Document [original]

Electronic and vibrational properties of

diamondoid derivatives, graphite, and carbon nanotubes

vorgelegt von

Master of Science (M.Sc.)

Christoph Tyborski

geb. in Berlin

von der Fakultät II - Mathematik und Naturwissenschaften

der Technischen Universität Berlin

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften

- Dr. rer. nat. -

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr. Michael Lehmann

Gutachter: Prof. Dr. Axel Hoffmann

Gutachter: Prof. Dr. Ralph Krupke

Gutachterin: Prof. Dr. Janina Maultzsch

Tag der wissenschaftlichen Aussprache: 12. Oktober 2018

Berlin 2018

"Das Volumen des Festkörpers wurde von Gott geschaffen, seine Oberfläche aber wurde

vom Teufel gemacht." - Wolfgang Pauli

Zusammenfassung

In dieser Arbeit werden vibronische und elektronische Eigenschaften von Diamantoiden,

Graphen, mehrlagigem Graphen, Graphit und Kohlenstoffnanoröhren untersucht. Alle

Untersuchungen basieren auf optischen Spektroskopiemethoden, wie Raman-Spektroskopie,

Photolumineszens-Spektroskopie, Photolumineszens-Anregungsspektroskopie und Absorp-

tionsspektroskopie, deren Ergebnisse mit computergestützten Rechnungen verglichen wer-

den. Obwohl die untersuchten Materialsysteme schon einige Zeit bekannt sind, gibt es

besonders im tiefen UV Bereich noch offene Fragen bezüglich deren optischer Eigen-

schaften. Es wird gezeigt, dass sowohl die Struktur der Kohlenstoffallotrope, als auch

die Hybridisierung des enthaltenden Kohlenstoffs, die optischen Eigenschaften im UV

Bereich bestimmen.

Im ersten Teil geht es um Diamantoide und deren Derivate. Große Fortschritte im Ver-

ständnis ihrer chemischen Reaktivität führten zu verschiedensten Funktionalisierungen

von den ansonsten weitestgehend inerten Diamantoiden. Gezielte Funktionalisierungen

bieten die Möglichkeit strukturbezogene Eigenschaften zu ändern. Beispielsweise hängen

optische Übergangsenergien stark von der Art einer funktionellen Gruppe ab, sodass diese

an mögliche Anwendungen angepasst werden kann.

Wir untersuchen im Besonderen Diamantoidoligomere, die aus verschiedenen Monomeren

mittels einer Kohlenstoffeinfachbindung oder -doppelbindung verbunden sind. Die Funk-

tionalisierung mit sp2Defekten, führt zu einer Verkleinerung optischer Übergangsenergien

von ∼6.5eV für unfunktionalisierte Diamantoide, auf ∼4.7eV für deren Derivate. Weit-

erhin zeigen wir, dass deren vibronische Eigenschaften denen von unfunktionalisierten

Diamantoiden ähneln, strukturbezogene Änderungen aber zum Auftreten neuer

Schwingungsmoden führen. Beispielhaft seien Dimer-Atmungsmoden, sowie hochlokali-

sierte C-H Streckschwingungen erwähnt. Diamantoide bilden van-der-Waals Kristalle,

die besonders hohe Bindungsenergien aufweisen. Deren periodische Anordnung führt zu

einer Ausprägung von kollektiven Eigenschaften wie einer vibronischen oder elektronis-

chen Bandstruktur. Mit resonanten Raman und Absorptionsmessungen zeigen wir, dass

in Diamantoid van-der-Waals Kristallen, die optischen Übergänge um ca. 1eV kleiner

sind als in vergleichbaren, isolierten Molekülen.

Im zweiten Teil diskutieren wir doppeltresonante Raman Streuprozesse in Graphit, Graphen

und Kohlenstoffnanoröhren (KN) im tiefen UV Bereich. Obwohl das Konzept der dop-

peltresonanten Ramanstreuung bereits vor mehr als 10 Jahren eingeführt wurde, ist es

bis heute nicht bekannt, wie sich die Streuprozesse bei sehr hohen Anregungsenergien

bis in den UV Bereich ändern. Wir werden sehen, dass die Intensitäten für die gut

verstandenen, doppeltresonanten Moden, wie die D,2Doder iTOLA (transversal op-

iii

tisch + longitudinal akustisch) bei Anregungsenergien zwischen 4.69eV und 5.46eV stark

zurückgehen. Anstelle dessen finden wir asymmetrische Ramanmoden, die wir der vibro-

nischen Zustandsdichte in zweiter Ordnung zuordnen. Die gemessenen Moden können

den zweifach entarteten LO/TO Phonon am ΓPunkt, den LA/LO und TO Phononen

am M Punkt, sowie dem Maximum des LO Phononenzweiges zugeordnet werden. Mit

computergestützten Rechnungen zeigen und erklären wir den Streumechanismus, der zu

den neuen Moden führt. Anregungen nahe der M Punkt Energie von ∼4.7eV führen

zur Aktivierung von doppeltresonanten Ramanmoden mit den kürzestmöglichen resonan-

ten Phonon Wellenvektoren. Eine prominente, dadurch entstehende Ramanmode, ist die

2DMode, die durch die sehr kurzen involvierten Phonon Wellenvektoren bei sehr hohen

Energien von ca. 3050cm−1zu finden ist. Weiterhin untersuchen wir die vibronische

Zustandsdichte in zweiter Ordnung von mehrlagigem Graphen. Wir können zeigen, dass

die Energien von den jeweils zweifach entarteten LO/TO Phononenzweigen am ΓPunkt

und die Maxima der LO Phononenzweige nur sehr wenig von der Lagenanzahl abhängen.

Im letzten Teil analysieren wir intensitätsschwache Ramanmoden von Kohlenstoffnanoröh-

ren. Wir konzentrieren uns dabei auf die Moden bei ca. 1950 cm−1und 2450cm−1in einer

Ensembleprobe. Beide Moden sind über einen großen Anregungsbereich von 1.71eV-

2.73eV dispersiv und zeigen jeweils verschiedene, ausgeprägte Bereiche. Mit Hilfe von

DFT (Dichtefunktionaltheorie) Rechnungen, ordnen wir deren Dispersionen einem En-

semble Effekt zu. Abhängig von der Anregungsenergie, sind verschiedene KN in Reso-

nanz, deren charakteristische vibronische Eigenschaften dann in Spektrum zu finden sind.

Besonders im Anregungsbereich zwischen 2.1eV-2.5eV, sind einerseits metallische, aber

andererseits auch halbleitende KN zu finden. Aufgrund ihrer unterschiedlichen elektron-

ischen Struktur, haben beide Arten verschiedene Durchmesser für gleiche optische Über-

gangsenergien. Dies führt zu starken Unterschieden in ihren vibronischen Eigenschaften,

wodurch sich beide Spezies im Ramanspektrum gut unterscheiden lassen.

Mit Hilfe von DFT und NOTB (non-orthogonal tight-binding) Rechnungen, können wir

die ∼2450cm−1Mode zu einem doppeltresonanten iTOLA Prozess in der Nähe des K

Punktes zuordnen. Im Gegensatz dazu, ist die Mode bei 1950 cm−1in den DFT Rech-

nungen eine doppeltresonante LOLA Kombinationsmode aus der Nähe des ΓPunktes.

Für die Zuordnung wurden die Dispersionen und Intensitäten aller 6×6 = 36 Kombi-

nationsmoden im Ausgangsmaterial Graphen verglichen. Im NOTB Ansatz jedoch, ist

die ∼1950cm−1Mode eine doppeltresonante TOZO (ZO: optische Schwingung nicht in

der Ebene von Graphen) Mode in der Nähe des K Punktes. Wir glauben, dass im DFT

Ansatz eine mögliche Beteiligung von ZO Phononen an Streuprozessen zu wenig berück-

sichtigt wird. Dieser nutzt die Elektron-Phonon Kopplungselemente des Ausgansmaterials

Graphen, wo der ZO Phononenzweig Raman-inaktiv ist. Deshalb ordnen wir die Mode

bei ca. 1950cm−1eher einer TOZO Kombination zu.

Wir zeigen einen erweiterten "Kataura plot", dh. eine Übersicht von optischen Über-

gangsenergien abhängig von dem Durchmesser einer KN. Die Erweiterung bezieht sich auf

Energien im tiefen UV Bereich bis hin zu den höchstmöglichen optischen π→π∗Übergän-

gen in Kohlenstoffnanoröhren. Einerseits bestimmen der "Quantum confinement"-, aber

auch der "Trigonal warping"-Effekt die Übergangsenergien im tiefen UV Bereich. Ander-

erseits können wir mit Hilfe von berechneten optischen Matrixelementen zeigen, dass es

verbotene Übergänge für Anregungen höher als die M Punkt Energie (∼4.7eV) gibt.

Dies trifft für keinen Übergang unterhalb der M Punkt Energie zu. Am Ende zeigen wir

eine Möglichkeit, wie mit Hilfe von KN und UV Licht, Siderit aus Eisen und Kohlenstoff

erzeugt werden kann. Alle Untersuchungen basieren auf der Analyse einer charakteristis-

chen Siderit Ramanmode, die systematisch mit der Variation von verschiedenen Parame-

tern wie verschiedenen Reaktionsgasen, verschiedenen Anregungsenergien und verschiede-

nen Belichtungszeiten durchgeführt wird.

Abstract

In this work, we present an analysis of vibrational and electronic properties of diamon-

doids, graphene, multilayer graphene, graphite, and carbon nanotubes. The realized stud-

ies are based on optical spectroscopy, such as Raman spectroscopy, photo luminescence,

photo luminescence excitation, and absorption spectroscopy, but also on in-depth model

calculations. Although the analyzed material systems have been partly known for many

decades now, fundamental optical properties, especially in the deep-UV, have not been

explored yet. We will see that both, structural characteristics of carbon allotropes, and

the hybridization of carbon have a large impact on the measured optical responses. All

findings are discussed with a deep analysis of supportive DFT (density functional theory)

computations, revealing new insights into the manifold properties of carbon allotropes.

The first part focuses on diamondoids and diamondoid derivatives which form a new

class of highly interesting materials. Recent developments in the functionalization of the

widely inert diamondoids possibly opened a new path of carbon to find its way in applica-

tions. The functionalization-caused altering of their electronic and vibrational properties

is, therefore, more and more of scientific interest. We focus our analysis on diamondoid

oligomers, that are diamondoid moieties connected by carbon-carbon single and carbon-

carbon double bonds. We will see that the introduction of sp2defects leads to a downshift

of optical transition energies from ∼6.5eV for pristine diamondoids, to ∼4.7eV for their

derivatives. We will further see that their vibrational properties can be classified by

the characteristic vibrations from carbon and hydrogen atoms, also know from unfunc-

tionalized diamondoids. However, structure-imposed vibrations, such as dimer-breathing

modes (DBM) or localized vibrational modes are reported that are unique for diamondoid

oligomers. Diamondoids tend to form stable van-der-Waals crystals with exceptionally

high binding energies. The ordered structures lead to a self-altering of optical transi-

tion energies. With resonant Raman and absorption measurements, we find that optical

transition energies in van-der-Waals crystals of two different diamondoid derivatives are

∼1eV than in isolated molecules.

The second part discusses double-resonant Raman processes in graphite, graphene, and

carbon nanotubes in the deep-UV range. Although the concept of double-resonant Raman

scattering was introduced more than a decade ago, it is today still unclear how the scat-

tering mechanism changes for very high excitation energies. We will see that for excitation

energies between 4.69 and 5.46eV, the intensities of the well-known double-resonant modes

such as the D,2Dmode, or the iTOLA mode (in-plane transverse optical, longitudinal

acoustic) in graphite are drastically quenched. Instead, we observe highly asymmetric

Raman peaks that we attribute to the second-order vibrational density of states. The

observed peaks are assigned to the degenerate LO/TO phonons at the Γpoint, LA/LO

and TO phonons from the M point, and to the overbending of the LO phonon branches.

We simulate the Raman spectra and explain the scattering paths which are different from

those of graphite under visible excitation. Excitations close to the M-point energy of

∼4.7eV lead to the activation of double-resonant Raman modes. Although exhibiting

very low intensities, we can observe the beginning of the 2Dmode involving the shortest

phonon wavevectors possible. We determine the 2Dmode to stem from close to the Γ

point exhibiting frequencies of around 3050cm−1. Further, the second-order vibrational

density of states from multilayer graphene is analyzed. We find that the frequencies of

the degenerate LO/TO phonons at the Γpoint and the LO phonon overbendings only

depend marginally on the layer number.

The last part focuses on low-intensity modes in the Raman spectra of carbon nanotubes.

In detail, we analyze the ∼1950 cm−1and 2450 cm−1modes in an ensemble sample that

were controversially discussed in the past. We find that both modes are dispersive over

an excitation range of 1.71eV-2.73 eV, partly exhibiting bimodal peak structures. With

the help of DFT + zone-folding calculations, we assign their dispersions to an ensem-

ble effect, i.e. each varied excitation energy is in resonance to a certain CNT species in

the ensemble sample, probing individual electronic and vibrational properties. The same

argument is true for the observed bimodal peaks structures. For excitation energies be-

tween 2.1 and 2.5eV, both metallic and semiconducting species are probed. Due to their

different structures of one-dimensional electronic bands, metallic and semiconducting car-

bon nanotubes have different diameters for likewise transition energies. Consequently, we

measure CNT specific vibrational properties with a monochromatic excitation which lead

to characteristic fingerprints in the Raman spectra. With the support of DFT + zone-

folding and non-orthogonal tight-binding calculations, we assign the 2450 cm−1band to

a double-resonant iTOLA mode originating from close to the K point. Instead, DFT +

zone-folding calculations indicate that the 1950cm−1band is a LOLA mode from close

to the Γpoint. The underlying assignment is done via the analysis of both the disper-

sions and intensities of all possible 6×6 = 36 combination modes in the initial graphene

structure. Non-orthogonal tight-binding calculations, however, indicate this mode to be a

TOZO (ZO: transverse optical, out-of-plane optical) combination mode from close to the

K point. As the DFT + zone-folding approach requires the initial electron-phonon matrix

elements from graphene, we believe that possible ZO contributions are underestimated in

this approach. In graphene, the out-of-plane ZO phonon mode is symmetry forbidden.

Therefore, we attribute the 1950 cm−1band to a TOZO combination from close to the K

point.

We introduce an "extended Kataura plot", i.e. we show a calculated plot on how the op-

tical transition energies from CNTs depend on their diameters up to the highest possible

π→π∗Γ-point transitions. We will see on the one hand that quantum confinement and

trigonal warping determine the transition energies, even in the deep-UV region. Based

on the calculations of optical matrix elements, we will see on the other hand, that several

optical transitions energies higher than the M-point energy are dipole forbidden. This is

not known from transitions from below the M-point energy. In the end, we discuss and

explain a way to utilize deep-UV light and carbon nanotubes to oxidize iron clusters to

vii

Siderite which we have explored in another systematic study. We base our statements

on the analysis of the Raman footprint of Siderite and show different spectra with var-

ied external parameters such as different reaction gases, different excitation energies, or

different exposure times.

viii

List of publications

1 Double-resonant LA phonon scattering in defective graphene and carbon nanotubes

Felix Herziger, Christoph Tyborski, Oliver Ochedowski, Marika Schleberger, and

Janina Maultzsch

Physical Review B, 90, 245431 (2014)

2 Raman spectroscopy of nondispersive intermediate frequency modes and their over-

tones in carbon nanotubes

Christoph Tyborski, Felix Herziger, and Janina Maultzsch

physica status solidi (b), 252, 2551-2557 (2015)

3 Beyond double-resonant Raman scattering: Ultraviolet Raman spectroscopy on

graphene, graphite, and carbon nanotubes

Christoph Tyborski, Felix Herziger, Roland Gillen, and Janina Maultzsch

Physical Review B (R), 92, 041401 (2015)

4 From isolated diamondoids to a van-der-Waals crystal: A theoretical and experi-

mental analysis of a trishomocubane and a diamantane dimer in the gas and solid

phase

Christoph Tyborski, Reinhard Meinke, Roland Gillen, Tobias Bischoff, Andre Knecht,

Robert Richter, Andrea Merlin, Andrey A. Fokin, Tetyana V. Koso, Vladimir N.

Rodionov, Peter Schreiner, Thomas Möller, Torbjörn Rander, Christian Thomsen,

and Janina Maultzsch

The Journal of Chemical Physics, 147, 044303 (2017)

5 Electronic and vibrational properties of diamondoid oligomers

Christoph Tyborski, Roland Gillen, Andrey A. Fokin, Tetyana V. Koso, Natalie

A. Fokina, Heike Hausmann, Vladimir N. Rodionov, Peter Schreiner, Christian

Thomsen, and Janina Maultzsch

The Journal of Physical Chemistry C, 121, 48 (2017)

6 Tunable quantum interference in bilayer graphene in double-resonant Raman scat-

tering

Felix Herziger, Christoph Tyborski, Oliver Ochedowski, Marika Schleberger, and

Janina Maultzsch

Carbon, 133, 254-259 (2018)

7 Double-resonant Raman scattering with optical and acoustic phonons in carbon

nanotubes

Christoph Tyborski, Asmus Vierck, Rohit Narula, Valentin N. Popov, and Janina

Maultzsch

Physical Review B, 97, 214306 (2018)

8 Reductive Diazotation of Carbon Nanotubes: An Experimental and Theoretical

Selectivity Study

Milan Schirowski, Christoph Tyborski, Janina Maultzsch, Frank Hauke, Andreas

Hirsch, and Jakub Goclon

accepted for publication in Chemical Science

9 Vibrational properties of single-bond diamondoid dimers

Christoph Tyborski, Tobias Hückstaedt, Tommy Otto, Roland Gillen, Andrey A.

Fokin, Tetyana Koso, Lesya V. Chernish, Pavel A. Gunchenko, Peter Schreiner, and

Janina Maultzsch

in preparation, based on Chapter 4.3

10 Electronic and vibrational properties of [2](1,3)Adamantano[2](2,7)pyrenophane

Tao Xiong, Christoph Tyborski, Tobias Hückstaedt, Stefan Kalinowski, Janina

Maultzsch, Paul Kahl, Philipp Wagner, Ciro Balestrieri, Jonathan Becker, Heike

Hausmann, Graham Bodwell, Peter Schreiner, and Peter Saalfrank

in preparation, partly based on Chapter 4.5

Contents

1 Introduction 2

2 Structure and electronic properties of graphene, carbon nanotubes, and

diamondoids 6

2.1 Carbon-Carbonbonds ............................. 6

2.2 Structure of graphene and tight-binding approximation . . . . . . . . . . . 8

2.3 Structure and electronic properties of carbon nanotubes . . . . . . . . . . . 10

2.3.1 Electronic band structure of carbon nanotubes . . . . . . . . . . . . 12

2.3.2 Optical matrix elements in carbon nanotubes . . . . . . . . . . . . 14

2.4 Structure and fundamental properties of diamondoids . . . . . . . . . . . . 17

3 Experimental methods 20

3.1 First-order Raman scattering - Raman spectroscopy . . . . . . . . . . . . . 20

3.2 Double-resonant Raman scattering . . . . . . . . . . . . . . . . . . . . . . 23

3.2.1 Characteristic Raman spectra of graphite, graphene, carbon nan-

otubes, and diamantane . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Experimentalsetup............................... 27

3.3.1 Second harmonic generation . . . . . . . . . . . . . . . . . . . . . . 28

4 Diamondoid derivatives 30

4.1 Double-bond diamondoid oligomers - vibrational properties . . . . . . . . . 30

4.2 Double-bond diamondoid oligomers - electronic properties . . . . . . . . . 37

4.3 Single-bond diamondoid dimers - vibrational properties . . . . . . . . . . . 44

4.4 Diamondoid van-der-Waals crystals . . . . . . . . . . . . . . . . . . . . . . 51

4.4.1 Electronic properties of chemically blended sp2/sp3diamondoid van-

der-Waalscrystals ........................... 51

4.4.2 Vibrational properties of diamondoid van-der-Waals crystals . . . . 53

4.5 Pyrene and diamondoids: Electronic and vibrational properties of

[2](1,3)Adamantano[2](2,7)pyrenophane . . . . . . . . . . . . . . . . . . . . 57

4.5.1 [2](1,3)Adamantano[2](2,7)pyrenophane - vibrational properties . . 57

4.5.2 [2](1,3)Adamantano[2](2,7)pyrenophane - electronic properties . . . 59

4.6 Summary .................................... 62

5 UV Raman spectroscopy of graphite, graphene, and multilayer graphene 64

5.1 UV Raman spectroscopy of graphite and graphene - a motivation . . . . . 64

5.1.1 Vibrational properties of graphene and graphite . . . . . . . . . . . 65

5.1.2 Beyond double-resonant Raman scattering: A UV analysis of graphite,

graphene, and carbon nanotubes . . . . . . . . . . . . . . . . . . . . 66

5.1.3 Raman scattering mechanism for deep-UV excitation energies . . . 71

5.1.4 Deep-UV Raman measurements in graphene and carbon nanotubes 75

5.2 The second-order vibrational density of states in multilayer graphene . . . 77

5.2.1 Layer-number determination and sample characterization . . . . . . 78

5.2.2 Temperature-dependent Raman measurements of the second-order

vibrational density of states . . . . . . . . . . . . . . . . . . . . . . 84

5.3 Summary .................................... 86

6 Carbon nanotubes - Low-intensity Raman modes and UV spectroscopy 88

6.1 Low-intensity Raman modes in carbon nanotubes . . . . . . . . . . . . . . 89

6.1.1 Low-intensity Raman modes - An introduction . . . . . . . . . . . . 89

6.1.2 Low intensity Raman modes - Experimental Raman spectra . . . . 92

6.1.3 Low-intensity Raman modes - DFT calculations . . . . . . . . . . . 92

6.1.4 Low-intensity Raman modes - Non-orthogonal tight-binding calcu-

lations.................................. 96

6.1.5 Low-intensity Raman modes - Dispersions and line shapes . . . . . 98

6.1.6 Low-intensity Raman modes - Intensity ratios . . . . . . . . . . . . 101

6.2 UV Raman spectroscopy: Extended Kataura plot and RBM measurements 104

6.2.1 Optical transitions energies above the M-point energy . . . . . . . . 104

6.2.2 Optical absorption above the M-point energy and UV Raman spectra108

6.3 Ultra-violet light assisted functionalization . . . . . . . . . . . . . . . . . . 113

6.3.1 Raman analysis of the functionalization . . . . . . . . . . . . . . . . 114

6.3.2 Explanation of the functionalization process . . . . . . . . . . . . . 117

6.4 Summary ....................................120

7 Conclusion 122

Acknowledgment 157

1. Introduction

Although being known for several centuries, the element carbon still evokes a tremendous

scientific interest. It seems from a physicist’s point of view, that its attention was even

rapidly growing in the last 30 years. The first successful isolation of a sp2conjugated,

complete two-dimensional carbon system is only 14 years ago, but its famous article "Elec-

tric Field Effect in Atomatically Thin Carbon Films" is already cited more than 42000

times1[1]. A lot of the interest can be understood from the fact that two-dimensional

systems were not considered to be stable at all. According to Mermin and Wagner, a

perfectly flat crystal could not exist due to thermal fluctuations in the range of the lattice

constant [2]. Irregular deformations in graphene, however, reduce its long-range symme-

try, leading to a stable crystal. The discovery of graphene can thus be seen as a starting

point for the search of other two-dimensional crystals, such as hexagonal boron nitride [3]

or MoS2[4], opening a complete new field of solid state physics.

On the other hand, a lot of the scientific interest of graphene can be understood from

the fact that it combines many extraordinary properties, such as a linear electronic band

dispersion that is otherwise only known from massless Dirac particle or surface states in

topological insulators [5], its ballistic charge carrier transport at room temperatures [6],

or its high mechanical stability of 42N/m [7]. All its characteristics make graphene a

promising candidate for applications in the field of electronics, sensors, composites, coat-

ings or membranes. However, graphene can also be considered as the starting material

for graphitic, one-dimensional carbon nanotubes (CNT) as well as other graphitic carbon.

CNTs can be seen as rolled-up graphene with a perfect, seamless contact at the graphene

edges, exhibiting diameters in the range of ∼nm [8]. The reduction of dimensionality

in carbon nanotubes leads to a confinement of their electronic states, while widely main-

taining the traits of graphene. Surprisingly, only their rolling-up angle and their diameter

determines whether a carbon nanotube is metallic or semiconducting. This aspect, in par-

ticular, distinguishes them from graphene, making carbon nanotubes a future candidate

for possible semiconductor applications.

Among many other experimental approaches, especially Raman spectroscopy was utilized

to gain access to both the electronic and vibrational properties of graphene and carbon

nanotubes. The electronic band structure of graphene opens the possibility to investi-

gate phonons with arbitrary wave vectors from a large area of the 1. Brillouin zone [9].

Together with the broad availability of compact Raman spectrometers, it is thus a com-

parably easy way to examine some of the peculiarities of graphene: At the one hand,

its linear electronic band structure can be probed, but also, the breakdown of the Born-

Oppenheimer approximation in graphene that leads to a strong Kohn-Anomaly [10]. In

1google scholar august 2018

1. Introduction

Raman experiments, these characteristics can especially be addressed by excitation ener-

gies in the near infrared and visible optical region. Combined with the limited availability

of laser light sources beyond the optical visible region, the focus on these specific excita-

tion energies in the past, can be well understood. In this work, we have instead extended

the Raman analysis of graphene, carbon nanotubes, and graphite to excitation energies

in the deep-UV (ultraviolet) region. By this approach, we gain access to regions of the 1.

Brillouin zone that are otherwise only accessible by way more complicated experimental

approaches. It is interesting to note that the Raman response in the deep-UV region, i.e.

away from the linear band structure, is systematically different compared to the response

in the visible region. All well-known, second-order, double-resonant Raman modes, such

as the Dor the 2Dmode, are not observable anymore. Instead, a new set of Raman peaks

appears that we attribute to the second-order vibrational density of states. The Raman

spectrum then resembles those from classical semiconductors like GaAs or ZnO [11–13].

By the careful choice of suitable excitation energies, we can manipulate the transition

of the "unique" Raman response from graphene, graphite, and carbon nanotubes to a

"conventional" Raman response known from other semiconductors. Until today, this tran-

sition has not been investigated yet and is again a new aspect of the rich characteristics

of graphitic materials. The deep-UV Raman experiments presented in this thesis lead to

a wider understanding of both Raman scattering in graphitic materials as well as their

optical and vibrational properties.

Two-phonon scattering in graphitic materials allows for the analysis of many different

phonon branches only by the careful choice of a certain mode in their Raman spec-

trum [14]. Compared to graphene, it has turned out that Raman spectra of carbon

nanotubes exhibit more Raman peaks, mainly due to their altered symmetries. Rolling

up a graphene sheet to a carbon nanotube, lifts the degeneracy of the Raman active E2g

mode on the one hand, but also leads to an altering of the strict Raman selection rules as

present in graphene. For instance, this offers a way to analyze out-of-plane ZO phonon

branches in carbon nanotubes, which is Raman forbidden in graphene [15,16]. Although

Raman spectra of carbon nanotubes have been analyzed for many years, several aspects

concerning peak compositions, peak dispersion, or the origin of certain Raman modes still

remain unclear. To some extent, this can be understood from the fact that typical CNT

samples often contain more than one CNT species. The total Raman spectrum is thus

a convolution of the intrinsic Raman response from each species, hindering an analysis

on the basis of pure experimental data. This, of course, opens room for interpretations

and leads to inaccuracies. Consequently, a thorough analysis of these ensemble samples

requires time-demanding and sophisticated simulations. However, also the theoretical ap-

proaches used to calculate the electronic and vibrational properties of CNTs, suffer from

inaccuracies. An often used approach is based on the zone-folding of graphene’s two-

dimensional reciprocal space in order to obtain the electron and phonon band structure.

This widespread a posteriori approach offers reliable results for prominent peaks in the

Raman spectrum of carbon nanotubes, but fails to describe structure-imposed Raman

modes, such as the radial-breathing mode (RBM), which are not present in graphene.

By combining sophisticated computational models, such as non-orthogonal tight binding,

with an extensive experimental analysis, we analyze low-intensity Raman modes in car-

1. Introduction

bon nanotubes and demonstrate the influence of ensemble effects on the measured Raman

spectra. Our results show that species-specific properties determine the Raman spectra in

terms of peak compositions and dispersion. Especially differences in the vibrational prop-

erties of metallic and semiconducting carbon nanotubes lead to distinguishable peaks of

the analyzed low-intensity Raman modes. Further, it has turned out that the zone-folding

approach underestimates out-of-plane ZO contributions in the Raman spectra of carbon

nanotubes. This leads to a wrong assignment of an apparent, low-intensity mode in the

Raman spectra of carbon nanotubes. However, with the help of a complementary compu-

tational approach, this somewhat misleading assignment can be rectified. These general

findings can be utilized to understand the very large variety of Raman data available in

literature to a greater extent. We believe our findings close a gap in the understanding

of low-intensity Raman bands in carbon nanotubes on the one hand, but also in the un-

derstanding of ensemble effects. Especially the latter might be used to recap established

Raman data widespread in current literature.

Besides graphene and carbon nanotubes, diamondoids can be seen as another extraor-

dinary carbon allotrope. The smallest diamondoid, namely adamantane, has been known

approx. 70 years longer than graphene or carbon nanotubes, but has not evoked such a

large scientific interest. It was first discovered 1933 in crude oil [17] and can be seen as the

smallest diamondoid possible, consisting of 10 carbon and 16 hydrogen atoms arranged

in a diamond-cage structure. Since then, it has found its way particularly in medicine

applications as drug carrier [18]. The scientific interest and possible applications, how-

ever, were back then limited to adamantane and the next two larger diamondoids, namely

diamantane and triamantane. This can be understood from the simple fact that higher

homologues were not available. Even today, higher homologues can be barely selectively

fabricated, as differences in the thermodynamic stabilities of intermediate states in their

synthesis are only very low. However, diamondoids experienced a renaissance in 2003

when Jeremy Dahl et al. [19] isolated different species of diamondoids up to undecaman-

tane (11 cages) from crude oil. The availability of mono-fraction species allowed for their

systematic analysis. For instance, it was found that the quantum confinement model has

a limited validity in lower diamondoids [20] or that diamondoid passivated surfaces have

a negative electron affinity [21]. Recent developments in diamondoid chemistry enabled

their functionalization with specific groups. This in turn opens a way to tailor their

electronic and vibrational properties, while maintaining their chemical inertness or their

robustness. One possible route is the introduction of sp2defects in a way that lower dia-

mondoids form dimers or trimers connected by a double bond [22]. Another way to form

dimers is the connection by a single bond, that can be seen as an extension of the initial

carbon cages [23]. The so obtained set of new diamondoid derivatives is both experimen-

tally and theoretically analyzed to a large extent in this dissertation. For instance, we find

that sp2defects in diamondoid dimers lead to a large redshift of their optical transition

energies, compared to their pristine counterparts. The localization of the introduced π

and π∗states further centralizes optically relevant orbitals between the diamondoid moi-

eties. This particular aspect reduces the sensitivity of their lower, optical transitions to

external influences such as neighboring molecules in diamondoid van-der-Waals crystals.

Our findings might therefore help to design certain diamondoid derivatives that, one the

1. Introduction

one hand exhibit desired electronic properties, but on the other hand, maintain those

traits when they are brought into certain applications in a solid phase.

2. Structure and electronic

properties of graphene, carbon

nanotubes, and diamondoids

In this chapter, we discuss the structure and electronic properties of sp2graphene, sp2

carbon nanotubes, and sp3/sp2diamondoids. At first, we discuss the element carbon and

its ability to form hybridized atomic orbitals as it is the basis of all analyzed materials in

this dissertation. Findings for the electronic structures of sp2systems are generalized for

periodic structures such as graphene or carbon nanotubes. We give a short introduction to

the optical absorption in carbon nanotubes necessary to understand the optical responses

in the ultraviolet range. In the end we introduce the structure and fundamental properties

of diamondoids.

2.1 Carbon-Carbon bonds

Although the element carbon and its different appearances are partly known for more

than two centuries, even in the recent past new configurations were found, namely carbon

nanotubes [8] and graphene [24]. These modifications are periodic, low dimensional, fully

unsaturated, and delocalized sp2carbon systems that were not expected to be stable for

decades [2]. The basis of their stabilities are local deformations that destroy the long-

range periodicity of the lattices [25,26], but also the comparably strong C=C double bond.

The ability of carbon to form different hybrid orbitals creates the foundation of its rich

modifications that are present in our daily environment. The carbon materials analyzed

in this work only contain sp2and sp3hybridized carbon. We will therefore reduce the

discussion on these two hybridizations and neglect the third type, the sp hybridization. It

is important to note that the energies of the πand π∗orbitals (responsible for the second

carbon-carbon bond in a carbon double bond) lie in between the energy gap of the σand

σ∗orbitals (responsible for the carbon-carbon bond in sp3configurations and the first

carbon-carbon bond in a sp2configuration). This fact is very important for all carbon

compounds analyzed in this dissertation as the low-lying orbitals determine the optical

responses in the visible and near ultraviolet region.

A hybridization can be understood as a linear combination of atomic orbitals that form

new, shared atomic orbitals. In the special case of carbon with its electron configuration

[He]2s22p2, the spherical 2sand the dumbbell-shaped 2porbitals are involved. Possible

are also other combinations of atomic orbitals with e.g. 4, 5, 6dorbitals in complexes with

2.1. Carbon-Carbon bonds

1s2

2s2

2p2

Ground state of

a carbon atom

1s2

sp3

1s2

sp2pz

sp2hybridization

sp3hybridization

Figure 2.1: The electronic states of

carbon core and valence electrons are

schematically plotted. Left: Electron

states of a carbon atom in the elec-

tronic ground state are shown. Right:

Electron states of hybridized carbon is

shown for a sp3hybridization (top) and

for a sp2hybridization (bottom).

transition metals, but also very similar to carbon, the 2sand 2porbitals in nitrogen [27].

The reorganization of atomic orbitals requires energy that is overcompensated by the

large bonding energies [28]. This in turn enables the occurrence of stable, covalent, inter-

atomic bonds. A schematic illustration of a sp2and sp3hybridization is given in Fig. 2.1.

Asp3hybridization leads to four equivalent hybrid orbitals that are mixed from 2sand

2porbitals in the ratio of 1:3 [28]. Their spatial direction is determined by the respective

contributions of 2px,2py, or 2pzorbitals within the hybrid orbitals [28]. In correspon-

dence to the σorbitals in diatomic orbitals, these hybrid orbitals are referred to as σ

orbitals or rather the atomic bonds are referred to as σbonds [28]. An equivalent atomic

orbital mixing occurs between carbon and hydrogen atoms present at diamond and di-

amondoid surfaces [19,29–31]. It is interesting to note that due to the 1sorbitals from

hydrogen, the electronic states of the C-Hσ-bonds lie energetically between those from

the C-Cσ-bonds with far-reaching consequences for the electronic structures of diamon-

doids [20,28,32]. They are discussed in more detail in Chapter 4.

Another hybridization, namely the sp2hybridization occurs from the mixing of a 2sand

two of three 2px2py2pzorbitals. This creates hybrid orbitals with a σcharacter aligned

in a joint plane, in strong contrast to the sp3hybridization [28]. The second bond that

originates from the out-of-plane porbital is usually referred to as πbond and accordingly,

the orbitals are referred to as π/π∗orbitals [28]. A schematic overview of their energy

levels is given in Fig. 2.1. The twofold chemical bonds of a sp2hybridization lead to a

comparably stiff and strong interatomic binding in carbon solids that explains the large

vibrational frequencies of ∼1600cm−1(∼198 meV) that we measure within this disser-

tation. It further squeezes the remaining bond partners in one plane which in the end

explains the two dimensional nature of graphene [24].

A heuristic and fast approach to understand the electronic structures of sp2systems was

introduced by Erich Hückel [33–35] in 1931. The suggested approach reduces the problem

to the out-of-plane pzπ/π∗-orbitals and parametrizes the Coulombintegral (Eigenener-

gies of the atomic orbitals) and the Resonanceintegral (interaction integral of neighboring

2.2. Structure of graphene and tight-binding approximation

carbon atoms) with empirical data usually obtained from other theoretical methods or

experiments [28, 33–35]. This approach only relies on next-nearest neighbors but still

achieves considerable results [33–35]. For instance, with the help of the "Hückel method"

it was possible to calculate the electronic structure of Benzol for the first time [33,34].

Its power and range can be estimated when we examine the outcome for sp2carbon

systems with a defined amount nof double bonds in a chain:

∆E∼4βsin π

2(n+ 1)!.(2.1)

Eq. (2.1) describes the approximated energy value for the HOMO (highest occupied

molecular orbital)-LUMO (lowest unoccupied molecular orbital) gap. βis a numeric

value exhibiting ∼250kJ/mol (∼2.5eV) [36]. It already indicates that in the limit of an

infinite chain of carbon double-bonds, as present in two-dimensional graphene, the energy

difference of the lowest optical transition vanishes. A finding that holds true even for a

material discovered in the 21th century and thus, almost 90 years after the theoretical

prediction.

The profound considerations made by Erich Hückel can be generalized to conjugated

carbon systems with arbitrary sizes. For instance, it can be applied to a periodic ar-

rangement of hexagons forming one-dimensional linear chains or fully two-dimensional

areas as present in graphene. The primary, discrete Eigenstates then turn into Bloch

functions and accordingly, the discrete Eigenenergies turn into energy bands. Nowadays,

this heuristic approach is widespread and also known as the tight-binding approximation,

which is discussed on the next pages [37,38].

2.2 Structure of graphene and tight-binding approx-

imation

A B

4π

3a0

2π

√3a0

k1k2

M K

(b)(a)

Figure 2.2: (a) A cutout of the real-space hexagonal graphene lattice is shown. a1and a2denote the

lattice unit vectors in real space, respectively. The unit cell is spanned by the lattice vectors, indicated

with the dashed lines. A and B show the carbon atoms within one unit cell. (b) The Brillouin zone of

graphene is plotted with the high-symmetry points Γ, M, and K. k1and k2denote the reciprocal lattice

unit vectors. Basic lengths between high-symmetry points are indicated.

As discussed above, the three σand one πbonds of the sp2hybridized carbon lead to a

triangular bond arrangement as shown in Fig. 2.2. If a sufficient amount of carbon atoms

2.2. Structure of graphene and tight-binding approximation

forms such a triangular network, a honeycomb structure appears that is characteristic for

the graphene lattice [24, 39]. Its unit cell contains two carbon atoms A and B with a

distance aAB of 1.422˚

A resulting in a lattice constant of a0=√3aAB ≈2.463 ˚

A [39], as

depicted in Fig. 2.2. In real space, the two-dimensional unit cell is spanned by the lattice

vectors a1and a2[40]:

a1=aAB

2(3,√3) a2=aAB

2(3,−√3).(2.2)

In reciprocal space, the Brillouin zone is given via:

k1=2π

3a0

(1,√3) k2=2π

3a0

(1,−√3).(2.3)

The Fourier transformation of the real-space honeycomb lattice is again a honeycomb

lattice in which the unit cell is rotated by 90◦[40]. Reciprocal lattice vectors k1and k2

are given in Fig. 2.2 and are plotted from the zone center (Γ) pointing to the middle of

a neighboring hexagon of the reciprocal space’s honeycomb structure. Particular high-

symmetry points of the unit cell, namely the Γpoint at the zone center, the K point at

the edge of a hexagon and the M point between two K point edges are marked in Fig. 2.2.

Further details can be found in Chapter 5, where the vibrational properties of graphene

and graphite are discussed.

Tight-binding approximation

A fast way to obtain the electronic band structure of graphene is the tight-binding ap-

proximation that goes back to Wallace in 1947 [41]. It considers only valence electrons

that are tightly bound to the atoms they belong to. The interaction with adjacent atoms

is supposed to be only weak in a way that the electron wave functions resemble those from

the atomic orbitals. Ensuing Bloch functions are then linear combinations of atomic wave

functions [41]. When only the next-nearest neighbors are taken into account, already rea-

sonable electronic band structures can already be achieved [41]. However, a considerable

improvement can be obtained, when several interacting neighbors are considered in the

calculations [38,40,41]. Only considering next-nearest neighbors in the easiest case, the

energy bands for the valence electrons in graphene have the form:

E±(k) = ±tq3 + f(k)−t0f(k)(2.4)

where tdescribes the nearest-neighbor hopping energy when the sub lattice is changed

(A →B in Fig. 2.2). t0is the hopping parameter when the sub lattice is not changed

(next-nearest neighbor). Both tand t0are empirical parameters that were found to exhibit

t= 2.7eV and 0.02t≤t0≤0.2t[38, 42]. f(k)can be divided in its two-dimensional

components [40]:

f(k) = 2cos (√3kya) + 4 cos(√3

2kya)cos (3

2kxa)(2.5)

A one-dimensional plot of the electronic band structure of graphene along the Γ-K-M

high-symmetry line is depicted in Fig. 2.3. The inset shows a magnification of the band

dispersion around the K point. It can clearly be seen that in the proximity of the K

2.3. Structure and electronic properties of carbon nanotubes

ΓK M

−8

−6

−4

−2

Electron wave vector

Energy (eV)

−1

Figure 2.3: A one-dimensional plot of

the electronic band structure of graphene

within a next-nearest neighbor approx-

imation is plotted for t= 2.7eV and

t0= 0.1eV, respectively [40, 42]. De-

noted are the high-symmetry points Γ,

K and M at the edges of the reciprocal

unit cell. The inset shows a magnifica-

tion of the linear band structure in the

proximity of the K point.

point, both the electron and hole bands are linear. This particular behavior is well

known from massless Dirac particles and is one of the reasons why graphene has evoked

such a great interest in the past. Electrons in graphene are therefore often referred to

as "massless Dirac particles" [43]. The electron band dispersion deviates from its linear

behavior around the K point for larger and smaller electron momenta. The analogy

to massless Dirac particles therefore only holds true in the direct proximity of the K

points [38,43,44]. The deviation from the linear behavior is anisotropic, i.e. the slope of

the electron bands in the K−Mdirection is smaller compared to the dispersion along the

K−Γdirection [38,43,44]. In graphene and graphite, this anisotropy of the electron band

structure is referred to as the "trigonal warping effect" with far-reaching consequences for

their electronic properties [40,44].

The one-dimensional electron band structure, as shown in Fig. 2.3, is used for various

calculations of double-resonant Raman spectra in graphene as discussed in the Chapters

5 and 6.

2.3 Structure and electronic properties of carbon nan-

otubes

A paper tube can be made when a sheet of paper is rolled up. In analogy, single-walled

carbon nanotubes can be thought of rolled-up graphene sheets that do not overlap at

the edges, i.e. they have a seamless contact. Since the rolling-up angle can be chosen

arbitrarily, there is an infinite amount of carbon nanotubes possible. Each one has an

unique rolling-up angle and consequential, unique electronic and vibrational properties.

The way a carbon nanotube is rolled up is defined by the chiral vector cmeasured in

units of the graphene lattice vectors:

c=n1a1+n2a2(2.6)

Rolling up a sheet of graphene can be seen as an introduction of a boundary condition that

quantizes the electronic states along the circumference of such a tube. It further reduces

the dimensionality in comparison to graphene from continuous 2-dimensional states to

constrained 1-dimensional states. The boundary condition reads as follows [45]:

m·λ=|c|=π·d⇔k⊥, m =2π

λ=2π

|c|·m=2

d·m. (2.7)

2.3. Structure and electronic properties of carbon nanotubes

a2chiral angle θ

armchair

zig-zag

circumference

(6,1)

Figure 2.4: The chiral vector c(circumference) of a (6,1) carbon nanotube is plotted onto the hexagonal

graphene lattice. The armchair and zig-zag high-symmetry directions are indicated by arrows and by the

edges of the graphene lattice (top and bottom: armchair, left and right: zig-zag). The chiral angle is θis

indicated, a1and a2are the lattice vectors of the hexagonal graphene lattice, respectively.

With mbeing the quantum number for each electronic state k⊥, m, also named the band

index. Electronic states in the direction of the tube axis are continuous and only depend

on the translational period a[45]:

kz=2π

a.(2.8)

The diameter din Eq. (2.7) can be parametrized by the chiral indices n1and n2[45]:

d=|c|

π=a0

πqn2

1+n1n2+n2

2.(2.9)

Eq. (2.7) shows that the quantized electronic states are more separated in kspace for

smaller tube diameters and vice versa. An example of the chiral vector cof a (6,1) carbon

nanotube plotted onto the hexagonal graphene lattice is depicted in Fig. 2.4.

Two high-symmetry chiral vectors and therewith two distinct directions can be found

in the honeycomb lattice structure of graphene. They are named the zig-zag and arm-

chair direction as given in Fig. 2.4. Any other direction in the honeycomb lattice can

be defined with a so called chiral angel θ, defined relative to the high-symmetry zig-zag

direction:

cosθ=n1+n2/2

qn2

1+n1n2+n2

.(2.10)

The six-fold symmetry of the graphene lattice only allows for distinguishable chiral an-

gles θbetween 0◦(armchair) and 30◦(zig-zag). All carbon nanotubes with chiral angles

θ6= 0◦and θ6= 30◦are referred to as chiral carbon nanotubes. If the first chiral index is

larger than the second chiral index, the helix of the respective tube is right-handed, if the

order of the indices is changed, the helix will change to be left-handed, accompanied by a

shift of the chiral angle to the interval between 30◦and 60◦[45]. The chiral angle of the

2.3. Electronic band structure of carbon nanotubes

depicted (6,1) tube in Fig. 2.4 (d∼5.1˚

A) exhibits θ∼7.6◦measured from the zig-zag

direction.

Depending on the carbon nanotube sample, typical diameters exhibit values between

6˚

A and 17˚

A [46–48]. The scientific progress on carbon nanotubes since their discov-

ery has lead to a broad availability of versatile samples. This especially includes highly

enriched samples containing just certain species of carbon nanotubes or even just one

chirality [49–51]. Ensemble samples, such as the HipCO sample, introduced and ana-

lyzed in Chapter 6, usually have a Gaussian diameter distribution and evenly distributed

chiralities [45, 52,53]. Carbon nanotube lengths up to 55.5 cm are reported in our days

resulting in length-to-diameter ratios of 132.000.000:1 [54,55], clearly showing their one-

dimensional character. Until now, there has no material been discovered that exhibits

higher aspect ratios. However, typical lengths of carbon nanotube reach from several

100nm up to several µm [56,57] with clearly lower aspect ratios. Still, carbon nanotubes

are considered to be one-dimensional material systems.

2.3.1 Electronic band structure of carbon nanotubes

A convenient approach to obtain the electronic band structure of carbon nanotubes is the

zone-folding approximation [38]. It requires the two-dimensional electronic band structure

of graphene in the first place and can be seen as an a posteriori approach with the

introduction of boundary conditions to the continuous electron wavevectors of graphene.

The so emerging one-dimensional reciprocal space can be constructed by the the reciprocal

space vectors kzand k⊥consisting of continuous lines kzequidistantly spaced by |∆k⊥|=

2/d, orthogonal to the direction of the continuous electron wave vectors kz[45,53]. Their

number, lengths, spacing, and their orientation depend on the chiral indices n1and n2

[45,53] as shown before. In this way, each carbon nanotube has a characteristic electronic

band structure and can either be metallic or semiconducting [58]. Via the family index

ν, carbon nanotubes can be classified into three different families [45,53,58]:

ν= (n1−n2)mod 3(2.11)

If one of the one-dimensional cutting lines crosses a K point in the reciprocal space of

graphene, the respective carbon nanotube will be metallic. For reasons of symmetry, this

always happens for ν= 0 [45, 53, 58]. If the closest subband to the K point lies in the

K−Γdirection, the family index will be ν=−1. If the closest subband to the K point

lies in the K−Mdirection, the family index will be ν= +1 [45,53]. In both cases, the

carbon nanotubes will be semiconducting. One third of all imaginable carbon nanotubes

are metallic, the other two thirds are semiconducting [45,53,58].

The alignment of subbands for three zig-zag carbon nanotubes belonging to the ν=

+1,−1,0families are plotted onto a contour plot of the electronic band structure of

graphene in Fig. 2.5. The spacing of subbands for different nanotubes increases for de-

creasing diameters, but due to the same directions of the chiral vector, their orientations

are the same. In case of the (9,0) tube, one cutting line passes the K point and thus

the (9,0) tube is metallic. Cutting lines responsible for the first optical transitions E11

are denoted. Slopes of the corresponding electron and hole bands depend on the one-

dimensional electron wave vector kzand can be followed in Fig. 2.5 on the right side.

2.3. Electronic band structure of carbon nanotubes

m=0

m=7

E11

m=0

m=8

E11

m=0

m=9

E11

(7,0)

ν= +1

-2.5

-2

-1.5

-1

-0.5

0.5

1.5

2.5

Energy (eV)

(8,0)

ν=−1

-2.5

-2

-1.5

-1

-0.5

0.5

1.5

2.5

Energy (eV)

(9,0)

ν= 0

-2.5

-2

-1.5

-1

-0.5

0.5

1.5

2.5

kz(π/T)

Energy (eV)

Figure 2.5: Left: Con-

fined electronic bands of a

(7,0), (8,0), and (9,0) car-

bon nanotube are plotted

onto the two-dimensional

contour plot of the elec-

tronic band structure of

graphene, respectively. All

carbon nanotubes belong to

different families νas in-

dicated in the right boxes.

The high-symmetry points

Γand Kare highlighted.

Band indices m of the sub-

bands are given for each

carbon nanotube. The sub-

bands responsible for the

lowest optical transitions

E11 are labeled. Right:

Electronic band structures

of the respective carbon

nanotubes are plotted. The

bands responsible for the

lowest optical transitions

E11 are plotted in blue.

Electronic bands are calcu-

lated within a next-nearest

neighbor tight-binding +

zone-folding approach.

2.3. Optical matrix elements in carbon nanotubes

Higher optical transitions generally occur for electron and hole bands with other band

indices mand are named by ascending indices Enn,i.e. E22, E33,. . . . Due to a lesser

confinement of electronic states, the k-space distance to the closest K point for the (8,0)

tube is smaller in comparison to the (7,0) tube, resulting in a smaller optical bandgap of

E(8,0)

11 = 1.28eV, compared to E(7,0)

11 = 1.392eV, obtained within a next-nearest neighbor

tight-binding + zone-folding approach. The K-point distance of the metallic nanotubes

is even larger, resulting in an optical bandgap of E(9,0)

11 = 2.914eV. For commercially

available nanotubes samples, the optical bandgaps are usually lower than 1eV [45,59,60].

Excitation energies in the visible optical range are therefore usually in resonance with the

E22 or E33 optical transitions.

The correlation between bandgap and diameter of a carbon nanotube with defined chiral

indices n1and n2was first given by Kataura et al. [59], simply referred to as the "Kataura

plot". In this plot, carbon nanotubes are arranged in branches reflecting both, quantum

confinement effects and trigonal warping [59]. The strong dependence of optical transi-

tion energies on the chiral indices n1and n2and the diameter is discussed in Chapter 6

where we show a theoretical Kataura plot extended by optical transition energies up to

the highest possible value.

2.3.2 Optical matrix elements in carbon nanotubes

Ensemble carbon nanotube samples contain a large variety of different carbon nanotubes.

These samples can be seen as a mixture of disjunct material systems that can individually

be probed by different optical excitation energies, as previously discussed. However, their

optical responses do not only depend on the optical excitation energy but also on their

coupling to optical light fields. We will see that the coupling strengths depend on the

chiral indices n1and n2and the particular optical transition probed by a certain excita-

tion energy. This is of great importance for the understanding of the optical responses of

ensemble samples as discussed in detail in Chapter 6. Only when each carbon nanotube in

an ensemble with its unique electronic properties is considered in the simulations, we are

able to understand the complex optical responses. We will therefore discuss the optical

matrix elements in carbon nanotubes on the next pages.

Following Ref. [61], the z-component of optical matrix elements as a function of the chiral

index (n1, n2), band index m, and the wave vector along the kzaxis in a tight-binding

approximation are given by:

Mn1,n2

e−r(m, kz) = 1

2√N|e(k)|h(n1−n2)cos Ψ3−(2n1+n2) cosΨ1+

(n1+ 2n2)cos Ψ2i(2.12)

2.3. Optical matrix elements in carbon nanotubes

E11 = 1.85 eV

m= 8

0 0.1 0.2 0.3 0.4 0.5

−1.5

−1

−0.5

kz(2π/a)

Mz(arb.units)

(9,9)

(12,0) (11,2) (10,4) (9,6)

(12,0) (11,2) (10,4)

(11,0) (10,2)

(9,4)

(8,6)

(11,0) (10,2)

(9,4)

(8,6)

0 5 10 15 20 25 30

1.2

1.4

1.6

1.8

chiral angle (◦)

|Mz|(arb.units)

E11 ν=−1

E22 ν=−1

E11 ν= 0

E22 ν= 0

a) b)

Figure 2.6: a) Calculated optical matrix elements of a (9,9) carbon nanotube tube as a function of

the electron wavevector are plotted. Different curves correspond to different electronic subbands of a

(9,9) tube. Highlighted in red is the band responsible for the optical matrix element of the lowest

optical transition with the band index m= 8. b) Calculated optical matrix elements of various carbon

nanotubes belonging to the ν= 0 and ν=−1families are plotted as a function of the chiral angle θ. All

carbon nanotubes have comparable diameters. Matrix elements of the first (E11, black) and second (E22,

red) optical transition are plotted, respectively. The matrix elements are obtained with a next-nearest

neighbor tight-binding approximation [61].

with

Ψ1=πm2n1+n2

N−2πn2

qkz(2.13)

Ψ2=πmn1+ 2n2

N+ 2πn1

qkz(2.14)

Ψ3=πmn1−n2

N−2πn1+n2

qkz.(2.15)

|e(k)|describes the energy of an optical transition as a function of the chiral index (n1, n2)

and the band index m. It is determined via:

|e(k)|=q3 + 2cos Ψ1+ 2 cosΨ2+ 2 cos Ψ3.(2.16)

The chiral indices n1and n2determine the values for Nand qaccording to the following

expressions:

N=n2

1+n1n2+n2

2and q=2N

nR,(2.17)

where R= 3 if (n1−n2)/3nis integer and R= 1 otherwise. nis the greatest common

divisor of n1and n2. Only light is considered that is polarized along the tube axis, i.e.

only transitions between electronic states with ∆m= 0 are allowed [61].

The so obtained optical matrix elements are plotted in Fig. 2.6. We show in a) the

electronic matrix elements of an armchair (9,9) tube as a function of the band index m

and the electron wavevector kz. Highlighted in red are the matrix elements for the optical

2.3. Optical matrix elements in carbon nanotubes

transition with the band index m= 8. The concrete electron wavevector responsible for

the optical matrix element of the E11 transition is determined via the bandgap of the re-

spective electron and hole bands [61]. This approach can be used to compare the optical

matrix elements of different carbon nanotubes with almost similar diameters of ∼9˚

A as

shown in Fig. 2.6 b). Keeping the diameter constant, enables the direct analysis of the

trigonal warping effect as the distance to the closest K points is almost similar for all

nanotubes within one family. The largest optical matrix elements are found for the E11,

ν=−1and E22,ν= 0 transitions accompanied by a decreasing coupling strength for

larger chiral angles θ. Those are the optical transitions that mainly stem from the prox-

imity of the high-symmetry K−Mdirection in graphene (compare Fig. 2.5). A larger

chiral angle moves the linear cutting lines away from the K−Mhigh-symmetry line, ac-

companied by a decrease of oscillator strength of the respective optical transitions [45,62].

In contrast, the lowest optical matrix elements are found for the E22,ν=−1and E11,

ν= 0 transitions, increasing for increasing chiral angles θ. Those mainly stem from the

K−Γhigh symmetry line in graphene and increase the more their direction is moved to

the K−Mhigh-symmetry line. Although not plotted in Fig. 2.6, it can generally be said

that optical matrix elements for tubes belonging to the ν=−1family are larger than

those from tubes belonging to the ν= +1 family. Depending on the optical transition,

metallic carbon nanotubes have larger/smaller optical matrix elements than those from

the ±1tubes [61] (compare Fig. 2.6).

The unlike couplings to optical light fields affect the Raman experiments performed within

this dissertation. Optical experiments alone are not able to display the correct distribution

of chiralities within an ensemble. The Raman experiments predominantly cover "inner"

optical transitions, i.e. those transitions that predominantly stem from the proximity of

the K−Mhigh-symmetry direction. This fact has far-reaching consequences for the

understanding of double-resonant Raman features in carbon nanotubes that are widely

discussed in Chapter 6. We also show calculated optical matrix elements for deep-UV

transitions i.e. E55, E66, E77,. . . transitions to understand the optical responses of car-

bon nanotubes in that spectral region.

2.4. Structure and fundamental properties of diamondoids

2.4 Structure and fundamental properties of diamon-

doids

ADA DIA TRIA [121] [1(2)3] [123]

| {z }

TETRA

Figure 2.7: Lower diamondoids are pictured, with increasing size from left to right. ADA, DIA, TRIA,

and TETRA stand for adamantane, diamantane, triamantane, and tetramantane, respectively. Numbers

in brackets denote the nomenclature for isomers of one species after Balaban and Schleyer [63].

In contrast to the trigonal carbon structure present in graphene and carbon nanotubes,

the sp3hybridization defines the tetrahedron-like arrangement of the carbon bonds in

diamondoids [28]. Each bond exhibits a bond angle of 109.5◦to its three neighbors.

The smallest possible formation of a closed carbon cage in which every carbon atom ex-

hibits a triangular pyramid bond structure consists of 10 carbon atoms and is named

adamantane [64]. It can be seen as a cage-shaped cutout of the diamond lattice [19].

The remaining dangling bonds are either saturated with one or two hydrogen atoms, de-

pending on the position of the carbon atom within the cage [19]. Its empirical formula

is therefore C10H16. By adding another diamond crystal lattice cage to adamantane, a

larger carbon molecule can be prepared that still conserves the tetrahedron diamond lat-

tice structure. This procedure can be repeated over again, resulting in the formation of

the homologous series of diamondoids [19]. Depending on the amount of containing face-

fused cages, the lower diamondoids are named adamantane (one cage), diamantane (two

cages), triamantane (three cages), tetramantane (four cages), and so on. Selected dia-

mondoids are shown in Fig. 2.7. In the case of diamantane and triamantane, the direction

in which the additional diamond cages are attached to adamantane does not change their

structures. However, already for tetramantane, there are three different ways to attach

a diamond carbon cage to triamantane, as depicted in Fig. 2.7. The amount of possible

isomers drastically increases for higher diamondoids [19]. To distinguish between differ-

ent isomers, Balaban and Schleyer suggested a nomenclature in which Arabian numbers

determine the direction new carbon cages are added starting from adamantane [63].

Bond lengths of the C-C and C-H bonds have values of 1.54˚

A and 1.112˚

A, respectively.

As a consequence, the diameter of an adamantane molecule exhibits ∼5˚

A. All diamond

molecules smaller than 1nm are named lower diamondoids, for diameters between ∼1

and 2nm they are named higher diamondoids, and large molecules with diameters from

∼2nm to µm are named nanodiamonds.

The availability of homogeneous species has made diamondoids to an object of broad

2.4. Structure and fundamental properties of diamondoids

scientific interest. Their gradually increasing size for instance allows for a step-by-step

analysis of quantum confinement effects. Interestingly, it was shown that the unoccu-

pied states in diamondoids do not exhibit a size-dependent blueshift as known for other

group IV semiconductors [20]. Their optical absorptions however are partly determined by

quantum confinement effects, leading to size-dependent band gaps of ∼5.8eV (Pentaman-

tanes) up to ∼6.6eV in adamantane [32,65]. For comparison, the indirect wide-bandgap

semiconductor diamond has an optical gap of 5.47eV [66]. It was found that the highest

occupied states are referred to C-Cσstates, spatially limited to the insides of the carbon

cages, whereas the lowest unoccupied states are rather diffuse C-Hσ∗states surrounding

the diamondoids [32,65,67]. The diameter of their average distribution is several times

larger than the confined C-Cσstates and thus quantum confinement predominantly ef-

fects the occupied electronic states. Due to their distinct symmetries, not all vertical,

optical transitions are dipole allowed [65]. For this reason, their optical responses are de-

termined by quantum confinement, superimposed by optical selection rules. For instance,

the HOMO (A1g)→LUMO (A1g) transition is symmetry-forbidden in diamantane [65].

The first allowed transition HOMO (A1g)→LUMO + 1 (A2u) is ∼0.6eV higher and

exceeds that of adamantane although the carbon cage has twice its size [65].

Vörös et al. theoretically demonstrated that the substitution of the hydrogen passivi-

sation with high-electronegativity atoms induce both, a charge transfer and symmetry

braking, significantly reducing the lowest optical transitions in diamondoids [68]. In case

of adamantane four sulfur atoms are necessary to redshift its optical gap to the visible

range whereas in [1(2,3)4] pentamantane, already one sulfur atom suffices for the same

downshift [68]. An experimental verification however is still absent. Downshifts of opti-

cal transition energies for other functional groups, such as -SH, -NH2, and -OH groups

are instead reported experimentally, but do not exceed energies lower than 1 eV [69,70].

Combining diamondoids to diamondoid oligomers with an introduction of sp2defects con-

stitutes another way to alter the electronic properties of diamondoids [22, 71–73]. The

fundamentally different structure of the double-bond related electronic states that lie in

between those from the σand σ∗in pristine diamondoids lead to fundamentally different

optical responses that are in detail discussed in Chapter 4.

The vibrational properties of pristine diamondoids up to [121321] heptamantane are

widely understood both experimentally and theoretically [30, 65, 74,75]. Their distinct

symmetries were used for a profound group-theoretic analysis with a great accordance

to experimental Raman spectra [30,74]. However, functionalizations significantly lower

their symmetries and therefore alter the allowed vibrational modes of chemically modified

diamondoids [72,73,76]. We will give a detailed introduction and analysis of the changes

induced by sp2defects in diamondoid oligomers in Chapter 4.

2.4. Structure and fundamental properties of diamondoids

3. Experimental methods

This chapter discusses inelastic light scattering in graphitic materials. We will introduce

first-order Raman scattering in molecules and periodic solid state bodies. In graphitic

materials, additional Raman features beyond the first-order scattering can be observed.

They are characterized by considerable phonon wave vectors involved in the higher-order

scattering process. This process, also named double resonant Raman scattering, is dis-

cussed in detail. In the end, we show the experimental setup and shortly discuss the

second harmonic generation of light.

3.1 First-order Raman scattering - Raman spectroscopy

A small fraction of light that interacts with matter can be inelastically scattered, caused

by atomistic vibrations in the material. In solid state bodies, these collective vibrations

are referred to as phonons. They can be seen as vibrations in which the entire lattice

is involved. This leads to the appearance of distinct phonon band structures in which

the energy of a lattice vibration depends on the wavelength of the collective atom dis-

placements. Instead, the lack of periodicity in molecules preludes the appearance of

phonons with an arbitrary wavelength. From a solid state physics point of view, phonons

in molecules always originate from the Γpoint, i.e. they are atomic vibrations with no

phonon wavevector involved.

The periodic, atomic vibrations affect the electric susceptibility of the analyzed mate-

rial system. This in turn allows for an access to the vibrational properties of materials

via optical spectroscopy methods. These spectroscopy methods with their large variabil-

ities on the one hand and the striking sensitivities of current spectrometers on the other

hand, can be utilized to map even smallest changes in optical responses. This also and,

in particular, counts for changes induced by atomic displacements.

Inelastic scattering involving atomic vibrations was the first time experimentally observed

by Raman 1928 [77]. Back then, it was called "A new type of Secondary Radiation" and

was theoretically predicted even several years earlier by Smekal [78]. In honor of its

discoverer, inelastic light scattering is today usually associated with the Raman effect.

Although the effect is referred to as light scattering with phonons, it is on a microscopic

level, rather the interaction of charge density coupling to light. To understand the Raman

effect within a classical description, we assume an incoming light field:

E(r, t) = Ei(ki, ωi)cos (kir−ωit)(3.1)

3.1. First-order Raman scattering - Raman spectroscopy

An incoming electromagnetic wave with a wavevector kiand frequency ωiinduce a po-

larization within the interacting material:

P(r, t) = Pi(ki, ωi)cos (kir−ωit)(3.2)

The induced polarization P(r, t)as a function of the driving light field is a material specific

physical quantity, also referred to as the electric susceptibility χ:

Pi(ki, ωi) = χ(ki, ωi)Ei(ki, ωi)(3.3)

The susceptibility χin turn slightly varies as a function of periodic atomic displacements,

such as phonons in a lattice. In the following the displacements of the atoms are described

by normal coordinates Qk:

Qk(r, t) = Qk(q, ω0)cos (qr −ω0t),(3.4)

where ω0describes the frequency of a lattice vibration that belongs to the eigenvector.

The susceptibility χcan further be expanded in a Taylor series depending on the normal

coordinates of a lattice vibration:

χ(ki, ωi,Qk, m) = χ0(ki, ωi)+X

k ∂χ

∂Qk!0

Qk(r, t)+X

k, m ∂2χ

∂Qk∂Qm!0

Qk(r, t)Qm(r, t)+. . .

(3.5)

where the sums run over all normal coordinates. The first term χ0describes the suscep-

tibility without distortions from atomistic displacements. In the second term, an atomic

displacement along the normal coordinate Qkis considered, defining the first-order ex-

pansion term. Further, higher-order terms can be considered in order to describe atomic

displacements along more than one normal coordinate.

Inserting the nonlinear susceptibility [Eq. (3.5)] into the general expression for the polar-

ization as given in Eq. (3.3) yields:

P(r, t, Q) = P0(r, t) + Pind(r, t, Q)(3.6)

P0describes the undistorted polarizability. For reasons of simplification, Pionly considers

the first order of the nonlinear susceptibility:

Pind(r, t, ω0) = ∂χ

∂Qk!0

Qk(r, t)Ei(ki, ωi)cos (kir−ωit)(3.7)

= ∂χ

∂Qk!0

Qk(q, ω0)cos (qr −ω0t)Ei(ki, ωi) cos(kir−ωit).(3.8)

The product of two cosine functions in Eq. (3.8) can be rewritten using addition theorems

via:

Pind(r, t, ω0) = 1

2 ∂χ

∂Qk!0

Qk(r, t)Ei(ki, ωi)

·[cos ((ki+q)r−(ωi+ω0)t+ cos((ki−q)r−(ωi−ω0)t)] .(3.9)

Three components of the induced polarizability can be distinguished that specify the mea-

surable optical response. The first term P0only depends on the frequency ωiof the driving

3.1. First-order Raman scattering - Raman spectroscopy

~ω0

~ωi~ωi~ωi

~(ωi+ω0)~ωi~(ωi−ω0)

ν0= 0

ν0= 1

ν0= 2

ν= 0

ν= 1

ν= 2

|1>

|0>

r-r0

Energy (arb.units)

~ωi~(ωi−ν1)

(8,0) CNT

kz(π/T)

Energy (arb.units)

~ωi~(ωi−ω0)

Anti-Stokes Rayleigh Stokes

b) c)

kz= 0

Molecule Carbon nantoube

Figure 3.1: a) Anti-stokes, Rayleigh, and Stokes scattering are schematically plotted. The energy of

the incoming photon is ~ωi; the phonon energy is denoted by ω0. E0denotes an electronic ground state,

E1,2,3are excited states. b) Stokes scattering is illustrated for a resonant excitation in a molecule. |0>

and |1>are Morse potentials exemplary for the ground and the first excited electronic state of a simple

molecule [79]. νand ν0are vibrational states within the electronic ground and excited state, respectively.

c) Resonant Stokes scattering is illustrated for a (8,0) zig-zag carbon nanotube (CNT).

light field. This elastically scattered light is usually referred to as Rayleigh scattering.

Pind contains two components in which the frequency of the scattered light is shifted by

the phonon energy ω0. In case there is a constructive interaction (ωi+ω0), the Raman

scattering is referred to as Anti-Stokes Raman scattering, in case there is a destructive

interaction (ωi−ω0), the Raman scattering is referred to as Stokes scattering.

As can be seen in the expression for the induced polarizability Pind in Eq.(3.9), the

nonlinear susceptibility contains its derivative after the normal coordinates of an atomic

vibration. As a consequence, only those vibrations contribute to a Raman signal, that

change the polarizability. The second derivative can be seen as a higher-order process

in which two phonons are involved. They can either correspond to the same atomic vi-

bration or can correspond to different atomic vibrations as generally described in Eq.(3.5).

A schematic overview of resonant Raman and Rayleigh scattering is given in Fig. 3.1

a). The excitation occurs from an electronic ground state E0to an excited state E2. A

more detailed view on a resonant Stokes Raman process in a molecule and in a (8,0) car-

bon nanotube is shown in Fig. 3.1 b) and c). Both ground and excited electronic states

in molecules exhibit a vibronic fine structure as denoted with νand ν0in Fig. 3.1. In

3.2. Double-resonant Raman scattering

K M K’

~ωi~(ωi−ων−ωµ)

+qν

−qµ

←− −→

Γ Γ

Energy

π∗

Figure 3.2: A representation of a second-order, two-phonon Raman scattering process in graphene is

illustrated. The scattering occurs along the Γ−K−M−K0−Γhigh-symmetry direction. As both the

Kand the K0points are involved, the scattering process is also named an intervalley Raman process.

The involved phonon energies are ωνand ωµwith the momentum q, respectively. Illustrated is a process

in which both electrons and holes are scattered.

diatomic molecules, the electronic states are Morse potentials [79] that get much more

complex in larger molecules [80]. The energy spacing of vibronic states in excited elec-

tronic states (ν0in Fig. 3.1) is usually smaller as the nuclei separation is larger in the

excited states accompanied by reduced restoring forces for the atomic vibrations [80].

This fact is of great importance as we measure shifted bond lengths via the analysis of

altered Raman frequencies in Chapter 4 [81]. Only the momenta of the incoming and

outgoing light are involved in such a process. The small momentum difference due to the

resonant scattering is transferred to the nuclei of the molecule.

The conservation of momentum during a first-order Raman process in carbon nanotubes

requires that both the excited electron and remaining hole do not change their momen-

tum. In case of a (8,0) tube, the absorption and emission therefore both occur at kz= 0

as depicted in Fig. 3.1. Only for reasons of visibility, the emission arrow (blue) is slightly

shifted.

All Raman measurements performed within this dissertation are Stokes processes, i.e. we

always measure reduced energies of the re-emitted photons.

3.2 Double-resonant Raman scattering

The Raman signature of graphitic sp2carbons contains several high-frequency modes ex-

hibiting energies way above those from individual lattice vibrations. Intriguingly, some of

these modes even show anomalous shifts as a function of the laser excitation energy that

cannot be explained only by first-order Raman scattering. First observations of the un-

usual shift rate of a certain Raman peak were done by Tuinstra et al. [82]. It was related

to defects in the graphite lattice which is why this mode is today named the defect mode,

or simply the Dmode [82,83]. The exact origin of this peak however was unclear for a

long time until Thomsen and Reich gave a conclusive explanation [9]. They suggested the

concept of double-resonant Raman scattering that explained the Dmode on the one hand

but was also capable to explain various high-frequency peaks in the Raman spectrum of

graphite on the other hand [9,45,53]. It was further successfully applied to other graphitic

sp2materials, such as graphene, multilayer graphene, and carbon nanotubes [45,53].

3.2. Double-resonant Raman scattering

Double-resonant Raman scattering is a second-order process in which either two phonons

or one phonon and one defect are involved. In quantum mechanics the lattice vibrations

(phonons) can be treated as a perturbation of the light-matter interaction [53]. This

fourth-order perturbational approach yields an expression in which the scattered photon

rates depend on the square of a detuned Lorentzian oscillator [9, 14, 45, 53]. Following

authors in Ref. [14] the two phonon Raman cross section in graphene can be written as:

Iµν

q(L, γ)∝

X

k,e,h,...

hkπ|Dout|kπ∗ihkπ|∆H−q,µ|k+q, π∗ihk+q, π∗|∆Hq,ν |kπ∗ihkπ∗|Din|kπi

(L−π∗

k+π

k−~ων

−q−~ωµ

q−iγC

2)(L−π∗

k+q+π

k−~ων

−q−iγB

2)(L−π∗

k+π

k−iγA

−X

k,e,h,...

hk+qπ|Dout|k+qπ∗ihkπ|∆H−q,µ|k+q, πihk+q, π∗|∆Hq,ν |kπ∗ihkπ∗|Din|kπi

(L−π∗

k+q+π

k+q−~ων

−q−~ωµ

q−iγC

2)(L−π∗

k+q+π

k−~ων

−q−iγB

2)(L−π∗

k+π

k−iγA

+···

.(3.10)

ωµand ωνdenote different phonon energies with respect to their momentum q. All elec-

tron and hole states in kspace are summed up for each individual phonon momentum q.

The first term considers electron-electron scattering where an electron is first scattered

by the −qνphonon and then backscattered by the qµphonon. The same can occur both

to electrons and holes as considered in the second term: The electron is scattered by

the −qνphonon with a consecutive backscattering of the hole by the qµphonon [14].

Involved in the complex scattering process can either be just electrons, just holes, or both

electrons and holes. Further, the succession of involved phonons in each individual scat-

tering process can vary allowing for overall eight different scattering processes [14]. π∗

and πdenote the energy of the electrons in the valence bands and holes in the conduc-

tion band, respectively. The broadening γcan be seen as the inverse lifetime of individual

electronic states involved and explicitly depends on the electron wavevector k, but is

often considered to be constant. The optical matrix elements (hk(±q)π|Dout|k(±q)π∗i

and hkπ∗|Din|kπi) describe the coupling strength to optical light fields. Electron-phonon

coupling elements are expressed via hk±qπ|∆H±q,µ,ν |k±qπ∗i. Both can be found in

Ref. [14].

A large Raman cross section is achieved when one of the denominators gets very small.

This happens when two or more of the intermediate electronic states within this higher

order process are real states. In this case, the particular Raman mode is resonantly en-

hanced and exceeds those with other phonon momenta involved by magnitudes [9, 14].

For this reason, this process is often referred to as a double-resonance, or even to a triple

resonance process.

In contrast to first-order Raman scattering, double-resonant Raman scattering can in-

volve phonons with an arbitrary momentum. The conservation of momentum in a double

resonance Raman process is preserved, when two phonons with likewise momenta, but

3.2. Characteristic Raman spectra of graphite, graphene, carbon nanotubes, and

diamantane

with opposite signs contribute:

~ks=~ki+~(±qν,µ

1∓qν,µ

2)(3.12)

The requirement of the conservation of momentum ks≈kiautomatically implies the

following relation for the phonons involved: |q1| ≈ |q2|. As a consequence, zone-center

phonons with very small momenta, as well as zone-boundary phonons with the largest

momenta possible can possibly contribute to the Raman spectrum. This fact is of par-

ticular interest to understand the density-of-states related Raman spectra as discussed in

Chapter 5.

The second-order Raman process as described in Eq. (3.10) can be divided into four

steps as illustrated in Fig. 3.2. (i) At first, an electron/hole pair with momentum kis

created in the π/π∗bands. (ii) The excited electron is scattered into a different electronic

state with momentum k+q. (iii) The same occurs to the hole that is scattered to an

electronic state with momentum k−q. (iv) In the end, electron and hole recombine

by emitting a photon with a reduced energy ~ωs=~(ωi−ων−ωµ)[14]. The process

illustrated in Fig. 3.2 is called an intervalley process as two distinguishable Kand K0

points are involved. Scattering can also occur within one Dirac cone; it is then referred

to as intravalley scattering [14].

The introduced concept of double-resonant scattering could finally explain the shift rates

of Raman modes in graphitic sp2materials. A certain laser excitation energy determines

the electron momentum and therefore the k-space distance to another real electronic

state. As phonons mediate between these two unlike electronic states, their momentum

necessarily needs to be different for different laser excitation energies in a second-order

Raman process. Depending on the phonon dispersion, the phonon energies involved in

such a process are therefore different for each individual laser excitation energy.

We show calculated Raman spectra of low-intensity Raman modes in carbon nanotubes

in Chapter 6. They are modeled via the transition matrix Tfi[q]with I∝ |Tfi[q]|2, equiv-

alent to the fourth-order perturbational approach as shown in Eq. (3.10) [84]. Different

to Eq. (3.10), electron, holes, and phonon bands for each individual tube are altered

according to the zone-folding constraints as introduced in section 2.3.1.

3.2.1 Characteristic Raman spectra of graphite, graphene, car-

bon nanotubes, and diamantane

Raman spectra of carbon nanotubes, graphite, graphene, and diamantane for an excitation

energy of εL= 2.73eV are plotted in Fig. 3.3. All modes in diamantane, the radial-

breathing mode (RBM), and the G mode are first-order Raman scattering processes. The

other modes can be referred to second-order Raman modes. The G mode is the most

pronounced peak in the Raman spectra of graphite, graphene, and carbon nanotubes.

It is referred to a C=C stretch vibration and is characteristic for all graphitic carbons.

This mode can already be observed, if only two double bond sp2hybridized carbon atoms

are present in the analyte. The D, and 2Dmodes are breathing modes of entire sp2

3.2. Characteristic Raman spectra of graphite, graphene, carbon nanotubes, and

diamantane

C-H vibrations

C-C vibrations

200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200

Raman shift cm−1

Diamantane

RBM D

2D’

TOZO iTOLA

εL= 2.73 eV

Intensity (arb.units)

HipCO carbon nanotubes

Graphite

Graphene

Figure 3.3: Raman spectra of HipCO-produced carbon nanotubes, graphite, graphene (top), and dia-

mantane (bottom) are plotted for an excitation energy εL= 2.73 eV. Characteristic modes for carbon

nanotubes and graphite are indicated. The TOZO and iTOLA modes are discussed in detail in Chapter

6. Characteristic modes in diamantane are generally assigned by the atom species involved. A detailed

analysis can be seen in Chapter 4. RBM stands for radial-breathing modes. All diamantane modes, the

RBM, and the G mode are first-order Raman processes, the D,2D,2D0, TOZO, and iTOLA modes are

double-resonant, second order Raman modes.

hybridized carbon hexagons [82]. In case of the Dmode, its intensity depends on the

defect density and therefore, it is often referred to as the defect mode [14,82,85]. The low-

intensity, double-resonant Raman modes TOZO (transverse optical, out-of-plane optical)

and iTOLA (in-plane transverse optical, longitudinal acoustic) are in detail discussed in

Chapter 6. The structure-induced altering of vibrational properties of carbon nanotubes,

leads to generally more complex Raman spectra. For instance, curvature effects lift the

degeneracy of the two-fold degenerate E2gmode in graphene, leading to various G modes

in their Raman spectra (compare Fig. 3.3). Raman modes in diamantane are generally

assigned by the species of atoms involved in the vibrations (C: carbon, H: hydrogen).

Because of lower masses, the high-frequency modes are usually attributed to the C-H

vibrations. A closer analysis of the vibrational properties in diamondoids and diamondoid

derivatives is given in Chapter 4.

3.3. Experimental setup

Microscope/Sample

pre monochromator 1 and 2

Spectrometer

Fred 90c Argon laser BBO crystal for

SHG

CCD

”Ferkel”

Reflective grating

* 1800/mm and 3600/mm

changeable gratings

Figure 3.4: The experimental Raman setup is schematically plotted. Key optical elements are named.

SHG and CCD stand for second harmonic generation and charge-coupled device, respectively.

3.3 Experimental setup

The key requirement for an experimental Raman setup is the suppression of the elastically

scattered light of the exciting laser. This can be achieved via a notch filter, or via the

application of pre monochromators. The latter has the advantage of flexibility, i.e. it

can continuously be adjusted according to the laser wavelength. In our case, we have

used a HORIBA T64000 triple monochromator that has two pre monochromators and a

final spectrometer. The pre monochromators are coupled in a subtractive configuration,

very efficiently suppressing these parts of the spectrum that are not required. The final

resolution of the scattered light is carried out in the spectrometer.

For all measurements in the UV range, we have used a Coherent 90c Fred Argon ion laser

supporting frequency doubling. We could use all fundamental wavelengths from 454.4nm

up to 528.7nm with the possibility of frequency doubling of each individual line. Different

BBO (beta-Barium borate) crystals were available exhibiting different cutting angles to

achieve phase matching for each laser line [86]. The so covered photon energies range

from 4.69eV up to 5.46eV.

All measurements with UV excitation energies have been done with the HORIBA T64000

as schematically plotted in Fig. 3.4. To achieve resolutions up to 1.5cm−1, we could change

the 1800/mm to a 3600/mm grating in the ultraviolet region. The final signal was collected

by a Symphony Cryogenic Back Illuminated UV sensitive CCD Detector. The incident

UV light beam was focused by a Thorlabs LMU-40X-UVB CaF2objective, providing spot

sizes up to 1µm in diameter in the applied UV range. Raman measurements in the optical

visible region have been performed with a LABRAM HR800 spectrometer. In contrast

3.3. Experimental setup

to the T64000 spectrometer, it has a notch filter providing better signal-to-noise ratios

but lacking the possibility to use variable light sources. All measurements have been

performed in a backscattering geometry.

3.3.1 Second harmonic generation

One of the key features that is discussed in this dissertation is the ultraviolet optical

response of graphitic materials. Although there is much progress in the field of compact

UV laser LEDs, continuous-wave, narrow-bandwidth, and high-intensity light sources for

wavelengths down to 200nm are still not commercially available. A convenient and well

established workaround for the generation of deep-UV radiation is the second harmonic

generation (SHG) utilizing the fact that laser light sources in the desired visible range

(∼450 −600nm) are very well available.

The already introduced polarization Pcan be expanded as follows [86,87]:

P=0X

χ(n)En=0χ(1)E1+0χ(2)E2+. . . (3.13)

χ(1) denotes the linear susceptibility, χ2is the second-order term. In general, even

higher order contributions can be important for nonlinear optic effects, such as the Kerr-

nonlinearity [88] (χ(3) effect). However, the relevant second harmonic generation is a χ(2)

process that is forbidden in systems with an inversion symmetry [86]. For strong light

fields, the absorption of photons leads to excitations that are above their locally parabolic

potentials. This in turn can activate additional restoring forces responsible for higher-

order terms in the electric susceptibility [86].

Again, the light can be seen an oscillating wave:

E(t) = E0sin (ωt)(3.14)

resulting in the second-order polarization:

P2=0χ(2)E2=0E2

0χ(2) sin2(ωt) = 0E2

0χ(2)

2· 1−cos(2ωt)

2!.(3.15)

It can be seen that Eq. (3.15) contains two contributions: a constant term and a second

term that explicitly depends on the doubled frequency 2ωresponsible for the higher-order

radiation.

The secondary radiation can only be emitted if all oscillators from an excited medium in-

teract constructively [86]. This can solely be achieved when both the fundamental and the

second harmonic light beam have the same refractive indices, although their wavelengths

are different: nω∼n2ω. Therefore, nonlinear crystals are used for second harmonic gen-

eration that have different refractive indices both for the ordinary and extraordinary light

beams. For instance, the refractive indices of the utilized BBO crystals for the ordinary

and extraordinary rays are no= 1.679 (488nm) and ne= 1.632 (244 nm), respectively [89].

3.3. Experimental setup

4. Diamondoid derivatives

Parts of this section are published in Refs. [81] and [73]

This chapter discusses the electronic and vibrational properties of functionalized diamon-

doids. Especially in the focus are diamondoid derivatives consisting of two or more lower

diamondoids that are covalently connected. Analyzed are various structures: On the one

hand chemically blended sp2/sp3hydrocarbons are discussed in which diamondoids are

connected by carbon double bonds. The carbon double bonds are either attached directly

to the diamondoids or they are between two carbon atoms forming small chains connect-

ing the oligomers. On the other hand, various diamondoid dimers are analyzed that are

connected by single carbon bonds. In contrast to the double-bond oligomers, they are

fully saturated hydrocarbons with fundamentally other electronic properties. Further, a

graphene/adamantane composite, namely [2](1,3)Adamantano[2](2,7)pyrenophane is an-

alyzed. It can be seen as a bridge between a fully saturated diamondoid and a fully

unsaturated graphene flake with interesting properties merged from both constituents.

The nearly unlimited availability of carbon and the variety of its allotropes make car-

bon even in the 21th century to an object of a broad scientific interest. Recent develop-

ments in the functionalization of diamondoids seem to offer a whole new path for carbon

derivatives to find their way in applications. The essential requirement however is a deep

understanding of the electronic and vibrational changes that are accompanied by the

modification of their structures. It may be even more important to first understand the

self-caused altering of electronic properties as diamondoids tend to form very stable van-

der-Waals crystals with unusually high binding energies. The periodic arrangement in a

crystal lattice leads to electronic and vibrational properties different from those in isolated

molecules or in solutions diamondoids are available otherwise. For this reason, we also

discuss the electronic properties of several diamondoid van-der-Waals crystals and report

newly observed collective vibrational modes that are referred to librations in diamondoids

van-der-Waals crystals.

4.1 Double-bond diamondoid oligomers - vibrational

properties

An interesting way to tailor properties of diamondoids is the introduction of unsaturated

bridges between neighboring sp3cage units [22]. These bridges can be seen as surface

sp2defects and lead, due to their fundamentally different nature of the chemical bond,

to widespread changes of their electronic and vibrational structure. Recently, a large

4.1. Double-bond diamondoid oligomers - vibrational properties

Figure 4.1: The analyzed sp2/sp3hybrid diamondoid oligomers are plotted. They are namely: a) 3,10-

bis-(2-adamantylidene)diamantane, b) 3-(2-adamantylidene)diamantane, c) 1,3-(bis-4-diamantyl)prop-1-

ene, and d) [4.4](1,3)adamantanophan-trans,trans-1,8-diene. For reasons of simplicity, the hydrogen

atoms are not plotted.

variety of diamondoid oligomers have been synthesized [22]. They consist of lower dia-

mondoids, such as adamantane or diamantane and are connected either directly with a

double bond or via carbon chains containing the C=C double bond. Simplified structures

of the molecules are plotted in Fig. 4.1.

Another group of available cagelike hydrocarbons are the Pentacycloundecanes

(Trishomocubanes) [90]. One isomer, namely the Pentacyclo[6.3.0.02,4.03,10,05,9] is the

only structure without three- or four-membered rings or other strain-inducing compo-

nents [90]. It was therefore also used to fabricate sp2/sp3dimers [91] comparable to those

with diamondoid constituents and is closer analyzed in section 1.4.1 and Refs. [72,92].

The introduction of sp2defects as a connection between lower diamondoids results in

a far-reaching change of their vibrational properties. On the one hand, the diamondoid

oligomers exhibit lower symmetries compared to their constituents (Tdfor adamantane

or D3dfor diamantane [30]). This should generally result in much more complex Raman

spectra in terms of the pure amount of visible Raman modes. On the other hand, we

expect new, characteristic Raman modes that originate from the specific structure of the

diamondoid derivatives.

A possible perspective to approach the vibrational properties of the diamondoid oligomers

is the linear combination of distinctive, characteristic vibrational modes of their con-

stituents. The sp2/sp3blended oligomers can be seen as a combination of pristine adaman-

tane/diamantane and ethylene, located between the diamondoids. Therefore, we expect

characteristic Raman modes from both ethylene and adamantane/diamantane in the Ra-

man spectra of the oligomers. For instance, reasonable are breathing-like modes that are

already known for diamondoids [30] or other carbon sp2systems such as carbon nan-

otubes [93], or Fullerenes [94].

Adamantane has 72 vibrational modes, 11T2+7T1+6E+1A2+5A1of which the T1and A2

are not Raman active, leaving 22 possible Raman modes [30]. In contrast, diamantane has

96 vibrational modes, 11A1g+6A1u+5A2g+10A2u+16Eg+16Eu. In this case only the A1g

4.1. Double-bond diamondoid oligomers - vibrational properties

C-C-C bend/wag C-C stretch CH2twist/rock CH2wag

CH2scissors

diamantane

adamantane

εL= 4.82 eV

BLM

200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400

Raman shift cm−1

Intensity (arb.units)

C-H stretch

2800 2900 3000

Figure 4.2: Raman spectra of diamantane (top) and adamantane (bottom) for an excitation energy

of 4.82 eV are given. Characteristic modes and their frequency ranges are indicated. BLM stands for

breathing-like mode.

and Egspecies are Raman active, resulting in 27 possible Raman modes in the spectrum

of diamantane [30]. Not all Raman-active modes are intense enough to be observed in an

experiment, explaining the lower amounts of modes in Fig. 4.2. Ethylene has the point

group D2hand 12 vibrational modes: 3Ag+2B2g+2B1u+2B3u+B3g+Au+B2uof which the

Ag, B2g, and B3gspecies are Raman active, resulting in six possible Raman modes [95].

The spectrum of diamondoids can be divided into characteristic vibrations as denoted in

Fig. 4.2. They can be separated by carbon-carbon vibrations, namely C-C-C bend/wag

and C-C stretch modes and carbon-hydrogen vibrations such as CH2twist/rock, CH2wag,

or CH2scissor modes [30,74]. The latter generally have higher frequencies due to the lower

masses involved in the vibrations. A species-characteristic mode is the breathing-like mode

"BLM", found at around ν∼600 −700 cm−1as indicated in Fig. 4.2. In lower unfunc-

tionalized diamondoids, it is a fully symmetric vibration whose frequency increases with

a decreasing size and vice versa [30]. Its determination is therefore useful to character-

ize diamondoid species in an unknown sample. The functionalized diamondoid oligomers

exhibit lower symmetries than their constituents: The lowest symmetry has the 1,3-

(bis-4-diamantyl)prop-1-ene dimer with the C1point group. [4.4](1,3)adamantanophan-

trans,trans-1,8-diene has C2symmetry, 3,10-bis-(2-adamantylidene)diamantane has C2h

symmetry, and 3-(2-adamantylidene)diamantane exhibits CSsymmetry.

By comparison to DFT computations we can separate the Raman modes into three

classifications: (i) Raman modes from the unfunctionalized diamondoids, (ii) Raman

modes from ethylene, and (iii) Raman modes from the relative movements of the en-

tire monomers within the compounds. Further, compounds in which the double bond is

directly attached to the monomers ([4.4](1,3)adamantanophan-trans,trans-1,8-diene and

3-(2-adamantylidene)diamantane) are separated from those where the double bond is

connected to a carbon atom that does not belong to a diamondoid cage. The additional

carbon atom enlarges the vibrational degree of freedom of the ethylene centers, leading

to several highly localized Raman modes that are equivalent to those of pristine ethylene.

4.1. Double-bond diamondoid oligomers - vibrational properties

*2*3

*4*5

DIA

BLM

Intensity (arb.units)

*1*2

ADA

BLM

ADA+DIA

BLM

500 1000 1500 2000 2500 3000

ADA+DIA

BLM

500 1000 1500 2000 2500 3000

Raman shift cm−1

*3*4

DIABLM

500 1000 1500 2000 2500 3000

Raman shift cm−1

Intensity (arb.units)

Figure 4.3: Raman spectra of diamondoid oligomers (blue) compared to unfunctionalized diamondoids

(black) for an excitation energy of εL= 4.82 eV are given. In case the oligomers consist of different

diamondoids, the sum of the constituents is given. Characteristic modes are labeled with asterisks and

are summarized in Table 4.1. BLM, ADA and DIA stand for breathing-like mode, adamantane, and

diamantane, respectively.

4.1. Double-bond diamondoid oligomers - vibrational properties

The reduced symmetries cause a lift of degenerated vibrational modes that can be seen in

Fig. 4.3, where a comparison of Raman spectra from the analyzed diamondoid oligomers

and unfunctionalized adamantane and diamantane is shown. In case the oligomers have

two different constituents, the sum of their single contributions is given. All Raman

spectra of the chemically blended oligomers show additional modes in the region of the

carbon-carbon bend/wag, stretch and CH2twist/rock, wag modes that mainly origi-

nate from degenerated vibrational modes in the unfunctionalized constituents. Also, new

carbon-carbon bend/wag, stretch modes of carbon atoms in the chains connecting single

constituents are found. The characteristic BLM is a carbon-carbon stretch vibration that

is, except for the 1,3-(bis-4-diamantyl)prop-1-ene, downshifted compared to the unfunc-

tionalized diamondoids due to the larger inertiae. However, a top (4-position) functional-

ization [as in the case of the 1,3-(bis-4-diamantyl)prop-1-ene] only marginally affects the

BLM frequencies [76]. The 4-position carbon atoms are not deflected during an oscillation

of the BLM and thus inductive effects of the attached molecules lead to a slight upshift

of the BLM rather than a downshift due to the additional mass attached [76].

The compounds generally exhibit all vibrational modes that are known from the un-

functionalized diamondoids [30,74]. Frequencies of C-C-C bend/wag/stretch modes reach

from ∼150−900cm−1and thus, compared to ∼300−900cm−1in pristine diamondoids, a

larger region of frequencies is covered. This mainly accounts for the low-frequency region

due to the generally larger inertia involved. CH2scissor modes around 1440cm−1are

highly localized vibrations at the cage edges and are not affected by a functionalization.

Both in the pristine and functionalized diamondoids, they exhibit the same frequencies

(compare Fig. 4.3). The same accounts for the carbon-hydrogen stretch vibrations found

at around 2900cm−1. As in the unfunctionalized diamondoids, they can be separated into

symmetric contributions (around 2850 cm−1) and anti-symmetric contributions (around

2915cm−1). However, for all oligomers, an additional carbon-hydrogen stretch mode is

found 50cm−1above the highest anti-symmetric stretch vibration in pristine diamondoid

(labeled with ∗5in Fig. 4.3). These modes are referred to highly localized vibrations

close to the C=C bridges. The sp2defects cause a local stiffening slightly increasing the

frequency of surrounding C-H modes.

We further observe new modes that are not found in pristine diamondoids: A C=C stretch

mode (∗4in Fig. 4.3), a C=C torsional mode (∗3in Fig. 4.3), a C=C rotational mode

(∗1in Fig. 4.3), and a newly observed breathing-like mode (∗2in Fig. 4.3) in which the

containing diamondoids oscillate towards each other. We will refer to that mode as dimer-

breathing mode (DBM) [73]. The C=C stretch and torsional modes originate from the

ethylene moieties and have a direct equivalent in ethylene (D2hpoint group): The sym-

metric Agstretching mode (∼1720cm−1) and a hydrogen-carbon-hydrogen out-of-plane

wagging mode (B2g) around 954cm−1. Both in the computations and measurements we

find these vibrations at lower frequencies, around 1660cm−1and around 754cm−1for the

stretching and torsional modes, respectively. The masses attached to the ethylene moi-

eties lead to a decrease of their frequencies, compared to ethylene.

The ethylene rotational modes exhibit frequencies around 200cm−1both in the com-

4.1. Double-bond diamondoid oligomers - vibrational properties

Table 4.1: Overview of experimental and theoretical frequencies of characteristic Raman peaks from

the double-bond diamondoid oligomers. In the upper part, we show Raman modes that all diamondoid

compounds have in common and compare them to the corresponding vibrational modes of ethylene [95].

In the lower part we compare Raman modes of dimers in which the double bond is not directly attached

to the diamondoid cage to the characteristic vibrational modes of ethylene. Modes marked with asterisks

can be found in the Raman spectra in Fig. 4.3. We used a GGA approach with the PBE functional and a

plain-wave basis set with pseudopotentials for the computations. Intermolecular, long-range interactions

are accounted for through the DFT+D3(BJ) method. All frequencies are given in cm−1. ADA and DIA

stand for adamantane and diamantane, respectively. Computations have been done by Roland Gillen [96].

Sample Mode type exp. calc. Mode

type

exp. calc.

ADA-2×{CH2-CH=CH-CH2}-ADA DBM ∗2285/- 281/138 BLM 712 689 / 705

DIA-CH=CH-CH2-DIA DBM ∗2177 168 BLM 715 705 / 711

ADA=DIA DBM ∗2200 191 BLM 698 /779 689 /772

ADA=DIA=ADA DBM ∗2171 162 BLM 683 /785 731 /772

Ethylene B2g944 Ag1621

ADA-2×{CH2-CH=CH-CH2}-ADA C=C

bend/twist

∗3

803 811 C=C

stretch

∗4

1666 1663

DIA-CH=CH-CH2-DIA C=C

bend/twist

∗3

773 812 C=C

stretch

∗4

1667 1664

ADA=DIA C=C

bend/twist

∗3

731 725 C=C

stretch

∗4

1661 1653

ADA=DIA=ADA C=C

bend/twist

∗3

743 725 /694 C=C

stretch

∗4

1660 1639

Ethylene B3u952 B3g1234

ADA-2×{CH2-CH=CH-CH2}-ADA - 967 - 1303

DIA-CH=CH-CH2-DIA - 968 - 1261

ADA-2×{CH2-CH=CH-CH2}-ADA ethylene

rotation ∗1

189 194

DIA-CH=CH-CH2-DIA ethylene

rotation ∗1

231 236

4.1. Double-bond diamondoid oligomers - vibrational properties

C=CR

−→

←−

←

→

←−

−→

←

−→ ←−

B2g

∗3

B3u

B3g

∗4

rot.

∗1

DBM

∗2

Figure 4.4: Left: Localized vibrational patterns of ethylene centers in two diamondoid dimers. R

stands for diamantane [1,4-(bis-4-diamantyl)but-2-ene] or a carbon atom with an attached adamantane

([4.4](1,3)adamantanophan-trans,trans-1,8-diene). The irreducible representations of the equivalent vi-

brations in ethylene are given next to the simplified chemical structures. Corresponding frequencies of

both ethylene and the diamondoid oligomers are summarized in Table 4.1. "rot." and "DBM" stand for

(hindered) rotations of the entire ethylene centers and dimer breathing mode, respectively. Asterisks

correspond to the modes in the Raman spectra of Fig. 4.3. Right: Several characteristic Eigenmodes

(rot. and DBM) of two dimers are illustrated, taken from Ref. [73].

putations and measurements but only in the dimers where the carbon double bond is not

directly attached to the diamondoid cages. It is a hindered rotation of entire ethylene

moieties where the hydrogen atoms stay in the same plane as the carbon atoms. Since

pure rotations are not Raman active, this rotational mode does not have an equivalent in

pristine ethylene. The dimer-breathing modes are oligomer specific modes that can nei-

ther be found in ethylene nor in diamondoids. Their frequencies are around 200cm−1

because of the comparably high masses of entire carbon cages. The ring-like struc-

ture [4.4](1,3)adamantanophan-trans,trans-1,8-diene has two BLMs with frequencies of

138cm−1and 285cm−1of which only the latter was experimentally accessible. The broad

shoulder on the low-energy sides of the C=C peaks (∼1660cm−1in Fig. 4.3) is attributed

to amorphous carbon due to a sample degeneration under UV exposition [97]. An overview

of some characteristic displacement vectors can be seen in Fig. 4.4.

In the computations, we have used experimentally obtained geometrical structures of

the van-der-Waals crystals, as shown in Ref. [91]. It has been found, that depending

on the diamondoid derivatives, the unit cell of the van-der-Waals crystals contain ei-

ther two or four molecules. Therefore, in the computations, we find several similar C=C

stretching/torsional modes and carbon-hydrogen stretching modes that are localized in

different molecules within the unit cell of the van-der-Waals crystals. In case of the ring-

like [4.4](1,3)adamantanophan-trans,trans-1,8-diene the modes are further split into in-

tramolecular in-phase and out-of-phase vibrations. However, the separation is ∼1.5cm−1

4.2. Double-bond diamondoid oligomers - electronic properties

and should therefore not be resolvable in a standard experiment. Table 4.1, summarizes

several characteristic sp2/sp3diamondoid oligomer vibrational modes with their corre-

sponding experimental and calculated Raman frequencies. All vibrations are compared

to those of ethylene; the irreducible representations are given in Fig. 4.4 next to the

patterns.

4.2 Double-bond diamondoid oligomers - electronic

properties

The electronic structures of sp2/sp3diamondoid oligomers are only marginally explored.

By valence photoelectron spectroscopy in the gas phase, the highest occupied electronic

states of various compound have been measured by Tobias Bischoff (birth name Zim-

mermann) et al. [98]. It was found that the ionization potential (IP) is much smaller

compared to those of unfunctionalized diamondoids or single-bond diamondoids. Fur-

ther, it was found that the ionization potential does not depend on the size of the dimers,

but is rather fixed in a small energy region around 7.2eV [98]. This stands in contrast

to the experimentally explored IPs of unfunctionalized diamondoids that are determined

by quantum confinement effects, i.e. the IP redshifts with increasing size of the diamon-

doids [98]. This even counts for the single-bond diamondoid dimers [98].

Optical transition energies of a trishomocubane dimer and a diamantane dimer in the

solid phase have been explored by Reinhard Meinke et al. [72, 92]. Via the analysis of

selectively enhanced Raman intensities of C=C stretch vibrations, it was found that the

optical transition energies of both systems are ∼1.5eV downshifted, compared to their

unfunctionalized counterparts. These findings agree well with theoretical predictions by

Shiladitya Banerjee et al. [65,71], who computed optical absorption energies and resonance

Raman spectra within a time-dependent correlation function approach. They find that

the optical transitions in chemically blended diamondoid oligomers are downshifted by

∼2eV compared to their unfunctionalized counterparts [65,71]. The optical responses are

generally determined by the π→π∗transitions with oscillator strengths of f∼0.8and

higher [65,71]. In all computed oligomers, the HOMOs are the πorbitals, but in none of

them the first excited state has a π∗character. They are always σ-like excited states with a

diffuse localization towards the outer periphery of the molecules [65,71]. The consequential

low overlap of HOMOs and LUMOs results in very low oscillator strengths f∼0.004 of the

direct HOMO→LUMO transitions (compare Fig. 4.5) [65,71]. In the computations, the π∗

orbitals are higher excited electronic states, namely the LUMO+2 in the anti-diamantane

dimer (anti-diamantylidenediamantane), the LUMO+1 in the syn-diamantane dimer (syn-

diamantylidenediamantane), or the LUMO+2 in the trishomocubane dimer [65, 81, 92].

These are clear predictions from theory and until now they have not been experimentally

confirmed.

Various molecular orbitals of three different diamondoid oligomers are plotted in Figs. 4.5

and 4.6. The computations are done within the General-gradient approximation (GGA)

using the PBE (Perdew-Burke-Ernzerhof) functional [99] implemented in Material studios

6.0. The π-like character of the highest occupied molecular orbitals can be clearly seen for

4.2. Double-bond diamondoid oligomers - electronic properties

a) π−Orbital

HOMO

b) π∗−Orbital

LUMO+2

HOMO-2

LUMO

Figure 4.5: Equi-energy surfaces of the electron density from molecular orbitals are plotted. The dia-

mondoid derivative is the 3,10-bis-(2-adamantylidene)diamantane trimer. Orbitals are named according

to their energetic levels. πand π∗orbitals are labeled. Lobes corresponding to negative values of the elec-

tron wavefunctions are indicated in red, green lobes correspond to electron wavefunctions with positive

values. DFT computations are done within a GGA approach using the PBE functional [99], implemented

in Material studios 6.0.

all computed oligomers. Instead, the LUMO for the 3,10-bis-(2-adamantylidene)diamantane

is rather diffuse and located at the C-H bonds of the diamantane centered in the middle of

the molecule. Within the computational approach, the π∗orbitals are higher molecular or-

bitals, namely the LUMO+2 for 3,10-bis-(2-adamantylidene)diamantane, the LUMO+1

for 4.4](1,3)adamantanophan-trans,trans-1,8-diene, and the LUMO+2 for the 3-(bis-4-

diamantyl)prop-1-ene.

The excited electronic states S1,2,...,n of diamondoid oligomers generally exhibit other

geometries than those of the electronic ground states S0because of different potential

energy surfaces of each individual state [65,71]. This behavior can be generally observed

in molecules and was first extensively discussed in the publication "Electronic Spectra

and Electronic Structure of Polyatomic Molecules" by Gerhard Herzberg [100]. The ge-

ometries of electronic states in the smallest molecule containing a double bond, namely

ethylene, was analyzed to a large extent [101–103]. For instance, it was found that the

electronic ground state exhibits D2hsymmetry whereas the geometry in the first excited

state changes to a D2dsymmetry, accompanied by a torsion of neighboring hydrogen

atoms and an elongation of the C=C double bond [101–103]. To a certain extent, the

analyzed diamondoid oligomers can be understood as a combination of two diamondoids

and ethylene forming the centers in all molecules. We therefore generally expect that elec-

tronic and vibrational properties of the ethylene moieties are transferred to the complex

diamondoid molecules. And in fact, a comparable elongation of the C=C double bond

in the excited π∗state was found in the diamondoid oligomers both experimentally and

theoretically [71,81]. The increase of bond length was found to exhibit ∼0.1˚

A in good

4.2. Double-bond diamondoid oligomers - electronic properties

a) π−Orbital

HOMO

b) π∗−Orbital

LUMO+1

c) π−Orbital

HOMO

d) π∗−Orbital

LUMO+2

Figure 4.6: Equi-energy surfaces of electron density from πand π∗orbitals are plotted. They belong

to the following dimers: [4.4](1,3)adamantanophan-trans,trans-1,8-diene dimer [a) and b)] and 1,3-(bis-

4-diamantyl)prop-1-ene [c) and d)] dimer. Orbitals are named according to their energetic level. Lobes

corresponding to negative values of the electron wavefunctions are indicated in red, green lobes correspond

to electron wavefunctions with positive values. DFT computations are done within the GGA using the

PBE functional [99] implemented in Material studios 6.0.

agreement to the computations [71,81]. The main reason is the displaced electron den-

sity from the center to the periphery of the double bond in the corresponding excited state.

The geometry change of molecular orbitals in the excited state can be used experimen-

tally to determine the π→π∗transition. Analyzing the Raman intensity of a certain

vibrational mode in which the equilibrium conformation of the molecule is distorted along

its normal coordinate, gives insights to the geometry of the molecule [97,104]. The Ra-

man intensity of a vibrational mode is resonantly enhanced when its change in geometry

resembles the change in geometry of electronic states going from a ground to an excited

state [65, 71, 97, 104]. For instance, ethylene has a twisted conformation in the excited

state and therefore the C=C stretch vibration and a torsional oscillation are resonantly

enhanced [104,105]. As a consequence, a careful analysis of the Raman intensity of the

C=C stretch vibration in diamondoid oligomers allows for an experimental access to the

energy of the π→π∗transition. We have therefore performed resonant Raman measure-

ments in which we have varied the laser excitation energy and analyze the intensity ratios

of the C=C stretch vibrations to the breathing-like modes for each excitation energy.

In the unfunctionalized diamondoids, the breathing-like modes have a fully symmetric

character that is partly conserved in the diamondoid oligomers. Such a normalization

reduces the systematic error occurring from the orientation of the molecules. Further, the

breathing-like modes are not resonantly enhanced when resonantly probing the π→π∗

transition [65,71] and are therefore a suitable candidate for the normalization. This rel-

ative normalization has two advantages: It cancels out the different spectral sensitivities

4.2. Double-bond diamondoid oligomers - electronic properties

εL= 5.32 eV

εL= 2.33 eV

εL= 1.49 eV

C=C stretch

vibration

200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800

Raman shift cm−1

Intensity (arb.units)

Figure 4.7: Raman spectra of 1,3-(bis-4-diamantyl)prop-1-ene for three different laser excitation energies

are plotted. The characteristic C=C stretch vibration is indicated.

of the experimental setup and the systematic error due to sample degradation. All mea-

sured samples are not stable under UV irradiation causing a huge intensity drop of the

Raman signal after exposure, making an absolute normalization via CaF2or diamond not

feasible. However, this relative normalization contains the assumption that all Raman

modes suffer from an equal intensity loss under UV irradiation. Further, we believe that

the degradation does not influence our results as they were reproducible for all measured

samples.

Fig. 4.7 shows Raman spectra of the 1,3-(bis-4-diamantyl)prop-1-ene dimer for three

different excitation energies. A clear resonant enhancement of the characteristic C=C

stretch vibration can be observed when tuning the laser excitation energy from the near

infrared to the ultraviolet spectral range. In contrast, for lower excitation energies, all

Raman active vibrational modes below the C=C stretch vibration, i.e. below ∼1660cm−1

gain importance. It is therefore recommendable to choose very low laser excitation en-

ergies when only the characteristic Raman active vibrational modes of the sp3moieties

within the blended diamondoid oligomers are of interest. This behavior corresponds well

with computed Raman spectra of various sp2/sp3diamondoid oligomer from Banerjee et

al. [65,71].

In Fig. 4.8 resonant (εL= 4.82 eV) and non-resonant (εL= 2.33 eV) Raman spectra

of all measured sp2/sp3diamondoid oligomers are plotted. Simplified structures of the

oligomers are given as insets. For all samples, a clear enhancement of the intensities from

C=C stretch vibrations can be observed when resonantly excited with εL= 4.82eV. Fur-

ther, two other vibrational modes are resonantly enhanced: the C=C bend/twist modes

and the dimer breathing modes DBM, as indicated in Fig. 4.8 d). In both cases, the

vibrational patterns are accompanied by an elongation of the C=C double bond and are

thus resonantly enhanced. However, the enhancement factor of these modes is significant

but lower compared to that of the C=C stretch vibration since their eigenvectors only

have a projection in the direction of the C=C double bond. Other modes, such as CH2

twist and wag vibrations (∼1100 −1350cm−1) are also resonantly enhanced. In the

4.2. Double-bond diamondoid oligomers - electronic properties

BLM

Intensity (arb.units)

εL= 4.82 eV

εL= 2.33 eV

×0.15

BLM

×0.33

BLMDBM

bend/twist

200 400 600 800 1000 1200 1400

×0.15

1600

BLM

200 400 600 800 1000 1200 1400

Raman shift cm−1

×0.2

1600

BLM

200 400 600 800 1000 1200 1400

Raman shift cm−1

Intensity (arb.units)

×0.1

1600

Figure 4.8: Raman spectra of 1,3-(bis-4-diamantyl)prop-1-ene [a)], [4.4](1,3)adamantanophan-

trans,trans-1,8-diene [b)], 3-(2-adamantylidene)diamantane [c)], 3,10-bis-(2-adamantylidene)diamantane

[d)], and syn-diamantylidenediamantane [e)] for two different laser excitation energies as given in the

figure are plotted. BLM stands for breathing-like mode. The intensities of the C=C stretch mode for

a laser excitation energy of εL= 4.82 eV are scaled for clarity. Simplified chemical structures of the

diamondoid derivatives do not contain hydrogen atoms.

4.2. Double-bond diamondoid oligomers - electronic properties

Intensity (arb.units)

1.5 2 4.5 5 5.5 1.5 2 4.5 5 5.5

1.49 eV

2.33 eV

4.7 eV

4.82 eV

5.00 eV

5.08 eV

5.21 eV

5.25 eV

5.32 eV

5.42 eV

5.46 eV

Raman shift (arb.units)

Intensity (arb.units)

1.5 2 4.5 5 5.5

Excitation energy (eV)

Figure 4.9: Top: Normalized Raman intensities of the C =C stretch vibrations of the 1,3-(bis-4-

diamantyl)prop-1-ene are plotted. For a better overview, Raman peaks are shifted by 30 cm−1, re-

spectively. Middle and bottom: Normalized Raman intensities of the C =C stretch vibrations for all

measured diamondoid oligomers. The simplified structures are given in the figures.

4.2. Double-bond diamondoid oligomers - electronic properties

computations, we find several localized modes with a strong deflection of carbon atoms

forming the C=C double bond. These vibrations cause a displacement of the double bond

carbon atoms along its direction and thus these modes are also resonantly enhanced for

suitable excitation energies.

The intensity progression of the C=C stretch vibrations for various excitation energies

of the measured diamondoid oligomers can be seen in Fig. 4.9. The upper part shows

normalized and horizontally shifted Raman spectra of the C=C stretch mode for the 1,3-

(bis-4-diamantyl)prop-1-ene for various laser excitation energies. The lower parts only

show the normalized intensities of the C=C stretch mode of the other measured systems.

For all oligomers, an intensity increase can be observed from ∼4.7eV that directly corre-

sponds to the π→π∗transition. This value is in good agreement to Refs. [65,71,72,81]

where resonance Raman spectra for various diamondoid dimers and trimers are computed

and experimentally determined. By these measurements, we can determine the adiabatic

E0−0transition, i.e. the energy difference between the πand π∗states [65,71] to exhibit

4.7eV. Excitation energies above the E0−0transition lead to a strong increase of absorption

and therefore to an increase of the Raman intensity of the C=C stretch vibration [65,71].

The π→π∗transition energy does not depend on the size or on the structure of the

diamondoid oligomer (compare Fig. 4.9). With these results, we show that the πorbitals

(HOMO) and π∗orbitals in chemically blended sp2/sp3diamondoid oligomers are highly

localized at the C=C double bond and that the influence of the surrounding constituents

to the π→π∗transition energy is only marginal due to screening effects. These findings

agree well with those from Ref. [98], where the measured ionization potentials of sp2/sp3

diamondoid oligomers are not affected by the size or shape of the molecules.

The high localization of both the πand π∗orbitals can exemplary be seen in Figs. 4.5 and

4.6 for 3,10-bis-(2-adamantylidene)diamantane and [4.4](1,3)adamantanophan-trans,trans-

1,8-diene. Equi-energy plots of electron densities from other sp2/sp3diamondoid oligomers

can be found in Refs. [22,72,98].

In comparison to unfunctionalized diamondoids, optical transition energies of the chem-

ically blended oligomers are donwshifted by at least 1.5eV. The nature of the optical

transitions are single-orbital π→π∗transitions with very high oscillator strengths up to

fα∼0.83 [65,71]. These values are approximately one magnitude higher than those in

adamantane or diamantane [65,71].

In summary, we can conclude that the nature of the optical responses of chemically blended

oligomers are fundamentally different compared to unfunctionalized diamondoids. We find

that the optical absorption is downshifted by ∼1.5eV and originates from π→π∗rather

than from σ→σ∗transitions.

In the beginning, we introduced the Hückel method that is capable of giving an esti-

mation of the energies of the lowest optical transitions in sp2carbon systems, namely:

∆E∼4βsin π

2(n+ 1)!(4.1)

When we want to apply this expression to the sp2/sp3diamondoid oligomers, we need to

insert n= 1, as the longest sp2chain only consists of one double bond. Further inserting

4.3. Single-bond diamondoid dimers - vibrational properties

β= 2.5eV [36], we find a transition energy of ∆E∼5eV. This value is already very close

to our experimentally obtained value of 4.7eV for all diamondoid oligomers and again

shows the power of the Hückel method.

4.3 Single-bond diamondoid dimers - vibrational prop-

erties

Another approach to form diamondoid dimers was recently shown by the Peter Schreiner

group in Gießen [23,106]. In contrast to the oligomers as shown before, the diamondoid

dimers, have a single-bond between their single constituents [23,106]. A highly interest-

ing and somewhat surprising aspect of the single-bond dimers is the exceptionally long

carbon-carbon distance between their monomers that exhibits up to 1.704˚

A in case of a

diamantane-triamantane dimer [23,106]. It is the so far longest reported bond length in

alkanes [23]. The average value of unstrained carbon-carbon bonds instead is found to ex-

hibit 1.54˚

A [107]. It is commonly assumed that larger bond lengths are accompanied by a

considerable bond weakening making them unfeasible to appear [108,109]. However, due

to steric repulsions and ring strain, longer carbon-carbon bonds can be observed in various

large molecules that often contain highly electronegative atoms [23,107,108,110,111].

The stability of the diamondoid dimers, as proposed in Ref. [23, 106], however, arises

from attractive dispersion interactions. The strong carbon-carbon bond elongations are

achieved by a shifting of the energy balance in favor of attractive dispersion interactions

exceeding the repulsive dispersive contributions [23]. The attractive interaction mainly

occurs between the intramolecular HC···CH surfaces in the proximity of the central

carbon-carbon bond [106]. An increase of interaction is observed, when the area of op-

posite surfaces is enlarged, accompanied by an increase of the observed bond length, as

shown for DIA-DIA (1.647 ˚

A), ADA-TRIA (1.659 ˚

A), and DIA-TRIA (1.704˚

A) [23]. Al-

though not yet fabricated, even larger bond lengths are predicted for diamondoid dimers

containing larger monomers [23].

The impact of the structural characteristic on electronic or vibrational properties is still

only barely investigated. Measurements of the ionization potential (IP) of various single-

bond diamondoid dimers indicate that quantum confinement effects determine their elec-

tronic structures, i.e. larger IPs are found for smaller dimers and vice versa [32, 98].

DFT computations show that in case of homo dimers, the HOMOs are symmetrically

distributed with no distinction between single CC-bonds [98]. Instead, in case of hetero

dimers, the HOMOs seem to be rather localized in the larger moieties of the dimers, i.e.

their electronic properties rather equal those from the the larger moieties [98]. Their

vibrational properties, however, are till today not yet investigated.

In the following, we will show Raman spectra of various single-bond diamondoid dimers

and compare characteristics in their vibrational properties to those from pristine dia-

mondoids and double-bond diamondoid dimers. We compare our findings with DFT

calculations on a PWC (Perdew-Wang) level.

In Fig. 4.10 we show Raman spectra of various single-bond diamondoid dimers, namely

1-(1-adamantyl)adamantane (C2Vpoint group), 2-(2-adamantyl)adamantane (C2point

4.3. Single-bond diamondoid dimers - vibrational properties

400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000

Intensity (arb.units)

εL= 1.96 eV

∗2

∗1

BLM

400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000

Raman shift cm−1

Figure 4.10: Raman spectra of single-bond diamondoid dimers are plotted for an excitation energy of

εL= 1.96 eV. The simplified chemical structures are given in the spectra, but do not contain hydrogen

atoms. The +sign means that the spectrum is the sum of the respective pristine diamondoids as given in

the figure. BLM stands for breathing-like mode. ∗1and ∗2indicate CH2scissors and CH stretch modes,

respectively.

4.3. Single-bond diamondoid dimers - vibrational properties

group), 1-(1-diamantyl)diamantane (C2Vpoint group), 2-(1-adamantyl)triamantane (CS

point group), 2-(1-diamantyl)triamantane (C1point group), and

2-(1-diamantyl)[121]-tetramantane (C1point group), enumerated from the top to the

bottom [106]. The first three are homo dimers in which the constituents (adamantane,

diamantane) are the same. In contrast, the other two dimers are hetero dimers.

We find many similarities in the Raman spectra of single-bond dimers when we compare

them to pristine diamondoids: The spectra can again be divided into different spectral

regions that can be attributed to characteristic vibrations in pristine diamondoids (com-

pare Fig. 4.2). As also true for the double-bond oligomers, the addition of diamondoid

cages via new carbon bonds leads to a symmetry reduction that lifts the degeneracy of

vibrational modes in the unfunctionalized diamondoids [30,73,74,76,92]. As a result, the

Raman spectra contain more distinguishable modes. This can especially be seen for the

widely delocalized C-C-C bend/wag and C-C stretch modes between ∼300 −800 cm−1

and for the CH2twist/rock modes between ∼900 −1250 cm−1. C-C-C bend/wag modes

can due to generally larger inertiae of the dimers, also be found at smaller frequencies

(also see Fig. 4.11). Instead, the rather localized CH2scissor (∼1450cm −1) and C-H

stretch modes (∼2950cm−1) seem to be widely unaffected by a functionalization (∗1and

∗2in Fig. 4.10).

Characteristic breathing-like modes (BLM) can be observed for all diamondoid dimers. In

terms of intensity, they are usually the most pronounced peaks in the C-C stretch region

(∼600 −900cm−1) and can be therefore well identified [30,73,76]. Within our DFT cal-

culations, we always find two different breathing-like modes in the dimers. They are either

localized in one or the other monomer of the molecules. In the case of homo dimers, the vi-

brations are split into in-phase and out-of-phase vibrations with almost equal frequencies.

Hetero dimers, instead, have localized breathing-like modes with different frequencies,

according to their constituents [30]. However, our calculations indicate that the fully

symmetric character of the breathing-like modes gets lost under the functionalization.

This especially counts for the larger counterparts of the dimers, i.e. triamantane and

[121]tetramantane, where we observe a mixing with other C-C-C bend and wag modes.

In our Raman measurements, we only observe one pronounced breathing-like mode for

each dimer although some contain two different diamondoid species. Instead, in a simple

summation of the respective Raman spectra of pristine constituents, two breathing-like

modes with clearly different frequencies are observable (compare Fig. 4.10). We tenta-

tively attribute the absence of a clear, second BLM in the experimental Raman spectra

to the mode mixing, accompanied by a loss of Raman intensity that is also known from

large unfunctionalized diamondoids [30]. Our calculations indicate that both BLMs in the

dimers are upshifted after a functionalization, but only a few cm−1. The experimentally

obtained upshifts therefore rather fit to the high-frequency BLMs in the dimers, i.e., to

the smaller diamondoids moieties. Both the calculations and experiments indicate that

the BLM of the larger diamondoid moieties are intensity-wise more affected by a func-

tionalization.

Equivalent to the double-bond diamondoid oligomers, we find high-frequency CH stretch-

ing modes that cannot be observed in pristine diamondoids. They are marked with ∗2

4.3. Single-bond diamondoid dimers - vibrational properties

ADA(2-2)ADA

ADA(1-1)ADA

ADA-TRIA

DIAO-DIAO

DIA-DIA

DIA-TRIA

DIA-TETRA

100 120 140 160 180 200 220 240 260 280

Raman shift cm−1

Intensity (arb.units)

ADA(2-2)ADA

ADA(1-1)ADA

ADA-TRIA

DIAO-DIAO

DIA-DIA

DIA-TRIA

DIA-TETRA

1.2

1.3

1.4

1.5

1.6

1.7

Bond length (nm)

Figure 4.11: Low-frequency Raman spectra of single-bond diamondoid dimers are plotted for an exci-

tation energy of εL= 1.96 eV in a). ADA, DIA, TRIA, and TETRA stand for adamantane, diamantane,

triamantane, and [121]tetramantane, respectively. DIAO-DIAO is a 1-(1-diamantyl)diamantane dimer

in which the 5th position [112] carbon atoms are substituted by oxygen, respectively. b) The calculated

bond lengths of the shared carbon bond is plotted for the different diamondoid dimers that are measured.

Calculations are done in Dmol3 [113] on a PWC [114] level, implemented in Material studios 6.0.

in Fig. 4.10 but cannot be observed for the (adamantane) homo dimers. The experimen-

tally obtained frequencies are ∼3050cm−1and thus ∼50cm−1larger than those in the

double-bond dimers. Our calculations show that these localized vibrations only occur

between the diamondoid moieties. Two opposite facets of the diamondoid moieties are

due to a strong van-der-Waals interaction approximately aligned. Therefore, they create

surfaces with pronounced repulsive forces for the hydrogen atoms of opposite diamon-

doid moieties [23,106]. In case of the DIA-DIA and DIAO-DIAO dimers, the opposite

surfaces are less aligned compared to dimers with larger diamondoid moieties. The re-

ciprocal repulsion forces for the "inner" CH stretch vibrations are consequently slightly

less pronounced accompanied by a lower frequency upshift. Both in the experiments and

calculations, the highest CH stretch vibrations in DIA-DIA and DIAO-DIAO dimers are

∼30cm−1smaller than in larger diamondoid dimers. In the calculations, we again find

in-phase and out-of-phase CH stretch vibrations of which the latter have slightly higher

frequencies. Localized CH stretching modes that do not point in the direction of the

neighboring, intramolecular diamondoid moieties are not affected by the additional re-

pulsion forces. As a consequence, they are found at the same frequencies as in pristine

diamondoids, both in the experiments and calculations. The (adamantane) homo dimers

do not have pronounced, opposite facets (compare insets of Fig. 4.10). As a consequence,

we cannot observe high-frequency CH stretching modes experimentally or in the calcula-

tions. In fact, the geometry optimized structure of the 1-(1-adamantyl)adamantane shows

that both monomers are rotated by an angle of 60◦around the shared carbon bond. This

increases the distance of the hydrogen atoms of the neighboring adamantane monomers

and therefore reduces the reciprocal repulsion forces.

These findings stand in clear contrast to those from the double-bond dimers. The stiff

double bond aligns the diamondoid monomers in a way that the analyzed oligomers do not

form pronounced, opposite facets. The observed increase of individual CH stretch vibra-

tions is rather explained by a local stiffening [73]. This effect is smaller than the influence

of the additional repulsion forces in the single-bond dimers. Therefore the high-frequency

4.3. Single-bond diamondoid dimers - vibrational properties

Table 4.2: Overview of experimental and calculated frequencies of characteristic Raman peaks from

the single-bond diamondoid dimers. ADA, DIA, TRIA, and TETRA stand for adamantane, diaman-

tane, triamantane, and [121]tetramantane, respectively. DBM and BLM refer to dimer-breathing and

breathing-like mode. We have used a LDA approach and the PWC [114] functional within the Dmol3

package implemented in Materials Studios 6.0. Asterisks belong to the Raman modes as marked in

Fig. 4.10. All frequencies are given in cm−1.

Sample Mode type exp. calc. Mode type exp. calc.

ADA(1-1)ADA DBM - 255 BLM 786 772

ADA(2-2)ADA DBM - 195 BLM 775 760

DIA-DIA DBM - 209 BLM 735 722 /757

DIAO-DIAO DBM - 201 BLM 751

DIA-TRIA DBM - 213 BLM 716 697

DIA-TETRA DBM - 190 BLM 712 722

ADA(1-1)ADA ∗2CH stretch - -

ADA(2-2)ADA ∗2CH stretch - -

DIA-DIA ∗2CH stretch 3017 209

DIAO-DIAO ∗2CH stretch 3020 201

DIA-TRIA ∗2CH stretch 3050

DIA-TETRA ∗2CH stretch 3048 190

CH stretch vibrations in the single-bond dimers exhibit 50cm−1higher frequencies than

those in double-bond dimers.

Our optimized geometries do not explicitly consider a dispersion correction that is sup-

posed to be necessary for a correct description of the analyzed alkanes [23, 106]. We

have rather chosen a functional that, without a dispersion correction, reproduces the

very long central carbon-carbon bond considerably well. However, sophisticated DFT

calculations and NMR (nuclear magnetic resonance) measurements show the existence of

various conformational isomers of all analyzed diamondoid moieties [106]. Due to strong

van-der-Waals interactions, the central C-C bond rotation barriers are shifted from ∼7

to ∼33kcalmol−1(∼17.4meV to ∼82.3meV), depending on the dimers [106]. There are

non-degenerate but isoenergetic rotation paths for various conformers found [106]. As a

consequence, multiple conformational states can be populated that are not covered by

our DFT calculations. Outcomes from our calculations can therefore deviate from the

experimental findings.

We have plotted the central carbon-carbon bond length of the analyzed diamondoid

dimers, ordered by their sizes in Fig. 4.11b). In agreement to Refs. [23, 106], the cen-

tral carbon-carbon bond length gets longer, the larger the diamondoids monomers are.

Fig. 4.11 also contains an artificial (homo) diamantane dimer in which the 5th [112] carbon

atoms are substituted by oxygen atoms. The electronegativity causes a localization of the

4.3. Single-bond diamondoid dimers - vibrational properties

HOMO to the functional group [68,69]. This in turn reduces the repulsive interaction be-

tween the diamondoid moieties, explaining the smaller carbon-carbon distance compared

to the same structure without substituted carbon atoms.

In addition to the main carbon-carbon bond, also the low-frequency Raman modes of the

diamondoid dimers are affected by their sizes as shown in Fig. 4.11 a). Following the low-

ermost to the top spectrum, we observe a downshift of Raman modes, i.e., we generally

observe Raman modes with lower frequencies for larger diamondoid dimers. Our DFT cal-

culations indicate that in the analyzed spectral region, we mainly find rocking, stretching,

and combined rocking/stretching vibrations of the whole diamondoid monomers within

the molecule. As a consequence, their frequencies depend on the particular inertiae and

exhibit generally lower frequencies for larger diamondoid monomers involved. This can

exemplary be seen for the DIA-DIA and the DIAO-DIAO dimers of which the latter is

due to substituted oxygen atoms slightly heavier. However, the vibrational characters of

low-frequency modes are not affected by the chemical modification and, thus, they are

slightly shifted to lower frequencies. The same behavior can be observed for d-16 adaman-

tane in which all hydrogen atoms are replaced by deuterium [74].

Assimilable to the double-bond dimers, we find a dimer breathing mode (DBM) in the

calculations for the single-bond diamondoids. However, we cannot assign a certain peak

in the experimental low-frequency Raman spectra to the DBM. Within our DFT cal-

culations, we find many Raman active, low-frequency modes up to ∼300cm−1that

can experimentally not convincingly be distinguished from each other. This is in clear

contrast to double-bond dimers, where a resonant intensity enhancement under UV exci-

tations can be used to identify dimer-breathing modes [71–73]. The calculations further

show that the breathing-like character of the DBM is altered compared to the double-

bond dimers. The very stiff double bond defines a pronounced direction for the DBM

resulting in clear, opposite vibrational pattern of the monomers within the diamondoid

molecules. Instead, the BLM in single-bond dimers contains rotational, rocking, and

stretching vibrations to variable extents. The highest symmetry of the analyzed dimers

has the 1-(1-adamantyl)adamantane dimer (C2Vpoint group) in which the principal axis

is along the central carbon-carbon bond (compare Fig. 4.11). The vertical mirror plane

allows for a straight dimer-breathing mode resembling those from the double-bond dimers.

The effect of repulsive interaction as a function of the distance between diamondoid dimers

and the mode mixing superimpose the influence of the diamondoid masses to the DBM

frequency. Calculated DBM frequencies, as given in Table 4.2, therefore deviate from the

simple f∼m−1proportionality.

Comparison of the vibrational properties of single- and double-bond diamon-

doid oligomers

The single and double-bond oligomers share many characteristics in their vibrational prop-

erties. These are mainly those vibrations that appear in pristine diamondoids, such as

the C-C-C bend/wag/stretch, CH2twist/rock/wag/scissors, and CH stretch vibrations.

All characteristic modes can be found in the functionalized diamondoid derivatives, but

with partially altered frequencies. The more the carbon cage is involved in the vibrational

patterns, the more their frequencies are affected by a functionalization. The arrangement

4.3. Single-bond diamondoid dimers - vibrational properties

in dimers allows for dimer breathing modes in which the monomers oscillate against each

other both in the single- and double bond dimers. However, the stiffer double bond de-

fines the vibrational direction causing slightly higher frequencies and a lower mode mixing

with other carbon cage vibrations for the double-bond oligomers. Further, due to their

fundamentally different electronic structures, the DBM in double-bond oligomers can be

resonantly enhanced with excitations energies above ∼4.7eV making them distinguish-

able from other Raman modes. The ethylene centers in double-bond oligomers exhibit

characteristic and very localized vibrational modes that cannot be observed in the single-

bond dimers.

In both types of the functionalized diamondoids, we find high-frequency CH stretch vi-

brations that are separated by ∼50 cm. The alignment of the monomers within the

diamondoid dimers is determined by the van-der-Waals interaction in case of the single-

bond dimers and by the comparably stiff C=C double bond in the double-bond oligomers.

As a consequence, diamondoid facets are well aligned within the single-bond molecules.

This in turn leads to additional repulsive forces that increase the frequencies of CH stretch

vibrations that point in the direction of opposite diamondoid moieties. This effect ex-

ceeds the slight frequency increase caused by a local stiffening in case of the double-bond

dimers.

4.4. Diamondoid van-der-Waals crystals

4.4 Diamondoid van-der-Waals crystals

Parts of this section are published in Refs. [81,92]

4.4.1 Electronic properties of chemically blended sp2/sp3dia-

mondoid van-der-Waals crystals

"Matter will always display attraction" used to be a basic principle of the 1910 Nobel prize

winner Johannes D. van der Waals [23,115]. The van-der-Waals interaction, that is often

referred to as London dispersion, is an interaction between polarizable particles/matter in

which (induced) electrostatic dipoles cause attractive forces [116]. Usual binding energies

exhibit between 5 and 50meV being responsible for the ability of trees to carry water to

leaves that are more than 10m above the ground or to simply write with a pencil.

It was found that the crystallization of super-saturated solutions of purified diamondoids

in acetone leads to macroscopic, transparent crystallites [19]. They have a well-defined,

long-range order with melting points about 540K for the lowest C10H16 adamantane [67].

This value is considerably (∼200K) higher than those of other hydrocarbons with com-

parable molecular weights due to a strong van-der-Waals interaction of adjacent molecules

within the crystal lattice [67]. In comparison to diamond, the van-der-Waals crystals from

adamantane, diamantane, and triamantane are direct isolators with band gaps between

6 and 7eV, decreasing for an increasing size of the diamondoid constituents [67]. Their

optical properties are therefore difficult to experimentally access by usual laser setups. As

discussed in the previous section, sp2/sp3blended diamondoid oligomers have redshifted

optical transition energies making them suitable candidates for optical experiments on

diamondoid van-der-Waals crystals. By these measurements, we can determine the effect

of van-der-Waals interaction to the optical properties of the analyzed crystals.

The arrangement of diamondoids in periodic van-der-Waals crystals leads to new, col-

lective properties such as the formation of an electronic band structure or a phonon dis-

persion. Depending on the extension of electronic orbitals in the isolated molecules, their

counterparts in the crystalline phase are different due to the overlap of adjacent molecular

orbitals. This in turn changes the energetic levels of conduction and valence bands and

generally reduces the band gap of the crystals compared to the isolated molecules [67,81].

For a closer analysis of the diamondoid van-der-Waals crystals, optical absorption spectra

of isolated molecules in the gas phase have been taken. The measurements have been

conducted at the U125/2-NIM beam line [117] at the BESSY II synchrotron radiation

facility (Helmholtz-Zentrum Berlin, Germany) [118] with the nice help of the AG Möller

from the TU Berlin.

In Fig. 4.12 absorption spectra of a trishomocubane dimer (PCU)2and a diamantane

dimer (DIA)2from molecules in the gas phase are plotted. A significant absorption in-

crease starts from around 5.6eV (plotted with dashed arrows) for both dimers. In good

agreement to the absorption increase, a resonant enhancement of the Raman intensity of

the C=C stretch vibration is also observed from around 5.6eV (plotted with a red arrow

in Fig. 4.12). In contrast, a resonant enhancement of the Raman intensity of the C=C

stretch vibration in the crystalline phase is observed around 4.9eV, i.e. at lower energies.

In both cases, the strong absorption is caused by a π→π∗transition as shown for dia-

4.4. Diamondoid van-der-Waals crystals

Trishomocubane dimer

(PCU2)

4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0

Energy (eV)

Abs./Int. (arb.units)

Raman (gas phase)

Raman (crystalline)

absorption (gas phase)

4.5 5.0 5.5 6.0 6.5

Energy (eV)

Abs./Int. (arb.units)

Raman (crystalline)

absorption (gas phase)

DIA=DIA

(DIA2)

1.4 1.8 2.2

Figure 4.12: Absorption spectra of isolated molecules from a trishomocubane dimer [(PCU2)] and a

diamantane dimer [(DIA2)] in the gas phase are given (continuous lines). Black dots represent normalized

Raman intensities of the C=C stretch vibration in the solid phase. Red dots represent normalized Raman

intensities of the C=C stretch vibration in the gas phase. Data is partly taken from Ref. [92]

mondoid oligomers in the previous section and by DFT computations in Refs. [65,71,92].

Due to the good agreement of Raman intensity and the optical absorption for (PCU)2,

we believe that we observe the same behavior in case of the diamantane dimer as shown

in Fig. 4.12. Again, the resonant enhancement of the Raman intensity of the C=C in the

solid phase begins at around 1 eV lower energies than the absorption in the gas phase.

DFT computations of the analyzed van-der-Waals crystals indicate that optical tran-

sition energies are downshifted compared to the isolated counterparts as can be seen in

Ref. [81]. Since the experimental approach is only sensitive for optical transitions from the

πto the π∗orbitals, experimental statements can only be derived for the corresponding

valence and conduction bands in the crystal. In both crystals, the valence bands originate

from πorbitals that have a lower dispersion compared to those from the σorbitals [81].

Their high localization between the diamondoid moieties leads to a low interaction with

orbitals from adjacent molecules and thus the dispersion is only weak [81]. The σand

σ∗orbitals, instead, are rather diffuse and outspread throughout the whole diamondoid

moieties leading to a large interaction and thus a comparably high dispersion in the equiv-

alent electron/hole bands in the crystals [67,81]. Our DFT calculations indicate that the

π∗orbitals are responsible for the conduction band in the PCU2and for the cb+3 band in

the DIA2van-der-Waals crystal [81]. Their computed downshifts in the crystals exhibit

550meV for PCU2and 400meV for DIA2, slightly lower than in the experiments [81].

The reason of the discrepancy might be the well-known underestimation of optical transi-

tion energies in DFT computations. The computed downshift of the direct valence band

→conduction band transition in the DIA2van-der-Waals crystals is another ∼500 meV

lower as the conduction band originates from σ∗orbitals that are much more affected by

the neighboring molecules [81]. However, since the chosen experimental approach is not

sensitive for these optical transitions, we cannot verify this outcome from the DFT compu-

tations. Calculated band structures and further information can be found in Refs. [81,92].

The formation of van-der-Waals crystals can be used to tailor the electronic properties

4.4. Diamondoid van-der-Waals crystals

of diamondoid moieties. It is an indirect way to downshift optical transition energies

of the primary constituents. In principle, the induced downshift should depend on the

extension of molecular orbitals forming the valence and conduction bands, as shown for

the chemically blended sp2/sp3diamondoid derivatives. By a certain functionalization,

the localization of HOMO and LUMO and their electronic characters can be changed

drastically [68,69,98]. This allows to tailor the overlap of adjacent molecular orbitals and

therefore to tailor the downshift of optical transition energies in a diamondoid van-der-

Waals crystal. Computations of sulfur-functionalized diamondoids indicate, that both

HOMO and LUMO are highly located at the periphery of the carbon cages accompanied

by a strong downshift of optical transition energies to the optical visible region [68]. A

further introduction of sulfur atoms does not reduce the optical band gap, but leads to

a slight upshift due to a reduced dipole-induced charge transfer from the sulfur atoms

to the carbon cages [68]. However, the extension of electron orbitals towards the outer

periphery further increases when more sulfur atoms are added. This might increase the

van-der-Waals interaction between adjacent molecules in a crystal and thus the downshift

of optical transition energies. Therefore, in such a crystal the optical band gap might

decrease although the intrinsic optical transition energies increase when introducing more

than four sulfur atoms to the diamondoid cages [68].

This general approach might be used to create diamondoid crystals that have tunable

bandgaps from the optical visible up to ultraviolet spectral region.

4.4.2 Vibrational properties of diamondoid van-der-Waals crys-

tals

The first Raman studies on diamondoid van-der-Waals crystals were performed 40 years

ago [75, 119]. The already available knowledge of certain crystal structures allowed for

their group theoretical analysis in a way that the Raman spectra from lower diamondoid

crystals were understood even 40 years ago [120–122]. It was found that only diamantane

has Raman active low-frequency modes at room temperatures [120, 122]. Adamantane

and triamantane instead appear in a pre-melting plastic phase [119] with orientationally

disordered molecules. Only for temperatures below 100K (adamantane) or 20 K (tria-

mantane), an ordered phase can be observed via pronounced peaks in the low-frequency

Raman spectrum [119]. Although the crystal structures of higher diamondoids up to a

hexamantane are known in our days, there are no Raman spectra available in the liter-

ature. The same accounts for the chemically blended sp2/sp3diamondoid oligomers as

discussed before.

The low-frequency Raman spectra of diamondoid van-der-Waals crystals are character-

ized by translatory displacements of entire diamondoid molecules or hindered rotational

displacements around their axes of inertiae (librations) [119]. In case of diamantane, a

group theoretical analysis at room temperature yields:

Γopt

trans =Au+Eu+ 2Tu(4.2)

Γopt

lib =Ag+Eg+ 3Tg,(4.3)

of which only the librations (lib) are Raman active, i.e. their consideration is important

for a comprehensive understanding of the low-frequency Raman regime.

4.4. Diamondoid van-der-Waals crystals

Table 4.3: Overview of crystal structures of diamondoid van-der-Waals crystals as given in the top row.

The numbers in brackets refer to the space group IT numbers, respectively. a, b, c, α, β, and γare the

lattice constants with the corresponding lattice angles. All lengths are given in ˚

A. Data is provided by

Franziska Emmerling [123].

[1234]Pentamantane [121]Tetramantane Triamantane

monoclinic monoclinic orthorhombic

space group P 21/c (14) space group P 21/c (14) space group Fddd (70)

a= 7.7075,b= 12.5845 a= 7.6567,b= 7.9611 a= 12.6769,b= 17.9695

c= 17.7808 c= 12.7999 c= 21.9399

α= 90◦,β= 100.142◦α= 90◦,β= 104.881◦α= 90◦,β= 90◦

γ= 90◦γ= 90◦γ= 90◦

units = 4 units = 2 units = 16

In order to understand the vibrational properties of diamondoid van-der-Waals crystals to

a greater extent, we have started a cooperation with Franziska Emmerling [123] from the

BAM (Bundesanstalt für Materialforschung). With a X-ray powder diffraction analysis,

we could identify some crystallographic structures that are not available in the database

of the Cambridge Crystallographic Data Centre (CCDC) or could be identified with a

higher accuracy. They are listed in Table 4.3. Crystal structures of other chemically

blended sp2/sp3diamondoid crystals can be found in Ref. [91]. The structures are used

for DFT calculations of both their vibrational and electronic structures and are partly

still in progress. One of the purposes is the understanding of the low-frequency Raman

spectra, as shown in Fig. 4.13.

In agreement to group theory based conclusions, we do not find any low-frequency Ra-

man modes for adamantane or triamantane van-der-Waals crystals [119]. Instead, up to

[12314]hexamantane, all other unfunctionalized diamondoid van-der-Waals crystals ex-

hibit pronounced Raman modes in the low-frequency region. The same accounts for the

chemically blended sp2/sp3diamondoid oligomers. DFT calculations indicate that in-

tramolecular (internal) vibrations in diamondoids occur at ∼200cm−1, depending on

their sizes [30,73,119]. The experimentally observed low-frequency Raman modes can be

therefore solely attributed to intermolecular translational displacements or to librations.

Interesting aspects of the van-der-Waals crystals that can be answered with the help of

DFT calculations are the influences of size and shape of the diamondoid monomers to the

low-frequency vibrations. Although the inertiae of the monomers noticeably increase go-

ing from adamantane up to [12314]hexamantane, we do not find a pronounced downshift

of low-frequency Raman modes. A simple explanation for the observed trends might be

that an increase of diamondoid intertiae is accompanied by an increase of the diamon-

doid surfaces and therefore their interaction surfaces. The larger inertness might cancel

a frequency upshift due to an enlarged interactions resulting in only marginal frequency

shifts of the low-frequency Raman modes.

4.4. Diamondoid van-der-Waals crystals

ADA

DIA

TRIA

[121]TETRA

[123]TETRA

[1(2)3]TETRA

εL= 1.96 eV

20 40 60 80 100 120 140 160

Raman shift cm−1

Intensity (arb.units)

[1234] [1213]

[1212]

[12(3)4]

[12(1)3]

[1(2,3)4]

[12314]HEXA

(PCU)2

(DIA)2(Z)

(DIA)2(E)

DIA-C=C-DIA

ADA=DIA=ADA

ADA-2×(C-C=C-C)-ADA

20 40 60 80 100 120 140 160

Raman shift cm−1

Figure 4.13: Low-frequency Raman spectra

of pristine diamondoids (a)-(b) and sp2/sp3di-

amondoid oligomers (c) for an excitation en-

ergy of εL= 1.96 eV are plotted. ADA, DIA,

TRIA, TETRA, and HEXA stand for adaman-

tane, diamantane, triamantane, tetramantane,

and hexamantane, following the nomenclature

given by Balaban and Schleyer [63]. In (b), all

but the top spectra belong to pentamantanes.

PCU stands for polycyclic undecane (compare

previous section and Ref. [92]). (Z) and (E) de-

note the anti−, and syn−Isomers of diaman-

tane.

In contrast, the diamondoid oligomers have intramolecular (internal) low-frequency Ra-

man modes, such as the dimer-breathing modes [72,73]. C-C-C vibrations are due to the

attached masses generally shifted towards lower frequencies, explaining the rich Raman

features in Fig. 4.13 c). However, even the functionalized diamondoids have pronounced

crystal structures and therefore their low-frequency Raman spectra consist of both inter-

molecular and intramolecular vibrations.

Outlook

With the help of the DFT computations, we want to understand the low-frequency vi-

brational properties of the diamondoid van-der-Waals crystals. Furthermore, we want

to understand their electronic structures. It was shown in Ref. [67] that the conduction

bands in (unfunctionalized) diamondoid crystals originate from very diffuse C-Hσ∗or-

bitals. The conduction band minima and valence band maxima occur at the Γpoint (up

to triamantane), i.e. the diamondoid crystals are direct insulators [67]. Bulk diamond

instead is an indirect insulator, in which the conduction band originates from C-Cσ∗

orbitals. Further, it was shown that the bandgap in diamondoid crystals decreases with

4.4. Diamondoid van-der-Waals crystals

increasing size of the diamondoid moieties [67]. Larger diamondoids than triamantane,

however, have not yet been analyzed. Interesting questions are from our perspective if

a diamondoid crystal exists that has a smaller bandgap compared to bulk diamond and

how different isomers of tetra- and pentamantanes determine the electronic structure of

the crystals.

4.5. [2](1,3)Adamantano[2](2,7)pyrenophane - vibrational properties

Figure 4.14: The ground

state molecular struc-

ture of [2](1,3)adaman-

tano[2](2,7)pyrenophane is shown.

The remaining bonds are saturated

with hydrogen, but are not shown

for reasons of visibility.

4.5 Pyrene and diamondoids: Electronic and vibra-

tional properties of

[2](1,3)Adamantano[2](2,7)pyrenophane

In the last part of this chapter, we want to discuss a recently fabricated hydrocarbon

that to some extent brings both diamondoids and graphene together. It is an aggregated

sp2/sp3hybrid structure of sp2-hybridized pyrene and sp3-hybridized adamantane con-

nected by small carbon chains, namely [2](1,3)Adamantano[2](2,7)pyrenophane [124], see

Fig. 4.14. The hybrid structure unifies the electron-acceptor and electron-donor abilities

of bended sp2and diamondoid moieties, respectively [21,124,125]. When arranged in the

ways as suggested in Ref. [124, 126], the actual non-polar carbon allotropes develop a

considerable dipole moment pointing to the sp2moieties. This trait is for instance desir-

able for a controlled self-assembly on suitable surfaces [127]. Widely covered surfaces by

adamantane containing compounds could then be used for high-brilliance electron beam

sources [21,128]. The electronic properties of diamondoids within the compounds, how-

ever, should not be altered too much to maintain their desired properties. It is therefore

necessary to analyze their electronic properties with respect to similarities to pristine di-

amondoids. In the previous sections we could see that the chemically blended sp2/sp3

diamondoid oligomers exhibit both electronic and vibrational properties of the ethylene

moieties and the diamondoids attached to each other. The same behavior appears in case

of [2](1,3)Adamantano[2](2,7)pyrenophane, as we will see on the next pages.

4.5.1 [2](1,3)Adamantano[2](2,7)pyrenophane - vibrational prop-

erties

The ground state molecular structure of [2](1,3)Adamantano[2](2,7)pyrenophane exhibits

a CSsymmetry being significantly reduced in comparison to adamantane (TD) and pyrene

(D2h) [124,129]. For that reason, we generally expect more complex Raman spectra in

terms of the amount of allowed Raman active modes. The bending of the pyrene moiety

in the ground state structure as shown in Fig. 4.14 further suggests that especially the

in-plane C=C stretch vibrations should be altered compared to pristine pyrene. C-H vi-

brations at the edges should however be way less affected by the bending.

In the following, we compare the experimental Raman spectra to DFT calculations on a

LDA level in DMol3 with the PWC functional [113,114]. General assignments are done

4.5. [2](1,3)Adamantano[2](2,7)pyrenophane - vibrational properties

ADA+Pyrene

ADA

Pyrene

200 400 600 800 1000 1200 1400 1600

Raman shift cm−1

Intensity (arb.units)

2800 3000

[2](1,3)adamantano[2](2,7)pyrenophane

∗1

∗2∗3

∗4

∗5

Figure 4.15: Raman spectra of [2](1,3)Adamantano[2](2,7)pyrenophane, adamantane (ADA), pyrene,

and ADA + pyrene are plotted for an excitation energy of εL= 1.96 eV.

via a careful analysis of each respective vibrational eigenmode.

Similar to the Raman spectra of the previously analyzed diamondoid oligomers, the Ra-

man spectrum of [2](1,3)Adamantano[2](2,7)pyrenophane can again be decomposed in

characteristic C-C cage, C=C, and C-H vibrations. Frequency-wise, characteristic C-H

vibrations such as the C-H stretch (∼3000cm−1) and CH2scissor (∼1450 cm−1) modes

are widely unaffected in the artificial compound. In particular, this can be seen in the

bottom spectra of Fig. 4.15, where the spectra of pristine adamantane and pyrene are

added. Due to the stiff carbon double bonds, pyrene generally has high-frequency C-H

stretching modes that are maintained in the [2](1,3)Adamantano[2](2,7)pyrenophane com-

pound [130]. The same counts especially for the lowest C-H stretch vibration, denoted

with ∗5in Fig. 4.15. In adamantane this mode is a fully symmetric breathing mode of

hydrogen atoms that widely maintains its character in the measured compound [74]. C-C

cage vibrations can again be found between ∼400 and 900 cm−1. The carbon chains

between adamantane and pyrene can vibrationally-wise be seen as an enlargement of the

sp3carbon cage of adamantane. For this reason, primary vibrations of the carbon cage

are not substantially altered but extended by new modes in the frequency range C-C cage

modes are usually found [30, 73, 74]. The well-known breathing-like mode BLM widely

maintains its character and its frequency is slightly upshifted in the measured compound

(∗1in Fig. 4.15). Four C=C stretch vibrations are found at around 1600 cm−1both in

the experiments and calculations, denoted with ∗4in Fig. 4.15. They can directly be

referred to one Aand three AgC=C stretch modes in pyrene (D2hpoint group) [130].

Their frequencies, however, are downshifted due to the bond bending. We find seven

4.5. [2](1,3)Adamantano[2](2,7)pyrenophane - electronic properties

State ∆Evert f

1 3.4623 0.0034

2 3.6718 0.0503

3 4.2282 0.1860

4 4.2665 0.0010

Table 4.4: Vertical transition en-

ergies ∆Evert in eV and oscillator

strengths ffor [2](1,3)Adaman-

tano[2](2,7)pyrenophane are

shown for the first four excited

states. Calculations are done on a

B3LYP/TVZ level of theory. Data

is taken from Refs. [129,131].

other pronounced C=C modes in which the carbon atoms are deflected towards oppo-

site carbon atoms and not along their shared C=C bonds as in the case of C=C stretch

modes. Calculated frequencies range from 1340cm−1up to 1490 cm−1. In the experimen-

tal spectra, these modes are found at slightly lower frequencies. The highest and lowest

frequencies are denoted with ∗3and ∗4and Fig. 4.15. This is a clear difference to the

diamondoid oligomers as previously discussed, since all vibrational modes between ∼900

and 1450cm−1could only be referred to C-H vibrations.

Another deviation to the vibrational modes of pristine diamondoids can be found in

the range below 500cm−1. Between ∼250 and 500cm−1, we find collective vibrational

modes in which the entire carbon rings are involved. Further, below ∼250cm−1many

rotational modes of the entire adamantane moieties can be found, accompanied by a

considerable stretching of the carbon chains between the pyrene and adamantane. Es-

pecially in the low-frequency region, we generally find an extensive mode mixing of the

pyrene moiety with C-C cage modes of adamantane. This, however, does not count for

the high-frequency modes above ∼1400cm−1, where the vibrational modes are rather

distinguishable.

In the previous section, we showed that the vibrational modes of double-bond diamondoid

oligomers can to some extent be understood as a linear combination of the vibrational

modes of the ethylene moieties and diamondoids. The same occurs in [2](1,3)Adaman-

tano[2](2,7)pyrenophane. Characteristic modes of both pyrene and adamantane are widely

maintained and are only slightly altered in their frequencies. This can, in particular, well

be seen when we compare the Raman spectrum of [2](1,3)Adamantano[2](2,7)pyrenophane

to the sum of the Raman spectra of pyrene and adamantane in the bottom spectra of

Fig. 4.15. However, the structure-induced change of vibrational modes leads especially

in the low-frequency spectrum to a considerable mode mixing and allows for adaman-

tane rotational modes that are only to a much lesser extent present in the double-bond

diamondoid oligomers [73].

4.5.2 [2](1,3)Adamantano[2](2,7)pyrenophane - electronic prop-

erties

A very large dipole moment was theoretically found for [2](1,3)Adamantano[2](2,7)

pyrenophane in the ground state exhibiting 2.3 D [124]. It is an interesting question if

that large dipole moment is conserved in excited electronic states of the compound or

4.5. [2](1,3)Adamantano[2](2,7)pyrenophane - electronic properties

λ= 264 nm

λ= 325 nm

∆E= 176 meV

250 300 350 400 450 500 550

Wavelength (nm)

Intensity (arb.units)

Figure 4.16: The PL (photo luminescence) spectrum of [2](1,3)Adamantano[2](2,7)pyrenophane is plot-

ted for two excitation wavelengths as denoted in the figure. Arrows indicate the energy difference between

maxima in the vibrational progression of the PL spectrum.

if the direction of the dipole moment can be reversed in higher electronic states. This

could for instance offer new approaches for the self-assembly of diamondoids on various

surfaces [127]. The reversal of dipole moments might be understood from the fact that

the lower-lying excited electronic states are π∗states of the pyrene moiety, whereas very

high excited states are σ∗states localized in the adamantane moieties, accompanied by a

considerable shift of charge concentration.

Xiong et al. have calculated ground and excited states geometries with individual vertical

transition energies within DFT on a B3LYP/TVZ level of theory [129]. Vertical transi-

tion energies and oscillator strengths for four excited states are listed in Table 4.4. Due

to the significantly lower symmetry of the compound, no optically forbidden transitions

were found [129]. The first allowed optical transition is considerably lower compared to

adamantane [32] and those from double-bond diamondoid oligomers [71–73]. Interest-

ingly, the first three excited states are accompanied by an increase of dipole moment of

up to 18.22% and a slight change in the direction of the internal field [129].

In contrast to lower diamondoids, the low-lying electronic states of the [2](1,3)Adaman-

tano[2](2,7)pyrenophane compound allow for the experimental investigation with com-

mercially available light sources [32]. All following experiments are performed in the solid

phase, i.e., the compound is measured in a van-der-Waals crystals. We therefore expect

slightly lower transition energies compared to the isolated molecules [67,81].

For excitation energies of εL= 4.69 eV (264 nm) and εL= 3.81 eV (325nm), we have

found a strong luminescence beginning from ∼420nm (2.95 eV) followed by a vibrational

progression, as shown in Fig. 4.16. Following Kasha’s rule [132], we attribute the mea-

sured onset of the luminescence to the E0−0transition, i.e. the optical transition from

the vibrational ground state in the first excited state S1to the vibrational ground state

in the electronic ground state S0[65,71]. The following maxima are referred to optical

transitions from the vibrational ground state in the first electronic excited state to higher

vibrational states in the electronic ground state (compare Fig. 3.1 in Chapter 3). They

are separated by ∼176 meV proportional to the dominant vibrational mode that alters

the structure of the electronic ground state. In the calculations, the E0−0transition is

found at around 3.23eV [129], slightly larger than in the experiments. The applied com-

putational approach tends to overestimate transition energies as a comparable difference

4.5. [2](1,3)Adamantano[2](2,7)pyrenophane - electronic properties

PL PLE a)

2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4

Energy (eV)

Intensity (arb.units)

∆E= 151 meV

PLE

”inverted” PL

3.2 3.4 3.6 3.8

Energy (eV)

Intensity (arb.units)

200 400 600 800 1000 1200 1400 1600 1800

Raman shift cm−1

Intensity (arb.units)

εL= 3.81 eV

εL= 1.96 eV

Figure 4.17: Photo luminescence a), photo luminescence excitation b), and Raman spectra c) of

[2](1,3)Adamantano[2](2,7)pyrenophane are plotted. b) contains an inverted photo luminescence spec-

trum for a better comparison to the photo luminescence excitation spectrum.

of ∼0.3eV between calculations and measurements was also found for the double-bond

diamondoid oligomers [71,73].

To investigate the first excited state of the artificial pyrene + adamantane compound,

we have performed photoluminescence excitation (PLE) measurements in the solid phase.

This spectroscopic method is widely utilized in solid state physics but is rather unusual

in molecular physics. It gives insights to the optical absorption of materials as the in-

tensity of the luminescence directly corresponds to the strength of optical absorption.

For molecules that are rather present in a gas phase, other spectroscopic methods, such

as UV/Vis spectroscopy are more suitable giving a direct access to the optical absorp-

tion. However, since the [2](1,3)Adamantano[2](2,7)pyrenophane at room temperatures

is present as small crystallites, PLE measurements can likely be applied.

A PLE spectrum vs. a PL spectrum of [2](1,3)Adamantano[2](2,7)pyrenophane is plot-

ted in Fig. 4.17 a). We find a considerable absorption starting from ∼3.1eV up to

3.8eV. The PLE spectrum also features a vibrational progression with maxima indicated

by small lines in Fig. 4.17. Their energy spacing ∆E exhibits ∼151meV, lower than

in the vibrational progression of the PL spectrum. In more detail, this can be seen in

Fig. 4.17 b), where the inverted PL spectrum is plotted against the PLE spectrum. The

smaller energies of vibrational states can be understood from the fact that in absorp-

tion measurements, we observe vibrational states in the first excited electronic state S1.

Higher electronic states of molecules are accompanied by a considerable change of ge-

ometry [101–103] that usually leads to an altering of vibrational frequencies. In the first

electronic state, the C=C bond lengths in the pyrene moiety are extended reducing restor-

4.6. Summary

ing forces and thus reducing the frequencies of relevant vibrational modes [129]. UV/Vis

spectra of [2](1,3)Adamantano[2](2,7)pyrenophane in various solvents are available that

show the same absorption onset but with a reduced spacing of vibrational states [124].

The fluorescence does not seem to depend on the solvent and is similar to that from

[2](1,3)Adamantano[2](2,7)pyrenophane measured in the solid phase [124].

The measured absorption of [2](1,3)Adamantano[2](2,7)pyrenophane in cyclohexane ex-

hibits a large increase from ∼3.5eV up to higher energies, that we cannot observe in the

PLE measurements. The increase of absorption correlates with an efficient absorption

in the second excited electronic state S2[124,129] that we can not probe with the PLE

measurements. Internal conversion processes [28] allow for non-radiative relaxation chan-

nels when excited in higher electronic states that do not contribute to the S1→S0optical

transition that we cover by the PLE measurements. However, when excited to the first ex-

cited electronic state, all absorbed photons should contribute equally to the fluorescence,

i.e, PLE spectroscopy can be seen as a measure of an optical absorption into the first

excited electronic state [132]. It is interesting to note that both the absorption and fluo-

rescence spectra of [2](1,3)Adamantano[2](2,7)pyrenophane are very similar to those from

crystalline pyrene [133]. Except for the lifting of degenerate electronic states, the artifi-

cial compound does not substantially alter the electronic properties of the pyrene moiety.

Both the vibrational and electronic properties of [2](1,3)Adamantano[2](2,7)pyrenophane

are widely disjunct combinations of adamantane and pyrene.

Calculated Raman spectra indicate that C=C stretch vibrations should generally be en-

hanced under a resonant excitation [129]. It is often observed that those vibrational

modes that are responsible for the vibrational progression in the fluorescence spectra, are

enhanced under a resonant excitation [71,97,104,129]. From the PL spectra, we obtain

an energy of ∼1415cm−1(176 meV) corresponding well with the energies of the C=C

stretch modes found in the DFT calculations. In fact, authors in Ref. [129] found that

the geometry change of the first excited state agrees to the vibrational pattern in which

the carbon atoms in the unbent carbon rings are deflected (compare Fig. 4.14). These

modes should also be very well observable for lower excitation energies, explaining the

large intensities of several modes between 1250 and 1400 cm−1in the Raman spectra plot-

ted in Fig. 4.15. Contradictory to the calculations, we observe a large intensity increase of

the high-frequency C=C stretch modes in the Raman spectra when excited resonantly, as

shown in Fig. 4.17. However, for an excitation energy of εL= 3.81 eV, we are already close

to the second excited electronic state exhibiting another geometry, compared to the first

excited electronic state. As a consequence, other vibrational modes might be resonantly

enhanced.

4.6 Summary

In summary we have analyzed the vibrational properties of various diamondoid deriva-

tives. We have shown that they can be understood as linear combinations of the vibra-

tional properties from the constituents involved. In particular, these are lower diamon-

doids and ethylene for the double-bond oligomers as well as adamantane and pyrene for

4.6. Summary

[2](1,3)Adamantano[2](2,7)pyrenophane. Since the connecting bond in the single-bond

dimers can be seen as an extension of the carbon cage, their vibrational properties resem-

ble those from the diamondoid species involved. Although the characteristic vibrational

modes of the constituents are widely unaffected in the oligomers, we find new, structure-

induced vibrational modes. On the one hand, this can be understood from the fact that

the symmetry is clearly reduced. On the other hand, the new structures give rise to ad-

ditional modes such as the dimer breathing mode (DBM) present in all oligomers or the

high-frequency C-H stretching modes.

If a C=C double bond is existent in the diamondoids derivatives, the optical transi-

tion energies are substantially downshifted. In fact, they resemble those from the ethy-

lene moieties in case of the double-bond dimers or those from pyrene in [2](1,3)Adaman-

tano[2](2,7)pyrenophane, widely unaffected from the diamondoid species attached. By res-

onance Raman measurements, we could determine the π→π∗transition energies of C=C

double bond connected oligomers to exhibit ∼4.7eV . This value is a bit less than 2eV

smaller than in pristine, lower diamondoids. In [2](1,3)Adamantano[2](2,7)pyrenophane,

the onset of the fluorescence was found at around 2.95 eV approximately resembling the

value in crystalline pyrene.

In the crystalline phase, diamondoids and diamondoid derivatives form very stable van-

der-Waals crystals. We find that the arrangement in a periodic lattice structure leads to

a downshift of the π→π∗transition energies of around 1 eV in diamondoid derivatives

containing a carbon double bond. It further allows for collective, low-frequency phonon

modes in which entire diamondoid moieties are involved.

5. UV Raman spectroscopy of

graphite, graphene, and multilayer

graphene

Parts of this chapter are published in Ref. [52].

5.1 UV Raman spectroscopy of graphite and graphene

- a motivation

Raman spectroscopy has established itself as a very common technique to analyze fun-

damental properties of carbon allotropes, in particular, of graphene, graphite, and car-

bon nanotubes [45, 53,134, 135]. From a simple point of view, it is surprising that the

Raman effect in graphene can be experimentally observed since there is no effective scat-

tering volume. However, graphene exhibits a reasonable, constant optical absorption of

πα = 2.293% (where α=e2/~cdenotes the fine structure constant) for photon ener-

gies between 0.2eV and 1.2eV [136]. The optical absorption for higher energies deviates

from the constant behavior due to considerably lower dispersions of electron and hole

bands [137–139]. A lower dispersion along the K−Mdirection leads to a higher density

of states and thus, to a substantial increase of the optical absorption up to 4×πα from

a single sheet of graphene [137–139].

Both in graphene and graphite, excited charge carriers undergo an efficient coupling to op-

tical phonons. The main reason is a strong Kohn Anomaly in both materials [10,140–142].

It causes a drastically enhanced electron-phonon coupling because of the reduced ability of

electrons from the Fermi surface to screen the vibration of the carbon nuclei [10,140,143].

This in turn leads to a discontinuity of the derivative from certain phonon branches

accompanied by a substantial softening of phonon frequencies close to the K and Γ

points [10,14,140,144]. Both the reasonable optical absorption and the strong electron-

phonon coupling allow for the analysis of optical and acoustic phonons in these materials

explaining the widespread usage of Raman spectroscopy.

A lot of interest has been put in the fundamental understanding of first- and second-

order Raman processes in graphene, multilayer graphene, and graphite. Due to the broad

availability of compact spectrometer and laser systems in the visible spectral range, many

of the available publications concerning inelastic light scattering show experiments where

a limited region of electron wave vectors around the Dirac points in graphene/graphite are

5.1. UV Raman spectroscopy of graphite, graphene, and carbon nanotubes

500

1000

1500

Phonon frequency cm−1

AΓK M

E2g

E1u

Raman active

IR active

1584 cm−1

1592 cm−1

Figure 5.1: Left side: The structure of the graphite lattice with indicated eigenvectors of the Raman-

active E2g(top) and IR-active E1u(bottom) modes is illustrated. Right side: The one-dimensional

phonon dispersion of graphite is plotted along the high-symmetry lines. The red circle corresponds to the

zone-center E1uand E2gvibrations as plotted on the left side. Calculations are done with a DFT/LDA

approach and are provided by Roland Gillen [96].

probed [9,14,145–147]. This can also be well understood from the fact, that the electron

dispersion in the proximity of the Dirac cones is linear - a behavior that is otherwise only

known from massless Dirac particles or from surface states in topological insulators [5,148].

Since the two-dimensional electron dispersion is nearly isotropic in the proximity of the

Dirac cones, in a double-resonant Raman process, the linearity can directly be probed by

laser excitation energies below 2eV [38]. Electrons with momenta beyond the proximity

of the Dirac cones exhibit a sub linear behavior with a substantially lower dispersion in

the K−Mdirection compared to the K−Γdirection [38]. This peculiarity in the electron

band structure and its effect on the Raman spectra has been discussed in literature exces-

sively [9,14,45,144,145,149,150]. An experimental and theoretical discussion of Raman

processes away from the "usual" scattering paths, i.e. DR Raman processes that happen

close to the M-point exciton, or far away from the proximity of the K points, are even

today scarce in literature.

In this chapter, we will discuss and explain these "unusual" scattering paths. We will

see that these processes will include phonons from all over the 1st Brillouin Zone of the

analyzed materials. This aspect is a novelty in the analysis of DR Raman processes

in graphene/graphite. By the careful selection of laser excitation energies, we can lift

the usual restriction of phonon wave vectors fulfilling multiple resonance conditions in

a double-resonant Raman process. This leads to fundamentally new insights into the

vibrational properties of graphite, graphene, and carbon nanotubes.

5.1.1 Vibrational properties of graphene and graphite

The unit cell of the bulk material graphite consists of two single, two-dimensional planes

of sp2hybridized carbon. They are arranged in a way that every second carbon atom is

located above the center of a carbon six-ring from the adjacent layer. Their interlayer

spacing is around 3.35˚

A, i.e. around 1.5 times the distance of a carbon-carbon double

bond (2.461 ˚

A) [39,151]. This stacking is also referred to as Bernal-stacking or simply as

5.1. UV Raman spectroscopy of graphite, graphene, and carbon nanotubes

AB-stacking. In other words, graphite consists of two sub lattices that are horizontally

shifted by the length of a carbon-carbon bond along its fixed direction and vertically

shifted by ∼1.5×the length of the carbon-carbon double bond. A scheme of the ar-

rangement of the two sub lattices can be seen on the left side in Fig. 5.1. The emerging

crystal has a D4

6hsymmetry and contains four atoms in the unit cell [152]. They are re-

sponsible for twelve phonon branches that are, due to the structure of the lattice, pairwise

almost degenerate. An one-dimensional plot along the high-symmetry lines can be seen

in Fig. 5.1.

The inversion center of graphite allows for new vibrational patterns compared to its single-

layer constituent graphene (D6h). The well known vibrational modes in graphene split up

to in-phase and out-of-phase motions when changing the layer [153, 154]. For instance,

this results in the IR activity of graphite or allows for Raman-active, low-frequency shear

modes that originate from a zone boundary acoustic mode in the first place [153, 155].

The twelve allowed normal modes for q= 0 in graphite read as:

ΓGraphite

vib = 2A2u+ 2B2g+ 2E1u+ 2E2g(5.1)

originating from the in-phase (Γgraphene

vib NA1g) and out-of-phase (Γgraphene

vib NB1g) combi-

nations with the allowed normal modes of graphene [154]. The two double degenerate E2g

modes are Raman active and responsible for the well-known G-band and the shear modes

in graphite [82,155,156]. The E1uand A2umodes are IR active whereas the B2gmodes

remain optical inactive. The vibrational patterns of the high-energy E2gand E1umodes

are illustrated in Fig. 5.1.

Based on Raman measurements in graphite, experimentally data is available for the LA,

TA, TO, and LO phonon branches with gaps for phonon wave vectors between the high-

symmetry points, i.e. far away from the high-symmetry points in the phonon disper-

sion. Other experimental approaches such as inelastic neutron scattering techniques [157],

electron energy loss spectroscopy [158], and high-resolution electron energy-loss spec-

troscopy [159,160] also added insights into the out-of-plane ZO and ZA phonon branches.

Deeper insights into the phonon dispersions of graphite were given by x-Ray Raman scat-

tering several years later [152,161]. These measurements could finally identify the unlike

dispersions of the TO and LO phonon branches and therefore identify the phonon branch

responsible for the Dand 2Dband in graphite, graphene, and carbon nanotubes [152,161].

This decisive assignment has finally led to a convincing and broad understanding of the

second-order Raman scattering processes in the named carbon allotropes.

5.1.2 Beyond double-resonant Raman scattering: A UV analy-

sis of graphite, graphene, and carbon nanotubes

Double-resonant Raman scattering is an essential concept for the understanding of the

complex Raman spectrum of graphite [9]. It explains the large variety of modes that are

usually visible in graphene and graphite as shown in Fig. 5.2 (bottom spectrum).

First-order processes such as the G-band (compare Fig. 5.2) are visible throughout a wide

range of laser excitation energies [162,163]. Instead, second-order processes such as the D

and 2Dbands [9,14,164], the iTOLA [14,149,165–167], or other low-intensity bands [14]

5.1. UV Raman spectroscopy of graphite, graphene, and carbon nanotubes

εL= 5.08 eV

Graphite

G-band

Intensity (arb.units)

εL= 2.33 eV

Graphite

G-band

DiTOLA

1400 1600 1800 2000 2200 2400 2600 2800 3000

Raman shift cm−1

Figure 5.2: Raman spectra of graphite for a deep-UV and visible light excitation energy are plotted.

The double-resonant D, iTOLA, and 2Dmodes disappear for an excitation energy of εL= 5.08 eV (upper

spectrum).

like the D’ band [14,85] or the LA defect band [168] are not present in a Raman spectrum

when the excitation energy is chosen to exhibit 5.08 eV as shown in Fig. 5.2.

In the visible optical range, it further occurs that due to sharp resonances of reciprocal

scattering processes, the intensity of second-order Raman modes exceeds that of the first-

order process associated with the G-band [141, 169, 170]. This even happens although

there is a strong Kohn Anomaly of the degenerate LO/TO phonon branches at the Γ

point in the phonon dispersion of graphite [10,140]. For deep-UV excitation energies this

behavior can not be observed anymore [52,162,163].

In the following, we will discuss and explain the drastic changes in the Raman spectra as

shown in Fig. 5.2 and extend the discussion of Raman spectra from graphite to a larger

set of laser excitation energies.

There has been a long-standing discussion about the phonon scattering paths of the disper-

sive Raman peaks in the spectra of graphene and graphite. The reason is the anisotropic

electronic band structure of graphene/graphite allowing for various scattering paths with

a broad range of different possible phonon wave vectors involved. In a simplified, one-

dimensional scattering scheme, the different scattering paths can be associated with so

called "inner" and "outer" processes when the optical excitation occurs along the K−M

direction ("inner") or along the K−Γdirection ("outer"). An example for an outer pro-

cess can be seen in Fig. 5.3 (b). As a consequence of the symmetry, the phonons involved

then stem from the Γ−Kdirection ("inner") or from the K−Mdirection ("outer").

Herziger et al. gave a more general definition of "inner" and "outer" processes in the

two-dimensional electronic band structure of bilayer graphene [144,171] that can also be

applied to graphene or graphite. According to the authors, a process is referred to as "in-

ner" when the resonant phonon wave vectors come from a circular sector of ±30◦around

5.1. UV Raman spectroscopy of graphite, graphene, and carbon nanotubes

qouter

←ΓK M K0Γ→

−2

Energy (eV)

(a) (b)

εL= 5.08 eV

Figure 5.3: (a) A two-dimensional plot of the electronic band structure of graphene with the high-

symmetry points Kand Mis shown. The red lines correspond to an equi-energy surface of εL= 5.08 eV

whereas the triangle-shaped contour with edges in the middle between the K−Mhigh-symmetry line

corresponds to an energy surface of εL= 3 eV. (b) An One-dimensional plot of the electronic band

structure of graphene is plotted. Indicated are optical excitations with an energy of εL= 5.08 eV and the

length of a resonant phonon wave vector that mediates between two real electronic states in a double-

resonant Raman process.

the Γ−Khigh-symmetry direction in the proximity of the Kpoints and from a circular

sector of ±30◦around the K−Mhigh-symmetry direction for "outer" processes [144,171].

Authors in the most cited article concerning double-resonant Raman scattering in graphene

and graphene layers claim that dominant processes are those where resonant phonon wave

vectors mainly stem from the K−Mdirection [164]. This interpretation, however, was

done only by a tentative assignment to calculated phonon frequencies but without the

support of sophisticated two-dimensional Raman calculations. Authors in Refs. [84,172]

also claim that the optical absorption occurs in areas with the "lowest curvature" in the

electronic band structure which translates in optical excitations along the K−Γhigh-

symmetry direction. Instead, with better agreement to experiments and with the support

of extensive two-dimensional Raman calculations, the initial and somewhat general as-

signment was superseded by several authors [14,144,145,149,173,174]. On the one hand,

it was shown that the initial and simplified approach to calculate the second-order Ra-

man spectra given by Thomsen et al. [9] was well justified. On the other hand, the full

two-dimensional calculations, revealed that the main contributions of resonant phonon

wave vectors originate from the K−Γhigh-symmetry line, i.e. the dominant processes

are "inner" process [14, 144]. "Outer" processes, however, also give rise to contributions

in second-order Raman modes but with generally lower intensities and unlike phonon

frequencies compared to the "inner" processes. For instance, they are responsible for the

asymmetric line shapes of the 2Dband or the iTOLA in graphene [14,149], but are gen-

erally less visible in carbon nanotubes as discussed in the next chapter.

A possible way to experimentally distinguish between both processes is to chose a laser

excitation energy that is above the M-point energy, i.e. higher than 4.7eV [136,138,174].

As illustrated in Fig. 5.3, this approach suppresses the inner scattering paths only allow-

ing outer processes. It is noteworthy that equi-energy surfaces in the two-dimensional

electronic band structure of graphene are fundamentally different for energies over the M-

5.1. UV Raman spectroscopy of graphite, graphene, and carbon nanotubes

iTOLA

2D’

2400 2500 2600 2700 2800 2900 3000 3100 3200 3300

Raman shift cm−1

Intensity (log.scale)

εL= 5.08 eV

εL= 2.33 eV

Figure 5.4: Raman spectra of graphite for an excitation energy of εL=5.08 eV (black) and εL=

2.33 eV (blue) are plotted in a logarithmic scale. The marked peak (*) corresponds to a Raman mode

from atmospheric nitrogen.

point energy compared to those below the M-point energy. Below 4.7eV, the equi-energy

surfaces correspond to the well-known triangles around the K points due to the trigo-

nal warping effect [38]. Instead, equi-energy surfaces for energies over 4.7eV are circles

around the Γpoints [compare Fig. 5.3 a)]. This topological change leads to a restriction

of possible scattering paths in a double-resonant Raman process. In detail, the dominant

"inner" scattering paths connecting unlike Kand K0points at regions with the highest

curvature are excluded due to the lack of real electronic states. As a consequence, Raman

modes associated with these scattering processes are not present in a Raman spectrum

anymore. This can clearly be seen in Fig. 5.2 where the excitation energy is 5.08 eV, i.e.

way above the M-point exciton.

However, we still find Raman modes besides the first-order associated G-band in graphite

for laser excitation energies above the M-point exciton but with a several magnitudes

lower intensity compared to the G-band. A corresponding Raman spectrum for an exci-

tation energy of εL= 5.08eV is plotted in Fig. 5.4. Five, characteristic and asymmetric

peaks between 2600 cm−1and 3300cm−1are found. None of these peaks exhibits the

typical Lorentzian line shape that is known from first-order or in a good approximation,

the "Baskonian" line shape that is known from second-order processes in graphite [175].

Further, we find a strong Raman band around 2331cm−1that is associated with a stretch

vibration of the nitrogen molecule 14N2from the atmosphere [176]. This high-frequency

peak can be used to calibrate the Raman spectra as it is independent from the laser ex-

citation energy.

The asymmetric line shapes indicate that the nature of processes responsible for the Ra-

man modes are fundamentally different compared to those present in the spectrum for

excitation energies in the visible optical range. In fact, if we compare the Raman peaks

to the phonon dispersion of graphite, we can assign these peaks to the second-order vi-

brational density of states [52]. Regions in the phonon dispersion exhibiting a very flat

slope give rise to a high density of states being responsible for the peaks as shown in

Fig. 5.4 and Fig. 5.5. These regions are namely the overbending of the LO phonon branch

around the Γpoint, the two pairs of TO/LO phonon branches at the Γpoint, the TO

5.1. UV Raman spectroscopy of graphite, graphene, and carbon nanotubes

2LOmax.

overbending

M point

2LA/LO

M point

2TO

2E2g2E1u

εL= 5.08 eV

2600 2700 2800 2900 3000 3100 3200 3300

Raman shift cm−1

Intensity (log.scale)

2600 2700 2800 2900 3000 3100 3200 3300

Raman shift cm−1

pDOS

200

400

600

800

1000

1200

1400

1600

Phonon frequency cm−1

1586

1590

1594 1416

1418

1420

1350

1360

AΓK M

Figure 5.5: Left: The measured (top) and calculated (bottom) second-order vibrational density of states

of graphite is plotted. The excitation energy is εL= 5.08 eV. Peaks are assigned by a comparison to the

calculated phonon dispersion of graphite (right). Insets show regions with a high density of states that

can be seen in the experiments. Calculations are done with a DFT + LDA approach and are provided

by Roland Gillen [96].

phonon branch at the M point, and the almost degenerate LA/LO phonon branches at the

M point. The respective first-order frequencies can experimentally be determined with

1626cm−1for the LO-overbending, 1592 cm−1for the Γpoint E1umode, 1578 cm−1for

the Γpoint E2gmode, 1408cm−1for the M point TO phonon branch, and 1344cm−1for

the M point LO/LA phonon branches. We find an excellent agreement of the experimen-

tal values with data in the literature [82,154,161,177]. Especially the calculated phonon

frequencies of the TO, LO, and LA phonon branches at the M point can be verified [161].

This is of particular interest since this region of the phonon dispersion is barely accessible

by optical spectroscopy.

Fig. 5.5 also contains the calculated vibrational (phonon) density of states named pDOS.

We find similar asymmetric peaks as in the experimental Raman spectra, but with dif-

ferent intensities. In the pDOS calculations, no coupling elements are considered, leading

to a strong underestimation of contributions from the Γpoint. Its well-known Kohn

Anomaly leads to a strong electron-phonon coupling giving rise to the E2gand E1uasso-

ciated second-order peaks in the experimental Raman spectrum [10,140].

The direct product of irreducible representations can be decomposed into a linear combi-

nation of irreducible representations belonging to the specific point group. Three examples

of the product of the relevant irreducible representations E2gand E1uwithin the point

5.1. Raman scattering mechanism for deep-UV excitation energies

group D4

6hare given below [178]:

E2g⊗E2g=A1g⊕A2g⊕E2g(5.2)

E1u⊗E1u=A1g⊕A2g⊕E2g(5.3)

E1u⊗E2g=B1u⊕B2u⊕E1u(5.4)

The first two products contain the fully symmetric representation A1gthat is always

Raman active. In fact, the direct product of irreducible representations always contains

the fully symmetric representation A1g[178]. The second order of a vibrational mode

should therefore always be Raman active and its experimental access is not anymore

restricted to the first-order scattering optical selection rules. It is consequently possible

to measure both Raman active and non-Raman active modes in a joint experiment as

shown for the E2gand E1umodes in graphite. Measuring IR active modes such as the

E1uotherwise requires more complicated experiments like reflectivity measurements at

infrared beamlines [177].

In contrast, the product of the representations E1uand E2gdoes not contain a Raman

active representation. We should therefore not be able to observe according second-order

combination modes which holds true for the experiments as can be seen in Fig.5.5

5.1.3 Raman scattering mechanism for deep-UV excitation en-

ergies

We will now turn the discussion to the mechanism of the scattering process and explain

the different Raman spectra when the laser excitation energy is tuned from the deep-UV

region down to the energy of the M-point exciton.

The well established concept of double-resonant Raman scattering proposed by Thom-

sen et al. [9] was initially used to understand and describe the second-order spectrum of

graphite for laser excitation energies in the visible optical range. Double-resonance refers

to two resonances, i.e. real electronic states that are namely the electron hole pair that is

created by the incoming photon matching the energy difference of valence and conduction

band (hν →a) and the scattered electron/hole finding another real state in the conduc-

tion/valence band (a→b) [9]. As a consequence of the dispersion of both electrons and

phonons, the consecutive backscattering with a phonon (b→c) is a non-resonant process

with an usually small detuning for excitation energies in the optical visible range [9]. The

scattering scheme can be seen in Fig. 5.6. For reasons of simplicity, we only explain the

incoming resonance and electron-electron scattering in detail. Equivalent statements can

be made for outgoing resonances and the predominant electron-hole scattering [14].

It is interesting to note that when the optical excitation approximately equals the M-

point transition energy (∼4.7eV) the flat dispersion of the valence band still impedes

double-resonant scattering for "inner" processes. Phonon momenta necessary for the dou-

ble resonance directly stem from the proximity of the Γpoint and exhibit very large

phonon energies of approximately 200 meV (compare Fig. 5.5). In a two phonon process

the energy difference between the initial and final state then adds up to ∼400 meV.

For such a large energy difference, the electron momentum mismatch of kinitial and kfinal

is comparably vast, resulting in a large detuning of the entire double-resonant process.

5.1. Raman scattering mechanism for deep-UV excitation energies

b’

c’

b”

c”

”Single-resonant“ scattering

”Double-resonant“ scattering

εL=

5.54 eV

hν

εL= 4.55 eV

-q

hν

←ΓK M K0Γ→

1.5

2.5

Energy (eV)

4.7eV

4.85 eV

5.0eV

25 eV

25 eV: Intensity ×5·105

2.33 eV

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

Phonon wave vector ˚

A−1

Intensity (arb.units)

Intensity ×115

4.7 eV

0.8 1 1.2 1.4

Figure 5.6: Top: Raman scattering schemes for excitation energies of εL=5.54 eV and εL=4.55 eV are

plotted into the one-dimensional electronic band structure of graphene. In a single-resonant process, the

intermediate states b, b’, and b” are not real states whereas in a double-resonant process, the intermediate

state (b) is a real state. Phonon momenta for various non-resonant processes are denoted with q and -q.

Bottom: Calculated phonon wave vectors in a second-order Raman process for various laser excitation

energies as given in the figure are plotted. The calculation is restricted to the high-symmetry lines with

no coupling elements considered as shown in equation 2.5. The inset shows resonant phonon wave vectors

for an intravalley scattering process. Electronic bands are calculated within a third-nearest neighbor

tight-binding approach.

The large mismatch of electron momenta can be seen in the double-resonant scattering

scheme in Fig. 5.6. In fact, due to the slope of the valence band, the mismatch of electron

momenta is maximal for excitations close to the M-point transition energy compared to

all other processes over the 1. Brillouin zone. Assuming a resonant scattering to a real

state in the valence/conduction band, it is ∆kinitial

final ≈1.5qresonant (compare Fig. 5.6). Al-

though the initial state is real, this aspect should reduce the intensity of the 2Dpeak in

graphene/graphite when excited with εL∼4.7eV.

The concept of double-resonant Raman scattering allows for more scattering processes

than its denotation suggests. It is a convenient concept since due to the very strong en-

hancement of subset scattering paths, all other possible scattering paths can be neglected

when explaining the Raman spectra of graphene or graphite [9,14].

Instead of being scattered to a real state, an excited electron in the valence band can be

scattered to a non-real, intermediate state and backscattered to an electron state exhibit-

ing the same momentum as that of the initially excited electron. We refer to such a process

5.1. Raman scattering mechanism for deep-UV excitation energies

2600 2700 2800 2900 3000 3100 3200 3300

Raman shift cm−1

Intensity (arb.units)

4.69 eV

4.82 eV

4.94 eV

5.08 eV

5.20 eV

4.7eV

4.85 eV

5.0eV

25 eV

Calculated Raman spectra

×80

Figure 5.7: Raman spectra of graphite for different laser excitation energies. The arrow indicates the

2Dmode for very small phonon wave vectors.

as a "single-resonant", second-order scattering process since only one real state is involved.

For instance, if we assume an incoming resonance, only the initial state is a real electronic

state. Various scattering paths are plotted in Fig. 5.6 entitled with "Single-resonant" scat-

tering. In such a process, the phonon wave vector can exhibit values stemming from all

over the 1. Brillouin zone and is not restricted to a small set of wave vectors connecting

real electronic states as apparent in a double-resonant process. None of these processes

is resonantly enhanced and thus, all of them contribute equally to a Raman spectrum.

As a result, the spectrum of single-resonant, second-order Raman processes is weighted

according to the vibrational density of states. In flat regions of the phonon dispersion, a

broad range of phonon wave vectors exhibits equal energies and thus, these contributions

add up to a considerable peak in the Raman spectrum. The asymmetry of the Raman

peaks directly reflects the slope of the measured phonon branches. In order to understand

the scattering process in more detail we have calculated the Raman spectra for excitation

energies close and above the M-point exciton energy by using the following equation [9]:

I ∝

α=1 X

a,b,c

(EL−Eai −iγ)(EL−Ebi −~ωα−iγ)

×1

(EL−Eci −2~ωα−iγ)

,(5.5)

All coupling elements Mare set to 1 and we restricted the scattering process only to the

high-symmetry directions as shown in Fig. 5.3. The sum in the absolute value contains all

possible states a, b, and c as denoted in Fig. 5.6. The outer sum considers the respective

5.1. Raman scattering mechanism for deep-UV excitation energies

εL= 5.08 eV

εL= 4.82 eV

2600 2700 2800 2900 3000 3100 3200 3300

Raman shift cm−1

Intensity (arb.units)

Exp.

Figure 5.8: Calculated (blue) and experimental (black) Raman spectra are given for excitation energies

of εL=5.08 eV and εL=4.82 eV. TO (transverse optical)phonon contributions are plotted in dashed lines,

LO (longitudinal optical) phonon contributions in continuous lines.

pairs of the TO, LO, and LA phonon branches.

Calculated resonant phonon wave vectors for excitations close and way above the M-point

energy are plotted in Fig. 5.6. Due to the lack of reliable broadening parameters for such

high excitation energies, we have assumed a broadening parameter of γ/2 = 120meV for

all calculated spectra in Fig 5.6. Considering a parabolic dependency of the broadening

parameter γon the laser excitation energy, as suggested by Venezuela et al. [14], we would

end up with a broadening factor of γ∼240meV. This very high value is also predicted by

Popov et al. [179], but an experimental verification is up to today still missing. However,

since we mainly discuss single-resonant, processes that do not include resonant interme-

diate states, the influence of the broadening factor is comparably low [180].

As a consequence of the flat bands of both electrons and holes in the proximity of the

M point, the resonant phonon wave vectors for such high excitation energies are short

and very broad (compare Fig. 5.6). When the excitation energy is tuned above the M-

point energy, the double-resonant Raman process vanishes accompanied by a reduction

of both the length of the resonant phonon wave vectors and their intensities. For an

artificially high excitation energy of 25eV, there is no distinct resonant scattering process

recognizable anymore. However, the scattering amplitude is small but nonzero leading to

contributions with very low intensities.

An optical excitation close to the M-point leads to intervalley scattering processes with

the shortest phonon wave vectors possible. Two-phonon Raman modes like the widely

discussed 2Dband will therefore be found at different phonon energies than usual for ex-

citation in the optical visible range. In fact, we find the 2Dband experimentally around

3050cm−1as shown in Fig. 5.7. A tuning of the laser excitation energy from 5.2 eV down

to 4.69eV leads to a decrease of the phonon frequency, accompanied by an increase of

2D-mode intensity. To some extent, we can speak of "the beginning" of the 2Dmode as

the phonons involved are zone-center phonons.

It is further interesting to note that intravalley scattering processes have the longest reso-

nant phonon wave vectors possible for excitations close to the M point. A corresponding

5.1. Deep-UV Raman measurements in graphene and carbon nanotubes

inset with a magnification factor of 115 is given in Fig. 5.6. Further, due to the trigonal

warping, the separation between the scattering paths M→Γand Γ→Mis also the

largest possible. In the Raman spectrum, both peaks should therefore be very well dis-

tinguishable due to a comparably large separation of their phonon frequencies. However,

in the experiments we are not able to see contributions of such a process that is usually

referred to the 2D0band with two LO phonons involved [14] and very well observable

for excitation energies in the visible range. In the discussed UV range, the correspond-

ing Raman peak was around 3200cm−1and therefore, we believe it is screened by the

second-order DOS peaks. To sum up, we can say that second-order Raman processes for

deep-UV excitation close to the M-point energy in graphite involve unusual phonon wave

vectors that lead to unusual Raman spectra.

In our calculations, we can distinguish between the phonon branches contributing to

the overall Raman spectrum for a certain excitation energy. Fig. 5.8 shows a comparison

of experimental and calculated Raman spectra where only the LO or the TO phonon

branches are considered, respectively. As can be seen, the DOS peaks in the experimental

Raman spectra can be assigned to both LO and TO contributions. The higher energy

components generally originate from the LO phonon branch whereas the lower energy

contributions stem from the TO phonon branch. An excitation energy near the M-point

exciton leads to reasonable DR contributions associated with the 2Dband but still ex-

hibiting very low intensities in the order of the not resonantly enhanced DOS peaks.

In the DFT computations, we find the maximum of the LO phonon branch at a phonon

wave vector of 0.46˚

A−1. The resonant phonon wave vector for an excitation energy of

εL= 4.82eV also exhibits q∼0.46˚

A−1. As a consequence, the overbending related

second-order Raman peak at around 3240cm−1is relatively enhanced for excitation en-

ergies around 4.82eV, both in the experiments and calculations.

5.1.4 Deep-UV Raman measurements in graphene and carbon

nanotubes

The introduced approach to investigate the vibrational density of states can also be ap-

plied to other sp2carbon systems such as graphene or carbon nanotubes as shown in

Fig. 5.9. Similar to graphite, graphene also exhibits a six-fold rotation axis with a hori-

zontal reflection symmetry but only consists of a single plane. Its point group is therefore

D6h[181]. The two atoms in the unit cell allow for six vibrational modes [153,181]:

ΓGraphene

vib =A2u+B2g+E1u+E2g(5.6)

The degenerate high-energy mode (E2g) does not split up into a phase and anti-phase

vibration as in graphite and as a consequence only one peak should be observed in the

second-order DOS Raman spectrum of graphene. This can nicely be seen in Fig. 5.9,

where the calculated and measured second-order vibrational density of states are shown.

Again, in the case of the calculated DOS, the E2g@ Γ peak exhibits a lower intensity com-

pared to other DOS peaks since within the simplified approach, the Kohn Anomaly [143]

is not considered. We can determine the energy of the E2gΓ−point vibration and the

maximum of the LO phonon branch to exhibit 1581cm−1and 1623cm−1, respectively [52].

5.1. Deep-UV Raman measurements in graphene and carbon nanotubes

CNT pDOS

Graphene pDOS

2400 2600 2800 3000 3200

Raman shift cm−1

Intensity (arb.units)

CNT

Graphene

εL= 5.08 eV

2600 2800 3000 3200

(c)(a)

(b)

Figure 5.9: Left side: Calculated second-order vibrational density of states (pDOS) of carbon nanotubes

(a) and graphene (b) are plotted. We have used the POLSYM code [182] for the calculations of the

pDOS from carbon nanotubes and have only considered the chiralities as listed in Table 6.1. An equal

distribution of all chiralities has been assumed. The pDOS of graphene has been calculated with a

DFT/LDA approach by Roland Gillen [96]. Right side: Experimental, second-order vibrational density

of states Raman spectra of an HipCO carbon nanotube sample (black) and graphene (blue) are plotted

for an excitation energy of εL= 5.08 eV.

Our findings have recently been confirmed by more complex ab initio calculations from

Valentin Popov where the second-order vibrational density of states peaks can be observed

for excitation energies above the πplasmon energy [179,183].

Compared to graphene and graphite, the UV Raman spectrum of CNTs only consists

of a single, broad, and asymmetric peak at around 3150cm−1. This experimental result

is also reflected in the calculated pDOS for the CNT ensemble in Fig. 5.9 and can be

understood from the fact that phonon frequencies in CNTs sensitively depend on the

nanotube diameter and chiral angle [16]. The broad diameter distribution of the HipCO

CNT ensemble (7 to 13 ˚

A) directly results in a broad range of phonon frequencies. For

instance, we infer a range of 2∆ω= 80cm−1for the second-order high-energy mode fre-

quencies [182], well explaining the experimentally and theoretically observed broad peak

centered at 3150 cm−1. In contrast to graphene or graphite, we do not observe distinct

peaks in our experimental Raman spectrum that are related to the LO overbending or

the pDOS at the M point. Again, these phonon frequencies also show a dependence on

tube diameter and chiral angle. The expected range of LO-phonon maxima in our CNT

sample is [182] ∆ωLO,max = 10cm−1. Thus, all contributions will add up to a broad

shoulder on the high-frequency side of the main peak. The broad Raman signal towards

lower wavenumbers is attributed to LO-, LA-, and TO-derived phonon bands from the M

point that cannot be separated in our experiments. The same counts for the calculated,

second-order vibrational DOS of the carbon nanotube HipCO ensemble sample (compare

Fig. 5.9). As already mentioned in the case of graphite, the calculated second order DOS

does not contain electron-phonon coupling elements. As a result, the relative intensi-

ties of apparent Raman peaks are not reproduced correctly. Further, the used POLSYM

code [182] underestimates the phonon frequencies, explaining the frequency gap between

calculations and experimental data in Fig. 5.9.

5.2. The second-order vibrational density of states in multilayer graphene

Outlook

We believe that the discussed approach can also be applied to other graphitic materials

possibly revealing new aspects of second-order Raman scattering. For instance, it could be

used to tune the quantum interference of different scattering processes in bilayer graphene

as recently shown by Herziger et al. [180]. The M-point transition energies of the π1/π∗

and π2/π∗

2bands are different, allowing to address the higher transition individually by

a suitable deep-UV laser energy [180]. This in turn eliminates a possible quantum inter-

ference between different scattering processes noticeably changing the line shape of the

2Dband in bilayer graphene [144,180]. The discussed approach might therefore help to

understand the Raman spectrum of bilayer (multilayer-) graphene to a larger extent.

5.2 The second-order vibrational density of states in

multilayer graphene

Although the phonon dispersions of graphene and graphite are well understood both

theoretically and experimentally in our days, there is still a lack of experimental data

concerning the phonon dispersions of multilayer graphene. First and second-order Raman

processes can provide a broad access to their phonon dispersions [144,155,164,184–187]

but are limited due to optical selection rules. It takes more advanced approaches to break

the symmetry of the multilayer systems in order to even get further insights into their

vibrational properties: Layer-dependent doping [188], arbitrary twisting angles of adja-

cent layers [189], or tensile and compressive strain [190]. However, it is experimentally

still unclear how for example the ZO related out-of-plane vibrations are affected by the

number of adjacent layers and their interactions or if the Kohn Anomaly affects different

Γpoint vibrations differently in multilayer graphene.

On the other hand, the phonon dispersions of multilayer graphene have been calculated by

various groups with DFPT (density-functional perturbation theory) approaches [191,192].

It is interesting to note that a DFT approach on the LDA (local density approximation)

level gives a reasonable agreement to experiments even without the explicit consideration

of a dispersion correction [156]. Instead, on the GGA (general gradient approximation)

level, the interlayer distance is too large by 30 %and therefore the Raman active shear

modes are predicted to exhibit way too small energies [156]. An a posteriori dispersion

correction is needed to overcome these insufficiencies [156].

We have tried to utilize the possibility to measure the phonon density of states to find

a new approach for a layer-number determination. Since every new layer in multilayer

graphene adds six phonon branches including one pair of degenerate LO/TO phonon

branches at the Γpoint, it should be possible to observe the new peaks in the DOS

Raman spectra. Measuring the second-order peaks again ensures Raman activity as dis-

cussed in the previous section.

On the following pages we discuss our findings and the challenges we had to encounter

for reliable measurements.

5.2. The second-order vibrational density of states in multilayer graphene

εL= 3.81 eV

2Dmode

vDOS peaks

Graphene

trilayer

many layers

Graphite

2600 2800 3000 3200

Raman shift cm−1

Intensity (arb.units)

εL= 3.81 eV

εL= 5.21 eV

Graphite

bilayer

2E2g

2E1u

2LO overbending

3100 3150 3200 3250 3300

Raman shift cm−1

Figure 5.10: Left side: Raman spectra of Graphene, trilayer graphene, many layer graphene, and

graphite for an excitation energy of εL=3.81 eV are plotted. Denoted is the 2Dband that was used for

a layer-number determination. vDOS stands for the vibrational density of states. Right side: Close-up

of the second-order density of states related Raman peaks as named in the figure are plotted. Shown

are spectra for graphite (εL= 3.81 eV and εL= 5.21 eV) and for bilayer graphene (εL= 3.81 eV). As a

protection layer, 30 nm Al2O3layer has been deposited onto the silicon wafer with the graphene samples.

5.2.1 Layer-number determination and sample characterization

A sample characterization is necessary before we can measure the second-order density of

states and assign the results to a multilayer graphene sample with a defined number of

adjacent layers. We have used mechanically exfoliated graphene on a SiO2/Si substrate.

Several techniques are available to determine the amount of graphene layers of a sam-

ple: An analysis of the 2Dline shape [164,193], optical contrast measurements [194,195],

an analysis of shear modes [155], or an analysis of stoke/anti-stokes, double-resonant

processes with out-of-plane phonons [187]. A first characterization was done via opti-

cal contrast measurements and was then verified by an analysis of the 2Dline shape.

Various spectra of the analyzed multi-layer graphene samples for an excitation energy of

εL= 3.81eV are plotted in Fig. 5.10. Note that the 2Dline shapes for the shown exci-

tation energy differ from those usually shown in literature. The characteristic low-energy

contributions originating from symmetric processes in bi- and trilayer graphene will be less

visible for very high excitation energies as the resonant phonon wave vectors of symmetric

and anti-symmetric get more similar and the TO phonon splitting decreases [144,193].

Also, mainly due to a higher electron-phonon-coupling, the electronic broadening for

higher excitation energy increases leading to a strong destructive quantum interference of

symmetric and anti-symmetric scattering processes [14,144,180]. This in turn pronounces

high-wavenumber, symmetric contributions explaining the asymmetric line shapes of the

2Dband as can be seen in Fig. 5.10 [144,164,180,193].

With the sample characterization we can now advance the discussion to the layer num-

ber influence on the second-order density of states as shown in Fig. 5.10. Plotted are

the second-order DOS peaks for bilayer graphene and graphite for an excitation energy

of εL= 3.81eV and for comparison, also the second-order DOS peaks of graphite for

an excitation energy of εL= 5.21eV. It can be seen that the second-order DOS peaks

of the 2E2gand 2E1umodes are except of their intensity ratios similar. In fact, these

5.2. The second-order vibrational density of states in multilayer graphene

1 layer

2 layer

3 layer

4 layer

5 layer

many layers

Graphite

εL= 3.81 eV

2900 3000 3100 3200 3300

Raman shift cm−1

Intensity (arb.units)

Figure 5.11: Raman spectra of graphene,

multilayer graphene, and graphite as named in

the figure are plotted. The laser excitation en-

ergy is εL= 3.81 eV. The red dashed lines de-

note the slope of DOS peaks originating from

the E2gvibration in graphene [153, 154] as a

function of the layer number. The blue dashed

line corresponds to the maximum of the LO re-

lated second-order peak in graphite [52] (com-

pare Fig. 5.5). As a protection layer, 30 nm

Al2O3layer has been deposited onto the sili-

con wafer with the graphene samples.

peaks should be observable for every excitation energy possible as the single-resonance

Raman process described in the previous section involves phonons with arbitrary wave

vectors. However, they tend to be screened by the 2D0mode [14,196] and due to their

single-resonance character, their intensities are magnitudes smaller compared to other

second-order Raman modes [9, 14, 196]. Compared to graphite, the DOS related peaks

in bilayer graphene are both downshifted to 3181cm−1for the higher and to 3150cm−1

for the lower mode. Due to the reduced symmetry of the bilayer graphene lattice (D3d),

the observed peaks can be related to vibrations with the irreducible representations Eg

(lower-frequency mode) and Eu(higher-frequency mode), different than those in graphite

as denoted in Fig. 5.10 [153,191,192]. According to DFT calculations on a GGA or LDA

level, the IR active, high-frequency anti-phase vibration in bilayer graphene (Eu) should

exhibit a lower frequency compared to the E1umode in graphite [191, 192]. We find a

difference of ∼5cm−1in the second-order measurements translating into a downshift of

∼2.5cm−1of the actual Γ-point frequency. This value is in very good agreement to DFT

calculations where values between 2cm−1and 3 cm−1are found [191,192]. In the calcula-

tions, the frequency of the Egmode in bilayer graphene is less affected compared to the

E2gmode in graphite but depending on the computational approach, a slight upshift is

predicted [191,192]. We instead also find a downshift of ∼2cm−1for the Egmode in

bilayer graphene in the experiments as shown in Fig. 5.10.

A more comprehensive picture on how the DOS related second-order Raman peaks change

as a function of the layer number is given in Fig. 5.11. The red, dashed lines indi-

cate how the average frequencies of the DOS related peaks evolve from the E2gmode

in graphene. The dashed, blue line indicates the maximum of the 2LO overbending fre-

5.2. The second-order vibrational density of states in multilayer graphene

quency in graphite (compare Fig. 5.5).

From the presented measurements we can derive several statements: (i) For all multilayer

samples, we observe two distinct, DOS related Raman peaks. This clear separation in-

dicates, that the frequencies of new vibrational modes added by each new layer are very

close to the initial Eu(1591cm−1) and Eg(1576 cm−1) modes in bilayer graphene. (ii)

We observe an upshift of the DOS peaks at around 3182cm−1and no significant shift of

the 3152cm−1DOS peak when the layer number is increased. (iii) The intensities of the

E2gΓ-point related second-order peaks increase, compared to the LO overbending peaks

for larger layer numbers. (iv) The maximum of the 2LO overbending frequencies only

marginally change as a function of the layer number as indicated with the blue, dashed

line in Fig. 5.11. (v) The FWHM of the 2LO overbending related peaks increase with

increasing layer numbers. (vi) Compared to the deep-UV Raman spectra, the 2LO over-

bending related Raman peaks exhibit a Lorentzian line shape, i.e. they can rather be

related to a double-resonant, second-order Raman process.

With the help of calculated phonon dispersions of graphene, AB bilayer graphene, and

ABA trilayer graphene, we can understand the experimentally obtained Raman spec-

tra as shown in Fig. 5.11. When the layer number is increased to three (black, trilayer

graphene), a new pair of TO/LO phonon branches at the Γpoint appears very close to

the Egmode in bilayer graphene (blue, bilayer graphene) as shown in Fig. 5.12 a). In

contrast to graphene, trilayer graphene does not have an inversion center anymore, the

six-fold rotational symmetry reduces to a three-fold rotational symmetry, but the hori-

zontal reflection plane is conserved [153]. The reduced symmetry translates into the D3h

point group for trilayer graphene allowing for the irreducible representations E0and E00

originating from the E2gmode in graphene [153]. The two E0modes found in trilayer

graphene are responsible for the highest and lowest pair of the three pairs of degenerate

TO/LO phonon branches at the Γpoint (compare Fig. 5.12). The third mode has the

irreducible representation E00 and is separated by less than 2cm−1from the lower E0mode

at the Γpoint [192]. All E0and E00 modes are Raman active [192].

In the used DFT + LDA approach, we also find the new pair of degenerate TO/LO

phonons at the Γpoint very close to the lower E0in good agreement to our experimen-

tal findings (compare Fig. 5.12). When the layer number is increased, our experimental

findings indicate that new pairs of degenerate phonon branches are very close to the E2g

and E1umodes in graphite, even closer than predicted in theory [191,192].

To understand the experimental Raman spectrum, we have calculated a Raman spectrum

of graphene for an excitation energy of εL= 3.81eV as shown in Fig. 5.12 d). We have

used the approach as suggested in Ref. [9], have set all matrix elements to 1 and have

only considered second-order processes involving LO phonons For comparison, we have

plotted a smoothed experimental Raman spectrum of graphene for the same excitation

energy next to the calculated Raman spectrum. We find a good agreement and thus it

it tempting to assign the experimental peak to a second-order, double-resonant Raman

process of the LO phonon branch. The bimodal peak structure in the calculated Raman

spectrum originates from the anisotropic slopes of the electronic bands due to the trigo-

nal warping effect. The double resonant process in which the initial electronic state is in

5.2. The second-order vibrational density of states in multilayer graphene

LO/TO phonon branches

1580

1585

1590

1595

Phonon frequency cm−1

Bilayer AB

Trilayer ABA

LO phonon branches

1570

1580

1590

1600

1610

1620

1630

Phonon frequency cm−1

Graphene

Bilayer AB

Trilayer ABA

3080 3120 3160 3200 3240 3280

Raman shift cm−1

Second-order DOS

Graphene

Bilayer AB

Trilayer ABA

3150 3200 3250 3300

Raman shift cm−1

Intensity (arb.units)

Graphene @ 3.81 eV

Calc. LO phonon @ 3.81 eV

Γ−→ K−→

ΓK

a) b)

d)c)

Figure 5.12: Top: Parts of the calculated phonon dispersions of graphene, bilayer graphene, and trilayer

graphene are plotted. a) The LO/TO phonon branches at the Γpoint from Bernal stacked bilayer

graphene (blue) and from Bernal stacked trilayer graphene (black) along the Γ−→ Kdirection are

plotted. b) The maxima of the LO phonon branches of graphene (red), Bernal stacked bilayer graphene

(blue), and Bernal stacked trilayer graphene (black) along the Γ−→ Kdirection are shown. c) High-

frequency parts of the, second-order vibrational density of states of graphene (red), Bernal stacked bilayer

graphene (blue), and Bernal stacked trilayer graphene (black) are plotted. d) An experimental Raman

spectrum of graphene is plotted (red). As a protection layer, 30 nm Al2O3layer has been deposited

onto the silicon wafer with the graphene sample. A corresponding calculated Raman spectrum for an

excitation energy of εL= 3.81 eV just considering the LO phonon branch of graphene is shown in black.

All matrix elements are set to 1. The phonon dispersions are calculated with a DFT/LDA approach,

provided by Roland Gillen [96].

the proximity of the K−Mdirection should be more dominant because of a reasonably

higher oscillator strength [197]. In the experiment, we should therefore only observe a

single peak in contrast to the simplified calculations. Its resonant phonon wave vector

is shorter, accompanied by a slightly higher phonon frequency as plotted in Fig. 5.12

d). The situation gets more complicated for a larger number of layers as in general due

to additional πand π∗electronic bands, more optical excitations are possible allowing

for various double-resonant scattering paths. In all of them, slightly different resonant

phonon wave vectors are involved, resulting in diverse Raman shifts. This somewhat gen-

eral trend can be followed when comparing the Raman spectra in Fig. 5.11. The peak at

around 3220cm−1broadens and gets more asymmetric when the number of layer increases.

The maxima of the LO phonon branches for the three calculated system are very close

as can be seen in Fig. 5.12 b). However, they differ in the phonon wave vector exhibiting

5.2. The second-order vibrational density of states in multilayer graphene

the maximum frequency [191]. This should be observable in a double-resonant Raman

process: The laser excitation energy required to observe the highest G0-mode frequency

should be different in graphene compared to bi-, tri- or multilayer graphene. However, this

is only a prediction from theory and, to our best knowledge, a corresponding experiment

has not been performed yet. The additional LO phonon branches will cluster between

the E2gand E1umodes in graphite when the layer number increases [191]. Therefore

the maximum frequency does not reasonably change as a function of the layer number,

but the vibrational density of states increases, especially for contributions slightly below

the maximum of the LO overbending. A corresponding plot can be seen in Fig. 5.12 c)

where the LO phonon related DOS peak (∼1630 cm−1) broadens as a function of the

layer number.

Experimental challenges with deep-UV measurements of graphene and related

compounds

It is well known that there is a reasonable interaction of graphene and related compounds

with UV light, especially in the presence of oxygen [198–201]. This has turned out to be

a problem as, except for graphite, the carbon related compounds simply disappear after a

short UV light exposition with an accompanying decrease of Raman intensity. This also

applies to functionalized and pristine diamondoids as discussed in Chapter 4 although

their bandgaps are partly above the used laser excitation energies. To circumvent this

problem, we have not just integrated the Raman intensity over time, but also over an area.

Resulting Raman spectra are then added up to increase the signal-to-noise ratios. For

instance, graphene is only stable for about 25s under an irradiation intensity of ∼1mW

independent from the laser excitation energy in the deep-UV region. In fact, we could

not find any differences in its stability for photon energies between 4.69eV and 5.46eV. In

contrast, graphite seems to be stable under UV irradiation. We however believe that also

graphite degenerates in the focus volume of the laser spot but there are always sufficient

graphite residuals left giving rise to a reasonable Raman signal. Although we have used

integration times up to 4h, we could not observe changes in the Raman spectra of graphite

throughout many samples.

Compared to graphene, multilayer graphene is supposed to be more stable in the face

of external influences, mainly due to a strain reduction [202, 203]. In our UV-Raman

experiments we have yet observed a fast degradation of the measured samples. We have

therefore tried to measure the multilayer graphene samples in a nitrogen and in an argon

atmosphere to avoid the contact to oxygen from the atmosphere. Nevertheless, we have

still observed a sample degradation but with a lower rate. A last attempt to stabilize

the samples has been done with the help of the Ralph Krupke group [204] from the KIT

(Karlsruhe Institute of Technology) where atomic-layer deposition (ALD) has been used to

deposit Sapphire (Al2O3) onto a pre-characterized Si/SiO2wafer with multilayer graphene

samples. The Sapphire layers were supposed to encapsulate the multilayer graphene from

extrinsic influences. We have tested thicknesses of 10nm, 20nm, and 30 nm.

Unfortunately, we have still not been able to stabilize the samples for a sufficient time

of ∼1.5h necessary for the deep-UV measurements. The Sapphire deposition however

reduced the degradation rate.

5.2. The second-order vibrational density of states in multilayer graphene

In the end, we have not been able to convincingly distinguish between Raman spectra

from multilayer samples in the deep-UV regions. In a degradation process, we remove

the uppermost layers first, followed by a degradation of the next layers. For instance,

this turns four-layer graphene into three-layer graphene, into bilayer graphene, and then

into single-layer graphene, making assignment of appearing Raman peaks to a certain

layer number unfeasible. To circumvent this problem, we have decided to measure in the

near-UV rather than in the deep-UV as shown for the graphite, graphene, and carbon

nanotubes samples. Although not discussed before, we have not observed a degradation

for excitation energies of εL= 3.81eV and a Sapphire layer thickness of 30nm even for

integration times up to 3 h and laser powers of ∼5mW. This has made layer-number

dependent density of states measurements possible.

5.2. Temperature-dependent second-order vibrational density of states of graphite

5.2.2 Temperature-dependent Raman measurements of the second-

order vibrational density of states

(a)

E1u

E2g

200 300 400 500 600 700 800

1570

1580

1590

Temperature (K)

Phonon energy cm−1

Giura et al.

DOS measurements

(b)

200 400 600 800

Temperature (K)

∆ωcm−1

ωE1u−ωE2g

Figure 5.13: (a) Temperature-dependent Raman shifts of the E1uand E2gmode in graphite are plotted.

Red and blue data points are taken from Ref. [177]. Circles correspond to density of states measurements

as described in the sections before. (b) The energy difference ∆ωof the E1uand E2gmodes as shown in

(a) is plotted.

The temperature-dependent frequency shifts of Raman-active lattice vibrations in sp2

carbon materials were reported in detail during the last two decades. A general down-

shift of Raman frequencies with increasing temperature is observed for all Raman modes

including the G-band, the D-band, and the radial-breathing mode (RBM) in carbon nan-

otubes [177, 205–210]. Reported temperature-dependent shift rates of the G mode in

graphite, HOPG (Highly oriented pyrolytic graphite), or carbon nanotubes slightly differ,

depend on the heating method, and usually exhibit χ∼2·10−2cm−1/K [177,205–210]. A

theoretical explanation for the temperature dependence of the E1uand E2gΓ-point vibra-

tions in graphite is given by Giura et al. [177]. According to the authors, the observable

downshift occurs due to a strong anharmonic, four-phonon interaction [177, 211]. This

effect even superimposes the lattice expansion that leads to an upshift of the Raman fre-

quency when the lattice is heated [177]. Different temperature-dependent shift rates are

reported for the E1uand the E2gmodes in graphite [177]. This difference is explained by

the unlike four-phonon anharmonic coupling of the respective modes to the out-of-plane

ZO’ branch [177]. However, the calculations can only be partly confirmed by expensive

experiments as shown in Ref. [177].

As discussed in the beginning of this chapter, the measurement of the second-order vi-

brational density of states allows for the contemporaneous measurement of both the IR

active E1uand the Raman active E2gmodes in graphite without the need of complicated

experimental setups [52, 177]. We can further analyze the shift rates of phonon wave

vectors away from the Γpoint that, to our best knowledge, have not been reported in

literature yet. By these measurements, we can more easily cover a larger temperature

region compared to the measurements as shown in Ref. [177]. We will show that the

temperature-dependent shift rates of the E1uand E2gmodes are more similar and do not

match for temperatures of 750K as theoretically predicted [177]. In fact even for tem-

5.2. Temperature-dependent second-order vibrational density of states of graphite

2LO overb.

3220

3240

3260

3280

Phonon energy (cm−1)

2E1u

2E2g

0 200 400 600

3120

3140

3160

3180

Temperature (◦C)

M point 2LA

200 400 600

2660

2680

2700

2720

M point 2TO/2LO

2780

2800

2820

2840

T= 415◦C

T= 27◦C2E2g

2E1u

2LO overb.

3000 3100 3200 3300

Phonon energy cm−1

Intensity (arb.units)

Figure 5.14: Right side: Raman spectra with density of states related Raman peaks of graphite are

plotted for two different sample temperatures as given in the figure. Left side: Temperature-dependent

shift rates of the second-order density of states peak as given in the figure. Compare Fig. 5.5.

peratures of almost 900K, the frequencies of the E1gand E2gdo not comply with each

other. We compare the temperature-dependent Raman shifts of the E1uand E2gmodes

in graphite to the data shown in Ref. [177] in Fig. 5.13. Solid, colored lines correspond to

a fit of the calculated temperature-dependent Raman shift calculated via the four-phonon

spectral function [177,211]. Black dots are measurements of the second-order density of

states. Comparing both data sets, we find that there is a good agreement of the experi-

mentally obtained shift rates for the E2gmode. In contrast, the shift rate of the E1umode

is weaker compared to the findings in Ref. [177] where the calculated slope exhibits twice

the value of the E2gmode. By linear fits, we find shift rates of χE2g∼1.7·10−2cm−1/K

and χE1u∼2.3·10−2cm−1/K. The blue Raman spectrum in Fig. 5.14 shows that for a

temperature of T ∼700 K, there is still a significant energy gap between the E1uand E2g,

contrary to the calculations in Ref. [177]. If we estimate the interception with the help

of the fitted linear functions, we find a very high value of T ∼3100K. This very high

temperature is still below the melting point of graphene [212,213] and might be realistic,

from that point of view.

Various contributions lead to the downshift of both the E1uand E2gvibrational modes.

They can be summarized as [177,211]:

ωph =ω0+ ∆ωle(T)+ ∆ω3p(T) + ∆ω4p(T) + O(~2)(5.7)

The lattice expansion (le) leads to a slight upshift of phonon energies when the temper-

ature is increased whereas the 3-phonon contribution (3p) slightly decreases the phonon

energies for temperatures above T ∼500K [177]. Both contributions are equal for the

E1uand E2gmodes. The largest contribution has the 4-phonon interaction giving rise

to the overall negative slope [177]. The remaining term is referred to other higher-order

contributions.

Since the lattice expansion contribution and the 3 -phonon coupling are the same for both

shift rates and since we find a good agreement of both calculations and experiments for

the E2gmode, our experimental results suggest that either the anharmonic 4-phonon

coupling for the E1umode is overestimated or the higher-order contributions are not neg-

5.3. Summary

ligible. A further reconsideration of the calculations might be needed in order to achieve

a better agreement to experiments.

The temperature-dependent shift rates of the M point 2LO/2TO, 2LA, and 2LO over-

bending phonon energies are χq6=0 ∼4.2·10−2cm−1/K, measured from the second-order

peaks i.e. they approximately exhibit the same values as found for the Γ-point vibra-

tions. From these measurements, we can say that the anharmonic coupling also counts

for vibrations away from the Γpoint. Vibrational modes with arbitrary phonon wave

vectors away from the Γpoint do not exhibit the same symmetry as group theory tells

for the high-symmetry points. The area of validity for the model proposed in Ref. [177]

is therefore larger than suggested in the article.

5.3 Summary

In summary, we have analyzed the ultra-violet Raman spectra of graphene, graphite, and

carbon nanotubes. Different to the widely understood second-order Raman spectra in

the optical visible range, we do not find any double-resonant Raman modes for excitation

energies higher than 4.7eV. Instead, we observe second-order vibrational density of states

related Raman peaks between ∼2650 and 3250 cm−1. On the one hand, deep-UV Raman

spectra contain peaks from very high momentum M-point phonons that are otherwise not

observable in Raman experiments. On the other hand, measuring the second order den-

sity of states, allows for the measurement of Raman inactive modes that are not visible

in first-order measurements. Ultra-violet Raman spectroscopy therefore gives insights to

the phonon dispersions of graphene, graphite, and carbon nanotubes that are otherwise

only accessible by more complicated experimental setups.

For an excitation energy close to the M-point energy of ∼4.7eV, we observe the begin-

ning of the 2Dmode for the lowest phonon momenta possible. In contrast to excitation

energies in the optical visible range, the TO phonons involved stem from the proximity

of the Γpoint with energies up to 1550cm−1.

By second-order vibrational density of states measurements in multilayer graphene, we

show that the layer-number dependent frequency shift of degenerate LO/TO phonons is

only small. The LO overbending related second-order peak in multilayer graphene also

depends only marginally on the layer number. By these measurements, we can show that

the Kohn Anomaly induced downshift of Γpoint LO/TO phonon frequencies in graphene

and graphite also counts for Raman inactive vibrational modes. Temperature dependent

Raman measurements show unlike shift rates of the E2gand E1umodes in graphite. How-

ever, we do not find a crossing of both phonon modes up to ∼900 K indicating that

the anharmonic phonon coupling for the E1umode is overestimated in corresponding

calculations.

5.3. Summary

6. Carbon nanotubes -

Low-intensity Raman modes and UV

spectroscopy

Parts of this chapter are published in Ref. [214]

This chapter discusses low-intensity Raman modes in the spectrum of carbon nanotubes

between 1800cm−1and 2500 cm−1. Although these modes are reported several times in

the literature, there is no uniform assignment available. The lack of a comprehensive

understanding is further confirmed by the lack of theoretical simulations in the denoted

spectral range.

We show measurements with a broad range of excitation energies and calculate double-

resonant Raman modes of an ensemble carbon nanotube sample with respect to each

individual tube. By these simulations we can assign and understand the dispersions,

peak intensities, and general compositions of double-resonant Raman modes. Especially

the quasi-continuous dispersions of double resonant Raman modes in ensemble samples

have not been theoretically addressed so far.

Raman spectra are simulated by a DFT + zone-folding and a non-orthogonal tight-binding

approach with apparent differences in the assignment of the mode at around 1950cm−1.

Both theoretical approaches are extensively discussed with respect to their initial assump-

tions necessary for the concrete calculations.

Even though carbon nanotubes have been spectroscopically analyzed for more than two

decades, detailed data for their optical responses in the UV range are still very scarce in

the literature. It is for instance widely unknown how their optical absorption changes for

excitation energies close to the M-point energy. With a tight-binding approximation +

zone-folding approach, we calculate optical transition energies and matrix elements for all

allowed transitions up to those stemming from the Γpoint. The so emerging "extended

Kataura plot" is discussed in detail as it contains novelties that are not present when only

the first optical transitions of carbon nanotubes are taken into account.

Coincidentally, we discovered how carbon nanotubes and UV radiation can be utilized to

oxidize iron clusters and form the mineral siderite. We explain the mechanism and discuss

its Raman footprint measured in a HipCO CNT sample.

6.1. Low-intensity Raman modes in carbon nanotubes

×13

1900 2000 2100

1800 2000 2200 2400 2600 2800 3000 3200

Raman shift cm−1

Intensity (arb.units)

×13

2300 2400 2500

2LO

D+G

Figure 6.1: Second-order Raman spec-

trum of an HipCO buckypaper carbon

nanotube sample. The laser excitation

energy is εL= 2.41 eV. Insets show a

magnification of the analyzed double-

resonant Raman modes.

6.1 Low-intensity Raman modes in carbon nanotubes

6.1.1 Low-intensity Raman modes - An introduction

The Raman spectra of carbon nanotubes are today well understood [45,53,134,135,215].

This counts, in particular, for the broad range of different carbon nanotube samples avail-

able where electronic and vibrational properties can differ significantly. In general, carbon

nanotube samples used in experiments can be differentiated by their growth mechanism

and further by their composition, i.e. ensemble samples or samples consisting of iso-

lated carbon nanotubes [38,216–222]. Ensemble samples can further be distinguished by

their constellations: Ensembles with a large variety of chiralities [15, 52, 220, 223, 224],

chirality-enriched ensembles [218,225, 226], or metallic/semiconducting enriched ensem-

bles [227–230]. In contrast, theoretical computations usually consider only particular

properties of carbon nanotubes such as optical matrix elements [61,197,231], electronic

properties [232–235], vibrational properties [16,93,232,236], or electron-phonon coupling

elements [80, 237, 238]. There is an obvious disparity between the Raman data that is

experimentally obtained and the rather discrete aspects that are theoretically discussed.

For the understanding of experimentally obtained Raman data of common samples, it

is necessary to consider each individual tube with respect to its unique properties. This

usually covers a large set of different electronic and vibrational properties rather than

only distinct aspects. On the next pages, we will first give an overview on low-intensity

Raman modes and then compare both spectra and calculations of two low-intensity Ra-

man modes in ensemble samples.

A lot of effort has been put in the understanding of the most pronounced modes in

the Raman spectra of carbon nanotubes such as the G+/G−modes, the Dmode, or the

radial-breathing mode RBM [215, 220, 229, 230, 238–241]. Instead, low-intensity Raman

modes have not been of such a great interest and even today some question are still

open. The intermediate-frequency range between the RBM (≈300 cm−1) and the D

mode (≈1300 cm−1) exhibits many dispersive low-intensity modes with non-convincing

explanations [242, 243]. A more conclusive explanation of the origin from the rich fea-

tures in the intermediate-frequency region was given recently [244]. Instead, the only

6.1. Low-intensity Raman modes in carbon nanotubes

non-dispersive, low-intensity mode found at around 850cm−1seems to be well under-

stood [15,16,245]. The spectral range between the Gbands (≈1600cm−1) and the 2D

band (≈2700cm−1) also exhibits many low-intensity dispersive Raman modes that are

not yet convincingly assigned. Two of them are highlighted in Fig. 6.1. Experimental

works available in literature, tentatively assign these combination modes only by their

dispersions in comparison to graphene. Corresponding Raman modes in carbon nan-

otubes have not been calculated before, impeding a concluding assignment. The Raman

mode at around 1950 cm−1is either attributed to a TO+LA or to a LO+LA combination

mode from the Γpoint. Fantini et al. [246] report a multipeak structure and assign the

peaks in covalently functionalized carbon nanotubes to originate from scattering with both

TO+LA and LO+LA phonons. Brar et al. [247] generally assign the mode to a TO+LA

combination whereas Ellis et al. [248] state the Raman band in acid treated double-wall

carbon nanotubes originates from scattering with LO+LA phonons. All articles suggest

that the combination mode is an intravalley, process i.e., a Γ−point process. The dis-

persive peak at around 2450cm−1in graphene and bilayer graphene was explained by a

TO+LA combination from the Kpoint, i.e. it is an intervalley process with a very good

agreement of both experiments and calculations [14,149,165,166]. Its asymmetry can be

well explained by the overall two-dimensional scattering process, involving phonon wave

vectors adjacent to the high-symmetry directions. A second peak towards higher energies

can be observed that was attributed to a LO+LA@ Γcombination with a much lower

intensity [14,149]. It is a quasi non-dispersive peak for the range of phonon wave vectors

that can be accessed by excitation energies in the optical visible range. The very weak

dispersion can be well understood from the fact that the TO and LA phonon branches in

the proximity of the Kpoint have almost similar slopes but with opposite signs [14,161].

Former assignments stated this process is a 2iTO phonon mode involving q=Kphonon

wave vectors [165–167]. However, we believe that this assignment is questionable, since

these processes should be canceled out due to destructive quantum interference [14,145].

In contrast to carbon nanotubes, there is an uniform assignment of Raman modes in

the spectral range between 1700 cm−1and 2100cm−1for other graphitic materials, such

as graphene and bilayer graphene. Calculated Raman spectra suggest many possible

modes: LOZO0@Γ, TOZO0@K, LOTA @Γ, LOLA @Γ, TOZO @Kand a TOTA@ Kin

bilayer graphene [249] whereas in graphene LOTA @Γ, LOLA @Γ, 2LA@K, TOTA @K,

and TOLA@Kcombination modes are suggested [14,149]. Hereby, the notation "@Γ"

refers to an intravalley process, i.e. the phonon wave vectors involved stem from the

proximity of the Γpoint. "@K" refers to an intervalley process, i.e. the phonon wave

vectors involved predominantly stem from the proximity of the Kpoints. Due to its

symmetry, optical out-of-plane vibrations are Raman forbidden in graphene lowering the

amount of possible Raman active combination modes compared to bilayer graphene. In-

stead, bilayer graphene has an inversion center between the layers, making out-of-plane

vibrations Raman active. This gives a reasonable contribution of combination modes with

ZO phonons in the spectral region between 1600 cm−1and 2700cm−1that have so far not

been considered in the Raman spectra of carbon nanotubes [14,249].

6.1. Low-intensity Raman modes in carbon nanotubes

1.71 eV 1.74 eV

1.77 eV

1.80 eV

1.91 eV

2.12 eV

2.14 eV

2.16 eV

2.18 eV

2.20 eV

2.34 eV

2.38 eV 2.41 eV

2.48 eV

2.5 eV 2.54

2.63 eV

2.67 eV

2.71 eV

2.73 eV

(a)

TOZO

1900 2000

Raman shift cm−1

Intensity (arb.units)

2.16 eV

2.20 eV

2.34 eV

2.38 eV

2.41 eV

2.48 eV

2.5 eV

2.54 eV

2.67 eV

2.63 eV

2.71 eV

2.73 eV

(b)

iTOLA

2350 2400 2450 2500

Raman shift cm−1

Figure 6.2: [(a) and (b)] Raman spec-

tra of the low-intensity Raman modes as

shown in Fig. 6.1 are plotted. Excita-

tion energies are given next to the spec-

tra. [(c) and (d)] Experimentally ob-

tained dispersions for the 1950 cm−1and

the 2450 cm−1Raman bands as a func-

tion of the laser excitation energy. Lines

are linear fits of the bimodal peaks fitted

with Lorentzian profiles as indicated in

the Raman spectra.

6.1. Low-intensity Raman modes - DFT calculations

6.1.2 Low intensity Raman modes - Experimental Raman spec-

tra

In Fig. 6.2, we show different Raman spectra of the low-intensity Raman modes as high-

lighted in Fig. 6.1. A broad range of laser excitation energies between 1.7−2.7eV is cov-

ered. We observe a clear bimodal peak structure for excitation energies between 2.14eV

and 2.5eV in case of the Raman mode at around 1950cm−1(left panel). A change in the

excitation energy results in a change of the peak composition. For higher and lower exci-

tation energies, i.e below 2.14eV and above 2.5eV, the line shape corresponds to a single

peak with a slight asymmetry. The low-frequency peak vanishes for increasing excitation

energies, whereas the peak towards higher wave numbers arises at 1.9eV and dominates

the Raman spectra up to 2.73eV. The average frequency difference between the two peaks

is around 55cm−1for excitation energies between 2.14eV and above 2.5eV. Both peaks are

highly dispersive, shifting towards higher frequencies with increasing excitation energy.

The right panel of Fig. 6.2 shows the low-intensity Raman mode at around 2450cm−1. For

excitation energies between 2.16eV and 2.73eV, we observe Raman peaks that exhibit two

contributions. They are separated by approximately 25 cm−1for all measured excitation

energies. Both peaks show a weak dispersion towards lower wave numbers increasing the

laser excitation energy.

The experimentally obtained Raman frequencies are plotted as a function of the excita-

tion energy in Fig. 6.2. In case of the 1950cm−1band, the twofold dispersions exhibit

73cm−1/eV and 92cm−1/ eV for the upper and lower branch, respectively. The exper-

imentally obtained dispersions are in contrast to a previous work where the reported

dispersion is found to exhibit 230cm−1/eV [135]. This extremely high value might be

understood from the fact that the authors of Ref. [135] did not consider the twofold con-

tributions of the Raman band as shown in Fig. 6.2: Not considering the ’step’ in the dis-

persion would increase the overall dispersion, but only up to approximately 140cm−1/eV

(compare Fig. 6.4). The remaining gap of roughly 100cm−1maybe originates from wrong

assignments of the modes in the first place. The experimentally obtained dispersion of

the 2450cm−1band is 22cm−1/eV in good agreement to the literature [14,135].

6.1.3 Low-intensity Raman modes - DFT calculations

In the following, we will show simulations of both dispersive Raman bands and discuss

their respective origins. We first assign the peaks and then discuss the overall line shape

of the modes. The quasi-continuous slopes of dispersive Raman bands was a subject

of discussion in the literature [135, 218, 220, 222, 251]. Measurements on chirality en-

riched samples suggest that carbon nanotubes probed within a range of ≈200meV near

their optical transition energies give rise to a reasonable Raman signal with no disper-

sion [218,222,251], due to the van-Hove-singularities in their electronic bandstructures.

Off-resonance excitations result in Raman peaks with much lower intensities and only

slight dispersions [218, 222, 251]. It was shown that because of the mixing of incoming

and outgoing resonances, the intensity maximum of the 2Dmode for a certain transition

energy occurs at Eii +~

2ω2Dand depending on the particular electronic structure with rea-

sonable contributions from lower transitions Ei−n i−n[222]. A similar behavior for other

double-resonant Raman modes can be seen in Fig. 6.3 where full Raman spectra of a

6.1. Low-intensity Raman modes - DFT calculations

1.0 eV

1.2 eV

1.4 eV

1.6 eV

1.8 eV

2.0 eV

2.2 eV

2.4 eV

2.6 eV

2.8 eV

εL=

(9,5) CNT DFT + zone-folding approach, absolute intensities

LO+LA @ ΓTO+LA @ K

(a)

Intensity (arb.units)

(b)

(9,5) CNT DFT + zone-folding approach, normalised intensities

1900 2000 2100 2200 2300 2400 2500 2600 2700 2800

Raman shift cm−1

Intensity (arb.units)

Figure 6.3: Raman spectra of a (9,5) carbon nanotube calculated in a DFT + zone-folding approach are

plotted. Considered are an intravalley LO+LA @ Γand an intervalley TO+LA @ K process. Calculations

are done by Rohit Narula [250].

(9,5) carbon nanotube are calculated within a DFT + zone-folding approach by Rohit

Narula [250]. The upper part shows Raman spectra with absolute intensities as a func-

tion of various excitation energies. According to the optical transition energies and the

respective phonon energies, different Raman peaks show resonances with no perceptible

dispersions although the excitation energies differ between 1.0 eV and 2.8 eV. This can

even clearer be seen in Fig. 6.3 (lower panel) where the same calculations are shown but

with a different normalization. In (b), the Raman spectra for each excitation energy are

normalized to 1 in contrast to (a) where only the most intense peak of all Raman spec-

tra is normalized to 1, i.e. in (a) the relative intensity ratios are correctly reproduced.

However, the artificial normalization in (b) reveals low-intensity modes that only show

marginal dispersions. The large variety of Raman peaks originates from the mixture of

contributions from different optical transitions. For each optical transition, a likewise

probing of confined phonon branches with different band indices contributes to the over-

all spectra as well as contributions from inner and outer processes [222,251]. For higher

excitation energies, intensities of the LO+LA @Γcombination modes generally exceeds

the TO+LA@Kcombination due to an increasing joint density of states, in agreement

to calculations for graphene [14].

We now advance the analysis from a single tube to an ensemble of carbon nanotubes.

To simulate Raman spectra of an ensemble sample, we first need to know its specific com-

position. We have therefore performed Raman resonance profiles of the RBM between

a range of 1.58eV and 2.73 eV of a HipCO sample. By a comparison of experimentally

6.1. Low-intensity Raman modes - DFT calculations

Table 6.1: List of carbon nanotubes considered for the calculations. Chiral indices were determined

experimentally by resonance Raman profiles of the RBM. Detailed spectra can be found in Ref. [52]

semiconducting metallic

(16,0) (12,2) (10,2) (10,6) (7,7) (12,0) (10,4) (9,9)

(9,1) (8,3) (8,6) (7,6) (9,3) (8,5) (9,6) (11,2)

(11,1) (7,5) (10,3) (9,5) (11,5) (12,6) (13,1) (10,7)

(11,4) (9,8) (14,1) (12,5) (13,4) (11,8)

(13,3) (11,7)

obtained optical transition energies and RBM frequencies to calculations, we can identify

individual CNTs with their chiral indices [93,232,234]. This approach was first suggested

by Hagen Telg and is also referred to as pattern recognition [220]. A list of the so found

chiral indices can be seen in Table 6.1. The diameter range is ≈7˚

A−13˚

A with both

semiconducting and metallic nanotubes [52]. With the used set of excitation energies,

we optically excite metallic (EM

11) as well as semiconducting (ES

22, ES

33, and ES

44) carbon

nanotubes.

Similar to the calculations of the (9,5) tube as shown in Fig. 6.3, we have calculated

Raman spectra of every individual carbon nanotube listed in Table 6.1. To simulate the

ensemble sample, we have then added up each individual spectrum. The resulting Raman

spectra are shown in Fig. 6.4. Individual Raman spectra in the first row are normalized to

1 and then summed up to one common Raman spectrum. This reveals low-intensity con-

tributions that are not visible in the bottom row where the Raman spectra of all carbon

nanotubes are summed up and then normalized to 1 for each excitation energy. In the

latter case, contributions from carbon nanotubes with a low absorption or low electron-

phonon coupling elements contribute less to the overall spectrum. Dot sizes in (b) and

(d) correspond to the peak intensities in (a) and (c). From the calculations in Fig. 6.4,

we can derive several statements: (i) We observe a highly dispersive, multimodal Raman

band at ≈2000cm−1. Its origin is an LO+LA combination from the Γpoint, i.e. it is

an intravalley process. The Raman bands also contain TO+LA contributions from the Γ

point but with a much lower intensity. (ii) We find a multimodal TO+LA combination

mode at around 2580cm−1. In contrast to the experiments, it exhibits a weak positive

dispersion of 70cm−1/eV and can be referred to an intervalley process. The weak disper-

sion both in the experiments and calculations can be attributed to the likewise dispersions

of the TO and LA phonon branches in the proximity of the Kpoint, but with different

signs [14,152,161]. (iii) The dispersion of the Raman bands from the ensemble measure-

ments originates from different carbon nanotubes that are in resonance to the excitation

energy. Each CNT adds an unique contribution depending on its chirality, the excitation

energy, and the lengths of the relevant phonon wave vectors involved in the second-order

Raman process. The relevant double-resonant Raman modes of a single carbon nanotubes

do not show a substantial dispersion. Thus, the dispersion can be explained by an average

Raman spectrum of all carbon nanotubes in the sample [218,222]. (iv) The zone-folding

calculations considered all 6 graphene derived phonon branches against each other for a

6.1. Low-intensity Raman modes - DFT calculations

(c)

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

1600

1800

2000

2200

2400

2600

2800

Laser energy (eV)

Raman shift cm−1

(d)

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

Laser energy (eV)

(a)

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

1600

1800

2000

2200

2400

2600

2800

Laser energy (eV)

Raman shift cm−1

(b)

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

Laser energy (eV)

TO+LA @ K

LO+LA @ Γ

Figure 6.4: Calculated Raman spectra in a DFT + zone-folding approach are plotted. The dot diameters

represent intensities of single peaks. In (a) and (b), the intensities of the combinations modes are

separately normalized to 1 and then composed to a spectrum. In (c) and (d), the combination modes

are summarized and then normalized to 1. According to the used approach, the 2450 cm−1mode can be

assigned to a TO+LA @ K and the 1950 cm−1mode to a LO+LA @ Γprocess as denoted in the figure.

Calculations were carried out by Rohit Narula [250].

total of 36 possible combinations modes. The assignment of combination modes was done

analyzing the Raman intensities and dispersions of all possible processes. As a conse-

quence, the calculations indicate that the combination modes LO+LA@Γ(1950cm−1)

and TO+LA@K(2450 cm−1) are responsible for the two low-intensity Raman bands

observed in the experiment, in agreement to former assignments in graphene [14]. This

explicitly excludes a possible TO+ZO@ Kprocess, as suggested by A. Vierck, due to

ultra-low electron phonon coupling elements. Joint phonon energies of such a process

would be close to those from a LO+LA@Γprocess and thus, it must be considered in

the discussion. However, the zone-folding approach is known to give incorrect results es-

pecially for the acoustic modes. For instance, it fails to predict the twiston acoustic mode

or the transition of the out-of-plane ZA acoustic mode into an optical phonon in carbon

nanotubes (RBM) [45]. Moreover, scattering processes with out-of-plane ZO phonons in

graphene are forbidden by symmetry [14] which does not necessarily counts for carbon

nanotubes [15, 16, 52]. Therefore, the contributions of a possible TO+ZO@Kprocess

might be underestimated in a DFT + zone-folding approach. (v) Calculated phonon

energies are systematically higher than the experimental values due to the well-known

underestimation of both electron and phonon energies on account of the employed LDA

6.1. Low-intensity Raman modes - Non-orthogonal tight-binding calculations

ΓM K Γ

500

1000

1500

phonon wave vector

Energy cm−1

1.70 eV

2.20 eV

2.73 eV

1900 2000 2100 2200

Raman shift cm−1

Intensity (arb.units)

LO+LA @ Γ

TO+ZO @ K

Figure 6.5: Left: The phonon dispersion of graphene along the high-symmetry directions of the 1.

Brillouin zone is plotted. Highlighted are parts of the LO/LA and TO/ZO phonon branches that are

used to calculate Raman spectra for various excitation energies on the right side. The calculations are

done to visualize the likewise dispersions of the two different processes and do not contain any coupling

elements [14]. In graphene, a TO+ZO process is forbidden by symmetry [14]. Within the energy range of

1.7 eV and 2.73 eV the LO+LA @ Γ(blue) and TO+ZO @ K(red) processes have comparable frequencies

and dispersions.

calculations. This explains the slightly too large frequencies and dispersions, compared to

experiments. Especially the energy of the TO and LO derived phonon branches is over-

estimated, leading to a positive instead of a negative dispersion in case of the TO+LA

combination mode.

The LO derived phonon branches in metallic carbon nanotubes exchange their frequencies

with those from the TO derived phonon branches at the Γpoint, compared to semicon-

ducting carbon nanotubes [215,252–257]. The reason is a strong Kohn Anomaly reason-

ably softening the LO derived phonon frequencies. The degeneracy of the TO/LO phonon

branches at the Γpoint in graphene is in metallic carbon nanotubes predominantly lifted

due to confinement effects [252] whereas in semiconducting tubes the degeneracy is rather

lifted due to curvature effects [252], explaining the fundamentally different Raman spectra

from both types of carbon nanotubes. However, the exchange of related phonon frequen-

cies only counts for a very limited region of phonon wave vectors close to the Γpoint [252].

Resonant phonon wave vectors probed in a LO+LA@ Γdouble-resonant Raman process

with excitation energies in the optical visible range are usually larger and thus our DFT

+ zone-folding computations indicate that the DR Γ−point processes involve the higher

frequency LO derived phonon branches. It is therefore sufficient to reduce the discussion

on the LO derived phonon branches for intravalley processes.

6.1.4 Low-intensity Raman modes - Non-orthogonal tight-binding

calculations

An assignment of the 1950 cm−1band only by its dispersion and frequency cannot con-

vincingly be done. As shown in Fig. 6.5, a LO+LA @Γ(blue) and a TO+ZO @K(red)

process have assimilable frequencies and dispersions when optically excited with usual

laser excitation energies in the optical visible range. The calculations do not contain any

6.1. Low-intensity Raman modes - Non-orthogonal tight-binding calculations

(11,0) @ 1.988 eV

(15,0) @ 2.208 eV

(9,9) @ 2.314 eV

(12,0) @ 2.504 eV

(16,0) @ 2.590 eV

(7,7) @ 2.764 eV

TO+ZO @ K TO+LA @ K

(a)

1800 1900 2000 2100 2200 2300 2400

Raman shift cm−1

Intensity (arb.units)

non-orthogonal tight-binding approach

TO+LA @ K

(c)

∼ −38 cm−1/eV

2300

2350

2400

Raman shift cm−1

M11

E22

E33

TO+ZO @ K

(d)

∼107 cm−1/eV

1.6 1.8 2 2.2 2.4 2.6 2.8 31900

1950

2000

2050

Excitation energy (eV)

M11

E22

E33

qres

(b)

-1 Γ1

−4

−2

Helical wave vector q

a

Energy (eV)

(8,4) tube

Figure 6.6: (a) Calculated Raman spectra of various carbon nanotubes excited at their E11 transi-

tion energies within a non-orthogonal tight-binding approach are plotted. Calculations have been done

by Valentin N. Popov [258]. The symmetry-based approach suggests that the 1900 cm−1band is a

TO+ZO @ K process as indicated in the figure; the 2400 cm−1band is assigned to a TO+LA @ K process.

(b) Illustration of the simplified scattering process used for the calculations in (c) and (d) is plotted. The

initial idea for that approach was brought up by Felix Herziger [259]. (c) and (d): Calculated phonon

frequencies of the TO+LA (c) and TO+ZO (d) combination modes for the EM

11, ES

22, and ES

33 transition

energies are shown. Considered are only "inner" processes and carbon nanotubes as shown in Table 6.1.

Indicated are the quasi-linear slopes of the dispersions, equivalent to those in Fig. 6.2.

coupling elements and therefore, the Raman intensities are not predicted correctly. A

TO+ZO@Kprocess is forbidden in graphene due to symmetry reasons [14]. The large

line width of the LO+LA@Γprocess originates from the very large joint density of states

probing the overbending of the LO phonon branch around the Γpoint in the 1st Brillouin

zone.

Since the DFT + zone-folding approach requires the initial electron-phonon matrix el-

ements calculated for graphene, the coupling elements for carbon nanotubes might be

reproduced incorrectly. We have therefore also calculated double-resonant Raman spectra

of various carbon nanotubes within a non-orthogonal tight-binding approach. All calcu-

lations are done by Valentin N. Popov [258] and are summarized in Fig.6.6. A dispersive

mode at around 1950cm−1can be observed that is assigned to a TO+ZO @K combination

mode in the used approach. This assignment is a clear difference to the results of the

6.1. Low-intensity Raman modes - Dispersions and line shapes

DFT + zone-folding calculations as discussed before. The 2400 cm−1band is assigned to

a TO+LA@K combination in agreement to the assignment before. However, in contrast

to the DFT+zone-folding approach a slight negative dispersion is observed. This out-

come describes the experimentally observed negative dispersion better compared to the

other computational approach. The calculations explicitly contain the Kohn Anomaly in

metallic carbon nanotubes leading to both a strong softening of LO phonon frequencies

and an increase of the electron-phonon coupling [257]. Still, we do not find a LO+LA@ Γ

process in carbon nanotubes.

A final assignment of the band at around 1950 cm−1only based on the computations can-

not convincingly be done. Due to the well-known shortcomings of the DFT+zone-folding

approach describing curvature-induced changes of vibrational properties, a tentative as-

signment of the mode to a TO+ZO@K process as suggested within the non-orthogonal

tight-binding approach seems feasible [260]. However, we believe that more experimental

data is needed. For instance, it might help to perform measurements on isolated carbon

nanotubes as appearing Raman modes can be identified and more certainly assigned, com-

pared to ensemble samples. A contemporaneous measurement of both Stokes/Stokes and

Stokes/anti-Stokes, double resonant Raman processes as shown by Herziger et al. [187]

and suggested in Ref. [246] might help to distinguish between a LO+LA@ Γand a

TO+ZO@Kprocess. This idea utilizes the fact that due to the different phonon en-

ergies involved, the combined Stokes/anti-Stokes phonon energies of a LO+LA @Γpro-

cess should be systematically higher than for a Stokes/anti-Stokes TO+ZO@Kprocess.

Thus, an assignment could be done more easily and was then a strong argument for the

correct assignment of the ≈1950 cm−1band as discussed before.

6.1.5 Low-intensity Raman modes - Dispersions and line shapes

We now continue with a discussion of the line shapes of the measured, double-resonant

Raman modes. The bimodal peak structures of the observed modes originate from dis-

tinctive contributions from different species of carbon nanotubes. Due to the specific

ensemble of the HipCO sample, mainly semiconducting tubes are probed below εL≈

2.1eV (ES

22) and above εL≈2.5 eV (ES

33). In the intermediate range, both semiconducting

and metallic (EM

11) carbon nanotubes are probed. Different tubes having likewise transi-

tion energies but are either semiconducting or metallic necessarily need to have different

diameters. Compared to ES

22 (ES

33) transitions, metallic tubes with EM

11 transitions have

larger (smaller) diameters for similar transition energies and therefore different phonon

energies [45,59, 232, 234,259]. Hence, we attribute different peaks in the Raman bands

to carbon nanotubes with different diameters. In detail, due to the specific ensemble

analyzed, we can attribute Raman peaks to semiconducting tubes for excitation ener-

gies below εL≈2.1eV and above εL≈2.5eV. The Raman peaks for excitation energies

between 2.1eV and 2.5eV can be assigned to both metallic and semiconducting carbon

nanotubes. Thus, the origin of the observed multimodal band is the concurrent probing

of carbon nanotubes. Within our analysis we can explicitly exclude the assignment done

by Brar et al. [247] where the bimodal peak of the 1950cm−1band was attributed to both

LO+LA@Γand TO+LA @Γprocesses. To understand the bimodal peak structure in

more detail, we have also modeled Raman frequencies of TO+ZO@K and TO+LA @K

6.1. Low-intensity Raman modes - Dispersions and line shapes

(c)

ZO derived phonon modes

0.5 1 1.5 2 2.5 3

550

600

650

700

Transition energy (eV)

M11”inner” M11”outer”

S22”inner” S22”outer”

S33”inner” S33”outer”

(d)

ZO derived phonon modes

5 10 15 20 25

550

600

650

700

Diameter ˚

A

(b)

LA derived phonon modes

5 10 15 20 25

800

1000

1200

1400

(a)

LA derived phonon modes

0.5 1 1.5 2 2.5 3

800

1000

1200

1400

Phonon frequency cm−1

Figure 6.7: Calculated frequencies of the out-of-plane ZO derived and of the in-plane LA derived

phonon branches in carbon nanotubes are plotted. Optical transition energies and phonon frequencies

were obtained from a six-nearest-neighbor tight-binding approach using the POLSYM code [182]. For

the calculations of the frequencies, we have used the model proposed by Herziger et al. [259]. Different

colors stand for different optical transitions, circles are referred to "inner" processes whereas diamonds

are referred to "outer" processes.

processes with a different computational approach. The model was brought up by Felix

Herziger [259] and is only based on the symmetry of the hexagonal lattice and geometric

considerations. An illustration of the simplified scattering process is given in Fig. 6.6(b).

A resonant phonon wave vector qres mediates between equivalent minima in the electronic

band structure. Its length is qres = 2kand visualizes the usual assumption "qres = 2k" for

M point centered, double-resonant Raman scattering processes in carbon nanotubes [171].

The scattering pass always traverses the Γpoint due to the symmetry of the hexagonal

lattice. This approach does not include phonon energies and limits the scattering only to

extrema in the electronic bands. However, predictions for the frequencies of the Dmode in

carbon nanotubes are convincing and are very close to experimental results [171,259,261].

The model was initially introduced for the description of the Dband in carbon nanotubes

but it is also capable of predicting correct phonon frequencies of other Kpoint centered

scattering processes, such as the TO+ZO@K or the TO+LA @K process as discussed

above. Corresponding calculations can be seen in Figs.6.6 (c) and (d). We have only

considered "inner" processes and those tubes that we have found in the sample (com-

pare Table6.1). Electronic and phonon band structures, and therefore respective optical

transition energies and phonon frequencies, were obtained from a six-nearest-neighbor

tight-binding approach using the POLSYM code [182]. For the TO+ZO@ K process, we

observe frequency "steps" between ES

22/EM

11 and between EM

11/ES

33 transitions, indicated

with linear fits in Fig. 6.6 (c) and (d). In contrast, there are no clear "steps" observable

6.1. Low-intensity Raman modes - Dispersions and line shapes

in case of the TO+LA@K process. This effect is mainly caused by the opposite influence

of curvature to the in-plane LA derived and out-of-plane ZO derived phonon energies as

can be seen in Fig. 6.7. Overall, we find a good agreement to the experimental findings.

The geometry-based approach gives reasonable results describing both the dispersions

and line shapes of the TO+ZO@ Kand TO+LA@ Kprocesses. In fact, the so obtained

dispersions of ∼ −38 cm−1/eV (TOLA @K) and ∼107 cm−1/eV (TOZO @K) are very

close to the experimental values of ∼ −22 cm−1/eV and ∼80cm−1/eV (averaged). We

will therefore extent the discussion to a broad range of carbon nanotubes with diameters

between 5˚

A and 25˚

In the following, the discussion is reduced to the influences of chirality and diameter

only to the ZO and LA derived phonon branches as these components are different in the

TO+ZO@Kand TO+LA @Kprocesses. A comparable discussion of the TO derived

phonon branches in carbon nanotubes can be found in Refs. [171,259,261].

At first, only "inner" processes are discussed i.e. processes where the phonons predomi-

nantly originate from the K−Γdirection. ZO derived phonon energies in carbon nan-

otubes increase with an increasing optical transition energy [Fig.6.7 (c), circles]. In

general, the phonon energies follow the slope of the out-of-plane ZO phonon branch in

graphene [14], i.e. in a M centered double-resonance process, larger excitation energies

result in shorter resonant phonon wave vectors automatically leading to higher phonon

energies since the ZO phonon branch has its maximum at the Γpoint in the 1st Bril-

louin zone. Further, carbon nanotubes with smaller diameters generally exhibit higher

ZO-derived phonon energies for likewise optical transitions [Fig.6.7 (d)]. A clear sepa-

ration of phonon energies between the "inner" EM

11, ES

22, and ES

33 transitions can be ob-

served. The findings agree with those of the TO derived phonon branches in carbon

nanotubes [171,259,261], i.e. smaller carbon nanotubes generally have higher TO derived

phonon energies and the phonon energies of different optical transitions for likewise diame-

ters are separated by several cm−1. These general trends are enhanced in the TO+ZO @K

combination mode since both phonon branches follow the same proportionalities. In par-

ticular, the separation of phonon energies as shown in Fig.6.7 (d) are increased and cause

"steps" in the continuity of the TO+ZO @K dispersion as indicated with linear fits in

Fig.6.6 (d). In contrast, the energies of the LA derived phonon modes show an opposite

behavior [Fig.6.7 (b), circles]. Increasing excitation energies result in lower phonon ener-

gies since the LA phonon branch along the Γ−Khigh-symmetry line in graphene has its

maximum at the Kpoint [14]. Smaller diameters generally have lower phonon energies

with a comparably large overlap from different optical transitions [compare Fig. 6.7 (b)].

This stands in strong contrast to both the TO and ZO derived phonon modes where the

phonon energies are clearly separated.

The opposite behavior of the curvature-induced changes to the LA derived phonon modes

compared to TO derived phonon modes in carbon nanotubes leads to a softening of the

overall influence of the diameter to the TO+LA@ K combination mode. This explains

the uniform peaks in the experimental Raman spectra of the TO+LA@K combination

mode although the analyzed range of excitation energies is large and the sample contains

carbon nanotubes with many different chiralities.

100

6.1. Low-intensity Raman modes - Intensity ratios

2400 2500 2600 2700

Phonon energy cm−1

TO+LA phonon DOS (arb.units)

(11,7)

Intensity ×4.5

2400 2500 2600 2700 2800

(7,7)

Figure 6.8: Calculated joint TO+LA phonon density of states for a (11,7) and a (7,7) carbon nanotube

in a DFT + zone-folding approach is shown. The scale is the same for both spectra. Calculation are

done by Rohit Narula [250].

Using the same approach, we have also calculated phonon energies of "outer" processes,

i.e. scattering processes where phonons originate from the K−Mdirection (compare

Fig.6.5) indicated with diamonds in Fig. 6.7. In case of the ZO derived phonon energies,

the deviations to energies of the "inner" processes are only small since the slopes of the

ZO phonon branch in the K−Γand in the K−Mdirections are almost similar in

the relevant range of phonon wave vectors (compare Fig. 6.5). Instead, the LA phonon

branch increases in energy along the K−Mdirection (compare Fig. 6.5) resulting in

systematically higher phonon energies for the "outer" processes (indicated with diamonds

in Fig.6.7). Depending on diameter and the optical transition of the carbon nanotubes,

the gap between "inner" and "outer" LA derived phonon energies can be several hundred

cm−1. As a result "outer" TO+LA@K processes in the observed HipCO sample should be

observed at around 2600 cm−1and higher energies. This however can not experimentally

be observed. These findings agree with a computational and experimental analysis of

the TO+LA@K process in graphene [149]. By a comparison of measured joint phonon

energies and sophisticated computations, it was found the TO+LA combination mode is

an "inner" process, i.e. that the involved phonon stem from the K−Γdirection [149].

The absence of a double-resonant Raman mode at around 2600cm−1confirms again that

Raman spectra of carbon nanotubes are predominantly determined by "inner" scattering

processes.

6.1.6 Low-intensity Raman modes - Intensity ratios

We now continue with a discussion of Raman intensities of the measured combination

modes. The intensity progressions of low-intensity Raman modes in graphene are still

only barely discussed in the literature and are usually given as ratios in comparison to

the strong 2Dmode [14]. Still, exact experimental values are even today not available.

This also counts for carbon nanotubes samples where an accurate analysis of the intensity

progression of a particular Raman mode gets even more complicated. An exact calcula-

tion of the intensity progression of the measured HipCO sample is given in Figs. 6.4 (c)

and (d) regarding an LO+LA@ Γand a TO+LA@K combination process. Although the

non-orthogonal tight-binding approach suggests that a LO+LA@ Γprocess should not

101

6.1. Low-intensity Raman modes - Intensity ratios

TO+TA @ K

2LA @ K

1.49 eV

2.41 eV

2.54 eV

2.67 eV

2.73 eV = εL

TOZO @ K TOLA @ K

1900 2000 2100 2200 2300 2400 2500

Raman shift cm−1

Intensity (arb.units)

Figure 6.9: Second-order Raman spectra of

an HipCO buckypaper carbon nanotube sam-

ple for various excitation energies given next

to the spectra are shown. The arrow indicates

a possible TO+TA @ K or a 2 LA @ K combi-

nation for the observed Raman mode.

be visible in carbon nanotubes, general statements can still be derived from the DFT

calculations. The relatively low intensity of the TO+LA@K process for high excitation

energies is due to an increasing joint density of states (jDOS) of the LO+LA@ Γmode

within this computational approach. It can mainly be referred to the LO derived phonon

branch that has a very low slope in the probed Γ→Mdirection (compare Fig. 6.5).

Instead, the jDOS of TO+LA @K processes does not show a clear dependence comparing

a large range of phonon wave vectors. This behavior is shown in Fig.6.8 where the jDOS

of TO+LA@K process for a (11,7) and a (7,7) carbon nanotube are calculated in a DFT

+ zone-folding approach. Only for very large excitation energies above 3eV and higher

(and therefore lower frequencies of the TO+LA combination mode) the jDOS tentatively

decreases since the gradient of the LA phonon branch increases in the K→Γdirection.

In contrast, both the ZO and TO phonon branches run parallel in the region of phonon

wave vectors that is usually probed by excitation energies in the visible range (compare

Fig. 6.5). As a result the jDOS should be comparably high and widely independent from

the excitation energy.

In a double resonance process, only a very small range of resonant phonon wave vectors is

responsible for particular modes in the Raman spectra of carbon nanotubes [45,53,218].

Therefore, the jDOS of a certain combination mode only gives a qualitative trend on how

the corresponding Raman intensity evolves as a function of the excitation energy. Consid-

ering the slopes of the discussed phonon branches and therefore the jDOS, we would only

expect small deviations of the intensity ratios from a TO+LA@ Kand a TO+ZO@ K

process.

Besides the jDOS, only the respective electron-phonon coupling elements of the ZO and

the LA derived phonons are different that determine the intensity ratios of the discussed

combination modes. However, neither computational nor experimental values of the re-

spective coupling elements are available today and therefore we can not make reliable

quantitative statements. Since in graphene there is no coupling to the optical out-of-

plane phonons, we believe that the electron-phonon coupling of the ZO derived phonons

in carbon nanotubes should generally decrease for larger diameters.

In Fig.6.9, Raman spectra of the HipCO sample for various excitation energies are plot-

ted showing both the TO+LA@K and TO+ZO @K combination modes. The spectra

are normalized to the TO+ZO@K mode. For all spectra, the TO+ZO@ K has a larger

102

6.1. Low-intensity Raman modes - Intensity ratios

intensity. Between 2.41 eV and 2.73eV, the intensity ratios are likely the same and do

not depend on the excitation energy. In contrast, the intensity of the TO+LA@ K for

εL= 1.49eV is only very low. We tentatively attribute the low intensity to the comparable

low LA electron-phonon coupling elements of the carbon nanotubes probed. Interestingly,

the frequency of the TO+ZO@K mode exhibits ν≈1900cm−1although the laser excita-

tion energy is below 1.5eV. This value stands in contrast to that from the measurements

shown in Fig. 6.2 where the lowest frequency ν≈1875cm−1was found for a higher ex-

citation energy of 1.71eV. Considering the diameter range of 7˚

A to 13˚

A of the HipCO

sample, we start to probe the E11

Stransition with εL= 1.5eV. Again, a concurrent prob-

ing of different optical transitions in carbon nanotubes is only possible, when the tubes

involved exhibit different diameters. This again results in different phonon frequencies

and therefore to different measured Raman shifts.

All spectra showed an additional low-intensity Raman mode at around 2200cm−1. It

exhibits a small negative dispersion with intensities about two magnitudes smaller than

those from the TO+ZO@K or TO+LA @K combination modes. In the literature, this

mode in carbon nanotubes has not been reported before. Experimental data in graphene

is also not available. However, DFT calculations in graphene suggest the existence of

a TO+TA@ K and a 2LA@ K combination mode in the discussed range of phonon fre-

quencies [14]. Both are supposed to exhibit a negative dispersion in agreement to the

measurements. Experimental data for the D” mode (defect mode with LA phonons)

in carbon nanotubes and graphene is available where a dispersion of 100 cm−1/eV was

measured [168]. The dispersion for the corresponding second-order (2D00) process should

amount to ∼200 cm−1/eV, confirming the theoretical findings for the dispersion of the 2D00

mode in graphene [14]. However, this value is larger than the experimentally obtained

dispersion as shown in Fig.6.9. We therefore rather attribute this mode to a TO+TA @K

combination [14] as the calculated dispersion only exhibits ∼30cm−1/eV. A closer analysis

is necessary to make a final assignment.

103

6.2. Extended Kataura plot

6.2 UV Raman spectroscopy: Extended Kataura plot

and RBM measurements

Even in nowadays, the optical transition energies of carbon nanotubes in the UV range

are neither experimentally nor theoretically discussed. The topological change of equi-

energy surfaces in graphene when excited above the M-point energy also leads in carbon

nanotubes to interesting new findings.

On the next pages we discuss how we get access to higher optical transitions above the

M-point exciton, calculate the corresponding optical matrix elements and show calculated

absorption spectra. With the understanding of the respective optical absorptions, we show

and discuss UV Raman spectra of the radial-breathing mode from a HipCO sample.

6.2.1 Optical transitions energies above the M-point energy

The Kataura plot of carbon nanotubes gives a general overview on how optical transi-

tion energies in carbon nanotubes are affected by their diameters [59,262]. The diameter

in turn can be parametrized by the chiral indices and, therefore, the Kataura plot also

gives information on how the diameter determines optical transitions [59,262]. For the

usual range of optically visible transition energies, some general findings can be made:

(i) Quantum confinement determines the general dependence on the diameter, i.e. tubes

with smaller diameters have larger transition energies and vice versa [59,263]. (ii) The

transition energies are arranged in larger and smaller branches: The optical transition

index Eii is constant along a larger branch. Smaller branches only contain a certain

type of carbon nanotubes. Within such a branch, the chiral indices follow the relation:

2n1+n2=constant. (iii) Smaller branches appear pairwise with optical transitions rather

stemming from the K−Mdirection in graphene and transitions rather stemming from

the K−Γdirection. Transition energies from the K−Mdirection are systematically

smaller than from the K−Γdirection. (iv) There is a clear separation between metal-

lic and semiconducting nanotubes. The larger branches only contain either metallic or

semiconducting carbon nanotubes [59, 262]. A calculated Kataura plot can be seen in

Fig. 6.11.

Optical transitions in carbon nanotubes occur between van-Hove singularities [264,265]

that have a strong excitonic character [233,266–268]. The singularities in the absorption

spectra are a consequence of the one-dimensional electronic density of states [264,265] and

arise from electronic bands that locally exhibit no dispersion. Below the M-point exciton,

the areas where the band dispersions disappear are always minima in the electron bands

as can be seen in Fig. 6.10 where the linear electron bands of a (8,0) carbon nanotube

are plotted, calculated in a next-nearest neighbor tight-binding + zone-folding approach.

Red dots indicate the electron momentum for each subband where the optical transition

originates from. In the case of a zig-zag carbon nanotube, the minima are always lo-

cated at the k= 0 and every subband (n=8) exhibits an extremum where an optical

transition is allowed. However, depending on the chiral vector in a general CNT, not

every electron subband exhibits an extremum which limits the amount of allowed optical

transitions [45,53,178].

104

6.2. Extended Kataura plot

(8,0)

E11 ... E88

Γπ

Energy (eV)

Figure 6.10: Electron bands of a

(8,0) carbon nanotubes are plotted.

Red dots correspond to optical tran-

sition from E11 to E88. Note: Zig-

zag tubes where n1is even, always

have one electron and one hole band

with no dispersion. The optical

transition is therefore independent

from kz. Optical transitions higher

than the M-point energy (5.42 eV

within the used next-nearest neigh-

bor tight-binding + zone-folding ap-

proach), are located at maxima of

the combined electron/hole bands

rather than at minima as apparent

for excitations below the M point.

Electronic bands higher than the M-point energy do not have a minimum with a dis-

appearing gradient anymore, but possibly exhibit a maximum, depending on the chiral

indices n1and n2(maxima only refer to areas where the gradient disappears). This point

can be seen in Fig. 6.10 where the five highest bands exhibit a maximum, respectively.

The red line corresponds to a subband with the band index m= 5 that exhibits no dis-

persion (discussed later), but by definition also contains a maximum (minimum).

We have calculated all allowed optical transition energies of carbon nanotubes with chiral

indices n1,n2between 2 and 21 within a next-nearest neighbor tight-binding + zone-

folding approach and have explicitly allowed transitions between minima and maxima.

They are plotted in Fig. 6.11. Allowing transitions between maxima, we get access to

optical transitions higher than the M-point energy (5.42eV within the used approach).

Below an energy of 5.42 eV, we find the known arrangement of transitions energies as

apparent in the usual Kataura plot. However above 5.42eV, the arrangement drastically

changes. We again find larger and smaller branches but this time they seem to have an

opposite dependence on the diameter, i.e. larger diameters have larger transition energies

and vice versa. Black dots represent allowed optical transitions with |Mz|>0, red dots

represent forbidden optical transitions with |Mz|= 0 that are for reasons of completeness,

also plotted into the extended Kataura plot. Analyzing the chiral indices of the respec-

tive carbon nanotubes, we can derive several statements from the extended, calculated

Kataura plot:

•In contrast to optical transitions below the M-point, we find possible optical tran-

sitions that are forbidden by symmetry. (Indicated with red dots in Fig. 6.11.)

•Smaller branches do not appear pairwise as they do for optical transitions below

the M-point. All transitions originate from the proximity of the steep-slope Γ−K-

direction and not from twofold K−Mand K−Γdirections with different slopes

of the electron and hole bands.

105

6.2. Extended Kataura plot

2 4 6 8 10 12 14 16 18 20 22 24 26 28

Diameter ˚

A

Energy (eV)

Figure 6.11: The extended Kataura plot is shown. Black dots represent optical transitions of carbon

nanotubes with chiral indices (n1, n2) between 2 and 21 and optical matrix elements |Mz|>0. Red

dots represent optical transitions with no oscillator strength, i.e. optically forbidden transitions. Carbon

nanotubes are sorted by their diameter. All optical transitions above 5.42 eV correspond to transitions

above the M-point energy. A zoom-in of the inset can be seen in Fig. 6.12. Black arrows indicate starting

points of different smaller branches, respectively. The blue arrows connect starting points of smaller

branches in which the optical transition Eii is always the same. Calculations are done in a next-nearest

neighbor tight-binding + zone-folding approach.

106

6.2. Extended Kataura plot

•n1+n2=constant for all carbon nanotubes within a smaller branch (compare

Fig. 6.12). All semiconducting tubes within a smaller branch have the same transi-

tion Eii; the metallic carbon nanotubes generally have a lower transition E(i−1)(i−1).

The low-energy end of the branches always belong to a carbon nanotube with the

largest chiral angle. The high-energy end always belongs to a zig-zag tube, i.e. to

a tube with the smallest chiral angle.

•Larger branches consist of smaller branches where the optical transition Eii and the

sum n1+n2change by +1 for increasing diameters, respectively (compare Fig. 6.12

and solid, black arrows in Fig. 6.11).

•The chiral indices of neighboring, smaller branches change by +1 both for n1and

n2as indicated with dashed arrows in Fig. 6.11. (Compare Fig. 6.12.)

•We find a clear separation between carbon nanotubes that have the same optical

transition Eii as indicated with blue arrows. They follow the same trend as below

the M-point energy, i.e. smaller tubes have higher transition energies and vice versa.

For each new smaller branch the sum n1+n2changes by +1 for larger diameters.

(Follow the blue arrows in Fig. 6.11.)

•Due to reasons of symmetry, the extended Kataura contains M-point and Γ-point

transitions for each carbon nanotube. However, the Γ-point transition is forbidden in

every carbon nanotube. This is a direct consequence of the dipole-forbidden Γ−point

transition in graphene. Additionally, all optical transitions above the M-point are

forbidden in armchair tubes. In armchair tubes, all conduction and valence bands

for k= 0 have the same parity with respect to the σhmirror plane. Z-polarized

light, however, reverses the parity for the horizontal mirror plane σhand therefore,

independent from the band index m, all optical transitions above the M-point are

forbidden [61].

•Following the blue arrows in Fig. 6.11 (identical transition Eii but smaller diameters

for each new arrow), the energy separation between the arrows increases up to

∼13eV followed by decreasing gaps.

Although the Kataura plot looks different for energies above the M-point energy, we find

three major similarities with the usual Kataura plot: (i) Again, quantum confinement de-

termines the general dependence of transition energies on the diameter. This can be seen

following the blue arrows in Fig. 6.11 where the optical transition index Eii is kept con-

stant. Further, the increasing energy separation (up to ∼13eV) followed by a decreasing

energy separation reflects the ∝1/d dependence, but also the dispersion of the electronic

band structure in graphene. The overall gradient towards the Γpoint increases up to

∼13eV (Ec−Ev) from all directions but drastically decreases in its proximity [38]. (ii)

Smaller branches do not appear pairwise since all transitions from the K−Mdirection

are excluded. Transition energies within smaller branches show the same dependence on

the chiral angle as those from below the M-point energy. For the same optical transition,

higher chiral angles have lower transition energies and vice versa.

In contrast to the transitions below the M-point energy, the chiral angle dependence

107

6.2. Optical absorption above the M-point energy

n1+n2= constant

(5,5)

(6,4)

(7,3)

(8,2) (9,1)

(10,0)

(5,4)

(9,0)

(6,5)

(11,0)

(6,6)

(12,0)

6 6.577.5 8 8.5 9 9.5

Diameter ˚

A

Energy (eV)

E66

E77

E88

E55

E66

E77

Figure 6.12: A zoom-in of the ex-

tended Kataura plot as shown in

Fig. 6.11, is plotted. Different col-

ors stand for different optical tran-

sitions. Semiconducting tubes are

marked with circles, metallic ones are

marked with diamonds. For reasons

of completeness, the plot contains the

(5,5) and (6,6) armchair tubes al-

though all optical transitions above

the M-point energy are forbidden by

symmetry. The sum of n1and n2

is constant within one branch. Cal-

culations are done in a next-nearest

neighbor tight-binding + zone-folding

approach.

seems inverted. The inversion can be understood from the symmetry of the hexagonal

graphene lattice. The helical vector cuts the hexagons of the graphene lattice the closer to

the K points, the larger the chiral angle is. At the same time, the k-space distance to the

various Γpoints gets larger for larger chiral angles and vice versa. In other words, optical

transitions for large chiral angles are rather close to the K points, but far away from the Γ

points (translates into "lower" transition energies in both cases) and optical transitions for

small chiral angles are rather away from the K points and close to the Γpoints (translates

into "higher" transition energies in both cases). Corresponding illustrations can be found

in Refs. [45,53,62,171]. All statements from above can be followed in detail in Fig. 6.12

where we show a zoom-in of the extended Kataura plot as indicated with the rectangle

in Fig. 6.11. Chiral indices of the respective carbon nanotubes are indicated to highlight

their arrangements within different branches in the extended Kataura plot.

6.2.2 Optical absorption above the M-point energy and UV Ra-

man spectra

With a deeper understanding of the optical transition energies above the M-point exciton

in carbon nanotubes, we can extend the analysis to the optical absorption and can com-

pare these results to UV Raman spectra.

The absorption in carbon nanotubes has a strong excitonic character [233,266–268]. Ex-

citon binding energies can reach values up to 500 meV for small semiconducting tubes,

where the holes and electrons are significantly confined [45,62,233,266–268]. Larger diam-

eters lead to a reduced confinement, weakening the binding energies, but values usually

still reach several hundred meV. Even in metallic tubes, exciton binding energies are

observed that exceed kBT∼25 meV for the usual optical transitions probed, although

there is an efficient Coulomb screening of free electrons in metallic carbon nanotubes [62].

The experimental and theoretical analysis however usually covers excitation energies in

the optical visible range. It was therefore interesting to know, how the optical matrix

elements Mzand the consecutive absorption spectra of carbon nanotubes evolve in the

ultra-violet region.

108

6.2. Optical absorption above the M-point energy

(11,0) (10,2) (9,4) (8,6)

(12,0) (11,2) (10,4) (9,6)

0 5 10 15 20 25

0.5

1.5

Chiral angle (◦)

|Mz|(arb.units)

E55 ν=−1

E66 ν=−1

E77 ν=−1

E88 ν=−1

E44 ν= 0

E55 ν= 0

E66 ν= 0

E77 ν= 0

Figure 6.13: The magnitudes of dif-

ferent optical matrix elements |Mz(k)|

for several optical transitions Eii are

plotted as a function of the chiral an-

gle. The lowest optical transitions E44

(metallic) and E55 are the last ones be-

fore the M-point transition. The M-

point transitions for metallic tubes are

therefore E55 and E66 for semiconduct-

ing tubes. Triangles belong to tubes

with the family index ν=−1, dia-

monds belong to tubes with the family

index ν= 0. The tubes have approx-

imately the same diameter of ∼9˚

Calculations are done in a next-nearest

neighbor tight-binding + zone-folding

approach.

We have plotted the linear band structure of a semiconducting (8,0) tube in Fig. 6.10

and for every subband we have marked the electron momentum kzwhere an optical tran-

sitions occurs. In the case of a zig-zag tube, all optical transitions occur at kz= 0 (Γ

point) as can be seen in Fig. 6.10. This is also true for every optical transition higher than

the M-point energy in armchair tubes [45,61,62]. If n1is an even number, zig-zag tubes

further have a band with no dispersion as their cutting lines on the hexagonal graphene

lattice are parallel to a M−Mconnection line. The non-existing dispersion can be under-

stood from the fact that equi-energy surfaces of excitations above the M point in graphene

are circles around the Γpoint and triangles around the Kpoint for excitations below the

M-point energy. The curvatures have opposite signs, respectively. As a consequence, there

is a zero-crossing of the curvatures resulting in piecewise straight equi-energy surfaces. In

Chapter 5, Fig. 5.3, a corresponding contourplot of the two-dimensional band structure

of graphene with depicted equi-energy surfaces can be seen.

The following high density of states results in a very strong absorption of zig-zag nan-

otubes with an even n1when excited at the M-point transition energy [61,62].

From the calculated electron bands, we have also calculated the optical matrix elements

Mzof each allowed optical transition as shown in the extended Kataura plot (details are

given in Chapter 2). We have used the approach as given in Ref. [61], where the optical

matrix elements are calculated within a next-nearest neighbor tight-binding approach.

Some of them are plotted in Fig. 6.13, where we show the optical matrix elements for ex-

citations below, at, and two above the M-point energy for various carbon nanotubes. We

have chosen nanotubes with an approximate diameter of d∼9˚

A but with chiral angles

ranging from 0 to 25 degrees. The matrix elements for the highest transitions below the

M-point energy E55 (E44 for metallic tubes, drawn in black) have very high values close

to 2, decreasing for an increasing chiral angle. This reflects the trigonal warping effect

also apparent for optical matrix elements of lower transitions [61, 62]. At the M-point

transition, the respective matrix elements are systematically higher for all chiral angles

(drawn in red). In case of the (12,0) zig-zag tube, the maximum value of |Mz|= 2 is

reached, the values for tubes with higher chiral angles are again slightly lower. E77 and

109

6.2. Optical absorption above the M-point energy

(7,4)

0 2 4 6 8 10 12 14

Energy (eV)

Absorption (arb.units)

×46

8 10 12 14

(8,0)

0 2 4 6 8 10 12 14 16

×32

8 10 12 14

Figure 6.14: Calculated optical absorption spectra of a (7,4) and a (8,0) carbon nanotube are plotted

as a function of the photon energy. Black spectra correspond to a band-to-band absorption with ∆m= 0.

The red spectra are the sum of all allowed optical transitions, respectively. The linear absorption spectra

are calculated in a next-nearest neighbor tight-binding + zone-folding approximation. Insets show a

magnification of the absorption in the UV range.

E88 (E66 and E77 in metallic tubes; green and blue) are the first transitions above the

M-point energy with significantly lower coupling elements. It is known from graphene,

that the optical absorption drastically decreases for energies above the M-point energy

both due to a reduced electronic density of states and a lower oscillator strength [138].

The same occurs in carbon nanotubes. Again, we find higher values of |Mz|for smaller

chiral angles and vice versa. This can be understood from the fact that for the same

optical transition, the bandgaps for higher chiral angles are significantly larger (compare

Figs. 6.11 and 6.12). Equivalent transitions in graphene generally have lower oscillator

strengths [62,138]. From Fig. 6.13, we can see that the values for zig-zag tubes for the

first two transitions above the M-point energy are roughly one magnitude smaller com-

pared to the transition below and at the M point, i.e. the optical absorption is drastically

reduced. Although not plotted in Fig. 6.13, all higher transitions exhibit even smaller

optical coupling elements |Mz|.

From the optical matrix elements, we have calculated full absorption spectra to get further

insights into the optical properties of carbon nanotubes in the UV range. For the calcula-

tions, we have mainly followed Ref. [61] where the linear absorption in the rotating-wave

approximation is given by [61]:

α(ω)∼~e2

0ωX

k|Mz(k)|2γ

(ω−ωcv)2+γ2(6.1)

The wave functions for |Mz(k)|are given in Chapter 2, ωcv =ωc−ωv(c and v stand

for conduction and valence bands, respectively), and the broadening factor γwas set to

6meV [62]. The absorption α(ω)was then obtained by a numerical summation over kfor

every respective subband of the analyzed carbon nanotube.

Two absorption spectra for a semiconducting (8,0) and a metallic (7,4) CNT as a function

of the photon energy are plotted in Fig. 6.14. In case of the metallic (7,4) nanotube, the

110

6.2. UV Raman spectroscopy: RBM measurements

absorption peaks occur pairwise due to the trigonal warping effect. Except for the M-point

transition of the (8,0) tubes, all absorption peaks are asymmetric mirroring the respective

electronic band dispersion. The non-existing dispersion of the electron and hole bands

with m= 4 leads to a very strong absorption when the (8,0) tube is excited in resonance

with the M-point exciton. Including Coulomb-Coulomb interaction however, reduces the

M-point absorption considerably but it should still exceed the absorption from electronic

bands with other indices m, but also the absorption from other, chiral tubes [62].

The optical absorption of each transition above the M-point energy is more than one

magnitude smaller than for absorptions below the M point as can be seen in Fig. 6.14. The

closer the excitation occurs to the Γ-point energy (∼16.2eV in the used approach), the

lower gets the optical absorption. Band dispersions of electron and hole bands are reversed

for optical transitions above the M-point energy as we have considered maxima rather

than minima in the combined transition energies. As a result, the van-Hove singularities

for these transitions are also reversed as apparent in the insets of Fig. 6.14.

UV spectroscopy on carbon nanotubes

What can we learn from the analysis of both the extended Kataura plot and the UV

absorption spectra? We can summarize our findings as follows:

•The difference of optical transition energies of carbon nanotubes with comparable

diameters is generally larger for transition above the M point, compared to those

below the M point. One should consequently observe a smaller amount of tubes in

resonance to the laser excitation energy when a sample is excited in the deep-UV

region.

•Due to optically forbidden transitions, all armchair carbon nanotubes are excluded

from deep-UV spectroscopy.

•Because of symmetry reasons, all carbon nanotubes have the graphene M-point

transition in common. As a consequence, all tubes except for the armchair tubes

should be in resonance when excited with the M-point transition energy. However,

if apparent in a sample, the optical absorption of zig-zag tubes with an even n1

should dominate the overall absorption.

•Compared to excitations below the M-point energy, the optical absorption above

the M-point energy is more than one magnitude smaller. It further decreases with

an increasing photon energy. Hence, deep-UV spectroscopy of carbon nanotubes is

supposed to be way more difficult than for excitations in the optically visible range.

Deep-UV Raman measurements

We have performed Raman measurements in the deep-UV range on HipCO produced

carbon nanotubes. We have tried excitation energies in resonance with the M-point tran-

sition (∼4.7eV), but also higher ones. Some spectra are plotted in Fig. 6.15 b) where

the RBM and the G-mode for an excitation energy of 4.69eV and 5.08eV are shown.

Although observable for an excitation energy εL= 4.69eV, the RBM disappears for an

excitation energy of 5.08eV whereas the G mode is still visible. Even though spectra

111

6.2. UV Raman spectroscopy: RBM measurements

εL= 1.84 eV

εL= 2.73 eV

εL= 4.69 eV

180 200 220 240 260 280 300

Raman shift cm−1

Intensity (arb.units)

RBM

CaF2

250 300 350

Raman shift cm−1

4.69 eV

5.08 eV

G mode

1500 1600 1700

Intensity (arb.units)

Eii = 5.42 eV

(M point)

0 5 10 15 20 25 30

−2

−1.9

−1.8

−1.7

Diameter ˚

A

Mz(arb.units)

0.5

1.5

Figure 6.15: Raman spectra of a HipCO sample and optical matrix elements Mzare plotted. a) RBMs

of the HipCO sample for many excitation energies between 1.84 eV and 2.73 eV (black) are plotted. The

average energy difference between the the excitation energy is ∆εL∼45 meV. The RBM for the same

HipCO sample excited at the M-point transition (∼4.7eV) is plotted in blue. b) and c) RBM and G

modes of the HipCO sample for two UV excitation energies are shown. d) and e) Optical matrix elements

of carbon nanotubes for the M-point transition as a function of the diameter are plotted. Considered are

chiral indices (n1, n2) between 2 and 21. Labeled are optical matrix elements for nanotubes where the

sum of n1and n2equals 16.

are not shown, this is also true for higher excitation energies. We believe the absence

of the RBM for very high excitation energies is due to the drastically decreasing optical

absorption as discussed before. This effect is overcompensated by a very strong electron-

phonon coupling due to the Kohn-Anomaly in case of the G mode explaining its presence

throughout all deep-UV excitation energies [140, 143, 252]. Further G mode spectra of

other carbon nanotube samples and deep-UV excitation energies can be seen in Figs. 6.17

and 6.18.

We find a very large FWHM (full width at half maximum) for the RBM as shown in

Fig. 6.15b) exhibiting ∼45 cm−1. This very high value is not observed for any other

excitation energies as shown in Fig. 6.15 a), in agreement with former findings [269,270].

We generally attribute the large line width to the concurrent probing of various carbon

nanotubes. As discussed before, except for the armchair tubes, all carbon nanotubes have

112

6.3. Ultra-violet light assisted functionalization

the M-point transition from graphene in common, allowing for their contemporaneous op-

tical excitation. From that point of view, we might expect a broad and symmetric RBM

Raman peak that represents the diameter distribution of carbon nanotubes within the

sample, rather than small fractions of tubes in resonance to the laser excitation energy.

The frequency range of the RBM measured at εL= 4.69eV covers ∼230 −300cm−1as

denoted with black lines in Fig. 6.15. In contrast, for optically visible excitation energies

between 1.84eV and 2.73eV, we find RBMs between ∼190 −300cm−1,i.e. a larger

frequency range. Especially the lower frequencies can not be seen within the broad RBM

peak in the UV region. A reason for the somewhat smaller frequency range remains un-

clear. A first explanation might be the fact that absolute values of the optical matrix

elements |Mz|of M-point transitions in carbon nanotubes range from 0 up to the overall

maximum of 2, as plotted in Fig. 6.15. This also counts for the limited range of diameters

d∼8−12˚

A of the probed HipCO sample. A similar behavior can not be observed for

lower optical transitions in the visible range [61,62]. As a consequence, several chiralities

should be way less visible than others when contemporaneously excited at the M-point

transition. For instance, the branch where the sum of n1and n2equals 16 is marked

in Fig. 6.15, in which all nanotubes have low optical matrix elements |Mz|. Thus their

specific contribution to the optical absorption should be very small compared to other

CNTs.

6.3 Ultra-violet light assisted functionalization

A lot of effort has been put in the exploration of paths for a covalent functionalization of

carbon nanotubes [271–275]. It is a necessity for possible applications since by a subtle

functionalization a solubility in organic or aqueous solvents can be achieved [276], one

can separate metallic from semiconducting nanotubes [277], carbon nanotubes can be

charged [272,278], can be prepared for drug delivery [279] or they can be prepared for a

far more complicated functionalization such as the growing of embryonic rat-brain neu-

rons [280]. The development of possible reaction paths has defaulted to the broad and -in

our days- common knowledge of the functionalization of other large sp2system such as

fullerenes [271,281,282]. Due to their chemical inertness, their functionalization is rather

difficult and often requires aggressive chemicals such as strong acids, elemental fluorine,

or alkali metals [283,284].

Another possible path is suggested in Refs [199–201] where photo-assisted functionaliza-

tions of carbon nanotubes are shown both theoretically and experimentally. By a solvent-

free UVO (ultra-violet-ozone) treatment, esters, quinines, and hydroxyl functional groups

could be attached to the surface of multi-walled carbon nanotubes while conserving an

insignificant number of defects [201]. Authors in Ref. [200] simulated the UV-light ac-

celerated interaction of carbon nanotubes and oxygen. They found that an UV-assisted

excitation of the oxygen ground spin-triplet state to a higher spin-singlet state significantly

lowers the activation energy for the molecular-oxygen chemisoprtion to the nanotube [200].

According to the authors, a predominant functionalization happens at defects, such as

7−5−5−7or Stone-Wales defects, accompanied by a charge transfer from the oxygen to

the carbon nanotube [200]. The suggested methods are a convenient and straight-forward

way to functionalize carbon nanotubes without using aggressive chemicals.

113

6.3. Ultra-violet light assisted functionalization

t = 2400 s

G mode

Functionalization

1000 1200 1400 1600

Raman shift cm−1

Intensity (arb.units)

4.69 eV

5.08 eV

t = 95 s

t = 1140 s

t = 2280 s

1000 1200 1400 1600 1800

4.69 eV

Figure 6.16: UV Raman spectra of a HipCO buckypaper carbon nanotube sample with different ex-

citation energies a) and different exposure times b) are plotted. The G mode and the functionalization

mode are denoted.

Based on the analysis of Raman spectra, we show that a HipCO CNT sample can be

functionalized only by UV irradiation under standard ambient conditions. Further, we

show that the iron clusters in the HipCO sample are oxidized in the presence of UV light

and carbon nanotubes.

On the next pages we show experimental data of a newly found Raman mode that we

attribute to a CNT and photo-assisted oxygen functionalization of iron clusters. We show

experiments where we change external parameters such as the exposure time to UV-light,

potential reaction gasses, and the carbon nanotube samples. Further, we demonstrate

that the photo-assisted reaction depends on the laser excitation energy. For energies

larger than 4.7eV we can not observe a functionalization in the Raman spectra. In the

end we give an explanation and explain the newly observed Raman mode.

6.3.1 Raman analysis of the functionalization

In Fig.6.16 a) two Raman spectra of the same HipCO buckypaper carbon nanotube sam-

ple are shown. The integration times are the same, but the laser excitation energies are

different. One is close to the M-point energy (blue) [52,136,138] whereas the second exci-

tation energy is 400meV higher, i.e. clearly above the M-point exciton. In the first case,

a new Raman mode at around 1086cm−1can be observed that does not come up when

excited with a photon energy of 5.08eV, even for long exposure times of 40min. This also

counts for higher laser excitation energies. On the right hand side in Fig.6.16b), three

Raman spectra are plotted with integrations times of 2×95s, but with overall different

UV exposure times as denoted in the figure. The spectra belong to a time series in which

a new Raman spectra was taken every 190s (2×95s) from the same sample position,

i.e. one can follow the time-dependent progression of the change in the Raman spectra.

For longer exposure times, the intensity of the functionalization mode increases with a

simultaneous decrease of the G-band intensity. This can be seen more clearly in Fig. 6.17

where both the absolute intensities of the Raman modes [a)] and their intensity ratios are

plotted [b)]. The intensities refer to the area of the Raman peaks as shown in Fig.6.16

fitted with one Lorentzian peak, respectively.

114

6.3. Ultra-violet light assisted functionalization

Func. mode

G mode

0 1000 2000 3000 4000

Exposure time (s)

Intensity (arb.units)

1.7 mW

2.0 mW

2.7 mW

G band @ 5.08 eV

0 1000 2000 3000 4000 5000

1585

1590

Exposure time (s)

Raman shift cm−1

1.7 mW

2.0 mW

2.7 mW

0 1000 2000 3000 4000 5000

0.5

Exposure time (s)

Func. /G-band ratio

1.7 mW

2.0 mW

2.7 mW

εL= 4.69 eV

t = 3600 s

1500 1600 1700 1800

Raman shift cm−1

Intensity (arb.units)

Nitrogen

Oxygen

Figure 6.17: Raman intensities, intensity ratios, and Raman frequencies of the fitted functionalization

mode and the G mode are shown. All spectra are taken with a laser excitation energy of εL= 4.69 eV

under standard ambient conditions in ambient air. a) Raman intensities of the Func. mode (dashed lines)

and the G mode for three different laser excitation powers are shown. b) Intensity ratios of the Func.

mode and the G mode as a function of the exposure time are plotted. c) The Raman shifts of the G

band as a function of the exposure time are plotted for difference laser excitation powers. The dashed

line denotes the G band frequency at the beginning of the UV exposition, i.e. it corresponds to the G

band of a widely unfunctionalized carbon nanotube sample. The arrows frame the experimental error.

d) Raman spectra of the HipCO sample under UV exposition and irradiation times of t= 3600 s in an

oxygen atmosphere (blue) and in a nitrogen atmosphere (black) are plotted.

After exposure times of t≈750 s the intensity of the functionalization mode saturates

and stays approximately at the same level up to exposure times of 5000 s. In contrast,

the intensity of the G band decreases with increasing exposure times. This explains the

increasing Func./ G-band ratios in Fig. 6.17b). Within the range of probed laser powers,

the intensity ratios of the functionalization mode and the G band do not differ substan-

tially. All show the same trends as discussed above.

The averaged G band of the carbon nanotube sample upshifts as a function of UV expo-

sition time in ambient air. For all laser excitation powers, the lowest Raman frequencies

are found at the beginning of the exposition followed by an increase of ≈4cm−1(compare

Fig. 6.17). We find the same behavior for graphene as illustrated in Fig.6.18 d). An expo-

sition to UV light with exposition times up to 5000s results in an upshift of approximately

3cm−1with a contemporaneous decrease of the G-band intensity (Fig. 6.18). Instead, the

G-band frequency of graphite under UV exposition does not change as a function of the

115

6.3. Ultra-violet light assisted functionalization

exposition time [compare Fig.6.18 d)]. Further, the intensity drop of the G band in

graphite begins after longer exposition times and saturates after approximately t= 2500s

in contrast to graphene where a saturation of the intensity drop cannot be observed.

Previous Raman studies on graphene observed a light-induced oxidation accompanied

by a frequency increase of the G band [285,286]. The frequency shift is attributed to an

oxidation causing a p-doping in graphene [285–287]. From the shift of the G band, we

can estimate the p-doping increase to exhibit ∆n= 3.5×102cm−2. Authors in Ref. [286]

have found similar upshifts for the G band in graphene after a 10min irradiation with

UV light from a Hg lamp in an oxygen atmosphere corresponding to a hole doping of

n∼3.5×102cm−2. We therefore believe the frequency increase in the Raman spectra

of the HipCO carbon nanotube sample can also be attributed to an oxygen-induced p-

doping. To further analyze this assumption, we have performed likewise experiments in

different atmospheres. Namely, we have used Argon, Nitrogen, and Oxygen atmospheres

as shown in Fig. 6.18 a). Differently colored spectra correspond to spectra of a HipCO

sample excited with εL=4.69eV and exposition times of t= 3600 s, but in various atmo-

spheres as denoted in the figure. The functionalization mode at 1086cm−1exhibits the

largest intensity in the oxygen atmosphere. Instead, its intensity is comparably low in an

Argon or in a Nitrogen atmosphere even after long exposition times of t= 3600s.In an

oxygen atmosphere, a significant increase of the G band frequency can be observed. We

have repeated the measurements on various sample spots and show the G band after a

t= 3600s UV irradiation in an Oxygen atmosphere (blue) and in a Nitrogen atmosphere

(black) in Fig. 6.17 d). Similar to the spectra in Fig.6.18 a), the UV exposition in Oxygen

leads to an upshift of the G band.

Changing the sample to CoMoCat, Elicarb, or arc discharge produced carbon nanotubes

we cannot observe the functionalization mode even after UV (εL= 4.69eV) exposition

times of t= 3600s [Fig. 6.18 b)]. The same accounts for graphene and for graphite as

shown in Fig. 6.18 c).

In contrast to all other tested carbon nanotube samples, the HipCO produced carbon nan-

otubes contain iron as a catalyst for the synthesis [46,60,288]. The iron atoms are existent

as iron clusters catalyzing the disproportionation: CO +CO Fe

→CO2+C(SWNT)[46].

This reaction creates nanotubes that usually have open caps filled with the catalyst [271].

The direct contact to iron clusters might therefore enable possible UV-induced chemical

reactions.

For a further investigation, we have patterned the HipCO sample with UV light (εL=

4.69eV) as illustrated in Fig.6.19 c). One can clearly observe the difference in contrast

between the areas that have been illuminated and those that have not been illuminated.

In the optical visible range (εL= 2.71eV) we have then taken a Raman map of the illumi-

nated area and show a map of the D/G ratio as a function of the location on the sample

in Fig.6.19 a). Throughout the entire area, we observe a high D/G ratio of 0.5 and above.

However, we can not follow the pattern as visible in Fig.6.19 c). In contrast, the ratio

of the functionalization mode to the G mode does show the pattern illuminated via the

UV light [b)]. Only at illuminated spots, we observe high intensities of the functionaliza-

116

6.3. Ultra-violet light assisted functionalization

t= 3600 s

εL= 4.69 eV

1000 1200 1400 1600

Raman shift cm−1

Intensity (arb.units)

Oxygen

Nitrogen

Argon

t= 3600 s

εL= 4.69 eV

1000 1200 1400 1600

Raman shift cm−1

Intensity (arb.units)

Arc discharge

CoMoCat

Elicarb

t= 3600 s

εL= 4.69 eV

1000 1200 1400 1600

Raman shift cm−1

Intensity (arb.units)

Graphite

Graphene

0 1000 2000 3000 4000 5000

0.5

Exposure time (s)

G mode Int.

Graphite

Graphene

1582

1584

1586

1588

Raman shift cm−1

Graphite

Graphene

Figure 6.18: Raman spectra of carbon nanotubes, graphene, and graphite are shown. a) Raman spectra

of a HipCO sample, an UV exposition of t = 3600 s, and a laser excitation energy of εL= 4.69 eV with

different ambient gasses are plotted. b) Raman spectra of Arc discharge (black), CoMoCat (blue), and

Elicarb (red) carbon nanotubes are plotted for a laser excitation energy of εL= 4.69 eV and an UV

exposition of t = 3600 s under standard ambient conditions. c) Raman spectra of graphene (blue) and

graphite (black) are shown for a laser excitation energy εL= 4.69 eV and an UV exposition time of

t = 3600 s. d) Raman shift of the G mode under UV exposition and standard ambient conditions as

a function of the exposition time for graphite (black) and graphene (blue) are plotted. e) Normalized

intensities of the G mode under UV exposition and standard ambient conditions for graphite (black) and

graphene (blue) are plotted. Different spectra correspond to different spots on the sample.

tion mode at 1086cm−1as exemplary shown Fig.6.19 d) for an excitation energy in the

optical visible range. Zones with high func. mode/G mode ratios directly correspond to

the illuminated area as shown in Fig. 6.19 c). This means that the functionalization can

apparently be reproduced and is clearly UV-light induced. Moreover, as can be seen in

the D/G Raman map, the functionalization does not further increase the defect densities

in the carbon nanotubes. The already high defect densities usual for HipCO CNT are not

further increased at UV-illuminated areas.

6.3.2 Explanation of the functionalization process

In the following we summarize the conditions in which we observe the functionalization

and try to frame an explanation for its origin.

•The functionalization is UV induced and can only be initiated for an excitation

energy of εL= 4.69eV. This energy is close to the M-point energy in graphene

117

6.3. Ultra-violet light assisted functionalization

εL= 2.71 eV

Func. mode

D mode G mode

1000 1200 1400 1600 1800

Raman shift cm−1

Intensity (arb.units)

a) b)

c) d)

Position in x-direction (µm)Position in x-direction (µm)

2 4 6 8 10 12 14 16 18 204 8 12 16

Position in y-direction (µm)

1.18

0.78

0.38

1.59

1.12

0.65

0.18

D/G

ratio Func./G

ratio

Figure 6.19: a) A Raman map of the D/G ratio @ εL= 2.71 eV from a HipCO carbon nanotube sample

that was illuminated with UV (εL= 4.69 eV) Light (step size in x-direction: 5 µm, step size in y-direction:

8µm) is shown. b) A Raman map of the Func. mode/G ratio @ εL= 2.71 eV is shown after illumination

as in a). The measured area on the sample is the same as before. c) Optical image (50 ×magnification)

of the UV illuminated area on the HipCO nanotube sample. One can clearly see the irradiation pattern.

The depicted rectangle corresponds to the measured areas as shown in a) and b). d) Representative

Raman spectra of the HipCO sample showing the func. mode with a high intensity. They correspond to

areas in b) with a high func. mode/G ratio. The laser excitation energy is εL= 2.71 eV.

[52,136], accompanied by a strong optical absorption [136,138]. As a consequence

of symmetry, all but armchair carbon nanotubes contain the M point of graphene,

i.e. by an excitation energy of εL= 4.69 eV almost all carbon nanotubes within an

ensemble sample are in an optical resonance. Further, this laser excitation energy

equals the work function of graphene, graphite, and carbon nanotubes [38,289–291],

potentially providing charge carriers for a chemical reaction.

•For excitation energies higher than 4.69eV, a functionalization can not be initiated.

The optical absorption decreases considerably for energies higher than the M-point

exciton [138], reducing the possible amount of charge carriers for a chemical reaction.

•The functionalization can only be observed in HipCO samples. It can not be seen

in arc discharge, Elicarb, or CoMoCat carbon nanotube samples. Further, other

carbon allotropes do not show the functionalization mode. We refer this behavior

to the presence of iron in the sample.

118

6.3. Ultra-violet light assisted functionalization

•The defect densities in UV illuminated areas of the HipCO carbon nanotube sample

do not increase.

•The presence of oxygen is essential for the chemical reaction.

•An analysis of the G band of the carbon nanotubes suggests that a p-doping after the

functionalization occurs. This is a well-known behavior in graphene, an oxidation

causes a p-doping attended by an upshift of the G band.

•The func. mode intensity saturates after certain UV exposition times, accompanied

by a decrease of both G-mode and D-mode intensity. An advancing UV-assisted

destruction of CNTs limits the possible supply of charge carriers for a chemical

reaction, showing the clear necessity of the presence of carbon nanotubes for the

reaction.

A strong Raman band at around 1086cm−1is characteristic for different calcite group

structures [292–294]. The calcite group contains ionic minerals with the carbonate [(CO3)−2]

as the characteristic anion [295]. The anion is paired with comparably small cations, such

as Li, Na, Mg, Zn, Fe2+, or Cd forming a group of minerals with very similar crystal-

lographic traits [295]. The symmetry of all its members is the trigonal ¯

mpoint group

(D3din the Schönflies notation) [295] allowing for six normal modes of which two are

doubly degenerate [293]. One of them corresponds to a symmetric stretching vibration

of the carbonate groups having the irreducible representation A0

1being Raman active

and usually showing a strong, distinct Raman peak [292–294]. The vibrational pattern

is illustrated in Fig.6.20. It seems feasible that the strong Raman peak in the spectra

of the UV-irradiated areas on the carbon nanotube sample can be referred to originate

from one member of the calcite group, namely the siderite (FeCO3). The iron and carbon

residuals in the sample react under UV irradiation with oxygen and form the mineral

siderite. Therefore, by a photo-induced chemical reaction, we produce a mineral whose

strong Raman signal superimposes with that of the carbon nanotubes still present in the

sample. However, this does not explain the clear upshift of the G band when the HipCO

sample is UV-illuminated. We believe the upshift is also due to an oxidation of the carbon

nanotubes in agreement with former observations [285–287].

The reaction path for the formation of the siderite remains unclear. In the presence

of carbon nanotubes, it is clearly an UV-light induced process. The carbon nanotubes

may provide the charge carriers necessary to reduce the pentavalent iron in the iron pen-

tacarbonyl complex that is usually present in HipCO samples [46, 60,288]. This would

explain, why the strong Raman band at around 1086cm−1can only be observed when the

sample is excited with εL=4.69eV. An excitation energy of ∼4.7eV on the one hand leads

to a comparably strong absorption in carbon nanotubes but on the other hand it also is

the energy of the work function in carbon nanotubes. Since this strong Raman band is

not initiated for higher excitation energies, we believe that an electrophilic reaction causes

a reallocation of both carbon and oxygen from the iron pentacarbonyl [Fe(CO)5] to the

Fe clusters present in the HipCO sample [46,60,288].

Authors in Ref. [200] refer the oxidation of carbon nanotubes in the presence of UV light

to the more reactive singlet oxygen O2(1∆g)rather than to the ground spin-triplet state

119

6.4. Summary

Figure 6.20: The vibra-

tional pattern of the A0

1mode

in siderite (D3dpoint group)

is illustrated [293].

oxygen O2(3Σ−

g). This may also be a feasible explanation for the UV-induced formation

of the siderite. However, since we cannot observe the siderite Raman peak for excita-

tion above ∼4.7eV this explanation is questionable. It was recently shown, that the

UV-induced formation of the reactive singlet state oxygen O2(1∆g)states occurs between

∼230 −290nm with comparable yields [296]. Higher excitation energies than 4.69 eV

(264nm) would therefore provide similar amounts of excited oxygen molecules. Yet, we do

not observe the functionalization peak for higher excitation energies. Hence, the carbon

nanotubes are crucial for the UV-induced formation of the siderite in the HipCO sample.

Since the work functions of graphene, graphite, and fullerenes are very close to those from

carbon nanotubes, the UV-induced formation of siderite should also be possible with the

just listed carbon allotropes. It might be interesting to repeat the UV exposure measure-

ments with other carbon allotropes but with the same catalysts as needed for the growth

of HipCO carbon nanotubes. However, this is beyond the scope of the present work.

6.4 Summary

We analyzed the low-intensity, double-resonant Raman modes at around 1950cm−1and

2450cm−1in an ensemble carbon nanotube sample. With the support of DFT + zone-

folding calculations we modeled their dispersions, peak intensities, and general peak struc-

tures. We find that the complex dispersions can be understood from contributions of

individual carbon nanotubes close to the resonance of the laser excitation energies. De-

pending on chirality and diameter, each carbon nanotube adds an unique share to the

complex line shape of the low-intensity Raman modes. Since metallic and semiconduct-

ing tubes necessarily exhibit unlike diameters when both species are in resonance to the

laser excitation energy, their vibrational properties need to differ accordingly. As a conse-

quence, the contemporaneous probing of both species results in bimodal peak structures

of the analyzed double-resonant Raman modes. This effect is a clear ensemble effect and

especially occurs for the ∼1950cm−1band since the frequencies of the phonons involved

highly depend on the tube diameter.

The assignment of both low-intensity modes has been discussed controversially in the

past. Especially the 1950cm−1mode was only tentatively assigned without the support

of calculations. In a DFT + zone-folding approach, we find a LO+LA@ Γcombination

mode responsible for the mentioned mode. Instead, in a non-orthogonal tight-binding

approach, the 1950 cm−1mode is rather attributed to a TO+ZO @Kcombination. The

out-of-plane ZO phonon mode is symmetry forbidden in graphene resulting in very low

electron-phonon coupling elements in the zone-folding calculations. This in turn might

120

6.4. Summary

lead to an underestimation of its contribution to the mode at around 1950cm−1in com-

parison to the LO+LA@Γmode. The non-orthogonal tight-binding approach however

does not have this insufficiency as it is an ab initio approach and takes the CNT into

account without initial assumptions. We therefore tentatively the 1950cm−1band to a

TO+ZO@Kcombination mode. In contrast, the 2450cm−1mode is uniformly assigned

to a TO+LA@Kcombination mode in both computational approaches.

We have introduced the "extended Kataura plot" containing optical transitions way above

the M-point energy in carbon nanotubes. This is a clear contrast to the usual Kataura

plot that only contains optical transitions in the visible range, i.e. below the M-point

energy. By simulating the matrix elements, we find that above the M-point energy, sev-

eral optical transitions are forbidden - a clear difference to those transitions that are lower

than the M-point energy. In particular, these are all transitions from armchair nanotubes,

but also all transitions stemming from the Γpoint. Although the arrangement of optical

transitions above the M-point energy appears to be different compared to those from be-

low the M-point energy, they also follow the well known trends caused by the quantum

confinement and trigonal warping effect.

Raman measurements of the RBM (radial-breathing mode) for an excitation energy close

to the M-point energy show a broad peak that cannot be observed for excitation energies

in the visible range. We generally attribute the large FWHM to the concurrent probing of

many carbon nanotube chiralities since the M-point transition is except for armchair tubes

included in all carbon nanotubes. Higher excitation energies lead to a drastic quenching

of RBM intensity that we generally attribute to the low optical absorption for very high

excitation energies.

In the end we showed how UV light and carbon nanotubes can be used to fabricate

siderite. Its Raman footprint can only be observed in the presence of oxygen, photon

energies of εL= 4.69eV, and the iron catalyst as present in a HipCO carbon nanotube

sample. We believe the formation of Siderite is caused by two reasons: On the one hand,

CNT have a very strong optical absorption around 4.7eV. On the other hand, 4.7eV cor-

responds to the work function of CNT possibly providing the charge carriers necessary

for the oxidation of the iron clusters.

121

7. Conclusion

Diamondoid derivatives

By (resonance) Raman scattering experiments, we have turned the attention to zero-

dimensional diamondoid derivatives. In particular, we have focused on the effect of sp2

defects in the electronic structure of otherwise fully saturated diamondoids. The struc-

tural changes cause energetically low-lying πand π∗orbitals, resulting in optical transition

energies of 4.7eV that are around 2 eV smaller than in pristine diamondoids. Studying a

large variety of diamondoid oligomers connected by sp2defects, we have found the optical

transitions energies of 4.7eV to be independent of their sizes or structures. These exper-

imental outcomes are in a very good agreement with sophisticated DFT calculations on

the one hand, but also with absorption and valence photoelectron spectroscopy measure-

ments on the other hand.

Along with the lifting of the degeneracy of certain vibrational modes, new structural-

induced eigenmodes could be found in the vibrational Raman spectra of sp2diamondoid

oligomers. Some of them can directly be assigned to the ethylene sp2moieties, such as

the intensity-wise strong C=C stretch vibration, or a combined twisting/stretch mode

of the C=C double bond. However, the new structures of diamondoid oligomers lead to

unusual high-frequency C-H stretch vibrations but also to vibrational modes in which the

entire diamondoid moieties are deflected along their shared carbon-carbon double bond.

We refer these modes to Dimer breathing modes (DBM) that have only small frequencies

in the range of ∼190cm−1caused by the comparably large masses of the diamondoid

moieties. To sum up, Raman spectra of sp3diamondoid oligomers are generally more

complex compared to pristine diamondoids. The covered frequency range of Raman ac-

tive vibrations is further enlarged due to new, structural-induced vibrational modes.

In analogy, we find a similar altering of the vibrational characteristics in diamondoid

dimers that are connected by single-bonds. Again, high-frequency C-H stretch vibrations

are found that are caused by additional restoring forces of neighboring diamondoid moi-

eties within the dimers. This is a clear difference to the sp2diamondoid oligomers as their

comparably stiff double bond determines the angle and distance between the diamondoid

moieties. This, in turn, impedes the formation of pronounced facets that induce addi-

tional van-der-Waals forces in single-bond dimers. We again find Dimer breathing modes

in the frequency range around ∼200cm−1. However, due to the loose single bond, the

vibrational modes are slightly modified because of a considerable mode mixing with other

low-frequency modes.

Diamondoid and diamondoid derivatives form very stable van-der-Waals crystals at room

122

7. Conclusion

temperatures, accompanied by a self-altering of their electronic properties. With ab-

sorption and resonance Raman measurements, we could determine a downshift of optical

transition energies in sp2diamondoid dimer crystals of around 1eV, compared to isolated

molecules. This value, in particular, counts for the π→π∗transition that was experi-

mentally accessible. DFT calculations, however, indicate that the reduction of transition

energies related to the more widespread σ→σ∗transitions can even be larger. We believe

that this particular aspect in the electronic properties of diamondoids is of great impor-

tance. Thinkable applications in devices for instance, require not just isolated molecules,

but rather diamondoids in a solid phase.

Graphite, graphene, and multilayer graphene

Raman intensities of the well-known, double-resonant Raman modes such as the Dor

the 2Dmode in graphitic sp2materials are drastically quenched for excitation energies

in the range between εL= 4.69 -5.46eV. We, instead, find new Raman peaks that we

attribute to the second-order vibrational density of states. Optical excitations higher

than the M-point energy (∼4.7eV) lead to an absorption far away from the K−M

high-symmetry line, accompanied by a large decrease of oscillator strength. The reduced

absorption combined with a drastically reduced electron-phonon coupling then results

in a strong quenching of double-resonant Raman processes. In fact, their quenching is

so strong, that not resonantly enhanced second-order processes exceed their intensities.

These "single-resonant" processes involve two phonons with arbitrary phonon wave vectors

that can stem from all over the 1. Brillouin zone. However, it turned out that only those

phonon wave vectors contribute to a deep-UV Raman spectrum that have a large density

of states. In particular, these are the LO/TO phonon branches from the Γpoint, the LO

overbending area around the Γpoint, and the TO/LA as well as the LO phonon branches

at the M point. Especially the high-momentum M-point phonons are not accessible with

excitation energies from the visible spectrum.

Although inactive in a first-order Raman process, the E1umode in graphite gets active

in a second-order Raman process. As a consequence, we get a simultaneous access to

the Raman active E2gand Raman inactive E1umodes in graphite. We find analogies in

the deep-UV Raman spectra of graphene and carbon nanotubes, but with altered phonon

frequencies. This is even true for carbon nanotubes as we have measured an ensemble

sample. To summarize, deep-UV Raman measurements give direct access to areas in the

phonon dispersions of low-dimensional carbon materials that are otherwise only accessible

by more complicated experimental methods such as elastic neutron scattering.

For excitation energies close to the M-point energy, we observe a fingerprint of the weakly

emitting 2Dmode. In contrast to measurements in the visible region, it is a very low-

intensity peak found around 3050cm−1. Due to the optical excitation close to the M

point, the involved phonons have the smallest possible phonon wave vectors in a double-

resonant process. As a consequence, the phonons stem from the Γpoint rather than from

the proximity of the Kpoints as in the case of excitations in the visible range.

We have found that the second-order vibrational density of states in multilayer graphene

only depends marginally on the layer number. By these experimental findings we can

123

7. Conclusion

conclude that Γ-point frequencies of degenerate pairs of LO/TO phonons in multilayer

graphene are very close to those from graphite.

Carbon nanotubes

Low-intensity Raman modes at around 1950 and 2450cm−1have been analyzed in an

ensemble sample. For a large range of excitation energies, we have found that the ex-

plored modes are dispersive exhibiting bimodal peak structures. Simulating the Raman

modes in an ensemble sample with DFT + zone-folding calculations, we find that both

dispersion and bimodal peak structures are clear ensemble effects. Due to their diverse

electronic structures, metallic and semiconducting species have different diameters for

likewise optical bandgaps. Their diameters determine distinct phonon energies resulting

in pronounced Raman peaks that can be attributed to metallic or semiconducting tubes

when excited contemporaneously.

Especially the assignment of the 1950cm−1mode was discussed controversially in the past.

Our DFT calculations indicate that it can be referred to a LO+LA @Γprocess. However,

a non-orthogonal tight-binding approach rather attributes this mode to a TO+ZO @K

process. As the DFT + zone-folding approach requires the initial electron-phonon cou-

pling elements from graphene, we believe a ZO contribution to low-intensity Raman modes

is generally underestimated. The out-of-plane ZO mode in graphene is symmetry forbid-

den that not necessarily holds true for carbon nanotubes. Therefore, we tentatively assign

the mode at around 1950 cm−1to a TO+ZO@ Kprocess. In both approaches uniformly

assigned, the mode at around 2450cm−1is a TO+LA @Kcombination mode.

We have introduced an "extended Kataura plot" that contains optical transitions up to the

highest possible π→π∗transition. Although appearing contrary on a first view, the ar-

rangement of optical transitions as a function of the tube diameter follows the well-known

quantum confinement and trigonal warping effect. However, in contrast to transitions in

the visible range, we find that several transitions are forbidden. In particular, these are

all transitions at and above the M point from armchair tubes and all transitions from the

Γpoint. The optical matrix elements of transitions way above the M point are gener-

ally orders of magnitudes smaller than those below the M point. We believe this is the

reason why we do not observe a Raman signal of the RBM for excitation energies higher

than 4.7eV. Instead, a very broad Raman peak is observed for an excitation energy very

close to the M-point energy. Because of symmetry, all but armchair nanotubes have the

M-point transition in common which is why we excite a huge variety of nanotubes for

εL= 4.69eV. This, in turn, leads to a very large FWHM of the RBM.

With our measurements and calculations, we can generally conclude that excitations in

the visible range are more suitable for the characterization of a carbon nanotube samples

than those in the deep-UV region. On the one hand, the amount of carbon nanotubes

excited with a deep-UV excitation energy is generally smaller than for excitations in the

visible range. On the other hand, the optical absorption drastically decreases for deep-UV

excitation energies. We rather believe deep-UV Raman spectroscopy of carbon nanotubes

is a complementary characterization tool that reveals new aspects such as the second-order

vibrational density of states. However, deep-UV excitations may open a new way of car-

124

7. Conclusion

bon nanotube chemistry as shown for the light and carbon nanotube-induced formation

of Siderite. We have found that excitations close to both the absorption maximum and

the work function of carbon nanotubes in the presence of iron and atmospheric oxygen,

lead to the formation of iron-carbonate clusters.

In many ways, the Raman response from carbon nanotubes in the deep-UV region differs

from that in the visible region. Deep-UV Raman spectroscopy can especially be utilized

to observe phonons that are not accessible by visible light. In particular, this might be

interesting for isolated carbon nanotubes since ensemble effects, as in the explored case,

screen individual contributions.

125

List of Figures

2.1 Carbon sp2and sp3hybridization....................... 7

2.2 Structure of the real and reciprocal lattice of graphene . . . . . . . . . . . 8

2.3 One-dimensional electronic band structure of graphene . . . . . . . . . . . 10

2.4 The geometric structure of carbon nanotubes . . . . . . . . . . . . . . . . . 11

2.5 Electronic band structure of carbon nanotubes . . . . . . . . . . . . . . . . 13

2.6 Optical matrix elements of carbon nanotubes . . . . . . . . . . . . . . . . . 15

2.7 Structure of diamondoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1 Schematic first-order Raman scattering . . . . . . . . . . . . . . . . . . . . 22

3.2 Schematic double-resonant Raman scattering . . . . . . . . . . . . . . . . . 23

3.3 Overview of Raman spectra from carbon nanotubes, graphite, graphene,

anddiamantane................................. 26

3.4 Experimental Raman setup . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.1 Structure of double-bond diamondoid oligomers . . . . . . . . . . . . . . . 31

4.2 Characteristic Raman spectra of adamantane and diamantane . . . . . . . 32

4.3 Comparison of Raman spectra from double-bond diamondoid oligomers

andpristinediamondoids............................ 33

4.4 Characteristic vibrational modes of double-bond diamondoids . . . . . . . 36

4.5 Equi-energy surfaces of atomic orbitals from the 3,10-bis-(2-adamantylidene)diamantane

trimer ...................................... 38

4.6 Equi-energy surfaces of atomic orbitals from the [4.4](1,3)adamantanophan-

trans,trans-1,8-dienedimer........................... 39

4.7 Deep-UV and visible light Raman spectra of [4.4](1,3)adamantanophan-

trans,trans-1,8-diene .............................. 40

4.8 Deep-UV and visible light Raman spectra of double-bond diamondoid oligomers 41

4.9 Resonance profiles of the C=C stretching mode intensities from double-

bond diamondoid oligomers . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.10 Raman spectra of single-bond diamondoid dimers . . . . . . . . . . . . . . 45

4.11 Low-frequency Raman spectra of single-bond diamondoid dimers and bond

lengths of the connecting carbon-carbon length . . . . . . . . . . . . . . . 47

4.12 Absorption and Raman spectra of a double-bond diamantane and trishomocubane

dimers...................................... 52

4.13 Ultra-low frequency Raman modes of pristine diamondoids and double-

bond diamondoid oligomers . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.14 Structure of [2](1,3)Adamantano[2](2,7)pyrenophane . . . . . . . . . . . . 57

126

4.15 Raman spectra of crystalline [2](1,3)Adamantano[2](2,7)pyrenophane, adaman-

tane,andpyrene ................................ 58

4.16 Photo luminescence spectra of crystalline [2](1,3)Adamantano[2](2,7)pyrenophane 60

4.17 Photo luminescence, photo luminescence excitation, and Raman spectra of

[2](1,3)Adamantano[2](2,7)pyrenophane . . . . . . . . . . . . . . . . . . . 61

5.1 Phonon modes and phonon dispersion of graphite . . . . . . . . . . . . . . 65

5.2 Deep-UV and visible light Raman spectra of graphite . . . . . . . . . . . . 67

5.3 Double-resonant Raman process for deep-UV excitation energies in graphene 68

5.4 Deep-UV and visible light second-order Raman spectra of graphite . . . . . 69

5.5 Comparison of the calculated, second-order vibrational density of states to

experimental deep-UV Raman spectra of graphite . . . . . . . . . . . . . . 70

5.6 Schematic deep-UV, single-resonant scattering process in graphene . . . . . 72

5.7 Calculated and experimental deep-UV Raman spectra of graphite for var-

iousexcitationenergies............................. 73

5.8 Calculated and experimental deep-UV Raman spectra of the TO and LO

phononsingraphite............................... 74

5.9 Calculated and experimental vibrational density of states from graphene

andcarbonnanotubes ............................. 76

5.10 Near-UV and deep-UV Raman spectra of multilayer graphene . . . . . . . 78

5.11 Near-UV Raman spectra of the vibrational density of states from multilayer

graphene..................................... 79

5.12 Phonon dispersions, second-order vibrational density of states, and exper-

imental Raman spectra of bilayer and trilayer graphene . . . . . . . . . . . 81

5.13 Temperature-dependent shift rates of the E1uand E2gmodes in graphite . 84

5.14 Temperature-dependent shift rates of second-order density of states peak

ofgraphite.................................... 85

6.1 Experimental, second-order Raman spectra of HipCO carbon nanotubes . . 89

6.2 Experimental Raman spectra of the iTOLA and TOZO modes in carbon

nanotubes for various excitation energies . . . . . . . . . . . . . . . . . . . 91

6.3 Calculated Raman spectra of double-resonant Raman modes from a (9,5)

tube and various excitation energies . . . . . . . . . . . . . . . . . . . . . . 93

6.4 Calculated spectra of low-intensity, double resonant Raman modes of a

HipCOensemblesample ............................ 95

6.5 Phonon dispersion of graphene and calculated iTOLA and TOZO Raman

spectra for various excitation energies . . . . . . . . . . . . . . . . . . . . . 96

6.6 Non-orthogonal tight-binding calculated Raman spectra of iTOLA and

TOZO Raman modes in carbon nanotube . . . . . . . . . . . . . . . . . . 97

6.7 Calculated phonon energies of in-plane LA and out-of-plane ZO derived

phonon branches in carbon nanotubes . . . . . . . . . . . . . . . . . . . . . 99

6.8 Calculated joint density of states of the TO+LA phonon mode of a (11,7)

and (7,7) carbon nanotube . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.9 Second-order Raman spectra of HipCO carbon nanotubes for various exci-

tationenergies..................................102

6.10 Calculated electronic band structure of a (8,0) carbon nanotube . . . . . . 105

6.11ExtendedKatauraplot.............................106

6.12 Zoom-in of the extended Kataura plot . . . . . . . . . . . . . . . . . . . . 108

6.13 Optical matrix elements of carbon nanotubes for deep-UV optical transitions109

6.14 Calculated absorption spectra of a (7,4) and a (8,0) carbon nanotube . . . 110

6.15 Radial-breathing modes for visible and deep-UV excitation energies of car-

bonnanotubes .................................112

6.16 Deep-UV Raman spectra of a HipCO carbon nanotube sample with a newly

SideriteRamanmode..............................114

6.17 Functionalization/G-mode ratios and G-mode shifts in HipCO carbon nan-

otubes as a function of laser power and exposure time . . . . . . . . . . . . 115

6.18 Deep-UV Raman spectra of various carbon nanotube samples, graphene,

and graphite with an analysis of the Siderite Raman mode . . . . . . . . . 117

6.19 Raman maps of the D, G-mode, and Siderite Raman with an optical image

of the deep-UV illuminated HipCO sample . . . . . . . . . . . . . . . . . . 118

6.20 Vibrational pattern of the A0

1mode of Siderite . . . . . . . . . . . . . . . . 120

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Acknowledgment

I would like to thank many kind people who helped me on the way to finish my thesis.

In particular

my supervisor Prof. Dr. Janina Maultzsch with her great foresight ordering an UV

Raman laboratory. This, in particular, has given me the possibility to work on new

topics in the field of carbon materials. Also, for her constant and always very kind

support in various matters. I have enjoyed being in the AG Maultzsch.

Prof. Dr. Michael Lehmann for being the chairman of the thesis committee.

Prof. Dr. Ralph Krupke for reviewing this thesis.

Prof. Dr. Axel Hoffmann for reviewing this thesis.

Felix Herziger who constantly came up with clever and stimulating ideas and, in his

inimitable way, brought even the most complicated things to the point.

Roland Gillen for so many nice moments in New York, Boston, Athens, and of course

for so many deep, interesting, and inspiring physical discussions. Also, for the vast

theoretical support concluding with several joint articles.

Harald Scheel for his analytic expertise and the non-stopping support in my some-

times stubborn laboratory.

René Meinke for building up the UV laboratory and supporting me in my first

months.

Rohit Narula for many animated discussions concerning double-resonant Raman

scattering in graphene and carbon nanotubes, and of course for his support for our

joint article.

Valentin N. Popov for many stimulating discussions concerning double-resonant

Raman scattering in bilayer graphene, graphite, carbon nanotubes, and of course

for his kind support for our joint article.

The whole AK Peter Schreiner and in particular Andrey A. Fokin, Tetyana V. Koso,

Vladimir N. Rodionov, Natalie A. Fokina, Heike Hausmann, Paul Kahl, Philipp

Wagner, Ciro Balestrieri, Jonathan Becker, Graham J. Bodwell for the kind pro-

viding of the manifold diamondoid samples. Further, Peter Schreiner for the very

critical proof reading of our joint articles.

157

The whole AG Möller and in particular Robert Richter, Tobias Bischoff, André

Knecht, Andrea Merli, Torbjörn Rander for the beam line measurements at BESSY

II. Further, Tobias Bischoff and André Knecht for many fruitful discussion on dia-

mondoids.

Prof. Dr. Peter Saalfrank and Tao Xiong for a theoretical introduction to the

electronic structure of diamondoid derivatives.

Stefan Kalinowski for the support of several measurements in the "Gelbes Labor".

The very well organized and supportive Anja Sandersfeld to manage all the bureau-

cracy.

Our daily, very early (11:15) lunch group with many amusing and refreshing mo-

ments.

The office colleagues, Stefan Westerkamp, Felix Kampmann, and Stefan Kalinowski

for so many rememberable moments, deep and interesting physical, political and

humorous discussions.

My Bachelor and Master student Tobias Hückstaedt always coming up with well-

reasoned and interesting ideas.

The whole groups AG Maultzsch and AG Hoffmann for the nice atmosphere during

the last years, especially Dirk Heinrich, Harald Scheel, Sevak Kachadorian, Thomas

Kure, Felix Nippert, Ludwig Greif, Stefan Jagsch, Alexander Mittelstädt, Stefan

Kalinowski, Andrei Schliwa, Felix Kampmann, Markus Wagner, Narine Ghazaryan

and Michael Mayer.