111,99 €
Mehr erfahren.
- Herausgeber: Wiley-VCH
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
A first on ultrafast phenomena in carbon nanostructures like graphene, the most promising candidate for revolutionizing information technology and communication
The book introduces the reader into the ultrafast nanoworld of graphene and carbon nanotubes, including their microscopic tracks and unique optical finger prints. The author reviews the recent progress in this field by combining theoretical and experimental achievements. He offers a clear theoretical foundation by presenting transparently derived equations. Recent experimental breakthroughs are reviewed.
By combining both theory and experiment as well as main results and detailed theoretical derivations, the book turns into an inevitable source for a wider audience from graduate students to researchers in physics, materials science, and electrical engineering who work on optoelectronic devices, renewable energies, or in the semiconductor industry.
Sie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 543
Veröffentlichungsjahr: 2013
Ähnliche
Contents
Preface
1 Introduction – The Carbon Age
1.1 Graphene
1.2 Carbon Nanotubes
2 Theoretical Framework
2.1 Many-Particle Hamilton Operator
2.2 Microscopic Bloch Equations
2.3 Electronic Band Structure of Graphene
2.4 Electronic Band Structure of Carbon Nanotubes
2.5 Optical Matrix Element
2.6 Coulomb Matrix Elements
2.7 Electron–Phonon Matrix Elements
2.8 Macroscopic Observables
3 Experimental Techniques for the Study of Ultrafast Nonequilibrium Carrier Dynamics in Graphene Guest article by Stephan Winnerl
3.1 The Principle of Pump-Probe Experiments
3.2 Characteristics of Short Radiation Pulses
3.3 Sources of Short Infrared and Terahertz Radiation Pulses
3.4 Single-Color and Two-Color Pump-Probe Experiments on Graphene
Part One Electronic Properties – Carrier Relaxation Dynamics
4 Relaxation Dynamics in Graphene
4.1 Experimental Studies
4.2 Relaxation Channels in Graphene
4.3 Optically Induced Nonequilibrium Carrier Distribution
4.4 Carrier Dynamics
4.5 Phonon Dynamics
4.6 Pump Fluence Dependence
4.7 Influence of the Substrate
4.8 Auger-Induced Carrier Multiplication
4.9 Optical Gain
4.10 Relaxation Dynamics near the Dirac Point
5 Carrier Dynamics in Carbon Nanotubes
5.1 Experimental Studies
5.2 Phonon-Induced Relaxation Dynamics
5.3 Coulomb-Induced Quantum-Kinetic Carrier Dynamics
Part Two Optical Properties – Absorption Spectra
6 Absorption Spectra of Carbon Nanotubes
6.1 Experimental Studies
6.2 Absorption of Semiconducting Carbon Nanotubes
6.3 Absorption of Metallic Carbon Nanotubes
6.4 Absorption of Functionalized Carbon Nanotubes
7 Absorption Spectrum of Graphene
7.1 Experimental Studies
7.2 Absorbance and Conductivity in Graphene
Appendix A Introduction to the Appendices
A.1 Microscopic Processes in Carbon Nanostructures
A.2 Outline of the Theoretical Description
Appendix B Observables in Optical Experiments
B.1 Temporal and Spectral Information in Measurements
B.2 Intensity-Related Optical Observables
B.3 Specific Solutions of the Wave Equation for Graphene and Carbon Nanotubes
B.4 Differential Transmission
Appendix C Second Quantization
C.1 Lagrange Formalism for Particles
C.2 Lagrange Formalism for Fields
C.3 Quantization of Free Fields
C.4 Quantization of Interacting Fields
C.5 Electron–Phonon Interaction in Second Quantization
C.6 Many-Particle Hamilton Operator
C.7 Electron–Light Interaction
C.8 Electrons and Phonons in Periodic Solid-State Structures
Appendix D Equations of Motion
D.1 Hierarchy Problem
D.2 Macroscopic Observables
D.3 The Relevant Density Operator
D.4 Treatment of a Bath
Appendix E Mean-Field and Correlation Effects
E.1 Mean-Field Contributions (Hartree–Fock)
E.2 Coulomb Correlations in an Equation of Motion Approach
E.3 Correlation Contributions: Electron–Phonon Interaction
E.4 A More Systematic Way to Correlation Effects: Screened Electron–Electron Interaction
References
Index
Related Titles
Jiang, D.-E., Chen, Z.
Graphene Chemistry
Theoretical Perspectives
2013
ISBN: 978-1-119-94212-2
Fujita, S., Suzuki, A.
Electrical Conduction in Graphene and Nanotubes
2013
ISBN: 978-3-527-41151-1
Waser, R. (ed.)
Nanoelectronics andInformation Technology
Advanced Electronic Materials and Novel Devices
2012
ISBN: 978-3-527-40927-3
Monthioux, M.
Carbon Meta-Nanotubes
Synthesis, Properties and Applications
2011
ISBN: 978-0-470-51282-1
Jorio, A., Dresselhaus, M. S., Saito, R., Dresselhaus, G.
Raman Spectroscopy inGraphene Related Systems
2011
ISBN: 978-3-527-40811-5
Delhaes, P.
Solids and CarbonatedMaterials
2010
ISBN: 978-1-84821-200-8
Akasaka, T., Wudl, F., Nagase, S. (eds.)
Chemistry of Nanocarbons
2010
ISBN: 978-0-470-72195-7
Krüger, A.
Carbon Materials andNanotechnology
2010
ISBN: 978-3-527-31803-2
Guldi, D. M., Martín, N. (eds.)
Carbon Nanotubes andRelated Structures
Synthesis, Characterization, Functionalization, and Applications
2010
ISBN: 978-3-527-32406-4
Reich, S., Thomsen, C., Maultzsch, J.
Carbon Nanotubes
Basic Concepts and Physical Properties
2004
ISBN: 978-3-527-40386-8
The Authors
Dr. Ermin Malic
Technische Universität Berlin
Inst. für Theoretische Physik
Hardenbergstr. 36
10623 Berlin
Germany
Prof. Dr. Andreas Knorr
Technische Universität Berlin
Inst. für Theoretische Physik
Hardenbergstr. 36
10623 Berlin
Germany
All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.
Library of Congress Card No.:
applied for
British Library Cataloguing-in-Publication Data:
A catalogue record for this book is available from the British Library.
Bibliographic information published by the Deutsche Nationalbibliothek
The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.d-nb.de.
© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Boschstr. 12, 69469 Weinheim, Germany
All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.
Print ISBN 978-3-527-41161-0
ePDF ISBN 978-3-527-65877-0
ePub ISBN 978-3-527-65876-3
mobi ISBN 978-3-527-65875-6
oBook ISBN 978-3-527-65874-9
Cover Design Adam-Design, Weinheim
Typesetting le-tex publishing services GmbH, Leipzig
To
Kati
Yvette, Paula, and Hannah
Claudia and Noah
Preface
In this book, we review recent progress in the field of optics and ultrafast processes in carbon nanostructures. The focus lies on the intensive theoretical research on graphene and carbon nanotubes performed at the Technische Universität Berlin combined with a number of collaborations with experimental groups worldwide. Parts of the book are based on the habilitation theses of the authors.
The key for designing and engineering novel carbon-based optoelectronic devices is a microscopic understanding of their pronounced optical properties as well as of the ultrafast carrier relaxation dynamics. The presented density matrix formalism offers microscopic tools to reveal an optical finger print, which unambiguously characterizes each carbon nanostructure. Furthermore, it enables us to track the path of optically excited carriers towards equilibrium – resolved in time, momentum, and angle. It treats carrier–light, carrier–carrier, and carrier–phonon interactions on the same microscopic footing allowing a study of the coupled carrier, phonon, and coherence dynamics. Combined with tight-binding wave functions, it does not rely on further adjustable parameters. A comparison with modern high-resolution experiments leads to new insights into the underlying elementary processes, which is a crucial prerequisite for exploiting the exceptional application potential of carbon-based materials.
The main part of the book introduces the reader to the ultrafast nanoworld of graphene and carbon nanotubes including their unique properties and future perspectives. It offers a theoretical foundation based on equations derived within an in-depth appendix. Furthermore, it reviews recent experimental techniques on pump-probe spectroscopy accessing ultrafast carrier relaxation within a guest contribution by Stephan Winnerl. The combination of theory and experiment throughout the book as well the connection between the main results and detailed theoretical derivations in the appendix only requiring knowledge of basic quantum mechanics makes the book suitable for theoreticians and experimentalists, for researchers and graduate students, and for physicists and engineers.
We would like to use this opportunity to thank a number of people, who contributed to the accomplishment of this book. First of all, we would like to thank our students for their exemplary carbon research – without them, this book would not exist. Especially, we are grateful to Torben Winzer for his excellent work on the relaxation behavior of excited electrons in graphene; Eike Verdenhalven, Evgeny Bobkin, and Faris Kadi, who helped us reveal the optical finger print of carbon nanostructures; Christopher Köhler and Matthias Hirtschulz for shedding light on the nonequilibrium dynamics in carbon nanotubes; Stephan Butscher for his initial study on carrier relaxation in graphene. Special thanks go to Faris, who helped us type the comprehensive appendix. Furthermore, we thank Frank Milde, Torben, Eike as well as Marten Richter, Florian Wendler, and Gunnar Berghäuser for careful reading of various chapters of the book. Their suggestions contributed to a clearer and better presentation.
The progress of this work has substantially benefited from close scientific collaborations with experimental and other theoretical groups. Here, our thanks go to Stephanie Reich (Freie Universität Berlin), who introduced us to the fascinating world of carbon nanotubes, Manfred Helm (Helmholtz-Zentrum Dresden-Rossendorf) for new insights into the relaxation behavior of graphene electrons close to the Dirac point, Tony F. Heinz (Columbia University, New York) for many stimulating discussions on carbon nanotubes and graphene, Thomas Elsaesser (Max-Born Institut Berlin) and Jürgen Rabe (Humboldt Universität Berlin) for the joint high-resolution study on the carrier and phonon dynamics in graphene, Theodore E. Norris (University of Michigan) and John E. Sipe (University of Toronto) for the successful collaboration on current decay in graphene, Ulrike Woggon (Technische Universität Berlin) for revealing efficient relaxation channels in carbon nanotubes, Rolf Binder (University of Arizona) for fruitful discussions on many-particle screening in carbon nanostructures, Pablo Ordejon, Carlos F. Sanz-Navarro (CIN2, Barcelona), Peter Saalfrank, and Tillmann Klamroth (Universität Potsdam) for scientific collaboration on the ab initio description of functionalized carbon nanotubes, and last but not least Janina Maultzsch and Christian Thomsen (Technische Universität Berlin) for valuable input on carbon nanostructures since the beginning of our research. We also thank Vera Palmer and Anja Tschörtner from Wiley-VCH for their support.
Stephan Winnerl is thankful to a number of colleagues working on ultrafast spectroscopy, especially Sabine Zybell, Jayeeta Bhattacharyya, Oleksiy Drachenko, Martin Wagner, Dominik Stehr, Harald Schneider, Manfred Helm, Thomas Dekorsy, and Luke Wilson. Furthermore, he thanks Alfred Leitenstorfer, Rupert Huber, Jonathan Eroms, Sergey Mikhailov, and Sergey Ganichev for stimulating discussions and gratefully acknowledges collaboration with Milan Orlita and Marek Potemski on the carrier dynamics in graphene and Claire Berger, Walt de Heer, and Thomas Seyller for graphene samples. Moreover, he would like to thank Martin Mittendorff and Fabian Göttfert for their enthusiasm concerning the graphene experiments.
Finally, we thank the Deutsche Forschungsgemeinschaft (DFG), Einstein Foundation Berlin, DAAD, and German National Academic Foundation for the financial support of our research on carbon nanostructures.
Berlin, October 2012
Ermin Malic, Andreas Knorr
1
Introduction – The Carbon Age
The continuing trend towards miniaturization of optoelectronic devices leads to fundamental physical limits of conventional silicon-based materials. The search for new concepts has moved low-dimensional carbon nanostructures into the focus of current research [1–5]. They are represented by a variety of different metallic and semiconducting materials with unique optical, electronic, and mechanical properties [2, 3].
The main carbon material is graphite. It consists of multiple flat layers of sp2-hybridized carbon atoms arranged in a hexagonal lattice [6]. While the σ-bonds between the carbon atoms are very strong, the Van der Waals coupling between different layers is rather weak and can be easily broken. Therefore, graphite is a suitable material for example for pencils.
In 1985, a new carbon structure named fullerene was discovered [7]. Its most common form C60 is a spherical carbon molecule with a mean diameter of 0.68 nm, cp. Figure 1.1c. Since the charge carriers are spatially confined in all directions, fullerenes are zero-dimensional carbon nanostructures. For their discovery, Richard Smalley, Robert Curl, and James Heath obtained the Nobel Prize in Chemistry in 1996.
Carbon nanotubes (CNTs) represent another low-dimensional carbon nanostructure, which was found for the first time in 1991 by Iijima [8, 9] and has attracted large scientific and technological interest. Nanotubes are tiny, hollow cylinders constructed by rolling up a single layer of graphite, cp. Figure 1.1b. They have diameters in the range of one nanometer, while their length can reach several micrometers. As a result, they are prototypical one-dimensional systems, in which carriers can move freely only along the axis of the cylinder. Arising from a specific geometry quantum confinement, they can be either metallic or semiconducting with a tunable bandgap. This makes them excellent materials for various technological applications.
In 2004, graphene was discovered – a perfect two-dimensional carbon nanostructure consisting of a monolayer of carbon atoms [10]. Graphene can be considered as the basic carbon material and it can be rolled up into a one-dimensional carbon nanotube, wrapped into a zero-dimensional fullerene, and stacked into a three-dimensional graphite, cp. Figure 1.1. Up to its discovery, graphene was considered as an academic material [6] that could not exist in reality due to thermodynamic instabilities of two-dimensional structures on a nanometer scale [11–13]. However, Konstantin Novoselov and Andre Geim from the University of Manchester succeeded to mechanically exfoliate a single layer of graphite and revealed a completely new structure with exceptional properties. They were awarded the Nobel Prize for Physics in 2010.
Figure 1.1 Schematic illustration of different carbon-based nanostructures. (a) Graphene as a two-dimensional material consisting of a single layer of carbon atoms can be rolled up into (b) one-dimensional carbon nanotubes, wrapped into (c) zero-dimensional fullerenes, or stacked into three-dimensional graphite (multilayered graphene). (d) Exemplary carbon-based hybrid nanostructure consisting of a nanotube functionalized with merocyanine molecules. The latter creates a strong dipole field giving rise to changes in the optical and electronic properties of the nanotube. Figure parts (a) and (b) adapted from [14], part (c) from [15].
Meanwhile, also carbon-based hybrid materials consisting of a nanostructure functionalized with molecules have moved into the focus of current research [16–18], cp. Figure 1.1d. Carbon materials offer a variety of metallic and semiconducting substrates, which show a large sensitivity to changes in their surrounding environment. Thus, they are excellent structures for functionalization with molecules [17, 19], which offers the possibility to control and optimize certain properties. As a result, functionalization is a promising strategy to exploit the great application potential of carbon nanostructures [18, 20, 21].
1.1 Graphene
Properties Since its discovery [10], graphene has attracted tremendous interest in fundamental research and industry [5, 22]. Graphene has an exceptional band structure exhibiting crossing points between the valence and the conduction band at the six corners of the Brillouin zone [6]. Around these so-called Dirac points, the energy dispersion is linear giving rise to unique optical and electronic properties [3, 15]. The electrons behave like relativistic massless particles and can be described by a Dirac-like equation [15]. One consequence is for example the anomalous fractional quantum Hall effect [23].
Consisting of just a single layer of carbon atoms, graphene is the thinnest known material. Because of the sp2-bonds between the hexagonally arranged carbon atoms, it is also one of the strongest structures that we know of [24]. Furthermore, it is an excellent conductor of electricity and heat, which can be traced back to its excellent carrier mobility at room temperature. The carriers can move freely without scattering events over length scales of some hundreds of nanometers (ballistic transport) [3, 25, 26]. Moreover, graphene is flexible and almost transparent in the optical frequency range [27–29]. As a result, it is a promising material for applications in nanoelectronics and optoelectronics [3, 30].
Synthesis Graphene was first obtained by mechanical exfoliation [10]. Konstantin Novoselov and Andre Geim used a common adhesive tape to repeatedly split graphite crystals into increasingly small flakes, cp. Figure 1.2. Then, they transferred the thinnest flakes onto a SiO2 substrate of a specific width. Finally, they could visualize monolayers of carbon atoms under a simple optical microscope. The procedure is known as the scotch tape or drawing method, since the mechanical exfoliation resembles writing with a pencil. Such exfoliated graphene exhibits a high crystal quality resulting in a large carrier mobility. This method is ideal for producing samples for fundamental research. However, the sample size is too small (around 10 μm) for technological applications. Thus, other methods allowing large-scale exfoliation are required to exploit the great application potential of graphene [30]. The most promising growth technique for the mass production of large area graphene films is chemical vapor deposition (CVD). Graphene samples with a size of over 75 cm have already been reported [31]. However, the structural quality of the produced graphene layers is so far lower compared to exfoliated or epitaxially grown samples.
Figure 1.2 (a) Schematic illustration of the scotch tape method applied to mechanically exfoliate graphene from bulk graphite. (b) Thin flakes of graphite on a SiO2 substrate account for various colors, which represent their thickness reaching values from 100 nm (white flakes) to a few nanometers corresponding to just a few monolayers (black flakes). Figure adapted from [5].
The crystalline quality of graphene grown epitaxially on silicon carbide or metal substrates via high-temperature annealing is very high [32, 33]. However, the growth strongly depends on which side of the SiC graphene layers are grown. On the Si-face a single layer of highly doped graphene is formed on top of a buffer layer [34]. On the C-face rotationally twisted graphene multilayers are produced, which do not exhibit Bernal stacking order but are rotationally twisted against each other resulting in a negligible coupling and a graphene-like behavior [32]. A more detailed discussion of growth methods can be found in Section 3.4.
Applications Graphene is considered a very promising future material for nanoelectronics [3, 5, 30, 35]. It is characterized by huge carrier mobilities, optical transparency, flexibility, robustness, and environmental stability giving rise to a large variety of applications ranging from solar cells and light-emitting devices to touch screens, photodetectors, and ultrafast lasers [30, 35].
In particular, the observed ballistic transport and the resulting exceptional carrier mobility can be exploited to build highly efficient transistors. Recently, IBM researchers reported on a graphene-based field-effect transistor with an on–off rate of 100 GHz exceeding the speed of corresponding silicon transistors [36, 37], cp. Figure 1.3a. Since graphene is almost transparent and at the same time an excellent conductor it is also a promising material for transparent electrodes required in touch screens and displays. Its mechanical strength and flexibility make it more suitable than indium tin oxide [5, 30, 38].
The exceptional carrier mobility and absorption over a large spectral range from terahertz to ultraviolet also suggest application as ultrafast, spectrally broad photodetectors [35, 39]. Recently, terahertz emission and amplification was observed in optically pumped graphene demonstrating the feasibility of active graphene-based terahertz devices [30]. Furthermore, graphene can be applied as a broadband and fast saturable absorber in ultrafast laser systems [40, 41] due to its ultrafast carrier dynamics, broadband absorption, and efficient Pauli blocking – in contrast to conventional materials with a narrow spectral range requiring costly bandgap engineering. Finally, graphene has already been applied in composite materials to achieve improved mechanic and electric properties [42].
Challenges Despite the impressive progress in the short time since its discovery, there are still substantial challenges on the way to graphene-based nanoelectronics. In particular, further progress in epitaxial growth techniques is needed to obtain high-quality graphene films complying with industrial standards including sufficient reproducibility and control of specific features in graphene devices. Furthermore, for many applications it is necessary to induce a bandgap at the Dirac point. This can be achieved by cutting graphene into thin ribbons exhibiting a series of subbands due to the spatial confinement similar to carbon nanotubes [43, 44]. The bandgap can then be tailored by changing the width of nanoribbons. The other strategy is to use bilayer graphene, where a bandgap of up to 250 meV can be opened by applying an electrical field [45].
Figure 1.3 Prototypes of field-effect transistors based on (a) graphene (IBM) [36, 37] and (b) carbon nanotubes (Infineon) [46]. Source IBM and Infineon, respectively.
The key prerequisite for most applications in optoelectronics is a microscopic understanding of the character of optical excitations as well as of their ultrafast relaxation dynamics. Many important aspects of experimental data have not yet been fully complemented with theoretical studies on a microscopic footing. This aspect is addressed in detail in Chapters 4 and 7 of this book.
1.2 Carbon Nanotubes
Similarly to graphene, carbon nanotubes are one of the strongest known materials due to the covalent sp2-bonds between the carbon atoms [47]. At the same time, they are very light, because they consist of a single layer of carbon atoms. The extraordinary crystal quality gives rise to a ballistic conduction of current and heat at room temperature. As a one-dimensional material with an extreme length-to-diameter ratio, CNTs also exhibit pronounced optical properties [48]. In contrast to graphene, the further spatial confinement gives rise to a variety of metallic and semiconducting nanostructures with distinct physical properties. Depending on the chiral angle, one third of all possible nanotubes are metallic [1]. Furthermore, the bandgap of the semiconducting CNTs is tunable with the diameter of the tube. This variety of one-dimensional nanostructures with distinct physical properties accounts for the huge potential of carbon nanotubes for technological applications [2, 4, 49].
Synthesis Carbon nanotubes can be grown by chemical vapor deposition, laser ablation, and arc discharge [14]. All methods have in common that first a carbon plasma is generated and then metal catalysts are added, which induce the growth of nanotubes. The type of catalyst and the growth conditions determine the number of nanotube walls, their diameter, and length [14].
Figure 1.5a shows an SEM image of CNT bundles grown by arc discharge [50]. Each bundle consists of 20–100 single-walled carbon nanotubes, as indicated in the high-resolution TEM image in Figure 1.5b. To obtain single-walled CNTs, the bundles are broken apart by ultrasonification and the single tubes are coated by a surfactant to prevent them from rebundling. Furthermore, CNT forests can be produced via water-assisted chemical vapor deposition [51], cp. Figure 1.5c. Additionally, the CVD technique enables the growth of CNTs in well-defined positions on patterned substrates [51], as shown in Figure 1.5d.
While the diameter and the length of CNTs can be controlled during growth, the control of the chiral angle remains a major challenge in current research. Usually, CNT samples exhibit a homogeneous distribution of chiral angles. Separation of metallic and semiconducting carbon nanotubes was successfully achieved by using i.a. (i) density-gradient ultracentrifugation (DGU), which separates surfactant-wrapped CNTs by the difference in their density [52], (ii) alternating current dielectrophoresis, which exploits the difference in the relative dielectric constants resulting in an opposite movement of metallic and semiconducting tubes along the electric field gradient [53], and (iii) agarose gel electrophoresis, where the relative mobility through a gel is used for separation of CNTs with different molecular weights [54]. Recently, progress has also been made allowing enrichment of CNTs with a specific chirality for example by exploiting chirality-dependent wrapping of CNTs with the DNA leading to enrichments of up to 90% and more [55].
Figure 1.5 Carbon nanotubes obtained with different growth techniques including (a) CNT bundles grown by arc discharge [50], (b) cross section of such a bundle, (c) mm-thick CNT forests grown by water-assisted chemical vapor deposition [51], (d) CNTs grown in predefined places on a patterned substrate [51]. Figure taken from [14].
Applications The variety of one-dimensional metallic and semiconducting carbon nanotubes with a tunable bandgap makes CNTs promising candidates for various technological applications [2, 4, 49]. In particular, nanotube-based field-effect transistors have already been demonstrated based on room-temperature ballistic transport [4, 46], cp. Figure 1.3b. Furthermore, metallic CNTs can be used in integrated circuits to conduct current and high-speed signals [56]. Consisting of a single layer of carbon atoms, CNTs exhibit a large surface-to-volume ratio, which is favorable for application as chemical or gas sensors [57]. Functionalization of CNTs can be used to improve their sensitivity and selectivity to specific chemical substances or biomolecules [58].
The strength and flexibility of CNTs on the one hand and their extraordinary conductivity on the other hand suggest engineering of transparent, electrically conductive films of CNTs for application in touch screens and flexible displays [49, 59]. CNTs have already been used in composite materials to improve their mechanical, thermal, and electronic properties [60]. Their mechanical strength is already exploited in everyday items, such as cloths, sports gear, cars and so on.
Challenges Most technological applications require an almost perfect separation of metallic and semiconducting nanotubes. For some applications, even a separation by chiral angle is necessary. Therefore, further progress in growth and selection techniques remains the major challenge in current nanotube research [2].
Furthermore, the unambiguous characterization and identification of specific CNTs is an important issue. In this context, the well-pronounced optical transitions can be seen as an optical finger print for each individual nanotube. Optical spectroscopy methods, such as absorption, photoluminescence, Rayleigh and Raman scattering can be used for structural assignment of CNTs [61–65]. This aspect is addressed in detail in Chapter 6.
For applications in nanoelectronics, a microscopic understanding of the Coulomb- and phonon-induced ultrafast relaxation dynamics of optically excited carriers is required, in particular addressing different relaxation channels and their diameter and chirality dependence. Microscopic investigations on the carrier dynamics in CNTs are presented in Chapter 5.
2
Theoretical Framework
In this chapter, the basic theoretical methods applied in this book are presented. All investigations are based on a many-particle density-matrix framework – an established technique for quantum-mechanical treatment of solid-state many-particle systems [66–68]. Applying the Heisenberg equation, we obtain graphene/CNT Bloch equations describing the coupled dynamics of carrier and phonon occupations as well as of the microscopic polarization. The approach provides microscopic tools for the investigation of optical and electronic properties of low-dimensional nanostructures including excitonic effects in optical spectra as well as the ultrafast many-particle kinetics in a nonequilibrium situation. More details on the theoretical approach can be found in the comprehensive appendix. In particular, if the reader is not experienced with the theoretical description of many-particle effects within the second quantization, we suggest reading Appendix C.
2.1 Many-Particle Hamilton Operator
To obtain the graphene/CNT Bloch equations, we first need the many-particle Hamilton operator H, which determines the energy of a carrier–phonon system interacting with an electromagnetic field [68]. We apply a semiclassical approach treating the carriers and phonons quantum mechanically and the electrical field classically. The formalism is expressed in second quantization based on the introduction of Heisenberg field operators
(2.1)
(2.2)
with the electronic single-particle energy εl and the phonon energy . The carrier–field interaction Hc,f is expressed within the radiation gauge [69] as
(2.3)
with the elementary charge e0, the free electron mass m0, and the vector potential A(t). The strength of this carrier–light coupling is given by the optical matrix element . The contribution proportional to the square of the vector potential has been neglected, since it is small within the limit of linear optics and it does not contribute to the dynamics of nonequilibrium carriers after an optical excitation.
The carrier–carrier interaction Hc,c is given by
(2.4)
including the Coulomb matrix element . Finally, the last contribution Hc,p of the Hamilton operator describes the interaction between carriers and phonons and reads
(2.5)
with the carrier–phonon matrix elements .
The Hamilton operator becomes specific by inserting the dispersion relations and the coupling elements for the material to be investigated. In Sections 2.3–2.7, we discuss in detail these material-specific quantities in the case of graphene and carbon nanotubes.
2.2 Microscopic Bloch Equations
With the Hamilton operator H, we have microscopic access to the temporal evolution of an arbitrary quantity within the Heisenberg equation of motion [66]
(2.6)
In this book, we focus on the carrier and phonon dynamics in a nonequilibrium situation. Therefore, the quantities of interest are: (i) the coherence or microscopic polarization . (ii) the carrier occupation probability in the state k within the band λ, and (iii) the phonon occupation in the mode j and with the momentum q. Figure 2.1 illustrates these quantities in a system with a linear electronic band structure. First, an optical pulse described by the vector potential A(t) is applied to perturb the carrier system. In the strong excitation regime, a nonequilibrium carrier distribution is generated by lifting electrons from the valence into the conduction band. The microscopic polarization pk is a measure for the transition probability between the two bands. The optical excitation and the subsequent scattering dynamics changes the occupation probabilities in the involved bands. The phonon-induced scattering also changes the phonon occupation . Therefore, knowledge of the temporal evolution of . and is required to microscopically access the nonequilibrium carrier dynamics.
Figure 2.1 Illustration of the microscopic quantities of interest for describing the carrier and phonon dynamics in nonequilibrium. The optical excitation is described by the vector potential A(t), which lifts electrons from the valence (v) into the conduction band (c). A measure for the transition probability is given by the microscopic polarization pk. The Coulomb- and phonon-induced scattering changes the occupation probabilities in the involved bands as well as the phonon occupation . Figure adapted from [70].
Applying the Heisenberg equation of motion and exploiting the fundamental commutator relations for fermionic and bosonic operators, we obtain optical Bloch equations (cp. Appendix D [70, 71]):
(2.7)
(2.8)
(2.9)
Here, is the momentum-dependent energy difference between the valence and the conduction band, . A(t) is the Rabi frequency, corresponds to the experimentally accessible phonon lifetime [72, 73], and nB denotes the equilibrium distribution of phonons, which is given by the Bose–Einstein distribution at room temperature. The intraband contribution of the carrier–field interaction is proportional to . A(t) is nonlinear in the vector potential (cp. Eq. (2.7)) and since it is also connected to pk, it gives rise to the generation of higher harmonics in a strong-excitation regime [74]. Because of the symmetry of the valence and conduction band around the K point, the carrier occupations in the two bands are related via . Within the electron–hole picture, we obtain and . that is, electrons in the conduction band and holes in the valence band show the same dynamics.
Many-particle interactions in Eqs. (2.7)–(2.9) can be separated into a Hartree–Fock and a scattering part . cp. Appendix E. The many-particle interactions couple the dynamics of single-particle elements to higher-order terms describing the correlation between carriers, that is, . The equation of motion for the appearing two-particle quantities depends on three-particle terms . which couple to four-particle quantities, and so on. The resulting set of equations is not closed and an infinite hierarchy of quantities with increasing number of involved particles appears [66].
At some level, this infinite hierarchy needs to be systematically truncated. In this book, we apply the correlation expansion and consider only contributions from a certain order assuming that higher-order terms are negligibly small, cp. Appendix E for more details [75, 76]. This factorization technique leads to a closed set of equations for the single-particle elements. To give an example, a two-particle quantity is factorized into products of single-particle terms yielding
where denotes the correlation term beyond the Hartree–Fock approximation. Neglecting in the first order corresponds to the Hartree–Fock factorization or the mean-field approximation [66, 67]. Considering the second-order terms in the carrier–carrier and carrier–phonon interaction, that is, explicitly calculating the dynamics of and neglecting the three-particle correlation quantities, allows the investigation of two-particle scattering processes [77]. This is called second-order Born approximation [68].
2.2.1 Hartree–Fock Approximation
Here, we discuss the Hartree–Fock approximation applied to the Coulomb interaction. This approximation already describes well the formation of excitons [66], which are known to dominate the linear spectra of carbon nanostructures, as discussed in Chapters 6 and 7. Neglecting the carrier–phonon coupling for now, the Bloch equations read within the Hartree–Fock approximation:
(2.10)
(2.11)
We observe two different Coulomb-induced contributions to the dynamics of the microscopic polarization: the term proportional to pk (t) (first line of Eq. (2.10)) describes the repulsive electron–electron interaction giving rise to a renormalization of the single-particle energy in Eq. (2.7). The second Coulomb contribution is proportional to pk′(t) (second line in Eq. (2.10)) and induces a redistribution of the momentum. It leads to a renormalization of the Rabi frequency Ωk(t) in Eq. (2.7) accounting for the formation of excitons. The strength of the renormalization is given by the involved material-specific Coulomb matrix elements .
In a many-particle system, it is of crucial importance to take into account the intrinsic screening of the Coulomb potential. The Coulomb matrix elements entering the Bloch equations are screened via the momentum-dependent dielectric function ε(q) resulting in
(2.12)
The dielectric function ε(q) can be derived self-consistently within the presented equation of motion approach by taking into account higher-order contributions in the correlation expansion, cp. Appendix E.2.3.
The Coulomb-induced change in absorption spectra of carbon nanostructures is investigated in detail in Chapters 6 and 7.
2.2.2 Second-Order Born-Markov Approximation
The access to the carrier relaxation dynamics requires an extension of the Bloch equations beyond the mean-field level. The scattering contributions , and in Eqs. (2.7)–(2.9) are obtained by considering the carrier–carrier and the carrier–phonon interaction within the second-order Born–Markov approximation [68]. The introduced Markov approximation neglects quantum-mechanic memory effects and accounts for a conservation of energy, as discussed below.
This approach results in a microscopic Boltzmann-like kinetic equation for the carrier occupation probabilities
(2.13)
2.2.2.1 Detailed Balance
In equilibrium, the in- and out-scattering processes compensate each other leading to . Here, the carrier distribution ρl corresponds to a Fermi function f1 =f(ε1,T,u), which depends on the energy εl, the temperature T, and the chemical potential μ. As a result, in equilibrium we can derive a fixed relation between the in- and out-scattering rate with
(2.14)
The initial carrier distribution before the optical excitation as well as the final, thermalized carrier distribution fulfill this relation.
2.2.2.2 Markov Approximation
The presented Boltzmann-like kinetic equation is obtained by applying a Markov approximation. Within the second-order Born approximation, equations of motion for two-particle quantities need to be evaluated, cp. Appendix E. For reasons of simplicity, we neglect the quantum numbers in this paragraph. Applying the Heisenberg equation and considering only the free-carrier part and the Coulomb interaction yields
(2.15)
with the energy difference Δε= (ε3 + ε4 – ε1 – ε2) of all involved states and the in-homogeneous part Q(t) containing the integrals over all scattering and dephasing contributions, cp. Appendix E.2.2. The solution of this inhomogeneous differential
equation is given by
(2.16)
(2.17)
Here, we introduced a convergence factor γ → 0 describing the decay of the two-particle quantity C(t) due to higher-order correlation contributions. In the last step of Eq. (2.17), we exploited the representation of the delta distribution
and neglected the contribution from the Cauchy integral principal value PV, which is known to have a small influence on the carrier dynamics in graphene. By applying the Markov approximation, Boltzmann-like kinetic equations can be derived. Here, all memory effects are neglected and only scattering events are considered, which conserve the energy. As shown later, the Markov approximation can be problematic for the Coulomb-induced carrier–carrier scattering in one-dimensional nanostructures, such as carbon nanotubes, since here the number of possible energy-conserving processes is strongly reduced. Therefore, in Chapter 5, we also discuss non-Markov aspects of the carrier relaxation dynamics.
In some cases, the Markov approximation requiring a strict energy conservation suppresses important scattering processes, which are actually allowed within the quantum mechanics on an ultrafast time scale. Therefore, for graphene (cp. Chapter 4), we apply a softened Markov approximation, that is, we do not consider the limit γ → 0 in Eq. (2.17). We take into account the more realistic case for a many-particle system, where higher-order correlations decay with an intrinsic time constant γ of the system giving rise to a softening of the strict energy conservation for many-particle scattering processes [78, 189]. The rate γ reflects the decay of the microscopic polarization and can be microscopically derived within our approach. Applying a finite γ, the procedure corresponds to considering a Lorentzian as a representation of the delta function [79]
(2.18)
In the following, we discuss the Coulomb- and phonon induced relaxation channels as well as the many-particle dephasing within the discussed Markov approximation.
2.2.2.3 Coulomb-Induced Relaxation Channels
Here, the scattering rates are given by
(2.19)
The influence of the Pauli blocking is explicitly included in the terms
We distinguish between scattering events crossing the conduction and the valence band (interband scattering) and those within one band (intraband scattering). Because of the zero-bandgap, Coulomb-induced interband processes are expected to be very efficient in graphene and metallic carbon nanotubes. In particular, Auger-type processes bridging the valence and conduction band and giving rise to a change in the carrier density could play a crucial role for the relaxation dynamics, cp. Section 4.8. For these processes, it is important to take into account a softened energy conservation, as discussed above.
2.2.2.4 Phonon-Induced Scattering Rates
The carrier–phonon scattering channels are treated in an analogous way. The corresponding phonon-induced in-scattering rate reads
(2.20)
with the condition for the conservation of energy including the emission and absorption of phonons. The latter depends on the phonon occupation . while the phonon emission scales with and therefore can always take place. An excited electron scatters from the state into the state (λ, k). The momentum and energy conservation is fulfilled by emitting or absorbing a corresponding phonon. The out-scattering rate is obtained by substituting ρl↔ (1 – ρl) and nu↔ (nu + 1) in Eq. (2.20).
In analogy to the carrier–carrier interaction, we distinguish between phononinduced intra- and interband scattering. In the case of graphene with its unique linear band structure, these two types of scattering channels can be clearly separated for optical phonons with a constant energy . Because of the momentum and energy conservation, it can be easily shown that interband scattering can only take place for phonons with q ≤ qp with qp corresponding to the parallel scattering along the Dirac cone, cp. Figure 2.2. In contrast, intraband processes can only occur involving larger momenta q ≥ qp, cp. also the discussion in Section 4.5.
Scattering via phonons can be very efficient and lead to the generation of hot phonons. The absorption of the latter through the electronic system can give rise to a considerable slow-down of the relaxation dynamics [80]. Therefore, it is very important to go beyond the bath approximation and to explicitly consider the dynamics of the phonon occupation . In analogy to the carrier population in Eq. (2.13), we obtain the Boltzmann-like kinetic equation for phonon occupations:
(2.21)
Figure 2.2 Schematic illustration of phonon-induced (a) intra- and (b) interband processes with qp denoting parallel scattering along the Dirac cone.
with the phonon emission rate
(2.22)
The efficiency of phonon emission depends on the square of the carrier–phonon coupling element as well as on the occupation of the initial state . and the availability of an empty final state . The corresponding phonon absorption rate is obtained by substituting in Eq. (2.22).
2.2.3 Many-Particle Dephasing
Many-particle interactions do not only change the occupation probability of the involved states, they also induce an ultrafast dephasing of the microscopic polarization:
(2.23)
consisting of a nondiagonal uk(t) and a diagonal part γ2,k (t). The latter is given by the time- and momentum-dependent Coulomb- and phonon-induced scattering rates via
(2.24)
The off-diagonal dephasing is numerically demanding, since it couples to all coherences in the entire Brillouin zone yielding
(2.25)
The contribution stemming from the Coulomb interaction reads
(2.26)
with and – (+)in the delta function in the case of . For reasons of clarity, we introduced the abbreviation
and
The contribution of the carrier–phonon scattering to off-diagonal dephasing can be obtained in a similar way and reads
The influence of the off-diagonal dephasing on the ultrafast relaxation dynamics of nonequilibrium carriers is investigated in Section 4.4.
So far, we have calculated Bloch equations and its contributions within the Hartree–Fock and the second-order Born–Markov approximation for an arbitrary structure. In the following sections, the material-specific electronic and phonon band structure as well as the carrier–light, carrier–carrier, and carrier–phonon coupling elements for graphene and carbon nanotubes of arbitrary chiral index (n1, n2) are discussed.
2.3 Electronic Band Structure of Graphene
The electronic band structure of graphene plays an important role for understanding its unique properties. It directly enters into the derived Bloch equations and crucially determines for example the efficiency of relaxation channels.
2.3.1 Structure and Symmetry of Graphene
Graphene is a perfect two-dimensional crystal consisting of a single layer of carbon atoms arranged in a hexagonal lattice. Its unit cell contains two atoms A and B and is spanned by the two basis vectors a1 and a2, cp. Figure 2.3a. Their length nm corresponds to the graphene lattice constant. The reciprocal lattice including the first Brillouin zone is shown in Figure 2.3b. It contains the high-symmetry Γ, K, K′ and M points and is spanned by the reciprocal vectors k1 and k2.
A carbon atom contains four valence electrons, two 2s-electrons, and two 2p-electrons. They are sp2-hybridized, that is, the two 2s-electrons and one 2p-electron form strong σ-bonds between carbon atoms leading to the trigonal planar structure with the carbon–carbon distance [15] of 0.142 nm. The remaining 2p-electron occurs as a 2pz-orbital, which is oriented perpendicularly to the planar structure, and forms a π-bond with the neighboring carbon atoms. The σ-bands are completely filled and form a deep valence band [15]. The smallest gap between the bonding and antibonding σ-bands is approximately 11 eV, cp. Figure 2.4. Therefore, the majority of low-energy physical effects is determined by the π-bands. Since the overlap with other orbitals (2s, 2px, 2py ) is strictly zero by symmetry, 2pz-electrons forming the π-bonds can be treated independently from other valence electrons [1].
Figure 2.4 The band structure of graphene including the energetically high σ-bands. The emphasized π- and π*-bands cross at the Dirac point. They give rise to most characteristics of graphene. Figure adapted from [1].
2.3.2 Tight-Binding Approach
(2.27)
(2.28)
with the matrix element of the Hamilton operator and the overlap between the Bloch functions . The symmetry between the two atoms A and B restricts the number of different matrix elements by exploiting the relations . The solution of this system of two linear equations is obtained by evaluating the corresponding secular equation . Then, the eigenvalues read
(2.29)
with
The two eigenvalues describe the valence (+) and the conduction (–) band corresponding to the antibonding π and the bonding π* band, respectively.
To obtain the band structure of graphene, we need to evaluate the matrix and overlap elements appearing in Eq. (2.29). They can be determined within the nearest-neighbor tight-binding approximation, which already gives a good description of the electronic dispersion relation in graphene and carbon nanotubes [6, 81]. Assuming that the overlap of the nearest carbon atom neighbors is dominant, we obtain
with the carbon–carbon interaction energy and
(2.30)
describing the contribution from the three nearest-neighbor atoms connected by bj, see Figure 2.3a. In a similar way, we obtain with the overlap integral between two neighboring atoms . Furthermore, assuming the atomic wave functions ϕ(r) to be normalized yields and with a constant parameter ε0. Inserting these matrix elements into Eq. (2.29) leads to the band structure of graphene
(2.31)
The three tight-binding parameters ε0, γ0, and s0 can be extracted from experimental measurements of the Fermi velocity or from first-principle calculations. The tight-binding parameter ε0 is set to 0 eV, since the valence and the conduction band are known to cross at the K point. Commonly used values [1, 2, 82] for γ0 range from –2.7 to –3.3 eV. The overlap s0 is a measure for the asymmetry between the conduction and the valence band. The corresponding values [1, 2, 82] range between –0.07 and –0.13. However, measurements reveal a symmetry between the conduction and the valence band close to the K point. In this region, the overlap s0 can be neglected resulting in a simple expression for the dispersion relation in graphene
(2.32)
The nearest-neighbor contributions in e(k) can be further analytically evaluated by calculating the scalar product of the momentum k and the connecting vectors bi in Eq. (2.30) (cp. Figure 2.3a) resulting in
(2.33)
Figure 2.5 illustrates the band structure of graphene along the first hexagonal Brillouin zone. It reveals a maximal energy at the Γ point and a saddle-point at the M point of the BZ. Furthermore, the conduction and the valence band cross at the K and K′ points. As a result, graphene is a zero-bandgap semiconductor or a semimetal, where the Fermi surface consists of only six points. Focusing on the region around these Dirac points, Eq. (2.33) can be expanded for small momenta k resulting in a unique linear energy dispersion
(2.34)
The nearest-neighbor tight-binding approach gives a good description of the graphene band structure around the K point. At higher energies it deviates quantitatively from ab initio calculations [81]. To obtain a better description over the entire BZ, the approach can be improved by including third-nearest-neighbor interactions and their overlaps or by including the influence of energetically higher σ-bands [81, 83]. However, for the investigations of this book including the ultrafast carrier relaxation dynamics and the excitonic effects in optical spectra, the nearest-neighbor approximation is sufficient.
So far, we have calculated the eigenvalues of the Schrödinger equation. The corresponding eigenfunctions are determined by the coefficients
(2.35)
which can be obtained by exploiting the normalization of the wave function Ψ (k, r).
2.4 Electronic Band Structure of Carbon Nanotubes
The band structure of carbon nanotubes can be obtained with zone-folded tight-binding wave functions considering the periodic boundary condition arising from rolling up the graphene layer into a cylinder.
2.4.1 Structure and Symmetry of CNTs
(2.36)
with and the chiral angle
(2.37)
The unit cell of a nanotube is defined by a cylindrical surface with the width corresponding to the circumference and the length corresponding to the translational period a along the tube axis, cp. Figure 2.6. It is determined by the smallest graphene lattice vector a perpendicular to the chiral vector c resulting in [1]
(2.38)
(2.39)
2.4.1.1 Nanotube Symmetry
The symmetry of carbon nanotubes can be described by line groups [85]. In analogy to crystal space groups, this approach describes the symmetry of systems which are periodic in one direction. Every nanotube belongs to a different line group (except for armchair and zigzag tubes with the same n). The corresponding symmetry operations allow a construction of an entire nanotube starting from one single atom at the position
(2.40)
Then, the position of every other carbon atom can be obtained via the relation [1, 85]
(2.41)
The parameter r in Eq. (2.41) characterizes the screw axis operations. It is defined by [85]
with Fr[x] as the fractional part of x. Furthermore, ϕ(n) is the Euler function giving the number of positive integers less than n, that are coprime to n, that is, they have no common factor other than 1.
Figure 2.7 The construction of the exemplary (3, 3) armchair nanotube by applying the line group symmetries. The nearest neighbors of the reference atom at r000 are gray.
This procedure of constructing an entire nanotube is applicable for the analytic evaluation of matrix elements, cp. Section 2.6.
2.4.2 Zone-Folding Approximation
(2.42)
with the integer m ε (–q/2, +q/2] labeling nanotube subbands, that is, the perpendicular component is quantized in q allowed lines. As a result, the BZ of graphene is divided into allowed lines, which are parallel to the nanotube axis and equally separated by 2/d, cp. Figure 2.8. Their orientation is determined by the chiral angle ϕ. The length of the lines corresponds to 2π/a with a as the length of the CNT unit cell. The number of lines can be determined by the ratio of the graphene BZ area and the length of the lines multiplied by their separation. For high-symmetry CNTs, the unit cell is small resulting in a small number of long allowed lines. Assuming that the tube is infinitely long, the component of the reciprocal vector along the nanotube axis is continuous.
The basic assumption of the zone-folding approach is that curvature effects are negligible for optical and electronic properties of CNTs. This is a good approxi mation for tubes with a diameter larger than approximately nm [1, 86]. Then, the electronic dispersion relation of a nanotube can be approximated by the graphene band structure restricted to the allowed lines, as shown in Figure 2.8b. For CNTs smaller than 1 nm, hybridization effects might play an important role [87].
Figure 2.8 Illustration of the zone-folding approximation applied to the exemplary metallic (6, 0) zigzag carbon nanotube. (a) The contour plot of the conduction band of graphene is shown in the background with the dark color corresponding to low energies. Note the triangular shape of the contour lines around the K point. The horizontal white lines display the allowed wave vectors kz fulfilling the boundary condition for the (6, 0) nanotube. The orientation of lines, their length and separation are determined by the CNT chiral indices (n1, n2). (b) The corresponding band structure of the (6, 0) zigzag nanotube. The subbands are equally separated by 2/d in the reciprocal space. Their length is given by 2π/a with a denoting the length of the nanotube unit cell.
Applying the zone-folding approximation, the two-dimensional graphene wave vector k decouples into a continuous component along the nanotube axis and a perpendicular component . which is quantized according to the boundary condition . Exploiting the general relations and yields
(2.43)
Throughout this book, the CNT axis is set along the z-direction and the parallel component of the momentum is denoted as kz. Now, the band structure of arbitrary carbon nanotubes can be derived by inserting Eq. (2.43) into Eq. (2.30)
(2.44)
Depending on the chiral indices n1 and n2, the carbon nanotube can be semiconducting or metallic displaying crossing conduction and valence bands.
Figure 2.9 Band structure of the exemplary (a) (6, 0) zigzag, (b) (6, 6) armchair, and (c) (6, 1) chiral nanotubes. The numbers in (a) correspond to the subband index m. The first two high-symmetry CNTs are metallic, since a conduction and a valence subband cross at the K point of the BZ. The (6, 1) CNT is semiconducting and shows a large number of subbands with a relatively small BZ.
Figure 2.9
