in the interstitial regionand inside the muffin tins [17]. This requires to remove or at least relax therequirements on the interstitial region and the sphericity of the muffin tin.Thus the difference from LAPW is the potential

V (~r) =

∑~K V

~KI ei

~K~r ~r ∈ I∑∞l=0 V

lMT (~r)Yl(r) ~r ∈ S (7.9)

where VI is a constant interstitial potential, and VMT is the spherical muffintin approximation.

This method is of course much more computer intensive, but mostly yieldsa lot better results. However the choice of shape approximation dependsstrongly on the system being considered. Most notably, the spherical approx-imation inside the muffin tin, are good when the system consists of closelypacked atoms. But for a more open system much less approximations arerequired.

Chapter 8

Results

Now all the methods needed for atomistic calculations have been discussed.To write a program for applying them, would be highly unnecessary as thathas already been done. In this case ELK has been chosen to do the calcula-tions.

8.1 ELK

ELK1, or Electrons in ~k space, is a Unix program that applies, high precision,density functional theory (DFT), chapter 6, using an all electron full potentiallinearized augmented plane wave (FP-LAPW) basis, chapter 7.2, [6].

Elk uses atomic units [6], or

h = m = a0 = e = 1 (8.1)

where h is the (reduced) Plank constant, m electron mass, a0 the Bohr radiusand e the electron charge. Yielding some new constants; e.g. the atomiclength 0.52917720859 A, and the atomic unit of energy (Hartree) 27.21138386eV.

1http://elk.sourceforge.net/

48

http://elk.sourceforge.net/

CHAPTER 8. RESULTS 49

8.1.1 Periodic Structures and unit cells

ELK applies periodic structures as they are far more computers efficient,than trying to calculate every atom.

Graphene

Thanks to the periodic structures of ELK graphene can be completely de-scribed with a two carbon atoms unit cell, fig. 8.2, and also its mirror imagemirrored around the y axis. This unit cell are then repeated many times inthe x, y plane. To calculate the position of the two atoms, ELK uses thelattice in, fig. 8.1, where

Figure 8.1: The Lattice vectors for graphene, used by ELK

~a = e

−√300

, ~b = e

−√3/23/20

(8.2)

where e = 2.6834 · a0 and a0 = 0.52918 A, yielding the unit cell whit the

Figure 8.2: The graphene unit cell

atomic positions (1/3, 1/3) and (−1/3,−1/3).


Carbon nanotube

For carbon nanotubes the symmetry yields a unit cell that has four atoms,for both armchair and zigzag.

(a) Unit cell for armchair carbonnanotube

(b) Unit cell for zigzag carbon nan-otube

Figure 8.3: The unit cells for carbon nanotubes, armchair and zigzag respectively

Repeating in the “winding direction” yields a larger radius. Repeating inthe “z axis of the tube” direction yields a longer tube. Since the zigzagunit cell is half as long in the “winding direction”, twice as many unit cellsare needed to get about the same radius of the tube. I.e. The smallestarmchair carbon nanotube needs four unit cells, with 16 atoms, and has aradius r = 2.74452 A. The smallest zigzag carbon nanotube needs eight unitcells, with 32 atoms, and has a radius r = 3.21301 A. The reason they do nothave same radius, is due to the flatness of the unit cell, whereas the carbonnanotube is cylindrical.

Once the energies in the interval ε ∈ [−0.05, 0.05],has been calculated. Withε the compression/stretch in percent, i.e. ε = 0.05 is a length increase of thegraphene or carbon nanotube by 5%. The energies have then been plottedand a curve has been fitted using Mathematica.

The length of the carbon nanotube cell, z = 2.46067 A, is same in all cases.


8.2 Graphene

First calculations on graphene where made. As graphene has a small 2 atomcell, the computational time required is significantly lower than that of carbonnanotubes. The fitted the curve is

æ

æ

æ

æ

æ æ

æ

æ

æ

æ

æ

-0.04 -0.02 0.02 0.04exy

0.1

0.2

0.3

0.4

eV

Figure 8.4: Calculated values (dots), and fitted values (line) for graphene

E(ε) = −0.000341027 + 1.30976ε+ 135.527ε2. (8.3)

Using the method in [22];

C =1

a0

∂2E

∂ε2. (8.4)

Here a0 is the atomic distance, 2.565 A, and C is tensile stiffness, which canbe interpreted as Young’s modulus in 1D, yielding

C = 52.837 eVA−1

(8.5)

8.3 Carbon nanotubes

8.3.1 Zigzag

Only one radius of the zigzag kind was done, as it has 32 atoms in the smallestunit cell and thus much harder, than armchair of comparable diameter, tocompute. Its radius is r = 6.0717 a.u. = 3.21301 A.


Two different measurements was made, first stretching/compressing in thelength (z) direction, and also a radial (r) direction.

Length

æ

æ

æ

æ

æ æ

æ

æ

æ

æ

æ

æ

-0.06 -0.04 -0.02 0.02 0.04ezz

0.5

1.0

1.5

2.0

2.5

3.0

eV

Figure 8.5: Calculated values (dots), and fitted values (line) for zigzag carbonnanotube in the z direction

E(ε) = −0.0661374 + 20.394ε+ 890.859ε2. (8.6)

Inserting into eq. 5.12 yields,

Y = 0.832828 TPa (8.7)

Radial

æ

æ

ææ

æ

æ

æ

æ

æ

æ

æ

-0.04 -0.02 0.02 0.04ezz

1

2

3

eV

Figure 8.6: Calculated values (dots), and fitted values (line) for zigzag carbonnanotube in the r direction


E(ε) = −0.0393307 + 31.0678ε+ 805.113ε2. (8.8)

Inserting into eq 5.12 yields,

Y = 0.752667 TPa (8.9)

8.3.2 Armchair

Instead of producing one graph for each radius, all seven armchair computa-tions are put in the same.

Table 8.1: E(ε) = a+ bx+ cx2, for all seven armchair carbon nanotubes

r a b c2.74452 -0.04516 7.10954 385.3963.41653 -0.0647281 9.14267 525.7884.09065 -0.0763511 11.0539 658.0194.76598 -0.0197127 11.6389 777.4575.44211 0.00650584 13.5431 876.3056.11866 0.00632335 14.6676 1020.86.79564 0.0127205 12.2647 1126.37

æ

æ

ææ

æ ææ

æ

æ

æ

æ

à

à

àà à

àà

à

à

à

à

ì

ì

ì

ì ìì

ì

ì

ì

ì

ì

ò

ò

ò

òò

ò

ò

ò

ò

ò

ò

ô

ô

ô

ôô ô

ô

ô

ô

ô

ô

ç

ç

ç

ç

çç

ç

ç

ç

ç

ç

á

á

á

á

áá

á

á

á

á

á

-0.04 -0.02 0.02 0.04%

0.5

1.0

1.5

2.0

2.5

3.0

3.5

eV

EHeL

Figure 8.7: Calculated values (dots), and fitted values (line) for seven armchaircarbon nanotubes, with different radius, in the z direction


8.4 Y(r)

Doing all the computations for the armchair carbon nanotubes and plotting,to see if there are any trends that can be concluded.

Table 8.2: Y (r) for all seven armchair carbon nanotubes

r Y(r)2.74452 0.8435823.41653 0.9245124.09065 0.9663484.76598 0.9799665.44211 0.9673316.11866 1.002246.79564 0.995721

3 4 5 6Å

0.6

0.7

0.8

0.9

1.0

TPaYHrL

Figure 8.8: Young’s modulus as a function of the carbon nanotubes radius

8.5 In situ Si

Here one carbon atom has been replaced with a silicon atom. This is donein a try to represent imperfections.


æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

0.02 0.04 0.06 0.08e

-0.8

-0.6

-0.4

-0.2

0.0

eV

Figure 8.9

E(ε) = 0.0673842− 40.0609ε+ 387.588ε2. (8.10)

Inserting into eq 5.12 yields,

Y = 0.362339 TPa (8.11)

Chapter 9

Discussion and Conclusions

We set out to try and find a method to calculate the Young’s modulus forcarbon nanotubes, using ELK. Looking at table 8.2 and/or fig. 8.8, one canconclude that our data seems to be converging towards 1 TPa. This would fitwell with the results reported in [26]. The dip at the low radius is probablydue to excessive bending of the sp2 bonds. No single walled carbon nanotubewith radius as low as 2.74452 A has ever been seen free. They do, however,exist as the smallest tube in multi walled carbon nanotubes [44].

When calculating the radial Young’s modulus for the zigzag carbon nanotubethe same formula as for Young’s in the length direction has been used. Thismay give some insight into its value but probably needs a deeper analysis.For the in situ silicon try, we got better results than expected. Here thatmeans a result. A silicon atom is much larger than a carbon atom; thusthe geometry needs to be reworked. However as with the radial Young’smodulus, this was a first try just to try the capacity of ELK.

When setting up ELK, the lowest possible amount of base functions werechosen. This value was found by the process of trial and error, where thefirst tries generated nonsensical results. It is, however, not known if morebase functions would yield a better results.

The most important success with this thesis is that we have shown thatELK will yield good results when doing calculations on both graphene and

56

CHAPTER 9. DISCUSSION AND CONCLUSIONS 57

carbon nanotubes. Our results are very much in line with other theoreticaltechniques as well as empirical measurements. Even though we get goodresults, we have only calculated the smallest of the carbon nanotubes, upto r = 6.79564 A, whereas they can be as large as r = 1 000 A. Also, wehave only calculated armchair, except for the smallest kind of zigzag. Fora deeper analysis, all kinds of chirality need to be considered. In all cases(graphene, armchair, zigzag and in situ silicon) we can from the graphs easilysee that the geometry needs to be optimized, as the ground state energy doesnot coincide with ε = 0. Geometry optimization is a tool that was includedinto ELK during the period of this thesis. Thus, it could not be used here,as most computations were already done. Even though there are severalproblems they all have one requirement in common, more computer power.The two largest armchair carbon nanotubes required us to rent a dedicatedcomputational server in Linkoping.

Appendix A

Tables

The measured values from ELK is tabulated, ε % length change, a.u. (atomicunits/Hartree energy) and eV (electron Volts) centered around zero stretch-ing.

Graphene

Table A.1: Graphene

ε a.u. eV-0.05 -75.61322814 0.285369048-0.04 -75.6177778 0.161566503-0.03 -75.62102946 0.073084335-0.02 -75.62305723 0.017905907-0.01 -75.62390280 -0.005103223

0 -75.62371526 00.01 -75.62251962 0.0325350190.02 -75.62043104 0.0893681710.03 -75.61749445 0.1692768490.04 -75.61377474 0.2704953050.05 -75.60928947 0.392545709

58

APPENDIX A. TABLES 59

CNT

Zigzag

32 atom

Table A.2: Zigzag r = 3.21301, first set is in the r-direction second set in thez-direction

ε a.u. eV a.u. eV-0.06 -1209.489631560 2.068251299-0.05 -1209.548155450 0.4757352630 -1209.522722610 1.167798035-0.04 -1209.566189120 -0.014985853 -1209.550201950 0.420047166-0.03 -1209.578012020 -0.336703324 -1209.566573120 -0.025435025-0.02 -1209.579861420 -0.387028057 -1209.574284160 -0.235263094-0.01 -1209.572179240 -0.177985308 -1209.573185990 -0.205380369

0 -1209.565638400 0 -1209.565638400 00.01 -1209.551822270 0.375956017 -1209.551438260 0.386405460.02 -1209.533661340 0.870140054 -1209.536325500 0.7976445740.03 -1209.506907100 1.598159949 -1209.514377860 1.3948702310.04 -1209.474457980 2.481145409 -1209.486187700 2.1619634960.05 -1209.435576230 3.539171634 -1209.453245780 3.058358726


Armchair

16 atom

Table A.3: Armchair r = 2.74452

ε a.u. eV-0.05 -604.6358791 0.648928075-0.04 -604.6503867 0.254155523-0.03 -604.6593093 0.011359066-0.02 -604.6646030 -0.132689456-0.01 -604.6605507 -0.022421092

0 -604.6597267 00.01 -604.6562282 0.0951987820.02 -604.6491946 0.2865946210.03 -604.6412578 0.5025640820.04 -604.6287630 0.842565970.05 -604.6135572 1.25633732

20 atom


ε a.u. eV-0.05 -755.8662757 0.847424372-0.04 -755.8827632 0.398776898-0.03 -755.8948618 0.069556787-0.02 -755.9003103 -0.078702696-0.01 -755.9013197 -0.106170956

0 -755.8974180 00.01 -755.8939599 0.094099550.02 -755.8864564 0.2982799240.03 -755.8708426 0.7231541450.04 -755.8550808 1.1520535830.05 -755.8359296 1.673185462


24 atom


ε a.u. eV-0.05 -907.0866106 1.078958255-0.04 -907.1069206 0.526295865-0.03 -907.1223111 0.107498327-0.02 -907.1295182 -0.088616674-0.01 -907.1308404 -0.12459445

0 -907.1262616 00.01 -907.1220190 0.1154472890.02 -907.1125994 0.3717681850.03 -907.0931996 0.8996624740.04 -907.0734271 1.4376998330.05 -907.0500351 2.074229014

28 atom cell


ε a.u. eV-0.05 -1058.296061 1.44657-0.04 -1058.323703 0.694385-0.03 -1058.339574 0.262518-0.02 -1058.34807 0.0313159-0.01 -1058.352369 -0.0856437

0. -1058.349221 0.0.01 -1058.34208 0.1943340.02 -1058.328122 0.5741390.03 -1058.31017 1.062570.04 -1058.285735 1.727550.05 -1058.260015 2.42744


32 atom


ε a.u. eV-0.05 -1209.507211 1.589888777-0.04 -1209.535706 0.814490325-0.03 -1209.551562 0.383035603-0.02 -1209.564572 0.029025567-0.01 -1209.567714 -0.056470696

0 -1209.565638 00.01 -1209.555990 0.2625569280.02 -1209.540099 0.6949626890.03 -1209.521372 1.204562520.04 -1209.493793 1.9550010570.05 -1209.461496 2.833865897

36 atom


ε a.u. eV-0.05 -1360.723993 1.910879975-0.04 -1360.755428 1.055468354-0.03 -1360.779417 0.402704535-0.02 -1360.791191 0.082316069-0.01 -1360.798959 -0.129067947

0 -1360.794216 00.01 -1360.784202 0.2725067710.02 -1360.762819 0.8543585400.03 -1360.742420 1.4094460090.04 -1360.711391 2.2537934810.05 -1360.677136 3.185916442


40 atom


ε a.u. eV-0.05 -1511.819422 2.281690863-0.04 -1511.853823 1.345608646-0.03 -1511.881660 0.588121816-0.02 -1511.896482 0.1847838-0.01 -1511.906794 -0.095812643

0 -1511.903273 00.01 -1511.894116 0.2491787240.02 -1511.871428 0.8665459750.03 -1511.850448 1.4374535980.04 -1511.817407 2.3365438430.05 -1511.780682 3.335867766

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