Carbon Nanotubes: A Theoretical study of Young's modulus

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 and optical properties. They are harder than diamond yet flexible, have better electrical conductor than copper, but can also be a semiconductor or even an insulator. These ranges of properties of course make carbon nanotubes highly interesting for many applications. Carbon nanotubes are already used in products as hockey sticks and tennis rackets for improving strength and flexibility. Soon there are mobile phones with flexible screens made from carbon nanotubes. Also, car- and airplane bodies will probably be made much 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 carbon nanotubes. This has been done by performing density functional theory calculations. The unit cell has been chosen as to accommodate for tubes of different radii. This allows for modelling the effect of bending of the bonds between the carbon atoms in the carbon nanotubes of different radii. The results show that Young’s modulus decreases as the radius decreases. In effect, the Young’s modulus declines from 1 to 0.8 TPa. This effect can be understood because the bending diminishes the pure sp² character of the bonds.

These results are important and useful in construction, not only when using carbon nanotubes but also when using graphene. Our results point towards a 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 hours of discussions on how to make this the best thesis it can be. Also, Krister Svensson for his many insights into carbon nanotubes that he has shared with me. My fiancee deserves one huge thank you for her support during this whole ordeal. Finally to the rest of my family and my friends for making the hard times durable, and the good times even better.

II

Contents

1 Introduction 5

2 Carbon 7

2.1 Diamond . . . 8

2.1.1 Structure . . . 8

2.1.2 Properties . . . 8

2.1.3 Lonsdaleite . . . 8

2.2 Graphite . . . 9

2.2.1 Structure . . . 9

2.2.2 Properties . . . 9

2.3 Fullerenes . . . 10

2.3.1 Buckminsterfullerenes . . . 10

2.3.2 Carbon nanotubes . . . 10

2.3.3 Nanobuds . . . 11

2.3.4 Nanofoam . . . 11

2.3.5 Carbyne . . . 11

2.4 Graphene . . . 12

2.4.1 Electronic Properties . . . 12

2.4.2 Mechanical Properties . . . 13

2.4.3 Optical Properties . . . 14

3 Carbon nanotube 15 3.1 Discovery . . . 15

3.2 Categories of carbon nanotubes . . . 17

3.2.1 Chirality . . . 17

3.2.2 Single walled . . . 18

3.2.3 Multi walled . . . 19

3.3 Properties . . . 20 1

CONTENTS 2

3.3.1 Strength . . . 20

3.3.2 Electrical properties . . . 21

3.3.3 Thermal properties . . . 22

3.4 Extreme carbon nanotubes . . . 22

3.5 Future uses for graphene and carbon nanotubes . . . 24

3.5.1 Material additives . . . 24

3.5.2 Nanomechanics . . . 24

3.5.3 Nanoelectromechanical systems . . . 25

3.5.4 Solar cells . . . 26

3.5.5 Energy storage . . . 26

3.5.6 Biomedicine . . . 27

3.6 Toxicity and biocompatibility graphene and carbon nanotubes 29 4 Orbital Hybridization 30 4.1 spⁿ Hybridization . . . 30

4.1.1 sp Hybridization (linear) . . . 31

4.1.2 sp² Hybridization (trigonal) . . . 32

4.1.3 sp³ Hybridization (tetrahedral) . . . 32

4.2 π bond . . . 33

5 Mechanical properties 34 5.1 Tensile strength . . . 34

5.2 Young’s modulus . . . 34

6 Density Functional Theory 38 6.1 Born Oppenheimer Approximation . . . 39

6.2 The theorems of Hohenberg Kohn . . . 39

6.2.1 The first Hohenberg Kohn theorem . . . 40

6.2.2 The second Hohenberg Kohn theorem . . . 40

6.3 Kohn Sham equations . . . 41

6.4 Local Density Approximation . . . 42

7 Augmented Plane Wave Method 44 7.1 Linearized APW . . . 45

7.2 Full Potential Linearized Augmented Plane Wave . . . 46

8 Results 48 8.1 ELK . . . 48

CONTENTS 3

8.1.1 Periodic Structures and unit cells . . . 49

8.2 Graphene . . . 51

8.3 Carbon nanotubes . . . 51

8.3.1 Zigzag . . . 51

8.3.2 Armchair . . . 53

8.4 Y(r) . . . 54

8.5 In situ Si . . . 54

9 Discussion and Conclusions 56

A Tables 58

List of Tables

3.1 Mechanical properties . . . 21

8.1 E() = a + bx + cx² . . . 53

8.2 Y (r) . . . 54

A.1 E() - Graphene . . . 58

A.2 E() - Zigzag r = 3.21301 . . . 59

A.3 E() - Armchair r = 2.74452 . . . 60

A.4 E() - Armchair r = 3.41653 . . . 60

A.5 E() - Armchair r = 4.09065 . . . 61

A.6 E() - Armchair r = 4.76598 . . . 61

A.7 E() - Armchair r = 5.44211 . . . 62

A.8 E() - Armchair r = 6.79564 . . . 62

A.9 E() - Armchair r = 6.79564 . . . 63

4

Chapter 1 Introduction

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

- Graphene Flagship

When Iijima 1991 [16] made carbon nanotubes known to the broad scientific community, he set a revolution in motion. Albeit slow at first but from 2004, when Geim and Novoselov [28] successfully uncovered one free layer of graphene, there has been a boom seldom seen. This was accentuated by graphene being the first of EU’s flagships¹, granting e1 B over a ten year period, 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 mother of all graphitic allotropes. It can be wrapped up into zero-dimensional struc- tures known as fullerenes. Stacked layer upon layer to three-dimensional graphite. Rolled up into one dimensional carbon nanotubes, the structure we have chosen to investigate further.

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

1http://graphene-flagship.eu/

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CHAPTER 1. INTRODUCTION 6

thus they were discussed and researched for purely academic reasons. In 1991 Iijima published a high impact article [16]. Here he claimed to have seen carbon structures shaped like needles, this was not the first evidence that carbon nanotubes existed. Despite not being first, Iijimas carbon needles, triggered a first boom in the interest around low dimensional materials. This boom was enough for some to disregard Peierls and Landau’s results. In 2004 Konstantin Novoselov and Andre Geim [28], successfully extracted a single layer of graphene from the graphite in a pencil, using only scotch tape.

The ensuing second boom this created was immense, making carbon and its allotropes one of the biggest fields of research. The discovery yielded Novoselov and Geim the Nobel prize in physics 2010.

Despite many tries, no one is yet successful in unambiguously determining the Young’s modulus of carbon nanotubes. Reported values are ranging between 0.6-5.5 TPa [38]. However, they seem to be converging to 1 TPa [26]. This is our try to cast some light on the subject, and hopefully some new insights can emerge from our work.

Chapter 2 Carbon

Carbon only exists as single atoms in extreme temperature environments. In ambient environment carbon reacts with other atoms to stabilize, to form multi atomic compounds. The most important of these compounds must be attributed to organic chemistry and carbons rˆole in makeup the nucleic acids in DNA. Also, the proteins which are the building blocks for all life, thus making carbon the basis of life.

Now, a short description of how carbon bonds to carbon forming what is known 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 strongly insulating to nearly perfect conductors. Thermal, most thermally conduct- ing, and mechanical, diamond is the hardest naturally occurring material and graphite one of the softest, carbyne has the highest Young’s modulus ever measured.

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)

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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 each atom 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, but due to the strength of the carbon-carbon bonds, it is the hardest naturally occurring 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 modulus of 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 a hexagonal 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 material has been tested harder than diamond with indentation pressures of 152 GPa in 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 atom is 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 through weak van der Waals forces. This gives graphite its softness and its cleaving properties, 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, graphite conducts electricity. However only in the plane of each covalently bonded sheet, resulting in a lower bulk electrical conductivity for carbon than for most metals.

2.2.1 Structure

Graphite has a layered, planar structure. In each layer, the carbon atoms are arranged in a honeycomb lattice with atomic distance of 1.42 ˚A, and interplanar spacing 3.35 - 3.45 ˚A. The two known forms of graphite, hexag- onal (α) and rhombohedral (β), have very similar physical properties. The only difference is stacking, either with the atoms inline or a slight shift of the layers. The hexagonal graphite may be either flat or buckled. The α form can be converted to the β form through mechanical treatment and the β 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 to that 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 hexagonal packing, 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 twenty hexagons and twelve pentagons, with a carbon atom at each vertex of each polygon and a bond along each polygon edge. The buckyballs are fairly large

Figure 2.1: C₆₀molecule 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 is bonded with sp²hybridization, in a curved sheet that forms a hollow cylinder, see chapter 3.

CHAPTER 2. CARBON 11

2.3.3 Nanobuds

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

2.3.4 Nanofoam

Carbon nanofoam is a ferromagnetic allotrope discovered in 1997 [34]. It consists of a low-density cluster assembly of carbon atoms strung together in a loose three-dimensional web, in which the atoms are bonded trigonally in six- and seven membered rings. It is among the lightest known solids, with a 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]. A tensile stiffness, see chapter 5.1 of C = 95.56 eV/˚A ∼ 10⁹ Nm kg⁻¹, which is double the stiffness of graphene, and three times stiffer than diamond. There are many other interesting physical applications of carbyne that have been proposed theoretically, including nanoelectronic- and spintronic devices, also hydrogen 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 environments as shock compressed graphite, interstellar dust, and meteorites.

CHAPTER 2. CARBON 12

2.4 Graphene

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

- Prof. James Hone Columbia University

Graphene is a flat monolayer of carbon atoms tightly packed (Atomic distance 1.42 ˚A) into a two-dimensional honeycomb lattice, fig. 2.3, and is a basic building block for graphitic materials of all other dimensionalities. It is part of the zero dimensional fullerenes, rolled into one dimension nanotubes or stacked into three dimension graphite [9].

Figure 2.3: Honeycomb structure of Graphene

Graphene is the material of superlatives; conducts electricity 1 000 times better 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 relation for low energies becomes conical, which leads to that the electrons (and holes) has a zero effective mass, fig. 2.4, [26].

CHAPTER 2. CARBON 13

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

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

E = ¯hvF

q

k_x²+ k_y²

where the Fermi velocity v_F ∼ 10⁶ m/s, and the wavevector k is measured from 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’s acoustic phonons. Experimentally of course the quality of the graphene and substrate plays a rˆole for the electron mobility, which is why it is reported to 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 carriers can travel several µm before scattering, leading to a phenomenon known as ballistic transport. The resistivity of the graphene sheet is 10⁻⁶ Ω cm, which is less than the resistivity of silver, the lowest resistivity substance known at room temperature.

2.4.2 Mechanical Properties

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

CHAPTER 2. CARBON 14

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

- Nobel Committee, 2010

Measurements have shown that graphene has a breaking strength over 100 times 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 suspended graphene sheets has been measured. Graphene sheets, held together by van der Waals forces, were suspended over SiO₂ cavities where an AFM tip was used to test its mechanical properties [8]. Its spring constant was in the range 1 − 5 N m⁻¹ and the Young’s modulus was 0.5 TPa, which differs from that of the bulk graphite. These high values make graphene very strong and rigid.

2.4.3 Optical Properties

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

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

Chapter 3

Carbon nanotube

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

The chemical bonding of nanotubes is composed entirely of sp² hybridized bonds, similar to those of graphene. These bonds, which are stronger than the sp³ 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 of graphite as anode and have a high voltage current go through will create all kinds of fullerenes, among other carbon nanotubes. The main problem has been in detection, before 1931 when Knoll & Ruska invented the first transmission 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 usable when any of these properties are relevant.

Once the TEM was invented there has been different groups detecting nano scale carbon structures, but when Iijima in 1991 [16] made the broad scientific community aware of the potentials of carbon nanotubes a rush to understand the new material ensued. This has yielded Iijima a somewhat unearned reputation as the sole discoverer of carbon nanotubes.

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

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

1979 John Abrahamson presented evidence of carbon nanotubes at the 14th Biennial Conference of Carbon at Pennsylvania State University. The conference paper described carbon nanotubes as carbon fibers that were produced on carbon anodes during arc discharge. A characterization of these fibers was given as well as hypotheses for their growth in a nitrogen 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 images and XRD patterns, the authors suggested that their carbon multi-layer tubular crystals were formed by rolling graphene layers into cylinders.

They speculated that by rolling graphene layers into a cylinder, many different arrangements of graphene hexagonal nets are possible. They suggested 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 for

the production of “cylindrical discrete carbon fibrils”

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

CHAPTER 3. CARBON NANOTUBE 17

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

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, ~a₂ in the honeycomb crystal lattice of graphene. Where

~a₁ = 3 2,

√3 2

!

a_c−c, ~a₂ = 3 2, −

√3 2

!

a_c−c (3.1)

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

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

C_h = a√

n²+ nm + m² (3.2)

where a =√

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

π = a π

√n² + nm + m² (3.3)

CHAPTER 3. CARBON NANOTUBE 18

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 be ranging up to centimeters. The structure of a single walled carbon nanotube can be conceptualized by wrapping one sheet of graphene into a seamless cylinder.

Single walled carbon nanotubes are an important variety of carbon nanotube because most of their properties change significantly with the chirality, and this dependence is non monotonic. In particular, their band gap can vary from 0 to about 2 eV, and their electrical conductivity can show metallic or semiconducting behavior.

CHAPTER 3. CARBON NANOTUBE 19

(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 single walled 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 distance between graphene layers in graphite, 3.35−3.45 ˚A. Individual layers are single walled carbon nanotubes. Since different single walled carbon nanotubes have different electronic properties, multi walled nanotubes are almost always zero gap metals.

CHAPTER 3. CARBON NANOTUBE 20

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 carbon nanotubes, but their resistance to chemicals are significantly improved. This is 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 some double bonds, destroying the structure on the nanotube thus, modifying both its mechanical and electrical properties. In the case of double walled carbon nanotube, only the outer wall is modified.

3.3 Properties

3.3.1 Strength

Carbon nanotubes are among the strongest and stiffest materials discovered in terms of tensile strength and Young’s modulus respectively. This strength results from the covalent sp² bonds formed between the individual carbon atoms.

CHAPTER 3. CARBON NANOTUBE 21

Carbon nanotube have been tested to have a tensile strength of up to ∼ 100 GPa. 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 is 48 000 kN m kg⁻¹ is the best of known materials, compared to high carbon steel’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 at strains of approximately 5% and increases the maximum strain the tubes undergo before fracture by releasing strain energy. Due to the hollow struc- Table 3.1: Mechanical properties, a comparison between carbon nanotubes and other selected materials. max is the maximum extension before breaking the ma- terial. Values within brackets are theoretical maximums

Material Young’s modulus TPa

Tensile strength GPa

_max

%

SWNT 0.65-5.5 126 16-23

MWNT 0.2-0.95 >63 (300)

Stainless steel 0.186-0.214 0.38-1.55 15-50

Kevlar 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 forces them to undergo buckling when placed under compressive, torsional, or bend- ing stress. Single walled carbon nanotubes can withstand a radial pressure up to 25 GPa without permanent deformation. After that they undergo a transformation to superhard phase nanotubes [31]. The bulk modulus of superhard phase nanotubes is 462 to 546 GPa, even higher than that of a diamond’s 420 GPa.

3.3.2 Electrical properties

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

CHAPTER 3. CARBON NANOTUBE 22

• 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 the tube’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 is 2G₀, where G₀ = 2e²/h is the conductance of a single ballistic quantum channel.

3.3.3 Thermal properties

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

CHAPTER 3. CARBON NANOTUBE 23

• 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 tube in a MWNT.

Thinnest single walled carbon nanotube is 4.3 ˚A in diameter

CHAPTER 3. CARBON NANOTUBE 24

3.5 Future uses for graphene and carbon nan- otubes

In the near future there are several areas that will be greatly influenced by graphene and carbon nanotubes. Below are some areas discussed but of course 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 their exact properties like chirality and number of walls they are easy and cheap to produce. This makes them perfect for material additives, yielding stronger and lighter materials at a very low cost. In principle this is easy, just add carbon nanotubes to the mixture and the mixture has new and improved properties, reality however is not always that easy.

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

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

3.5.2 Nanomechanics

The field of nanomechanics applies the properties of nanomaterials to create machines on a nanometer scale. The problem, and benefit, here is of course that on the nano scale it is required to not only follow the normal mechanical rules but also those of the nanoworld.

CHAPTER 3. CARBON NANOTUBE 25

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 nanotubes has superior properties, together with them being almost frictionless bearings yields applications in the NEMS field that previously only could be dreamt of, i.e. electrical wires, nanomotors, switches and high frequency oscillators.

Wires and connectors

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

Nanomotors

Nanomotors made of carbon nanotubes generate a rotating motion via a suspended multi walled carbon nanotube with a rotor attached between two stator 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 of the intersections can read with a weak current and the state changed with a stronger current.

CHAPTER 3. CARBON NANOTUBE 26

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 low rate 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 of indium tin oxide as electrodes in organic solar cells. However, recent studies have used graphene flakes in a “polymer blend bulk heterojunction” solar cell [42]. This increases the donor to acceptor ratio, and with only a small amount 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 another possibility.

Multi walled carbon nanotubes are widely used in lithium ion batteries, this gives the batteries faster recharge time and higher storage rate. The fast recharge 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 specific capacitance with fast charging time at high current density. This makes graphene ideal for these kind of applications. [18].

CHAPTER 3. CARBON NANOTUBE 27

The main reason for not having electric cars yet is the recharge time of the batteries. 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 a fraction of that time, but wont hold nearly as much energy. This is where the graphene based supercapacitor comes in play, they can be recharged faster than 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, maintaining its mileage.

Gravimetric energy storage

Using carbon nanotubes mechanical properties as a way to store energy is done 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 about 20%, will yield an energy storage capacity three orders of magnitude higher than ordinary steel springs.

3.5.6 Biomedicine

Since carbon nanotubes have a high surface area, chemical stability, and rich electronic polyaromatic structure, they are able to adsorb or conjugate with a wide variety of molecules. This makes them perfect candidates for biomedicine 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 without metabolism 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 into the cells and destroying them, instead of treating the entire body.

CHAPTER 3. CARBON NANOTUBE 28

Biosensor

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

CHAPTER 3. CARBON NANOTUBE 29

3.6 Toxicity and biocompatibility graphene and carbon nanotubes

The big bump in the road for graphene and carbon nanotubes is the lack of research on its toxicity and biocompatibility. Also, the few results that exists are often inconclusive or contradictory. Until there are some thorough studies, most applications will not reach the market. Despite this, rater large problem, there are a plethora of applications proposed in the biomedical field alone [12].

One positive side note is that previous use of carbon based biomaterials show high biocompatibility. However, it is suggested that due to the presence of transition metal catalysts carbon nanotubes are some what toxic, and if they reach the organs they can induce inflammation or fibrotic reactions. Though, chemically functionalized carbon nanotubes for drug delivery have so far not demonstrated any toxicity.

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

Chapter 4

Orbital Hybridization

The solution to the schr¨odinger equation yields a region where the electron density is expected to be highest. This region is called an orbital. When Linus Pauling in the 1950s tried to calculate the orbits of atoms heavier than hydrogen, he failed; as everyone else. He, therefore, set out to create a mathematical model that could explain the different orbitals.

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

Consider the element whose bonding is of most interest to us: carbon. Its electronic configuration is 1s²2s²2p². 2s and 2p differ very little in energy;

therefore, the wave functions can mix when carbon is bonded.

4.1 sp

Hybridization

spⁿis 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 are obtained, called sp orbitals, made up of 50% s and 50% p character. The major 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) with opposite sign. The remaining two p orbitals are left unchanged, these two forms two π bonds.

The 180^◦angle that results from this hybridization scheme minimizes electron repulsion. The oversized front lobes of the hybrid orbitals also overlap better than lobes of unhybridized orbitals; the result is energy reduction due to improved bonding. This is the hybridization that gives rise to Carbyne, one of the materials harder than diamond.

CHAPTER 4. ORBITAL HYBRIDIZATION 32

4.1.2 sp

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 2p atomic orbital. The combination of these atomic orbitals creates three new hybrid orbital equal in energy-level. The hybrid orbital is higher in energy than the s orbital but lower in energy than the p orbital, but they are closer in energy to the p orbital. The new set of formed hybrid orbital creates trigonal structures, creating a molecular geometry of 120^◦.

The combination of an s orbital, and two p orbital from the same valence shell provides a set of three equivalent sp² hybridized orbital that point in directions separated by 120^◦. The directions of these new, hybridized orbital are the dictators of the spatial arrangement for bonding. The sp² hybridized orbital 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 sp

Hybridization (tetrahedral)

To achieve sp³ bonding, first promotion of one electron from 2s to 2p results in four singly filled orbitals. Then, the 2s orbital is hybridized with all three 2p orbitals. This makes four equivalent sp³ orbitals with tetrahedral symmetry, for electron repulsion minimization. Each has 75% p and 25% s character and occupied by one electron. The bond angles of a tetrahedron is 109.58^◦.

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-

CHAPTER 4. ORBITAL HYBRIDIZATION 33

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 explained by; a greater extension from the positive charge of the atomic nucleus. Also by a significantly less overlap between the p orbitals due to their parallel ori- entation. This is contrasted by σ bonds which form bonding orbitals directly between the nucleus of the bonding atoms, resulting in greater overlap. A π bond by itself is weaker than a σ bond, but π bonds are only found in combination with σ bonds, so the combination of the two bonds is stronger than either bond would be by itself.

Electrons in π bonds are sometimes referred to as π electrons. These π electrons are closest to the Fermi level and thus crucial for the electronic properties 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 object being 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 the Young’s modulus of a carbon nanotube is done.

First the definition of young’s modulus Y = σ

 (5.1)

where, σ = F/A with F applied force on the area A.  = ∆z/z with, z the length 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 A (5.2)

Secondly Hooke’s law

F = k∆z (5.3)

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

∆z = Y A

∆z . (5.4)

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

Z

F dz = 1

2k(∆z)² (5.5)

now combining eqs. (5.4, 5.5), yields E = 1

2 Y A

∆z (∆z)²

= 1

2Y A∆z.

(5.6)

Thus, the formula for young’s modulus can be written as Y = 2E

A∆z (5.7)

now using

∆z = ∆z

z · z = z (5.8)

yields

Y = 2E

Az². (5.9)

A can be chosen in several different ways, which probably is the reason for such 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)

CHAPTER 5. MECHANICAL PROPERTIES 36

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

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

• At first glance the radius of the carbon atom r_C = 70 pm, might be a good choice. But this will yield a Young’s modulus much larger than any 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 spread equally 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 modulus of a carbon nanotube is

Y = E

πrtz². (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 cells and the formula must thus be properly edited for that purpose.

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

CHAPTER 5. MECHANICAL PROPERTIES 37

r t

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

Chapter 6

Density Functional Theory

“The underlying laws necessary for the mathematical theory of large parts of physics and the whole of chemistry are thus com- pletely known, and the difficulty is only that the exact application of 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 solve many body problems. The central part of DFT is that it determines the electron density, with respect to spatial dependency, ρ(~r), and not the wave function, φ. The total energy of the molecule, E [ρ], is a functional¹ of the electron density, ρ(~r). Thus Density Functional Theory. It is also assumed that the density is slowly varying.

The electron density is calculated using ρ(~r) =X

i

φ^∗_iφ_i (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

X

i

∇²_~

Ri

M_i − ¯h² 2m_e

X

i

∇_~²_r

i

− e² 4π₀

X

i,j

Z_i

| ~R_i− ~r_j| + e² 8π₀

X

i6=j

1

|~r_i− ~r_j| + e² 8π₀

X

i6=j

Z_iZ_j

| ~R_i − ~R_j|

(6.2)

where m_e, ~r_i and M_i, ~R_i is the mass and position for the electron and nucleoid particles, respectively. This is not doable for any atomic system larger than Hydrogen atom, thus some approximations are needed.

6.1 Born Oppenheimer Approximation

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

φ = φ^electrons· φ^nuclear (6.3)

6.2 The theorems of Hohenberg Kohn

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

2only position and charge is here considered; most notably lacking is the spin

3a quantum mechanical theory for the electronic structure of many body systems de- veloped semiclassically shortly after the introduction of the Schr¨odinger equation.

CHAPTER 6. DENSITY FUNCTIONAL THEORY 40

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 for reducing the many body problem of N electrons with 3N spatial coordinates to 3 spatial coordinates, through the use of functionals of the electron den- sity. This theorem can be extended to the time dependent domain to develop time dependent density functional theory (TDDFT), which can be used to describe excited states [3].

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

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

D

φ| ˆO|φE

= O[ρ] (6.4)

6.2.2 The second Hohenberg Kohn theorem

The second Hohenberg Kohn theorem defines an energy functional for the system and proves that the correct ground state electron density minimizes this 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 of Hohenberg Kohn yields; the nuclear core can be said to be stationary, and

CHAPTER 6. DENSITY FUNCTIONAL THEORY 41

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

H[ρ] = T [ρ] + V [ρ] + V_ext(~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 of many 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 external potential V_ext.

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 using DFT [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] = Z

V_ext(~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

Z ρ(~r)ρ(~r⁰)

|~r − ~r⁰| d~rd~r⁰+ E_xc[ρ]. (6.8)

CHAPTER 6. DENSITY FUNCTIONAL THEORY 42

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

E_xc[ρ] = Z

ρ(~r)_xc[ρ]d~r, (6.9) where xc[ρ] is the exchange and correlation energy per electron of a uniform electron gas.

Applying the condition of slowly varying density Z

δρ(~r)d~r = 0 (6.10)

yields, Z

δρ(~r)



Vext(~r) +

Z ρ(~r⁰)

|~r − ~r⁰| + δT_s[ρ]

δρ(v~r) + µxc[ρ]



d~r = 0 (6.11) here

µ_xc= d

dnn_xc[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 only depend 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 electron gas model. Since the homogeneous electron gas model yields extremely good results, LDA is often synonymous with the homogeneous electron gas func- tionals.

The LDA exchange correlation energy is found by E_xc^LDA[ρ] =

Z

ρ(~r)_xc[ρ]d~r

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

E_xc = E_x+ E_c ,

CHAPTER 6. DENSITY FUNCTIONAL THEORY 43

so that separate expressions for E_x and E_c are sought. The exchange term often takes on a simple analytic form for the homogeneous electron gas. Only limiting expressions for the correlation density are known exactly, leading to numerous different approximations for _c.

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

V_xc^LDA(~r) = δE_xc^LDA

δρ = _xc[ρ] + ρ∂_xc[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. This manifests itself especially in electron rich atoms, falsely stating they are unable to bind some electrons, yielding an prediction of the atom being unstable.

Chapter 7

Augmented Plane Wave Method

The Augmented Plane Wave (APW) method is a method that uses the muf- fin tin approximation to approximate the electrons energy states in a crystal lattice. For the potential a spherical symmetry is assumed centered at the atomic nucleus, the electron can thus be considered as they where in a free atom. 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 the Schr¨odinger equation and matching the solutions to the spheres and intersti- tial region, then optimizing by using the variational method the augmented plane waves are constructed [3, 37].

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

φ^~k_K~(~r, E ) = (

e^{i(~k+ ~K)·~r} ~r ∈ I P∞

l=0

Pl

m=−lA_lmR_l(r)Y_m^l(θ, ψ) ~r ∈ S (7.1) where ~k is a Bloch vector in the first Brillouin zone, ~K the reciprocal lattice vector. R_l(r) is the solutions to the radial Schr¨odinger equations:

− 1 2r²

d dr



r²dRl(r) dr



+ l(l + 1)

2r² + V (r)



R_l(r) = E R_l(r) (7.2)

44

CHAPTER 7. AUGMENTED PLANE WAVE METHOD 45

which are solvable. Y_m^l(θ, ψ) is the spherical harmonic [41] and A_lm 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 spherical harmonic wave and the wave must be matched over the entire surface of S.

Thus, the expanded plane-wave becomes;

e^i~q·~r = 4π

∞

X

l=0 l

X

m=−l

i^lj_l(qr)Y_m^l∗(θ_~q, ψ_~q)Y_m^l(θ, ψ) (7.3)

where ~q = ~k + ~K, (r, θ, ψ) and (q, θ_~q, ψ_~q) are the spherical coordinates of ~r and ~q respectively. j_lis the Bessel function of order l. It is highly impractical to have a infinite summation, thus a maximum is chosen for l, called l_max. The coefficients A_lm are determined by requiring continuity over the entire sphere, thus equaling the expanded plane-wave and the spherical harmonic wave from eq. 7.1, fixes A_lm as;

A_lm = 4πi^le^i~q·~r

R(R) j_l(qR)Y_m^l∗(θ_~q, ψ_~q) (7.4) yielding,

φ_q~(~r) = 4πX

lm

i^l j_l(qR) R_l(R)



R_l(r)Y_m^l∗(θ_~q, ψ_~q)Y_m^l(θ, ψ) (7.5)

as wave function inside the sphere.

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

7.1 Linearized APW

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

CHAPTER 7. AUGMENTED PLANE WAVE METHOD 46

This matching is done by first calculating R(r, E ) using the ground state energy, 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 f⁰(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 R_l but also R˙_l yielding,

φ_~q(~r) =( e^i~q·~r ~r ∈ I

P

l,m

h

A_lmR_l(r, E₀) + B_lmR˙_l(r, E₀)i

Y_m^l(θ, ψ) ~r ∈ S (7.8) and the constants Alm, Blm are fixed by matching conditions. Now the energy dependence in the wave function has been eliminated, and they are perfectly smooth around the entire boundary. Alas, the trade off is the exactness of the 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 for obtaining the coulomb potential with a charge density is general, periodic and shape approximation free. This is as it sounds very complicated and any attempt in a derivation is material for a complete thesis. Thus, herein will only a brief discussion about the method and its usability be included. A reader that wants a fuller explanation should read [17] or go directly to the source papers [11, 40].

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

CHAPTER 7. AUGMENTED PLANE WAVE METHOD 47

charge density without any shape approximations in the interstitial region and inside the muffin tins [17]. This requires to remove or at least relax the requirements on the interstitial region and the sphericity of the muffin tin.

Thus the difference from LAPW is the potential V (~r) =

 P

K~ V_I^K~e^{i ~K~r} ~r ∈ I P∞

l=0V_{M T}^l (~r)Y_l(ˆr) ~r ∈ S (7.9) where V_I is a constant interstitial potential, and V_{M T} is the spherical muffin tin approximation.

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

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 that has 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 potential linearized 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 radius and e the electron charge. Yielding some new constants; e.g. the atomic length 0.52917720859 ˚A, and the atomic unit of energy (Hartree) 27.21138386 eV.

1http://elk.sourceforge.net/

48

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 image mirrored around the y axis. This unit cell are then repeated many times in the x, y plane. To calculate the position of the two atoms, ELK uses the lattice in, fig. 8.1, where

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

~a = e





−√ 3 0 0



, ~b = e





−√ 3/2 3/2

0



 (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).