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Optical absorption in carbon nanotubes Motavas, Saloome 2014

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Optical Absorption in CarbonNanotubesbySaloome MotavasA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2014c© Saloome Motavas 2014AbstractDue to their unique optical properties, carbon nanotubes have been widely investigated for usein photonic and optoelectronic devices and optical absorption and emission with nanotubes havebeen achieved in experiments. On the other hand, the structural characteristics of nanotubes, e.g.the chirality, diameter, and length, as well as other factors such as the polarization of the incidentlight, presence of a magnetic field and mechanical deformation can significantly affect the opticalproperties of these structures. Some of these effects have been theoretically studied at the tight-binding approximation level. However, a systematic first-principles-based study of nanotubes thataddresses these effects did not exist in the literature prior to the present work. This thesis aims atperforming such a fundamental study.We first describe a method for calculating the dipole moments and transition rates in nanotubes.This also enables the study of selection rules, based on which a modified set of rules is defined. Theprobability of absorption is studied in the full range of infrared-visible-ultraviolet. We show thatpi-σ*, σ-pi*, and σ-σ* transitions that are neglected in previous works are allowed and can lead tohigh probabilities of transition. We then investigate several effects caused by the curvature of thenanotube sidewall and their impacts on the optical properties. The overall effect is shown to notonly depend on the diameter, but also on the chirality of the nanotube. Through the study of thelight polarization effect, we show that the overall transition rate spectrum of the perpendicularlypolarized light is suppressed for smaller-diameter nanotubes in the IR/VIS range. In the UVregion, however, perpendicular polarization is generally absorbed at a higher rate compared toparallel polarization. Finally, we show how the absorption spectra of short nanotube segments canbe different from those of long nanotubes. We examine the effect of length on individual absorptionpeaks and also investigate the effect of spin on the optical properties of nanotube segments. Theiicalculation method described in this thesis and the results can be used to estimate the effects ofstructural and environmental factors on the optical absorption properties of nanotubes.iiiPrefaceThe work presented in this thesis is my original contribution, under the supervision of Prof. AlirezaNojeh and Prof. Andre Ivanov. I was the lead investigator, responsible for all the research,calculations and analyses carried out in this work, as well as writing and composition of manuscripts,with the help and guidance of my supervisors. Parts of this thesis that have been previouslypublished in journals or presented in conferences are listed below:• A version of Chapter 2 has been published in [S. Motavas, A. Ivanov, and A. Nojeh, “Op-tical transitions in semiconducting zigzag carbon nanotubes with small diameters: A first-principles broad-range study,” Physical Review B, vol. 82, p. 085442, 2010].Some of the results were also presented in [S. Motavas, A. Ivanov, A. Nojeh, “CurvatureEffects on Optical Transitions in Semiconducting Carbon Nanotubes with Small Diameters”,11th International Conference on the Science and Application of Nanotubes (NT10), MontrealQC, Canada, June 2010].• A version of Chapter 3 has been published in [S. Motavas, A. Ivanov, and A. Nojeh, “Thecurvature of the nanotube sidewall and its effect on the electronic and optical properties ofzigzag nanotubes,” Computational and Theoretical Chemistry, vol. 1020, pp. 3237, 2013].Some of the results were also presented in [S. Motavas, A. Ivanov, and A. Nojeh, “Effect ofvariations in carbon-carbon bond lengths on the optical absorption properties of different car-bon nanotubes,” 12th International Conference on the Science and Application of Nanotubes(NT11), Cambridge, UK, July 2011] and [S. Motavas, A. Ivanov, and A. Nojeh, “Tight-binding Model Including the Effect of Curvature for Calculating the Electronic and OpticalProperties of Small-Diameter Semiconducting Carbon Nanotubes,” 15th Canadian Semicon-ivductor Science and Technology Conference (CSSTC 2011), Vancouver BC, Canada, August2011].• A version of Chapter 4 has been published in [S. Motavas, A. Ivanov, and A. Nojeh, “Theeffect of light polarization on the interband transition spectra of zigzag carbon nanotubes andits diameter dependence,” Physica E: Low-dimensional Systems and Nanostructures, vol. 56,pp. 7984, 2014].Some of the results were also presented in [S. Motavas, A. Ivanov, and A. Nojeh, “Diameterdependence of the effect of light polarization on interband transitions in zigzag carbon nan-otubes,” 56th International Conference on Electron, Ion, and Photon Beam Technology andNanofabrication (EIPBN 2012), Waikoloa HI, USA, May 2012].• Parts of Chapter 5 were presented in [S. Motavas, A. Ivanov, and A. Nojeh, “Optical ab-sorption spectra of zigzag carbon nanotube segments: Length dependence and effect of spinpolarization,” 5th Workshop on Nanotube Optics and Nanospectroscopy (WONTON 2013),Santa Fe NM, USA, June 2013].vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix1 CHAPTER1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Structural and Electronic Properties of Carbon Nanotubes . . . . . . . . . . . . . . 41.3 Optical Properties of Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Motivation and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Contributions and Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . 121.6 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.6.1 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.6.2 Gaussian 09 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.6.3 SIESTA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20vi2 CHAPTER 2: Optical Interband Transitions and Selection Rules . . . . . . . . 242.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2 Methodology and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3 Discussion and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.4 Comparison with Tight-Binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 CHAPTER 3: Effect of the Curvature and Geometry . . . . . . . . . . . . . . . 433.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2 DFT vs. TB/ZF and σ-pi Rehybridization . . . . . . . . . . . . . . . . . . . . . . . . 443.3 Effect of the C-C Bond Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.4 Importance of Geometry Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 563.5 Mixing of d and p orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 CHAPTER 4: Effect of Light Polarization . . . . . . . . . . . . . . . . . . . . . . . 624.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695 CHAPTER 5: Effect of Spin and Length on the Optical Properties . . . . . . . 715.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.3 Effect of Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.4 Effect of Length on the Absorption Spectra . . . . . . . . . . . . . . . . . . . . . . . 815.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856 CHAPTER 6: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87vii6.2 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.3 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107A Gaussian 09 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 108A.1 Sample Gaussian 09 Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108A.2 Sample Cubegen Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110B SIESTA Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111B.1 Sample SIESTA Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111B.2 Optical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117C MATLAB Code for Calculating the Dipole Moment . . . . . . . . . . . . . . . . 118D Excitonic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120D.1 Binding Energy of Excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120D.2 Radiative Lifetime of Excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121viiiList of Tables2.1 Summary of the results for an (8,0) nanotube using the BLYP/6-31G method. Theallowed transitions from each valence band are determined and the dipole momentmagnitude (arbitrary units) and the photon energy required for those transitions arecalculated (transitions with the relative dipole moment squared of less than 0.005are not listed in the table for brevity). The parameters related to the selection rulesare listed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2 Lowest two optical transition energies and their ratio for an (8,0) nanotube. . . . . . 343.1 Comparison of the lowest two optical transition energies calculated with TB/ZF andBLYP/6-31G. The energy change compared to TB/ZF results is denoted by ∆. . . 503.2 The CC1 and CC2 bond lengths (parallel and perpendicular to the nanotube axis,respectively) after geometry optimization with BLYP/6-31G. CC1 = CC2 = 1.42 A˚before geometry optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.3 Comparison of the lowest two optical transition energies, calculated by BLYP/6-31G before and after the geometry relaxation. The energy change after geometryoptimization is denoted by ∆. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.4 The difference between the results obtained using DFT calculations after geometryoptimization and the TB/ZF results. . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.5 Comparison of the C-C bond length in graphene calculated with different methods. . 573.6 Comparison of the C-C bond lengths in an (8,0) nanotube calculated with differentmethods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57ix3.7 Comparison of the first two optical transition energies for an (8,0) nanotube thatis geometry optimized with different methods against experimental values. Theelectronic structure has been calculated using BLYP/6-31G in all these cases. . . . . 583.8 Comparison of the lowest two optical transition energies with and without includingthe d orbitals. ∆ is the energy difference. . . . . . . . . . . . . . . . . . . . . . . . . 594.1 Ratio of the transition rates for parallel and perpendicular polarization of light (W‖W⊥)in different energy ranges. For each polarization, W is the sum of all the transitionrates in the specified range for the nanotube under study. . . . . . . . . . . . . . . . 664.2 Ratio of the dipole moments squared for parallel and perpendicular polarizationof light (D2‖D2⊥) in different energy ranges. For each polarization, D2 is the sum ofthe square of the magnitudes of the dipole moments in the specified range for thenanotube under study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.1 Transition energies of peaks D, B, and C obtained by simulations and extrapolationof the data from nanotube segments. . . . . . . . . . . . . . . . . . . . . . . . . . . . 85xList of Figures1.1 Schematic of a graphene layer (a), a single-walled nanotube (b), and a multi-walled nanotube(c). c© 2004 IEEE [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Schematic of the chemical vapour deposition system. Used with permission from [7]. Copy-right c© 2013 Elsevier B.V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Schematic of a top-gate CNTFET with a single-walled nanotube as the transistor channel (a)and a scanning electron microscopy (SEM) image of the device (b). c© 2006 World ScientificPublishing Company [18]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Comparison of the ION vs. ION/IOFF in a CNTFET and a silicon-based transistor (90nmMOS). c© 2006 World Scientific Publishing Company [18]. . . . . . . . . . . . . . . . . . . 41.5 Schematic of a graphene layer and the chiral vector (Ch), lattice vectors (a1 and a2) andthe translation vector (T) of a carbon nanotube. An (n,m) nanotube has a chiral vectorof Ch = na1 + ma2. Reprinted (adapted) with permission from [22]. Copyright (2000)American Chemical Society. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.6 Band structure of graphene calculated based on the pi-orbital tight-binding (TB) model andthe corresponding symmetry points. Copyright (1998) Imperial College Press [23]. . . . . . 51.7 Energy subbands of (a) a (10,0) nanotube (semiconducting) and (b) a (9,0) nanotube (metal-lic) on the contour plots of pi-orbital energy dispersion of graphene. The first Brillouin zoneof graphene is shown by the white hexagons. Reprinted (adapted) with permission from [22].Copyright (2000) American Chemical Society. . . . . . . . . . . . . . . . . . . . . . . . . 6xi1.8 Calculated density of states for two SWNTs with approximately the same diameter, showingthe van Hove singularities and the zero DOS for the semiconducting SWNT (top) vs. thefinite DOS for the metallic SWNT (bottom). Used with permission from [24]. Copyright(2005) WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. . . . . . . . . . . . . . . . . 71.9 Schematic of VHS peaks in the density of states of a semiconducting CNT and the mech-anism of light absorption through excitation of an electron from the second valence to thesecond conduction band, followed by relaxation of the electron and emission over the E11energy gap. Used with permission from [25]. Copyright (2002) American Association for theAdvancement of Science. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.10 Emission spectrum of an array of SWNTs (a) and an individual SWNT (b). Copyright(2004) by the American Physical Society [32]. . . . . . . . . . . . . . . . . . . . . . . . . 91.11 Schematic of the electroluminescence mechanism in a CNT device, in which the injection ofelectrons and holes from opposite ends of the CNT and the radiative recombination of theseelectrons and holes results in light emission. Reprinted (adapted) with permission from [33].Copyright (2004) American Chemical Society. . . . . . . . . . . . . . . . . . . . . . . . . 91.12 Electron-hole pairs in a CNT device that are generated by the incident light and decayinto free electrons and holes under the applied external field. Reprinted (adapted) withpermission from [34]. Copyright (2003) American Chemical Society. . . . . . . . . . . . . . 101.13 Schematic representation of (a) the transmitter and (b) the receiver circuitries in a commu-nication model based on carbon nanotubes. c© 2008 IEEE [43]. . . . . . . . . . . . . . . . 122.1 DFT (BLYP/6-31G) calculation of (a) band structure and (b) dipole moment magnitudesquared (arbitrary units) for transitions between HOMO and the first 16 conduction sub-bands of an (8,0) nanotube, showing the allowed transitions to the 4th(A) and 6th (B)conduction sub-bands (a =√3a0, where a0 is the translational period along the tube axis). . 272.2 Molecular orbitals for the 5th conduction band (a) and 6th valence band (b) of an (8,0)nanotube calculated with BLYP/6-31G. Left: view along the tube axis. Right: view per-pendicular to the tube axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30xii2.3 (a) |H ′|2 and (b) W (transition rate) at the Γ point versus the energy of transition for an(8,0) nanotube, showing the maximum absorption at 1.42 eV (infrared) and relatively highabsorption probability in the visible and ultraviolet regions. . . . . . . . . . . . . . . . . . 332.4 Transition rate at the Γ point versus the energy of transition for (a) (10,0) and (b) (7,0)nanotubes, showing high probability of transitions in the infrared, visible and low UV regions.Maximum of absorption for (10,0) and (7,0) nanotubes is located at 0.79 eV and 3.03 eV,respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.5 Valence bands of an (8,0) nanotube with the pi-band TB method (plotted using the formulasin Ref. [80]) (a) and DFT (BLYP/6-31G) calculations (b). The dashed lines indicate bandsresulting from the σ or strongly hybridized pi-σ orbitals. The solid lines correspond to the piorbitals and resemble the TB bands. The Fermi level is located at zero. . . . . . . . . . . . 372.6 Conduction bands of an (8,0) nanotube with the pi-band TB method (plotted using theformulas in Ref. [80]) (a) and DFT (BLYP/6-31G) calculations (b). Conduction bands inthe DFT calculations are shifted down and have switched place (see the four lowest bandsnear kz= 0) compared to the TB bands. The dashed lines indicate bands resulting from theσ orbitals. The solid lines correspond to the pi orbitals. The Fermi level is located at zero. . 382.7 Magnitude squared of the dipole moment for (a) (8,0), (b) (10,0), and (c) (7,0) nanotubesat the Γ point with the TB and DFT (BLYP/6-31G) methods. . . . . . . . . . . . . . . . 402.8 Magnitude squared of the dipole moment for an (8,0) nanotube at the Γ point with the DFT(BLYP/6-31G and B3LYP/6-31G) and RHF/6-31G methods. . . . . . . . . . . . . . . . . 413.1 One unit cell of (a) a (4,0) (b) an (8,0), and (c) a (16,0) nanotube with diameters of ∼0.34,∼0.63, and ∼1.27 nm, respectively. The nanotube axis is vertical in all cases. . . . . . . . . 443.2 Electronic band structure of (a) a (4,0) (b) an (8,0), and (c) a (16,0) nanotube with TB/ZF(solid blue) and BLYP/6-31G (dashed red) calculations. The geometries of nanotubes arenot optimized. The singly degenerate band (α) is predicted for both band structures. Thefermi level is at zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46xiii3.3 Band structure of (a) a (4,0) (b) an (8,0), and (c) a (16,0) nanotube calculated with BLYP/6-31G, showing the repulsion of the singly degenerate pi and σ bands (dashed lines). Thegeometry of the nanotubes are not optimized. . . . . . . . . . . . . . . . . . . . . . . . . 473.4 Molecular orbitals of the (α) band at the Γ (kz=0) point for (a) (4,0), (b) (8,0) and (c) (16,0)nanotubes. The isovalue is 0.03 for (4,0) and (8,0) and 0.02 for (16,0). . . . . . . . . . . . 483.5 The molecular orbitals of a singly degenerate valence band at the Γ (kz=0) point (a) (4,0),(b) (8,0) and (c) (16,0) nanotubes.The isovalue is 0.03 for all three nanotubes. . . . . . . . 493.6 Electronic band structure of (a) a (4,0) (b) an (8,0), and (c) a (16,0) nanotube without(solid blue) and with (dashed red) geometry optimization . The geometries and the bandstructures are calculated with BLYP/6-31G. The fermi levels are based on the calculationsprior to the geometry optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.7 Plot of the difference between the first (a) and second (b) transition energies (∆E11 and∆E22, respectively) for BLYP/6-31G calculations (after geometry optimization) of (n,0)nanotubes and their corresponding TB/ZF values. Values for nanotubes with n(mod)3 =1 are shown in purple (empty) circles and the ones with n(mod)3 = 2 are shown in green(solid) circles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.8 Band structure of an (8,0) nanotube, optimized with HF/3-21G (solid blue) and BLYP/6-31G(dashed red) methods. The band structure of the nanotube is calculated with BLYP/6-31Gin both cases. The fermi level is based on the calculations with BLYP/6-31G geometryoptimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.9 Band structure with BLYP/6-31G (solid blue) and BLYP/6-31G(d) (dashed red) for (a)(4,0), (b) (8,0) and (c) (16,0) nanotubes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.1 Transition rate for an (8,0) nanotube (calculated with BLYP/6-31G) versus theenergy of transition for parallel (a) and perpendicular (b) polarization of light. pi-pi*transitions with significant transition rate magnitudes in the perpendicular spectrumare pointed out. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64xiv4.2 Wave functions for the 1st valence (a), 4th conduction (b), and 3rd conduction (c)bands of an (8,0) nanotube calculated with BLYP/6-31G. Left: view along the tubeaxis. Right: view perpendicular to the tube axis. . . . . . . . . . . . . . . . . . . . . 654.3 Transition rate (calculated with BLYP/6-31G) versus the energy of transition for a(4,0) (a), an (8,0) (b) and a (16,0) (c) nanotube. For each nanotube the spectraon the left and right correspond to the calculations with parallel and perpendicularpolarization of light, respectively. pi-pi* transitions with significant transition ratemagnitudes are pointed out in the spectra. . . . . . . . . . . . . . . . . . . . . . . . . 675.1 From top to bottom: 1 unit cell (32 toms and 4.26 A˚), 3 unit cells (96 toms and12.78 A˚), 5 unit cells (160 toms and 21.3 A˚) and 7 unit cells (224 atoms and 29.82A˚) of an (8,0) nanotube. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.2 Total energy of the system for different cut-off radii chosen for the 3s orbital of a 1-unit cell(8,0) nanotube segment terminated with hydrogen. . . . . . . . . . . . . . . . . . . . . . 735.3 Total energy of the system for different lengths of the vacuum along the nanotube axis for a1-unit cell (8,0) nanotube segment terminated with hydrogen. . . . . . . . . . . . . . . . . 745.4 Total energy versus spin for a 3-unit cell (8,0) nanotube segment calculated withSIESTA. The inset shows our calculation with Gaussian 09 (BLYP/6-31G method). 755.5 Total energy versus spin for a 3-unit cell (7,0) nanotube segment calculated withSIESTA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.6 Total energy versus spin for a 1-unit cell (8,0) nanotube segment calculated withSIESTA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.7 Total energy versus spin for a 1-unit cell (8,0) nanotube segment calculated in Gaus-sian 09 with HF (a), LSDA (b), and BLYP (c) methods and 6-31G basis set. . . . . 765.8 Total energy versus spin for a 5-unit cell (8,0) nanotube segment calculated withSIESTA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77xv5.9 Band structure of a zigzag nano-ribbon with the length equal to the 3-unit cell of azigzag nanotube, obtained by the LDA calculations (top) and the allowed k-pointsfor the (7,0), (10,0), and (8,0) nanotube segments with 3-unit cell length (bottom).Copyright c© 2003 The Physical Society of Japan [103]. . . . . . . . . . . . . . . . . 785.10 Absorption spectrum of a 3-unit cell (8,0) nanotube segment before (a) and after (b)considering the spin multiplicity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.11 Absorption spectrum of a 5-unit cell (8,0) nanotube segment before (a) and after (b)considering the spin multiplicity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.12 Zoomed in version of the absorption spectrum of a 5-unit cell (8,0) nanotube segmentbefore (a) and after (b) considering the spin multiplicity. . . . . . . . . . . . . . . . . 805.13 Optical absorption spectra of (8,0) nanotube segments with 1(a), 3 (b), 5 (c) and 7(d) unit cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.14 Transition energy versus the inverse of length (normalized) for peak B. . . . . . . . . 835.15 Optical absorption spectrum of a 15- (a) and 20- (b) unit cell (8,0) nanotube segment. 845.16 Optical absorption spectrum of an infinitely long (periodic) (8,0) nanotube. . . . . . 856.1 Kinked (a) and bent (b) nanotubes at several deformation angles. Copyright (2003) by theAmerican Physical Society [111]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91xviList of AbbreviationsB3LYP Becke3-Lee-Yang-ParrBLYP Becke-Lee-Yang-ParrCNT Carbon NanotubeCNTFET Carbon Nanotube Field-Effect TransistorCVD Chemical Vapour DepositionDFT Density Functional TheoryDOS Density of StatesEL ElectroluminescenceETB Extended Tight-BindingGGA Generalized Gradient ApproximationGTO Gaussian Type OrbitalHF Hartree-FockHOMO Highest Occupied Molecular OrbitalIR InfraredLCAO Linear Combination of Atomic OrbitalsLDA Local Density ApproximationxviiLSDA Local Spin-Density ApproximationLUMO Lowest Unoccupied Molecular OrbitalMWNT Multi-Walled NanotubePL PhotoluminescenceRHF Restricted Hartree-FockSoC System-on-ChipSTO Slater Type OrbitalSWNT Single-Walled NanotubeTB Tight-BindingUV UltravioletVHS Van Hove SingularityVIS VisibleZF Zone-FoldingxviiiAcknowledgementsI would like to express my deepest gratitude to my supervisors, Prof. Alireza Nojeh and Prof.Andre Ivanov, for their continued support, guidance, and patience throughout my studies at UBC.This work would not have been possible without their help. I would also like to thank all mylab-mates for their helpful discussions and feedback throughout this research. Lastly, I would liketo dedicate this work to my parents for their constant support and encouragement during my PhDyears.xixChapter 1CHAPTER1: Introduction1.1 Carbon NanotubesCarbon nanotubes (CNTs) are one-dimensional (1-D) nanostructures, which can be thought of assingle graphite layers (also known as graphene) that are wrapped around themselves and rolled upinto cylinders. While single-walled nanotubes (SWNTs) consist of only one layer of graphene, multi-walled nanotubes (MWNTs) comprise several graphene sheets that are wrapped into concentrictubes (Figure 1.1).Figure 1.1: Schematic of a graphene layer (a), a single-walled nanotube (b), and a multi-walled nanotube(c). c© 2004 IEEE [1].The most commonly used methods for synthesis of carbon nanotubes are arc discharge, laserablation, and chemical vapour deposition (CVD). In the arc discharge method, CNTs are producedthrough arc vaporization of carbon atoms into a plasma by a high current flowing between twocarbon rods [2, 3]. The deposit obtained from arc discharge contains a mixture of amorphouscarbon, CNT bundles and residual metal catalysts and requires extensive cleaning and purification.In laser ablation, a pulsed laser evaporates a graphite target at a high temperature [4]. Similarto arc discharge, laser ablation produces high amounts of impurities and nanotube bundles, whichrequire purification and cleaning treatments.1In the chemical vapour deposition method, CNTs are synthesized by thermal decompositionof a hydrocarbon gas (usually methane, ethylene, or acetylene) on the catalyst particles [5, 6].Growth is influenced by a variety of factors, such as the choice of catalyst, hydrocarbon source,temperature, as well as time, pressure and the flow rate. Figure 1.2 shows a schematic of theCVD system. Compared to the two previous methods, CVD allows the synthesis of nanotubes atmuch lower temperatures. Nanotubes obtained from CVD are in general cleaner and contain lessimpurities. Also, both direct and patterned growth of CNTs on a substrate are possible by thismethod, eliminating the need for further deposition and purification processes.Figure 1.2: Schematic of the chemical vapour deposition system. Used with permission from [7]. Copyrightc© 2013 Elsevier B.V.Due to their one-dimensionality and the strong carbon-carbon bonds in their structures, nan-otubes possess unique properties and potential for a variety of applications. They are widely con-sidered for energy storage applications such as hydrogen storage [8,9], lithium ion batteries [10,11]and fuel cells [12]. Several studies have reported the application of CNTs as components of biosen-sors and medical devices for DNA detection [13, 14], glucose sensing [15], fluorescent imaging [16],drug delivery [17], etc.Carbon nanotubes can withstand current densities as high as 109A/cm2 and allow the ballistictransport of carriers over distances as long as micrometers, which makes them ideal candidatesfor electrical interconnects [19]. Carbon nanotube field-effect transistors (CNTFETs) have alsobeen fabricated and examined in numerous experiments, showing promising characteristics andadvantages over traditional silicon-based devices [18, 20, 21]. An example of a CNTFET made ofa single-walled nanotube is shown in Figure 1.3. To compare the performance of this CNTFETwith the state-of-the-art silicon transistors, their on-current (ION ) as a function of ION/IOFF is2Figure 1.3: Schematic of a top-gate CNTFET with a single-walled nanotube as the transistor channel (a)and a scanning electron microscopy (SEM) image of the device (b). c© 2006 World Scientific PublishingCompany [18].compared against each other in Figure 1.4, showing the higher current density that can be achievedby CNTFETs for all ION/IOFF values at a given drain voltage (VDD).3Figure 1.4: Comparison of the ION vs. ION/IOFF in a CNTFET and a silicon-based transistor (90nmMOS). c© 2006 World Scientific Publishing Company [18].1.2 Structural and Electronic Properties of Carbon NanotubesSince there are infinite ways of rolling up a graphene layer into a tube, CNTs exist in differentdiameters and structures. The chiral vector (Ch), which is expressed in terms of a pair of indices(n,m), defines the angle around which the graphene sheet is wrapped and the orientation of thehexagons relative to the nanotube axis:Ch = na1 +ma2, (1.1)where a1 and a2 are lattice vectors of graphene (Figure 1.5). The translation vector (T) definesthe 1-D unit cell of the nanotube along the tube axis.Based on the chirality, i.e., the angle of the chiral vector with respect to the lattice vector a1 (θ),carbon nanotubes are divided into three main categories: armchair nanotubes (θ=30◦ or m=n),zigzag nanotubes (θ=0◦ or m=0) and chiral nanotubes (the rest). Many properties of CNTs suchas their electronic band structure are dependent on the chirality of the tube even for nanotubeswith the same diameter.Owing to the 1-D structure of CNTs, electrons are confined in the direction perpendicular to thenanotube axis and their movement is mostly allowed in the axial direction. This confinement leadsto the quantization of energy and the wave vector K in the circumferential direction, according to4Figure 1.5: Schematic of a graphene layer and the chiral vector (Ch), lattice vectors (a1 and a2) and thetranslation vector (T) of a carbon nanotube. An (n,m) nanotube has a chiral vector of Ch = na1 + ma2.Reprinted (adapted) with permission from [22]. Copyright (2000) American Chemical Society.the periodic boundary condition of K ·Ch = j2pi, where j is an integer. Therefore, one commonway to calculate the electronic band structure of a nanotube is to slice up the 2-D band structureof graphene (Figure 1.6) based on this boundary condition, which results in a set of 1-D subbandslabeled by j.Figure 1.6: Band structure of graphene calculated based on the pi-orbital tight-binding (TB) model and thecorresponding symmetry points. Copyright (1998) Imperial College Press [23].Figure 1.7 shows examples of such cutting lines on the graphene pi-orbital electronic energycontour plots. K is the point where the conduction and valence bands meet in the graphene5band structure (Fermi points). If any of the nanotube energy subbands passes through a Fermipoint, then the nanotube is metallic with zero band gap. It can be shown that all armchairnanotubes (m=n) and nanotubes with n-m=3p (p is an integer) are metallic while all other tubesare semiconducting [23].Figure 1.7: Energy subbands of (a) a (10,0) nanotube (semiconducting) and (b) a (9,0) nanotube (metallic)on the contour plots of pi-orbital energy dispersion of graphene. The first Brillouin zone of graphene isshown by the white hexagons. Reprinted (adapted) with permission from [22]. Copyright (2000) AmericanChemical Society.This approach is known as the zone-folding approximation, which offers a valuable qualitativeinterpretation of the nanotube electronic structure. However, for nanotubes with small diameter,this method fails to accurately predict the band structure due to the effect of curvature resultingfrom the sidewalls of small-diameter nanotubes. This will be further discussed in Chapter 2 and 3.6The density of states (DOS) is the number of electronic states per interval of energy at a givenenergy, which is a quantity that often plays an important role in the studies of carbon nanotubesand in particular their optical properties and applications. Because of the 1-D nature of nanotubesand the resulting discrete energy subbands in their band structure, their DOS is characterized byvan Hove singularity (VHS) points as shown in Figure 1.8. A semiconducting SWNT has a DOSof zero at the Fermi level (Ef ) while a metallic SWNT has a finite DOS at Ef due to its zero bandgap.Figure 1.8: Calculated density of states for two SWNTs with approximately the same diameter, showingthe van Hove singularities and the zero DOS for the semiconducting SWNT (top) vs. the finite DOS for themetallic SWNT (bottom). Used with permission from [24]. Copyright (2005) WILEY-VCH Verlag GmbH& Co. KGaA, Weinheim.71.3 Optical Properties of Carbon NanotubesThe unique electronic properties of carbon nanotubes, such as the sharp van Hove singularities intheir density of states, lead to interesting optical properties of these structures. Therefore, opticalabsorption and emission properties of CNTs have been widely investigated and their applicationsin optoelectronic and photonic devices have been studied. Various research groups have experimen-tally measured the optical absorption of CNTs and explored the structural dependence of thesespectra [25–29]. Figure 1.9 shows a simple schematic of sharp VHS peaks in the DOS of a semi-conducting nanotube and an example of the absorption and emission mechanism. Light emissionfrom nanotubes has been observed in photoluminescence (PL) and electroluminescence (EL) ex-periments [30–33]. Photoluminescence happens upon the excitation of electrons with an externallight source and the interband recombination of electron-hole pairs. In EL experiments, on theother hand, electrons are injected from one end and holes are injected from the other end of thenanotube. Radiative recombination of the electrons and holes results in light emission. Figure 1.10shows an example of photoluminescence from an array and an individual SWNT. A schematic ofthe electroluminescence mechanism in a CNT-based device is shown in Figure 1.11.Figure 1.9: Schematic of VHS peaks in the density of states of a semiconducting CNT and the mechanismof light absorption through excitation of an electron from the second valence to the second conduction band,followed by relaxation of the electron and emission over the E11 energy gap. Used with permission from [25].Copyright (2002) American Association for the Advancement of Science.8Figure 1.10: Emission spectrum of an array of SWNTs (a) and an individual SWNT (b). Copyright (2004)by the American Physical Society [32].Figure 1.11: Schematic of the electroluminescence mechanism in a CNT device, in which the injection ofelectrons and holes from opposite ends of the CNT and the radiative recombination of these electrons andholes results in light emission. Reprinted (adapted) with permission from [33]. Copyright (2004) AmericanChemical Society.9Photoconductivity and photodetection have also been examined in CNT devices [34,35]. Figure1.12 demonstrates an example of the photoconduction process in a CNT device: Incident photonsgenerate electron-hole pairs (excitons). These excitons can subsequently decay into free carriersand generate current.Figure 1.12: Electron-hole pairs in a CNT device that are generated by the incident light and decay intofree electrons and holes under the applied external field. Reprinted (adapted) with permission from [34].Copyright (2003) American Chemical Society.101.4 Motivation and ObjectivesAlthough numerous applications for nanotubes in optoelectronic devices can be envisioned, predict-ing the optical properties of these devices, such as the wavelength of operation, can be a challenge.Optical properties of nanotubes are highly dependent on their chirality and diameter [25–29,36–42].Other parameters such as length, deformation or defects have also been shown to affect the elec-tronic structure of CNTs, which consequently, impact their optical properties. External parameterssuch as the polarization of the incident light can also affect the absorption spectra.Also, while light emission and detection in CNT devices can be achieved relatively easily atrandom wavelengths, some applications may require the operation of the devices at a specificwavelength. An example is wireless communication between a CNT-based transmitter and receiver[43]. In this structure (Figure 1.13), the communication is accomplished by light emission fromthe transmitter upon the application of a DC voltage and absorption of the emitted light on thereceiver side and generation of current. Applications can be in wireless communication of System-on-Chip (SoC) devices [43] or in chemical and biological nanosensors [44–46] for precise targetingand sensing. However, for this structure to work properly, the emitter and detector must work atapproximately the same wavelength.In such applications, therefore, the ability to predict and control the wavelength of absorptionand emission is of high importance and theoretical calculations that can accurately take into accountthe structural and environmental parameters are essential. Hence, the goal of this work is to:(a) Perform a systematic theoretical study of the optical absorption in carbon nanotubes based onfirst-principles methods(b) Understand and evaluate the impact of several structural and non-structural parameters onthe optical properties11Figure 1.13: Schematic representation of (a) the transmitter and (b) the receiver circuitries in a communi-cation model based on carbon nanotubes. c© 2008 IEEE [43].1.5 Contributions and Thesis OrganizationThe contributions of this work, achieved through the course of pursuing the above goals are:• A comprehensive study of the optical properties of carbon nanotubes based on first-principlesmethods, which enables the redefinition of the selection rules for optical transitions and amore accurate calculation of the absorption spectra compared to the ones from the previouslyexisting methods• Evaluation of the effects of structural characteristics (such as diameter, carbon-carbon bondlength, chirality and length) of nanotubes on their optical properties, which can be used topredict the optical behaviour of nanotube-based devices or to select the nanotube with a rightgeometry in order to achieve the desired optical properties for a device• Evaluation of the effects of non-structural parameters (such as the polarization of the incidentlight) on the optical properties of nanotubes, which can be utilized to tune and control theproperties of nanotube devices without changing the internal characteristics of the device12• A study that goes beyond the infrared region and covers a wide range of frequencies (infrared-visible-ultraviolet), which allows the analysis of the optical behaviour of nanotubes in thevisible and ultraviolet frequencies and suggests their applications in these ranges.This thesis is organized as follows:Chapter 2 describes a method for calculating the transition dipole moments and transitionrates and a thorough investigation of the optical absorption mechanism in carbon nanotubes.Chapter 3 is dedicated to studying several effects caused by the curvature of the nanotubesidewall and their impacts on the optical properties.Chapter 4 discusses the effect of light polarization on the interband transition spectra ofnanotubes and the dependence of this effect on the nanotube diameter.Chapter 5 focuses on the effect of the nanotube length on the absorption properties and theeffect of electron spin associated with short nanotube segments.Chapter 6 is the concluding chapter which provides a summary of the thesis and the thesiscontributions, as well as future research that can be pursued based on the work presented in thisthesis.131.6 MethodologyWe employ the density functional theory (DFT) to perform an in-depth study of the optical ab-sorption properties of carbon nanotubes. The optical spectra of SWNTs in the visible-infraredregion have been studied using first-principles methods [37–39]. A quantitative analysis of theoptical spectra based on the tight-binding (TB)/zone-folding (ZF) band structure calculations hasbeen reported [40]. However, a detailed study of the optical transitions spectra with attentionto the transition mechanisms and molecular orbitals based on the first-principles methods and inthe broad range of ultraviolet-visible-infrared (UV-VIS-IR) does not exist in any previous work.The TB method 1 offers an intuitive interpretation of the electronic structure and is useful forthe qualitative description of the transition rates and absorption properties. However, the simpli-fications used in this method impose limitations and may result in significant inaccuracies. Forexample, pi-orbital TB does not treat the σ-pi rehybridization resulting from the curvature of thenanotube sidewall, which is pronounced especially in single-walled nanotubes (SWNTs) with diam-eters smaller than 1 nm [47]. It has been shown that it fails in accurately predicting the conductionbands in particular [48]. Also, since only pi orbitals are included in this approximation, the possibil-ity of other transitions that may occur in CNTs is neglected [49,50]. Finally, TB does not includemany-body exchange and correlation effects. In our work, on the other hand, curvature and many-body effects are automatically included in the calculations. These methods are computationallyexpensive compared to the TB approximation. However, they can provide more accurate molecularorbitals and band structures, which can result in a more realistic estimate of the transition dipolemoment, especially for CNTs with small diameters.In this section, we briefly talk about DFT. Then we go over the features of the two main softwarepackages used for the calculations in this project: Gaussian 09 (Revision D.01, Gaussian Inc.) [51]and SIESTA [52,53].1From here on, unless otherwise specified, by TB we mean to refer to the pi-orbital TB method with the zone-foldingapproach.141.6.1 Density Functional TheoryDFT is a quantum mechanical approach for calculating the ground state properties of many-bodysystems, which describes a system of interacting electrons with its density rather than its many-electron wave function. As a result, the many-body problem of N electrons with 3N spatial coor-dinates is reduced to 3 spatial coordinates of the electron density function, in principal leading toa much lower computational cost compared to fully-wave function-based methods such as Hartree-Fock (HF) .DFT is based on two Hohenberg-Kohn theorems [54] and typically implemented using theKohn-Sham formulation [55]. According to the first Hohenberg-Kohn theorem, the external po-tential and the ground state properties of a many-electron system can be uniquely determinedby its electron density. The second Hohenberg-Kohn theorem is based on the variation principle,saying that the correct ground state electron density minimizes the total energy of a system. Fi-nally, the Kohn-Sham equation reduces the many-electron problem to a non-interacting system ofelectrons moving in an effective potential. This effective single-particle potential consists of theexternal potential, the classical Coulomb repulsion (Hartree term), and the exchange-correlationpotential (which contains all the many-electron interactions and the correction to the kinetic en-ergy). The exchange-correlation potential and the Hartree term are both functions of the electrondensity. Therefore, the Kohn-Sham equations can be solved self-consistently by starting with aninitial density functional, calculating the effective potential and solving the Schro¨dinger equation,and subsequently, calculating the new charge density based on the wave functions obtained andrepeating this procedure until convergence.The exchange-correlation energy is not known in an exact form and approximations need to bemade for estimating it. The most common exchange-correlation energy functionals are obtainedwithin the following approximations:• Local density approximation (LDA)A class of approximations that treats an inhomogeneous electronic system as a locally homo-geneous one and assumes that the exchange-correlation energy at each point in space dependsonly on the electron density at that point, thus neglecting the variation of the electron density.15Some of the common correlation functionals within the local density approximations are:– Cole-Perdew (CP) [56]– Perdew-Zunger (PZ) [57]– Vosko-Wilk-Nusair (VWN) [58]The extension of this approximation to spin-polarized systems is known as local spin-densityapproximation (LSDA)• Generalized gradient approximation (GGA)This class of approximations takes into account not only the information about the the densityat a particular point in space, but also the gradient of the density at that point in order toaccount for the non-homogeneity of the charge density. Some of the most common exchange-correlation functionals based on the GGA method are:– Becke-Lee-Yang-Parr (BLYP) [59,60]– Becke-Perdew86 (BP86) [59,61]– Becke-Perdew-Wang91 (BPW91) [59,62]1.6.2 Gaussian 09Gaussian is a general purpose first-principles computer program that is capable of predicting ener-gies, geometries, electronic structures, vibrational frequencies and various other molecular proper-ties. The latest version of the program is Gaussian 09.Basis SetsBasis sets are sets of functions that are linearly combined in order to generate molecular orbitals.They are also used to build the overlap matrices for calculating the overlap of atomic orbitals.Basis sets are generally categorized into atom-independent basis sets and atom-centered basis sets.An example of the former are plane waves, which allow simple evaluation of the overlap matrixelements and systematic improvement of the basis sets by varying the plane wave cutoff (including16plane waves that have kinetic energies smaller than the cutoff energy). These basis sets are usedin calculations involving periodic boundary conditions and they are dependent on the volume ofthe periodic cell rather than the number of atoms. As a result, a high number of plane wavesmight be required and calculation time might become too long. For atom-centered basis sets suchas Gaussian basis sets, the computation cost depends on the number of atoms and number of basisfunctions for each atom. Therefore, they are generally more suitable for systems with big unit cellvolumes. To describe the molecular orbitals with the linear combination of atomic orbitals (LCAO)method, either Slater type orbitals (STOs) or Gaussian type orbitals (GTOs) can be used. Belowis a simple representation of the radial part of STOs and GTOs:STO(r) = Arle−ζr,GTO(r) = Brle−αr2,(1.2)where A and B are the normalization constants, l is the angular momentum quantum number andα and ζ are the orbital exponents, determining the diffuseness. Although the use of STOs leadsto more accurate results compared to GTOs, their calculations can become time-consuming andcomputationally expensive for large molecules. Calculations of GTOs are comparatively easier,taking advantage of the “Gaussian Product Theorem”, which guarantees that the product of twoGTOs is a linear combination of GTOs with a common center. This speeds up the calculationsby a few orders of magnitude compared to the calculations with STOs, but at the cost of losingaccuracy. However, it is found that linear combinations of GTOs can be used to improve theaccuracy of calculations and mimic the results of STOs.Minimal basis sets are composed of the minimum number of basis functions needed for repre-senting all the electrons on each atom. The most common minimal basis set is STO-nG, with nbeing the number of GTOs combined to approximate an STO. In the minimal basis set, all orbitalsare approximated to have the same shape and due to its simplicity the results do not lead to highaccuracy.17In the double-zeta basis set, on the other hand, each atomic orbital is expressed as the sumof two STOs with different spatial extents (ζ) and in different proportions. Each of the STOs inturn are approximated by a linear combination of primitive Gaussian functions. The triple andquadruple-zeta basis sets have three and four STOs, respectively. Having multiple STOs combinedallows for each basis-function to be defined separately and have a different shape depending on theparticular molecule, which provides more flexibility for the calculations. Since valence electrons arethe ones mainly in charge of the bonding, in many cases the multiple-zeta basis set is calculatedonly for the valence orbitals and the core electrons are described with a single STO. This basis setis called a split-valence basis set.A split-valence double-zeta basis set is typically shown with the m-npG notation, where m isthe number of primitive Gaussians for the basis function of each core atomic orbital and n and pare the number of primitive Gaussians comprising the two basis functions of the valence orbitals.Triple-zeta and quadruple-zeta basis sets are represented by m-npqG and m-npqrG, respectively.Some of the most common basis sets used in the Gaussian software are: STO-3G [63], 3-21G [64]and 6-31G [65]. Polarization and diffuse functions are generally denoted by “*” and “+” signs. Forexample, 6-31G* is a valence double-zeta polarized basis set that has six (ten) extra d-type Gaussianpolarization functions on each atom for Li through Ca (Sc through Zn).Input ParametersBelow are some of the sections in a standard Gaussian input (.com) file:• Route section: Method and basis sets also known as model chemistry (for example BLYP/6-31G), desired calculation type (for example, SP (single point energy) or Opt (Geometryoptimization) calculations and other options, such as the type and specifications of the resultsto appear in the output file• Title section: Brief description of the calculation• Molecule specification: Charge, spin and the initial coordinates of the atoms making up themolecular system to be studied18A sample of the Gaussian 09 input file can be found in Appendix A.1.Output DataAt least one output (.out) file is produced for each Gaussian job, which is an ASCII file, containingresults such as the geometry, energy and the frequencies. Some of the parameters to appearin the Gaussian output file can be assigned through the IOp keywords in the Route section ofthe input file. An IOp keyword sets internal options to specific values. For example, IOp(5/98)determines whether to save the eigenvalues and orbitals at all k-points or not. If the value is set to1 (IOp(5/98=1)), the data is written in the output file. In the case of IOp(5/98=0) (default) thisinformation is not saved. Another example is IOp(5/103), which writes the number of occupiedand unoccupied orbitals at each k-point to the output file.The results of the simulations are also saved in the checkpoint files (.chk) in a binary format andthey can be used as a starting point for other calculations. The formatted checkpoint file (.fchk)is generated from the .chk file with the purpose of being machine independent and usable by otherutilities and applications.Cubegen UtilityThe data in the Gaussian formatted checkpoint file can be used to create cube files (the molecularorbital in 3 dimensions) with the Cubegen utility. Some of the parameters that can be specified inthe input of Cubegen are listed below:• MO=N : the molecular orbital number N• fchkfile: name of the formatted checkpoint file from which the data is retrieved• cubefile: name of the output cube file to be generated• npts: number of points per dimension in the cube file.• format: format of the output file (h includes the header (default) and n does not)A sample of the Cubegen input file can be found in Appendix A.2.191.6.3 SIESTASIESTA (Spanish Initiative for Electronic Simulations with Thousands of Atoms) is a DFT-basedcomputer program and also a method for electronic structure calculation as well as moleculardynamics simulations.SIESTA also uses atom-centered basis sets and as a result, the computation cost mostly dependson the number of atoms and number of basis functions for each atom and not so much on the unitcell volume. This makes SIESTA suitable for calculations of systems with big cell volumes. Atom-centered basis sets used in SIESTA are numerical and can be of any form and shape, however,they need to be strictly localized (strictly zero beyond a cut-off radius). Linear scaling (O(N))operations, in which the computational cost scales linearly with the number of atoms, rely heavily onthe sparsity of the hamiltonians and overlap matrices that are obtained from these strictly confinedbasis sets. This distinguishes SIESTA from the other DFT methods with the computational loadof O(N3).SIESTA takes advantage of the pseudopotentials, where only valence electrons are included inthe calculations as opposed to all electrons. This replaces the atomic all-electron potential withan effective pseudopotential which acts on a set of nodeless pseudo wave functions rather than thecore wave functions. This, in turn, results in the reduction of basis set size and number of electronsinvolved in the calculations. The pseudopotentials used in SIESTA are of the norm-conservingtype, meaning that the norm of the pseudo wave function and the all-electron wave function isequal beyond the core cut-off radius. Periodic boundary conditions are assumed in all SIESTAcalculations. Therefore, to calculate a molecule, a periodic unit cell with large enough dimensionsneeds to be used in a way that the interaction between the molecule and its translated replicas canbe neglected.To run SIESTA, three main files are required: (1) the SIESTA executable, (2) the flexible dataformat (.fdf) file (which contains the input parameters), and (3) the pseudopotential (normally inthe .psf format) file for each atom used in the calculations.20Input ParametersSome input parameters involved in running the SIESTA simulations for calculating the opticalproperties of non-periodic nanotubes are:• General System descriptors: Includes the system name, system label (to be used for the filesgenerated in the output), number of atoms, and number of species.• ChemicalSpeciesLabel: Specifies a number for the chemical species in the simulation, theatomic number for them and the symbol. For example:%block ChemicalSpeciesLabel1 6 C2 1 H%endblock ChemicalSpeciesLabelThis input can also be used to define atoms of the same species with different basis sets orpseudopotentials.• PAO.EnergyShift: This is the energy shift that the orbital experiences due to the terminationat the cut-off radius. This block offers a general way to define the cutoff radii of the first-ζpart of all the orbitals in the simulation. It is considered only if the block PAO.Basis doesnot exist or when the radii for the first ζ are set to zero in the block PAO.Basis.• PAO.SplitNorm : Specifies the norm for the second-ζ part of the split-valence basis sets,according to which the split radius is defined. In case of multiple-ζ basis sets, 1/2, 1/4,...fractions of the SplitNorm are assigned to higher zetas. This parameter is considered onlywhen the block PAO.Basis does not exist or when the radii for zetas higher than one are setto zero in the block PAO.Basis.• PAO.Basis: Defines the basis sets (based on the finite-range pseudo-atomic orbitals (PAOs))for each of the chemical species. Multiple-ζ and diffuse orbitals can also be used. If the basisset is not specified for a species, other parameters such as PAO.EnergyShift, PAO.SplitNorm,21etc. will be used to automatically generate the basis set for that specific species. Some of theinformation to be specified in the PAO.basis block are as follows:– Label: Species label according to the block ChemicalSpecieslabel– l shells: Number of shells of orbitals with different angular momentum– n: Principal quantum number of the shell– l: Angular momentum of the shell’s orbitals– nzls: Number of ”ζ”s for the shell– rcls: Cutoff radius in units of Bohr for each ”ζ”• LatticeVectors: They are to define the periodicity of the system. For the non-periodic nan-otubes along the z-axis, the z-component of this vector defines the amount of vacuum neededto assure that the interaction of the nanotube with the translated replicas is negligible.• AtomicCoordinatesAndAtomicSpecies: Specifies the coordinates and species of each atom.• Exchange-correlation functionals: Consists of the type of the exchange-correlation functionals(XC.functional), such as LDA, and a compatible parametrization of that exchange-correlationfunctional (XC.authors) like CA (Ceperley-Alder) [66], which is equivalent to PZ (Perdew-Zunger) [57].A sample of the SIESTA input file can be found in Appendix B.1.Output DataBelow are some of the parameters to specify in the input file and the results corresponding to themthat are written in the output file:• Systemlabel.XV: Coordinates which are overwritten at every step.• Systemlabel.EIG: Hamiltonian Eigenvalues• SystemLabel.WFSX: The k-points, energies and coefficients of each wave function requestedin the input22• SystemLabel.DOS: Density of states• Systemlabel.EPSIMG: Imaginary part of the dielectric functionIn Appendix B.2, we explain how to use the ”optical” utility of the SIESTA program to obtainthe optical properties of structures.23Chapter 2CHAPTER 2: Optical InterbandTransitions and Selection Rules 22.1 IntroductionIn this part of the project, we study the optical transition mechanisms in (8,0), (10,0), and (7,0)nanotubes which are semiconducting zigzag CNTs with sub-nanometer diameters of ∼0.63, ∼0.78,and∼0.55 nm, respectively, in the broad range of ultraviolet-visible-infrared. The goal is to take intoaccount the possible effects of curvature and rehybridization on the optical transitions, selectionrules, and dipole moments, as well as attribute the spectral peaks to the relevant transitions.Optical transition energies in nanotubes have been previously calculated with extended TB (ETB)and first-principles methods [37, 67]. However, to our knowledge, transition dipole moments andselection rules for nanotubes have not been investigated by any method other than the pi-orbitalTB approach [68–72]. Here, we calculate the dipole moment for each of the transitions based onfirst-principles approaches and the density functional theory.The transition dipole moment is a vector quantity associated with the transition of electronsbetween two states, with the magnitude of the dipole determining the probability of this transition.The dipole moment is not only needed for calculating the transition rates and absorption spectra,but also provides valuable insight into interband transition mechanisms and selection rules. Tran-sition dipole moments for nanotubes have so far been reported only based on the TB approach.One can calculate the inter-band transition rates using the perturbation theory and Fermi′s golden2This chapter is a reproduction of the article [S. Motavas, A. Ivanov, and A. Nojeh, “Optical transitions insemiconducting zigzag carbon nanotubes with small diameters: A first- principles broad-range study,” PhysicalReview B, vol. 82, p. 085442, 2010]. Inclusion in thesis is permitted. Copyright (2012) American Physical Society.24rule. According to the golden rule, the transition rate, Wif , between an initial state, ψi, and a finalstate, ψf , can be calculated from the following [73]:Wif =2pi~|H ′if |2ρf , (2.1)where ~ is the reduced Planck constant, ρf is the density of states around the final state, and H ′ifis the absorption matrix element. In our case H ′if is the optical perturbation Hamiltonian matrixelement given by:H ′if = ie~mω√Icei(ωf−ωi−ω)tP.D, (2.2)where I, ω, and P are the intensity, angular frequency and polarization vector of the incident light,respectively, and i and f refer to the initial and final states. Electron mass and the elementarycharge are represented by m and e, respectively. c is the speed of light and  is the dielectricconstant. D is the electric dipole vector given by:D = 〈Ψf |∇|Ψi〉 (2.3)In this section, we focus on the z-polarized light with the nanotube lying along the z direction(parallel polarization).2.2 Methodology and Results(8,0), (10,0), and (7,0) SWNTs were relaxed by geometry optimization in the software packageGaussian 03 [74]. The diameters of these nanotubes are less than 1 nm in all three cases andwe expect curvature-induced effects to be pronounced in them. In order to mimic an infinitelylong nanotube, one unit cell of the nanotube with periodic boundary conditions was used. Theorbital energies and wave functions were then obtained using both Gaussian 03 and Gaussian 09with various levels of theory and basis sets. Here, we discuss the results obtained from DFTcalculations using Becke-Lee-Yang-Parr (BLYP) [59, 60] and Becke3-Lee-Yang-Parr (B3LYP) [59,2560, 75] exchange-correlation potentials, as well as those obtained from the restricted Hartree-Fock(RHF) method, using 6-31G basis sets. The discussions are mainly based on BLYP calculationsfor an (8,0) nanotube. The rest of the results are presented for comparison purposes.To calculate the dipole moment according to Eq. (2.3), the wave function for each valenceand conduction band was extracted from the simulations. The derivative of the wave function forthe studied valence band was calculated using the method of finite differences and the integralwas performed. An example of the MATLAB code for these calculations is provided in AppendixC. Approximately, 2 × 107 grid points with a spacing of 0.044 A˚ in all directions were used forcalculations in the 3-D space. The number of grid points in the x and y directions varied basedon the diameter of the nanotube. As an example, a 300 × 300 × 221 grid was used to discretizethe molecular orbitals for the BLYP calculations in an (8,0) nanotube. Further refining of thegrid did not lead to any significant change in the results. Gaussian gives the wave functions onlyat the Γ point (kz=0). Therefore, dipole moments were calculated only at this k-point, which inthe case of zigzag nanotubes turns out to be where the VHS occur. Since the density of states ismainly due to these singularities, the dipole moment at the Γ point is expected to have the mostsignificant effect on the overall rate of transition to a specific sub-band. The dipole moment wascalculated for transitions between all the possible combinations of valence and conduction bandsfor 12 valence and 16 conduction bands. As an example, Fig. 2.1(b) shows the squared magnitudeof the dipole for transitions between the highest occupied molecular orbital (HOMO) and all thefirst 16 conduction bands of an (8,0) nanotube. Figure 2.1(a) depicts the calculated band structurewith the BLYP method.26Figure 2.1: DFT (BLYP/6-31G) calculation of (a) band structure and (b) dipole moment magnitude squared(arbitrary units) for transitions between HOMO and the first 16 conduction sub-bands of an (8,0) nanotube,showing the allowed transitions to the 4th(A) and 6th (B) conduction sub-bands (a =√3a0, where a0 is thetranslational period along the tube axis).27As we can see from Fig. 2.1(a), in contrast with TB results, the band structure resulting fromthe first-principles calculations is not symmetric with respect to the Fermi level. As depicted inFig. 2.1(b), Dz ≈ 0 for all the transitions except those to the 4th (A) and 6th (B) conduction bands.The first transition, in fact, occurs between the HOMO and the 4th conduction band instead ofhappening across the band gap. Table 2.1 summarizes the results and analysis of an (8,0) nanotubefor the first 12 valence bands. Only simulation results with squared dipole value of more than 0.005(∼0.01% of the maximum value obtained from our simulations) are shown and other transitionsare not listed in the table for the sake of conciseness. The possible transitions are determinedand the square value of the dipole moments (arbitrary units) and the photon energy required foreach transition are shown. The selection rules have also been studied for these transitions and thesymmetry parameters have been determined.Dipole selection rules which govern light absorption and emission processes in CNTs only al-low electronic transitions between specific valence and conduction bands for light polarized alongthe tube axis. These selection rules can be derived based on the symmetry of nanotubes andthe symmetry-based quantum numbers. For achiral tubes (armchair and zigzag), the irreduciblerepresentations of the carbon nanotube symmetry groups are in the form of kXhm, where k is thewave vector along the nanotube axis, m is the quasi-angular-momentum related to the rotationalsymmetry, and h is the parity with respect to the horizontal mirror plane, σh, denoted by ”+” foreven states and ”-” for odd ones. X can be A or B for one-dimensional, E for two-dimensional,and G for four-dimensional representations. A and B also indicate the parity quantum numberwith respect to the vertical mirror plane, σv, where A corresponds to even and B to odd parity.28Table 2.1: Summary of the results for an (8,0) nanotube using the BLYP/6-31G method. Theallowed transitions from each valence band are determined and the dipole moment magnitude(arbitrary units) and the photon energy required for those transitions are calculated (transitionswith the relative dipole moment squared of less than 0.005 are not listed in the table for brevity).The parameters related to the selection rules are listed.Valence mv >mv h Conduction mc >mc h ∆>m Transition Energy (eV) |Dz|2v1 ±5 ∓3 +c4 ∓3 ∓3 - 0 pi-pi* 1.42 1.129c6 ±5 ∓3 - 0 pi-pi* 6.16 0.426v2 ±2 ±2 -c3 ∓6 ±2 + 0 pi-pi* 1.53 2.296c14 ±2 ±2 + 0 pi-σ* 10.06 0.005v3 ±1 ±1 -c2 ∓7 ±1 + 0 pi-pi* 2.67 3.540c12 ∓7 ±1 + 0 pi-σ* 10.25 0.222c13 ±1 ±1 + 0 pi-σ* 10.58 1.668v4 0 0 -c1 8 0 + 0 pi-pi* 2.73 3.886c8 8 0 + 0 pi-σ* 9.75 2.224v5 4 4 + c5 4 4 - 0 pi-pi* 5.67 0.999v6 8 0 - c11 0 0 + 0 σ-σ* 11.12 9.645v7 0 0 +c7 0 0 - 0 σ-pi* 9.93 4.934c15 0 0 - 0 σ-σ* 13.89 51.572v8 ±1 ±1 +c9 ±1 ±1 - 0 σ-pi* 11.31 1.935c16 ±1 ±1 - 0 σ-σ* 14.50 45.828v9 ±7 ∓1 - c12 ±7 ∓1 + 0 σ-σ* 12.45 10.131v10 ±3 ±3 +c4 ±3 ±3 - 0 pi-pi* 5.42 0.830c6 ∓5 ±3 - 0 pi-pi* 10.16 0.005v11 ±2 ±2 + c10 ∓6 ±2 - 0 pi\σ-pi* 12.69 0.318v12 ±6 ∓2 - c14 ∓ 2 ∓2 + 0 σ-σ* 15.52 9.736Because of its 0A−0 symmetry, z-polarized light preserves the angular momentum quantumnumber, m. However, the horizontal parity is reversed upon this interaction since z-polarized lightcarries odd σh and vertical parity is conserved because of even σv [76].In our studies, the type of the orbitals, their horizontal parity, and the angular momentumnumber, m, are determined by plotting the spatial distribution of the orbitals for each sub-band.Figure 2.2 shows example orbitals for two different sub-bands (c5 and v6) of the (8,0) nanotube.The out of plane (out of the nanotube surface) orbital in Fig. 2.2 (a) indicates a pi orbital whilethe in-plane orbital in Fig. 2.2 (b) suggests its origin to be mainly from σ bonds. In this work, theangular momentum number, m, for each orbital corresponds to half of the number of nodes of itswave function around the circumference.29Figure 2.2: Molecular orbitals for the 5th conduction band (a) and 6th valence band (b) of an (8,0) nanotubecalculated with BLYP/6-31G. Left: view along the tube axis. Right: view perpendicular to the tube axis2.3 Discussion and AnalysisBased on our observations, in order to explain all the allowed transitions, the angular momentumnumber, m, defined before, needs to be transformed to a modified angular momentum number, >m,according to the following:>m = m+>Mn (2.4)where n is the index of the nanotube, and>M is an integer determined in a way that the modifiedquantum number satisfies the following range criteria:>m ∈ (−n2,n2] (2.5)Table I shows that this modified angular momentum number, >m [which is reminiscent of thehelical angular momentum, m˜ (Ref. [76])], is conserved for all the transitions.30From Table 2.1, the maximum dipole moment in an (8,0) nanotube corresponds to the transitionfrom the 7th valence band to the 15th conduction band, which is a σ-σ* transition. In general,the calculated dipole moment for the σ-σ* transitions have much larger values compared to theother transitions. However, since these transitions normally happen over large energy gaps, theircorresponding transition rates is undermined (see Eq. (2.2)). Our simulations for (10,0) and (7,0)nanotubes indicate a similar trend. The maximum dipole moment for transitions between the first12 valence and 16 conduction bands of a (10,0) nanotube corresponds to the transition from the10th valence band to the 14th conduction band, and it happens from the 6th valence band to the14th conduction band for a (7,0) nanotube. Both of these transitions are of σ-σ* nature.Figures 2.3 (a) and (b) show |H|2 and W (transition rate) at the Γ point, according to Eqs. 2.1and 2.2, respectively, versus the required energy for the transitions in an (8,0) nanotube. Because ofthe E−2 dependence of the transition rate (E is the transition energy), the probability of transitionfor the higher energy transitions, that generally correspond to the σ-σ* transitions, is rather weak.Also, the density of states (DOS) has a significant effect on the strength of transitions. For example,although the second transition has a much higher dipole moment compared to the first one, itsprobability of transition is less than the first transition because of its lower DOS. The maximumof absorption happens for the first transition at around 1.42 eV, corresponding to the transitionbetween the 1st valence and 4th conduction bands and mapping to the infrared region. There isalso a high probability of absorption for 1.53 eV (infrared), and 2.67 and 2.73 eV (blue) radiation.Interestingly, there is also a relatively high absorption probability at around 12.45, 13.89 and 14.50eV, mapping approximately to the low ultraviolet (12.4-14.1 eV) range.The first 7 peaks in the absorption spectra are related to pi-pi* transitions, which cover theinfrared, visible and also the near UV region. As shown in Table 2.1, transitions from a valenceband to several conduction bands are possible. In fact, in some cases a transition from a valencepi-band happens to more than one conduction pi-band, in contrast with what is presumed in theTB approximation. For example, as shown in Fig. 2.1 (b), transitions from HOMO can be madeto either the 4th or the 6th conduction bands.Table 2.1 also reveals that optical transitions are not limited to pi-pi* transitions only; pi-σ*,31σ-pi*, and σ-σ* transitions are also allowed and can happen with a relatively high probability insome cases. For example, the three strongest peaks in the low UV region at 12.45, 13.89 and 14.50eV (Fig. 2.3) are all related to the σ-σ* transitions. These transitions can provide insight into thepossibility of optical absorption in the high-frequency regions, and the ultraviolet applications ofcarbon nanotubes.The first two peaks in our calculations closely follow the experimental data in [26]. Table 4.2compares the first and second optical transition energies and their ratio obtained from differentmethods. The energies obtained from BLYP calculations are closer to the experimental valuescompared to the B3LYP and RHF results. The E22/E11 ratio from BLYP results is equal to 1.07,close to the value of 1.09 from the local-density-approximation (LDA) calculations in [37] and insignificantly better agreement with the experimental results of 1.17 reported in [26] than the valueof ∼1.6 predicted by an improved TB model including third-order neighbours [25,77]. We also seethat the extended tight-binding model, which takes into account the σ bands and the curvatureeffects, provides a much better agreement to the experimental values compared to the pi-orbitalTB [67]. However, calculation of the dipole moment and study of the selection rules for nanotubeshave not been done beyond the pi-orbital TB.As noted in Ref. [37,38] and shown in Table 4.2 (BS results), excitonic effects that are ignoredin our calculations can qualitatively affect the optical absorption results. This may explain the∼8% discrepancy between our results and the experimental data. It has been shown that thequasiparticle corrections and electron-hole interactions affect the band gap and play a crucial rolein the optical absorption spectra of semiconducting carbon nanotubes [37,38].32Figure 2.3: (a) |H ′|2 and (b) W (transition rate) at the Γ point versus the energy of transition for an (8,0)nanotube, showing the maximum absorption at 1.42 eV (infrared) and relatively high absorption probabilityin the visible and ultraviolet regions.33Table 2.2: Lowest two optical transition energies and their ratio for an (8,0) nanotube.E11(eV ) E22(eV ) E22/E11BLYP 1.42 1.53 1.08B3LYP 2.12 2.26 1.07RHF 5.00 5.17 1.03LDAa 1.39 1.51 1.09GWa 2.54 2.66 1.05BSa 1.55 1.80 1.16TBb ... ... 1.60ETBc 1.30 1.62 1.25Experimentd 1.60 1.88 1.17aLocal-density-approximation (LDA), GW approximation, and Bethe-Salpeter (BS) equation, Ref. [37]bImproved tight-binding, Ref. [25,77]cExtended tight-binding, calculated according to Ref. [67]dRef. [25, 26]Figure 2.4 (a) and (b) show the transition rate at the Γ point versus the transition energyfor (10,0) and (7,0) nanotubes, respectively. The maximum of absorption for (10,0) and (7,0)nanotubes happens at 0.79 eV and 3.03 eV, respectively. Comparing the transition plots for thethree nanotubes shows a few similarities in their absorption spectra. The majority of transitionswith high probability happen in the infrared and visible range for the three nanotubes. Exceptin the range of ∼5-6 eV, no probability of transition is observed beyond the visible range up to 9eV. However, considerable transition probabilities exist in the low UV range, mostly as a result ofσ-σ* transitions. While the maximum absorption probability happens at the first transition andmaps to the infrared region for (8,0) and (10,0) nanotubes, the maximum transition rate for a (7,0)nanotube happens at 3.03 eV (violet) and with a significantly higher relative probability comparedto the maximum for the other two nanotubes. The reason for this high probability of transition canbe attributed to the nearly dispersionless band at the 4th conduction band in the band structureof the (7,0) nanotube, which results in a very high DOS and therefore hight transition rate.34Figure 2.4: Transition rate at the Γ point versus the energy of transition for (a) (10,0) and (b) (7,0)nanotubes, showing high probability of transitions in the infrared, visible and low UV regions. Maximum ofabsorption for (10,0) and (7,0) nanotubes is located at 0.79 eV and 3.03 eV, respectively.Optical spectroscopy measurements carried out on SWNTs have shown strong peaks in theinfrared and visible ranges and revealed the dependence of these peaks on the diameter and chirality35of the nanotubes [25–28]. The possibility of fluorescence and infrared photoluminescence forisolated SWNTs in aqueous suspensions or suspended SWNTs in air have also been reported [30,32].Photoconductivity experiments on SWNTs have also shown peaks in the infrared and visible ranges[34,35].Ultraviolet spectroscopy has revealed the dependence of the spectrum on the nanotube diameter[78]. UV absorption components in the optical spectra of carbon nanotubes have usually beenattributed to the pi-plasmon excitations [79]. pi-pi* transitions at the X (K = pia ) point of the bandstructure have also been suggested to be responsible for UV absorption [41]. Our results show thatpi-σ*, σ-pi*, and σ-σ* transitions can also contribute to the UV components of the spectra.2.4 Comparison with Tight-BindingTo understand the discrepancies between our dipole moment results and the ones obtained fromthe pi-orbital TB model (within the zone-folding scheme), we first compare our calculated bandstructure with the TB band structure (Figs. 2.5 and 2.6). In TB, the valence and conductionbands are symmetric with respect to the Fermi level as shown in Figs. 2.5 (a) and 2.6 (a), andevery corresponding valence and conduction bands have the same angular momentum quantumnumber and opposite horizontal parity. Therefore, transitions between each pair of conduction andvalence bands of a zigzag nanotube satisfy the selection rules. In reality, however, the pi-bandsare not the only bands contributing to the electronic dispersion relation of nanotubes. Especiallyfor nanotubes with small diameters, the σ-pi hybridization alters the band structure and electroniccharacteristics significantly.36Figure 2.5: Valence bands of an (8,0) nanotube with the pi-band TB method (plotted using the formulasin Ref. [80]) (a) and DFT (BLYP/6-31G) calculations (b). The dashed lines indicate bands resulting fromthe σ or strongly hybridized pi-σ orbitals. The solid lines correspond to the pi orbitals and resemble the TBbands. The Fermi level is located at zero.37Figure 2.6: Conduction bands of an (8,0) nanotube with the pi-band TB method (plotted using the formulasin Ref. [80]) (a) and DFT (BLYP/6-31G) calculations (b). Conduction bands in the DFT calculations areshifted down and have switched place (see the four lowest bands near kz= 0) compared to the TB bands.The dashed lines indicate bands resulting from the σ orbitals. The solid lines correspond to the pi orbitals.The Fermi level is located at zero.Our band structure results are consistent with other existing first-principles calculations of theband structure such as those reported in [48]. Valence bands from TB compare relatively well withthe pi bands from the DFT calculations, especially for the first few bands with higher energies.Bands resulting from σ bonds or strong pi-σ hybridization are shown with dashed lines in Fig.2.5(b). The discrepancy between the two band structures is more apparent for the conductionbands. As shown in Fig. 2.6(b), the conduction bands are shifted down, distorted, and haveswitched places and, overall, they cease to form a symmetric image of the valence bands, consistentwith calculations in [48]. This could be partially due to curvature and rehybridization effects. Sucha strong down-bending effect has also been confirmed experimentally [30, 31]. As the diameter of38the nanotube increases, these effects become less noticeable and the pi band structure approachesthe one predicted in TB. However, for small-diameter tubes, the discrepancies in the electronicband structures calculated from the first-principles methods and the TB approximation are at thecore of the inconsistencies between the resulting optical transition dipole moments.Figure 2.7 compares the dipole moment at the Γ point between the valence and conduction pi-sub-bands with the same index, resulting from our calculations using both TB and DFT (BLYP/6-31G) electronic structures. A similar trend is observed for the three nanotubes. The dipole momentsquare for each nanotube is normalized based on the TB value for the last sub-band and the sub-band indices are assigned in a manner consistent with the TB sub-band number assignment. Evenfor these sub-bands, although there is general agreement between TB and DFT results, there areclear and significant differences. Moreover, in contrast with the TB prediction, the dipole momentcalculated using DFT does not show a monotonic increase with the sub-band number.Neglecting the curvature effects can also affect the density of states and consequently the cal-culated probability of transition. For example, according to the TB calculations in [70] for (n,0)zigzag nanotubes with n even, there is a sharp peak in the absorption spectrum corresponding tothe transition to the conduction sub-band with m = n2 . This maps to the transition with the energyof 5.67 eV in Fig. 2.3(b) which does not indicate a very strong peak. The reason for the strongpeak in the TB calculations is the dispersionless band for m = n2 , which results in a high DOS. Ourband structure calculations for (8,0) and (10,0) nanotubes, however, show that this band is notcompletely dispersionless and its associated DOS is not as high as what is predicted by TB. On theother hand, our band structure calculations for the (7,0) nanotube result in a nearly dispersionlessband near the Γ point at the m = n−12 sub-band, which is not dispersionless according to the TBcalculations. This conduction band in fact leads to the maximum of absorption probability and asharp peak in the optical spectrum of the (7,0) nanotube because of its high DOS.We performed a similar comparison between our results based on different first-principles ap-proaches. This is shown in Fig. 2.8, where the three curves follow a similar behavior. Notably,B3LYP, which is a hybrid DFT method involving both BLYP and HF contributions, leads to valuesthat fall between the BLYP and RHF results.39Figure 2.7: Magnitude squared of the dipole moment for (a) (8,0), (b) (10,0), and (c) (7,0) nanotubes atthe Γ point with the TB and DFT (BLYP/6-31G) methods.400 1 2 3 4 5 6 7 801234567Subband Number|Dz|2  BLYPB3LYPRHFFigure 2.8: Magnitude squared of the dipole moment for an (8,0) nanotube at the Γ point with the DFT(BLYP/6-31G and B3LYP/6-31G) and RHF/6-31G methods.As discussed earlier, since the VHS for zigzag nanotubes happen at the Γ point, we expect thespectra in Fig. 2.3 and 2.4 to be very close to the overall spectra of these nanotubes. A comparisonwith other first-principles calculations for the optical absorption spectra of an (8,0) nanotube inthe 3-7 eV [41] and 0-8 eV [42] ranges confirms this point. The only peak missing in our calculationwithin this range is the one at around 4 eV that has been shown to occur at the X point of theband structure for a transition between the valence and conduction band with m = n2 . This mightbe partially due to the high density of states for these bands at the X point of the band structure,which results in a non-negligible transition rate.2.5 SummaryIn summary, the band-to-band transition dipole moment for (8,0), (10,0), and (7,0) zigzag nan-otubes was calculated at the Γ point for 12 valence and 16 conduction bands in a wide range ofwavelengths (infrared-visible-ultraviolet) using first-principles methods. We compared the results41with the conventional selection rules for nanotubes. We noted that modified angular momentumnumbers should be used in order to explain all the allowed transitions. We showed that besidesthe pi-pi* transitions, pi-σ*, σ-pi*, and σ-σ* transitions also contribute to the optical spectrum.In fact, the dipole moment magnitude was shown to be highest for σ-σ* transitions and some ofthese transitions resulted in a high probability of absorption in the low UV region. The trends indipole moments, selection rules, and transition rates were similar in all the three nanotubes. Weobserved high transition rates in the infrared, visible and even low UV ranges. The maximum ofabsorption at the Γ point for (8,0), (10,0), and (7,0) nanotubes occurred at approximately 1.42 eV,0.79 eV and 3.03 eV, respectively, and the strong peaks in the UV region were related to the σ-σ*transitions in all the three nanotubes. By using first-principles approaches, the curvature effectswere automatically taken into account. These effects could play an important role in the opticalproperties of nanotubes with small diameters. The results of this chapter have been publishedin [81].42Chapter 3CHAPTER 3: Effect of the Curvatureand Geometry 33.1 IntroductionBecause of the strong connection between the absorption behaviour and nanotube diameter, under-standing the effect of diameter on the electronic structure and absorption properties of nanotubes isessential. Measurements of the optical transition energies for nanotubes with small diameters [25]have yielded values significantly different from the theoretical values predicted with the tight-binding (TB) method [77]. This is mostly due to the fact that the results reported in [77] arebased on the zonefolding (ZF) approximation of graphene, where the curvature of the nanotubesidewall is neglected. The effect of this curvature becomes increasingly important as the diameterof nanotube decreases and, therefore, neglecting it can result in significant calculations errors forsmall-diameter nanotubes. Although general discussions about the importance of the nanotubesidewall curvature exists in the literature, a thorough study on the several effects arising from thiscurvature and their impacts on the optical transition energies is missing. Here, we study the impactof the nanotube sidewall curvature on the electronic and optical absorption properties by examininghow it affects σ-pi orbital rehybridization, the carbon-carbon (C-C) bond length, and the mixing ofd and p orbitals, separately. The goal is to investigate how and to what extent each of these effectscan impact the optical transition energies and to understand which of these effects are crucial inaccurately predicting the electronic and optical properties of carbon nanotubes.3This chapter is a reproduction of the article [S. Motavas, A. Ivanov, and A. Nojeh, “The curvature of the nanotubesidewall and its effect on the electronic and optical properties of zigzag nanotubes,” Computational and TheoreticalChemistry, vol. 1020, pp. 3237, 2013]. Inclusion in thesis is permitted. Copyright (2013) Elsevier.43We employed the density functional theory to calculate the electronic structure of zigzag nan-otubes. Geometry optimization, as well as the calculation of the orbital energies and wave functionswere performed in Gaussian 09. We obtained the transition rate spectra by calculating the dipolemoments and using Fermi’s golden rule as explained in Chapter 2. The polarization of light wasassumed to be parallel to the nanotube axis.Figure 3.1: One unit cell of (a) a (4,0) (b) an (8,0), and (c) a (16,0) nanotube with diameters of ∼0.34,∼0.63, and ∼1.27 nm, respectively. The nanotube axis is vertical in all cases.3.2 DFT vs. TB/ZF and σ-pi RehybridizationWe simulated (4,0), (8,0), and (16,0) nanotubes with significantly different diameters (∼0.34, ∼0.63,and ∼1.27 nm, respectively) in order to capture the effect of diameter and curvature (Figure 3.1).To separate the effect of bond length, in this section we present the results of DFT calculationson nanotubes prior to geometry optimization, i.e., nanotubes for which all the C-C bonds have44a length equal to that in graphene (1.42 A˚). In the next section, we compare the results aftergeometry optimization of nanotubes to show the effect of bond length.Rehybridization of the σ and pi orbitals, as one result of the nanotube sidewall curvature, cansignificantly affect the electronic structure of nanotubes with small diameters [47]. Figure 3.2shows the band structure of the (4,0), (8,0), and (16,0) nanotubes as calculated by TB/ZF (withhopping energy of 2.7 eV) and DFT calculations using BLYP exchange-correlation functional andthe 6-31G basis set. The difference between the two band structures increases progressively as thenanotube diameter decreases. In particular, the singly degenerate conduction band (α) in the DFTband structure is shifted down significantly for all three nanotubes compared to the one predictedby TB/ZF [47]. For an (8,0) nanotube, this downshifting places the (α) band in the band gap,resulting in a smaller gap (0.79 eV) compared to the one predicted by the TB/ZF method (1.27eV). This band is shifted down even further to below the fermi level for a (4,0) nanotube, leadingto metallic properties for this nanotube. As can be seen in Figure 3.2, the DFT valence bands aremostly shifted up compared to the TB/ZF bands.45Figure 3.2: Electronic band structure of (a) a (4,0) (b) an (8,0), and (c) a (16,0) nanotube with TB/ZF(solid blue) and BLYP/6-31G (dashed red) calculations. The geometries of nanotubes are not optimized.The singly degenerate band (α) is predicted for both band structures. The fermi level is at zero.46Fig. 3.3 shows the singly degenerate pi (α) and σ (β) bands that were mixed and then repelledfrom each other. It can be seen that the repulsion becomes stronger as the nanotube diameterbecomes smaller. The rehybridization effect is also apparent in the molecular orbitals. In Fig. 3.4we show the molecular orbitals of the (α) band at the Γ (kz=0) point for the three nanotubes. Theorbitals become more and more asymmetric with decreasing the diameter, showing more chargepopulation outside the nanotube compared to the inside due to the mixing of the σ and pi orbitals.In Fig. 3.5 we show a singly degenerate orbital in the valence bands of the three nanotubes thatis not significantly affected by the rehybridization effect. As can be seen, the charge populationinside and outside the nanotubes is almost identical. The change in the molecular orbitals affectsthe values of the dipole moments and consequently, together with the change in the transitionenergies and density of states, impacts the transition rate.Figure 3.3: Band structure of (a) a (4,0) (b) an (8,0), and (c) a (16,0) nanotube calculated with BLYP/6-31G, showing the repulsion of the singly degenerate pi and σ bands (dashed lines). The geometry of thenanotubes are not optimized.47Figure 3.4: Molecular orbitals of the (α) band at the Γ (kz=0) point for (a) (4,0), (b) (8,0) and (c) (16,0)nanotubes. The isovalue is 0.03 for (4,0) and (8,0) and 0.02 for (16,0).48Figure 3.5: The molecular orbitals of a singly degenerate valence band at the Γ (kz=0) point (a) (4,0), (b)(8,0) and (c) (16,0) nanotubes.The isovalue is 0.03 for all three nanotubes.49Table 3.1 shows a comparison of the first two optical transition energies obtained from ourDFT and TB/ZF calculations. It is important to note that not only the transition energies butalso the order of some of the transitions are different between the two methods. The reduction ofE11 values in the DFT calculations are mainly due to the upshifting of the first valence band. Thecorresponding conduction bands are almost at the same energy as the ones calculated in TB/ZF.∆E11 increases as the diameter decreases due to the stronger shift of the first valence band in smallernanotubes. We do not see the same diameter dependence for ∆E22. For an (8,0) nanotube theupshift of the second valence (v2) and downshift of the third conduction (c3) bands both accountfor the significantly smaller E22 (v2 → c3 transition) energy compared to TB/ZF energies. For a(4,0) nanotube, on the other hand, E22 is not significantly different between the two methods. Thisis due to the fact that the dispersionless second valence (v2) and second conduction (c2) bands inthe DFT band structure of a (4,0) CNT are both shifted up with respect to the TB/ZF bands byalmost the same amount. Since none of the transitions listed in Table 3.1 involve the (α) band, thedifference between transition energies obtained from the two methods (∆) is not as drastic as thedifference in the band gaps derived from them; nonetheless, the differences are still quite significantespecially for small-diameter nanotubes.Table 3.1: Comparison of the lowest two optical transition energies calculated with TB/ZF andBLYP/6-31G. The energy change compared to TB/ZF results is denoted by ∆.E11(eV ) Transition E22(eV ) Transition(4,0)TB/ZF 2.24 v1 → c1 5.40 v2 → c2BLYP/6-31G 0.80 v1 → c1 5.04 v2 → c2∆ -64.29% -6.67%(8,0)TB/ZF 1.27 v1 → c1 2.24 v2 → c2BLYP/6-31G 1.18 v1 → c4 1.78 v2 → c3∆ -7.08% -20.54%(16,0)TB/ZF 0.60 v1 → c1 1.27 v2 → c2BLYP/6-31G 0.57 v1 → c1 1.23 v2 → c2∆ -5.00% -3.15%503.3 Effect of the C-C Bond LengthAnother important consideration is that the C-C bond lengths in nanotubes are different from theones in a graphene sheet due to the curvature of the nanotube sidewalls. For zigzag nanotubes,this curvature results in two different bond lengths; one is the C-C bond length parallel to thenanotube axis (CC1) and the other is the one around the nanotube circumference (CC2), with CC2being longer than CC1 due to the weakening of the strained bonds wrapped around the nanotube[82,83]. Here, we explore the effect of bond length on the electronic structure as well as the opticaltransitions. To do so, we first optimized the geometry of the three nanotubes using BLYP/6-31G.Prior to geometry optimization, CC1 and CC2 were both set equal to the C-C bond length ingraphene (1.42 A˚). Table 3.2 shows the CC1 and CC2 bond lengths after geometry optimization.The CC2/CC1 ratio shows that the two bond lengths approach each other as the diameter of thenanotube increases.Table 3.2: The CC1 and CC2 bond lengths (parallel and perpendicular to the nanotube axis,respectively) after geometry optimization with BLYP/6-31G. CC1 = CC2 = 1.42 A˚ before geometryoptimization.Nanotube CC1(A˚) CC2(A˚) CC2/CC1(4,0) 1.399 1.501 1.073(8,0) 1.429 1.451 1.015(16,0) 1.436 1.440 1.003Figure 3.6 shows the comparison for the band structures of the three nanotubes before and aftergeometry optimization. Significant changes to some of the bands can be observed after geometryoptimization, and the values in Table 3.3 indicate substantial differences in the transition energiesfor almost all the cases, in the same order as or even bigger than the changes discussed in theprevious section (Table 3.1 values).51Figure 3.6: Electronic band structure of (a) a (4,0) (b) an (8,0), and (c) a (16,0) nanotube without (solidblue) and with (dashed red) geometry optimization . The geometries and the band structures are calculatedwith BLYP/6-31G. The fermi levels are based on the calculations prior to the geometry optimization.52Table 3.3: Comparison of the lowest two optical transition energies, calculated by BLYP/6-31Gbefore and after the geometry relaxation. The energy change after geometry optimization is denotedby ∆.E11(eV ) Transition E22(eV ) Transition(4,0)Before Relaxation 0.80 v1 → c2 5.04 v2 → c3After Relaxation 0.43 v1 → c2 5.43 v2 → c4∆ -46.25% 7.74%(8,0)Before Relaxation 1.18 v1 → c4 1.78 v2 → c3After Relaxation 1.35 v1 → c4 1.46 v2 → c3∆ 14.41% -17.98%(16,0)Before Relaxation 0.57 v1 → c1 1.23 v2 → c2After Relaxation 0.52 v1 → c1 1.23 v2 → c2∆ -8.77% 0We can see that after geometry optimization, E11 decreases even further for (4,0) and (16,0)nanotubes. This time, the reduction is due to the downshift of the first conduction band. On theother hand, the second transition energy of the (4,0) nanotube increases since the dispersionlessvalence band (v2) shifts to a lower energy. For an (8,0) nanotube, the opposite mechanism happens;the first valence band is shifted down, leading to a bigger value for E11 and the downshift of thec3 band results in the reduction of E22. Both in here and also in Table 3.1, we can obviously seetrends that are oscillating with diameter rather than having a monotonous diameter dependence.If we look at the overall effect of DFT calculations and bond length (Table 3.4), we notice thesame trend. This suggests the idea that the effect of curvature on the transition energies mightbe dependent on the zigzag nanotube index, n. To test this, we performed the calculations on afew more zigzag nanotubes and plotted their ∆E11 and ∆E22 versus the nanotube index (Figure3.7). The results in fact confirm the dependence on the nanotube index, where nanotubes withn(mod)3=1 show a different behaviour compared to the ones with n(mod)3=2.53Table 3.4: The difference between the results obtained using DFT calculations after geometryoptimization and the TB/ZF results.Nanotube ∆E11 ∆E22(4,0) -80.80% 0.56%(8,0) 6.30% -34.82%(16,0) -13.33% -3.15%54Figure 3.7: Plot of the difference between the first (a) and second (b) transition energies (∆E11 and∆E22, respectively) for BLYP/6-31G calculations (after geometry optimization) of (n,0) nanotubes andtheir corresponding TB/ZF values. Values for nanotubes with n(mod)3 = 1 are shown in purple (empty)circles and the ones with n(mod)3 = 2 are shown in green (solid) circles.553.4 Importance of Geometry OptimizationIn the previous section, we noticed a substantial difference between the optical transition energiesof nanotubes with and without geometry relaxation, which raises the question: what method canpredict the geometry of the nanotubes most accurately and what is the sensitivity of the electronicstructures and absorption properties on the method of relaxation? This is particularly importantsince for many nanotubes there are no experimental data on bond lengths. To address this issue,we calculated the electronic band structure of an (8,0) nanotube that is geometry optimized usingthe Hartree-Fock/3-21G and BLYP/6-31G methods (Figure 3.8). The band structure itself iscalculated with BLYP/6-31G in both cases and the only difference between the two is the exactgeometry of the nanotube. As we can see, the relative position of the bands does not depend onthe method of geometry relaxation. However, the shift in the bands, which is mostly pronouncedfor the conduction bands, can affect the calculated transition energies.0 0.2 0.4 0.6 0.8 1−4−202468Kz (pi/a)E (eV)Ef Figure 3.8: Band structure of an (8,0) nanotube, optimized with HF/3-21G (solid blue) and BLYP/6-31G(dashed red) methods. The band structure of the nanotube is calculated with BLYP/6-31G in both cases.The fermi level is based on the calculations with BLYP/6-31G geometry optimization56Table 3.5: Comparison of the C-C bond length in graphene calculated with different methods.Method C-C bond length(A˚)Experimental 1.421BLYP/6-31G 1.436991PW91PW91/6-31G 1.430848PBEPBE/6-31G 1.431983HSEH1PBE/6-31G 1.421419VSXC/6-31G 1.432008The HF method is known to underestimate the bond lengths while DFT approaches with GGAfunctionals generally overestimate bond lengths [84]. Consequently, one might think that hybridmethods can provide a closer estimate to the experimental values of bond lengths. To investigatethis, we performed a series of simulations using different functionals on graphene, for which theexperimental value of the bond length is available. Table 3.5 summarizes the C-C bond lengthsobtained for graphene. It can be seen that HSEH1PBE, which is a hybrid method, yields a morereasonable value (closer to the experimental value) for the bond length of graphene compared tothe others. By extension, we suggest that this method will be more accurate for nanotubes aswell. In Table 3.6 we compare the (8,0) nanotube C-C bond lengths calculated with HF, BLYP,and HSEH1PBE. We can see how the values obtained from the hybrid method HSEH1PBE fallbetween the values obtained from BLYP and HF (almost equal to the average of the two). Table 3.7shows the first two optical transition energies calculated for an (8,0) nanotube that is optimized byHSEH1PBE/6-31G, compared against the values obtained from BLYP/6-31 and HF/3-21 optimizednanotubes and experimental values.Table 3.6: Comparison of the C-C bond lengths in an (8,0) nanotube calculated with differentmethods.Method of Geometry Optimization CC1(A˚) CC2(A˚)BLYP/6-31G 1.429 1.451HF/3-21 1.403 1.425HSEH1PBE/6-31G 1.414 1.43357Table 3.7: Comparison of the first two optical transition energies for an (8,0) nanotube that isgeometry optimized with different methods against experimental values. The electronic structurehas been calculated using BLYP/6-31G in all these cases.Method of Geometry Optimization E11(eV) E22(eV)BLYP/6-31G 1.35 1.46HF/3-21 1.42 1.53HSEH1PBE/6-31G 1.37 1.53Experiment a 1.60 1.88aRef. [25, 26]583.5 Mixing of d and p orbitalsIncluding the d orbitals in the basis set results in the possibility of polarization and additionalflexibility in the shape of the molecular orbitals. The mixing of d and p orbitals, as one otherconsequence of nanotube curvature, has been discussed in some works [85]. Figure 3.9 showsthe band structures calculated using BLYP/6-31G and BLYP/6-31G(d) for (4,0), (8,0), and (16,0)nanotubes. While the improvement of data is expected by inclusion of the d orbitals, we wouldlike to investigate the level of improvement and its dependence on the diameter. Furthermore, wewould like to study the impact of this inclusion on the optical transition energies. We see fromFigure 3.9 that valence bands are in general more affected by inclusion of the d orbitals comparedto the conduction bands. Moreover, the effect is stronger for smaller-diameter nanotubes due tothe stronger mixing of the orbitals at higher curvature. The first two optical transition energiescalculated with BLYP/6-31G and BLYP/6-31G (d) are shown in Table 3.8. Interestingly, again anoscillating behaviour can be seen for both ∆E11 and ∆E22 as a function of nanotube index. Whilethe effect of d orbitals on some of the transitions might seem negligible, for certain transitions theenergy difference is comparable to if not larger than the difference caused by other effects discussedin this paper (cf. Table 3.4), and therefore, important to include.Table 3.8: Comparison of the lowest two optical transition energies with and without including thed orbitals. ∆ is the energy difference.E11(eV ) E22(eV )(4,0)BLYP/6-31G 0.43 5.43BLYP/6-31G(d) 0.42 5.31∆ -2.11% -2.23%(8,0)BLYP/6-31G 1.35 1.46BLYP/6-31G(d) 1.30 1.44∆ -3.57% -1.64%(16,0)BLYP/6-31G 0.52 1.23BLYP/6-31G(d) 0.51 1.20∆ -1.16% -2.53%59Figure 3.9: Band structure with BLYP/6-31G (solid blue) and BLYP/6-31G(d) (dashed red) for (a) (4,0),(b) (8,0) and (c) (16,0) nanotubes.603.6 SummaryWe investigated several effects caused by the curvature of the nanotube sidewall and their im-pacts on the electronic and optical properties of zigzag nanotubes. Although in the literature,the effect of curvature is mainly attributed to the σ-pi rehybridization, we showed that both theσ-pi rehybridization and bond length effects are crucial in the accurate prediction of the opticaltransition energies of small-diameter nanotubes. Transition energies predicted in DFT showed todiffer significantly from the ones in TB/ZF calculations (up to ∼ 64%) and the effect of bondlength, by itself resulted in substantial differences for most E11 and E22 energies (up to ∼46%).The overall effect on the transition energies showed to depend on the nanotube index, n, ratherthan the diameter. We confirmed this by plotting the differences against the nanotube index forseveral nanotubes. Nanotubes with n(mod)3=1 showed a different behaviour compared to the oneswith n(mod)3=2. For some of the transitions also, the effect of mixing of d and p orbitals due tothe curvature turned out to be as significant as the effect of σ-pi rehybridization and bond length.Finally, the importance of the effect of bond length prompted a study on the method of geometryoptimization. Among all the methods tested, our DFT calculations using HSEH1PBE functionalhad the most accurate prediction for the C-C bond length of graphene. The study presented inthis paper can provide an insight into the significance of several individual effects arising from thenanotube sidewall curvature and the level of improvement that including each of these effects canbring into the theoretical calculations. Parts of this chapter have been published in [86].61Chapter 4CHAPTER 4: Effect of LightPolarization 44.1 IntroductionDue to the one-dimensional nature of nanotubes, their optical absorption spectra depend on thepolarization of the incident light. For perpendicularly polarized light, the absorption spectrum isknown to be suppressed due to the depolarization effect [87], i.e., the reduction of the effectiveelectric field as a result of the induced charges on the nanotube walls that create a dipole againstthe external electric field. The polarization dependence of the optical absorption spectrum hasbeen shown experimentally for 4 A˚ nanotubes (nanotubes with diameter of ∼4 A˚) in zeolite [88].Theoretical calculations that take into account local-field effects (LFE) have also confirmed theweakening of the perpendicular polarization absorption spectra of 4 A˚ nanotubes due to the depo-larization effect [89]. The depolarization effect has also been observed in Raman spectroscopy ofCNTs [90–92].However, despite the depolarization effect, distinct peaks have been observed in the photolu-minescence spectra of individual single-walled nanotubes (SWNTs) under the perpendicular po-larization of light [93]. This suggests that although the depolarization effect (often referred to as“antenna effect” in the nanotube literature) is known to play a major role, other factors might alsobe influential in the polarization dependence of the absorption spectra of nanotubes. Hence, ex-ploring the effect of light polarization beyond the depolarization effect is important. Band-to-band4This chapter is a reproduction of the article [S. Motavas, A. Ivanov, and A. Nojeh, “The effect of light polarizationon the interband transition spectra of zigzag carbon nanotubes and its diameter dependence,” Physica E: Low-dimensional Systems and Nanostructures, vol. 56, pp. 7984, 2014]. Inclusion in thesis is permitted. Copyright (2014)Elsevier.62transition rates are at the root of optical absorption spectra. Therefore, the goal of this work isto investigate the effect of light polarization on the interband optical transition spectra of zigzagnanotubes independently of the antenna effect and to explore its dependence on the nanotubediameter.4.2 MethodologyWe employed a first-principles method to calculate the optical absorption spectra of zigzag CNTs forparallel and perpendicular polarizations of light. We studied the band-to-band transition spectrawithout including the LFE in order to examine the effect of light polarization on the dipole momentsand interband transitions separately from the depolarization effect. In order to investigate thediameter dependence of the polarization effect, we calculated the spectra for three zigzag nanotubeswith significantly different diameters. For this purpose, we chose (4,0), (8,0), and (16,0) nanotubeswith diameters of ∼0.34, ∼0.63, and ∼1.27 nm, respectively. Structural relaxation and energycalculations were performed in Gaussian 09. The calculation method is the same as in the previouschapters. We calculated the dipole moment for all the possible optical transitions between the first12 valence and 16 conduction bands, including both pi and σ bands. However, for the purposes ofcomparison, all the optical spectra are plotted only in the 0-11 eV range.4.3 Results and DiscussionFigure 4.1 shows the optical transition spectrum of an (8,0) nanotube for light polarized paralleland perpendicular to the nanotube axis. The overall transition rate spectrum for perpendicularlypolarized light is sparse compared to the one for parallel polarization. However, at photon energiesof 1.2 eV, 1.8 eV and 10.6 eV the transition probability is significant for perpendicular polarization.According to the selection rules, light with parallel polarization, due to its 0A−0 symmetry,preserves the angular momentum quantum number, but reverses the horizontal parity, σh. Perpen-dicularly polarized light changes the angular quantum number by ±1 while it preserves σh [94,95].We define the angular momentum quantum number, m, as half of the number of nodes that a63Figure 4.1: Transition rate for an (8,0) nanotube (calculated with BLYP/6-31G) versus the energyof transition for parallel (a) and perpendicular (b) polarization of light. pi-pi* transitions withsignificant transition rate magnitudes in the perpendicular spectrum are pointed out.wave function carries around its circumference. We have previously shown in Chapter 2 that if amodified angular quantum number, >m, is defined for an (n,0) nanotube so that >m = m+>Mn, allthe optically allowed transitions satisfy the ∆>m = 0 condition for parallel polarization of light (>Mis an integer determined such that >m ∈ (−n2 ,n2 ]). For perpendicular polarization also, we note thatall the allowed transitions follow the ∆>m = ±1 rule. For example, the first peak in the parallelabsorption spectrum of an (8,0) nanotube is the result of the transition between the 1st valenceband (with m = 5, >m = 3 and σh even) to the 4th conduction band (with m = 3, >m = 3 and σhodd) while the first peak in the perpendicular polarization happens between the 1st valence and the3rd conduction band (with m = 6, >m = 2 and σh even). Figure 4.2 depicts the molecular orbitalscorresponding to each of these bands. Nodes are defined as the points where the wave functionvalue changes signs (goes from grey to black or vice versa). The number of nodes are 10 (a), 6 (b),and 12 (c) and, therefore, the m number for these wave functions is equal to 5, 3 and 6, respectively.According to the pi-orbital tight-binding model, for semiconducting nanotubes, under the per-pendicular polarization of light, there exist only two band-to-band transitions with significant dipole64Figure 4.2: Wave functions for the 1st valence (a), 4th conduction (b), and 3rd conduction (c) bandsof an (8,0) nanotube calculated with BLYP/6-31G. Left: view along the tube axis. Right: viewperpendicular to the tube axis.moments (between the first and second VHS points) [69]. Our calculations, however, show multiplepi-pi* transitions with considerable dipole moment values for an (8,0) nanotube. Three of thesetransitions lead to peaks with noticeable strengths in the transition rate spectrum (at 1.2 eV, 1.8eV and 9.8 eV) after including the JDOS and transition energies. This discrepancy originates fromthe fact that only transitions with subband number differences of ±1 are considered in the TBmodel. Based on this and the conservation of horizontal parity, only transitions between b2n3 c5and b2n3 +1c subband numbers are possible for an (n,0) semiconducting zigzag nanotube, which areindeed the transitions between the first and second VHS points (2n3 is where the horizontal parity5(b c of a number is the floor of that number)65changes signs in zigzag nanotubes [96]).In our calculations, however, if the ∆>m = ±1 rule is considered, aside from the two transitionsabove, other transitions will also be allowed. Some of these transitions have small dipole momentvalues and/or happen over large energy gaps and, therefore, their corresponding peaks are notnoticeable in the transition rate spectra. On the other hand, some of them can result in strongpeaks in the absorption spectra. The transition at 9.8 eV, for example, happens between the 4thvalence band (with m = 8, >m = 0 and σh odd) and 9th conduction band (with m = 1, >m = 1 andσh odd). The other two strong peaks at 10.6 eV and 10.8 eV are the results of σ to pi* transitions,which are also not taken into account in the pi-orbital tight-binding model.To study the diameter dependence of the polarization effect, we also calculated the transitionrate spectra of (4,0) and (16,0) nanotubes for both polarizations (Figure 4.3). The ratio of thetransition rates for parallel and perpendicular polarizations (W‖W⊥) for the (4,0), (8,0), and (16,0)nanotubes is listed in Table 4.1, where W is the sum of all the transition rates in the specified range.We observe that the overall spectra (0-11 eV) are suppressed for the perpendicular absorption inall three nanotubes, although this weakening does not show a monotonic diameter dependence.However, the same comparison in the infrared/visible (IR/VIS) and ultraviolet (UV) regions showsclear trends.Table 4.1: Ratio of the transition rates for parallel and perpendicular polarization of light (W‖W⊥)in different energy ranges. For each polarization, W is the sum of all the transition rates in thespecified range for the nanotube under study.Range (4,0) (8,0) (16,0)0-11 eV 3.56 2.20 3.69IR/VIS (0-3 eV) 10.31 3.79 2.11UV (3-11 eV) 0.49 0.83 127.0166Figure 4.3: Transition rate (calculated with BLYP/6-31G) versus the energy of transition for a (4,0)(a), an (8,0) (b) and a (16,0) (c) nanotube. For each nanotube the spectra on the left and rightcorrespond to the calculations with parallel and perpendicular polarization of light, respectively.pi-pi* transitions with significant transition rate magnitudes are pointed out in the spectra.67The overall transition rate in the IR/VIS portion of the spectrum (0-3 eV) is stronger forparallel polarization, and theW‖W⊥ratio decreases as the nanotube diameter increases. Note thatthe depolarization (antenna) effect is not included in these calculations. Thus, one may concludethat the antenna effect is only partially responsible for the suppression of the absorption spectrawith perpendicular polarization in the IR/VIS range and part of this overall weakening can be dueto the role of the interband transitions, directly. For (8,0) and (4,0) nanotubes, in this range thepeaks are in general stronger for the parallel light compared to the perpendicular light. For a (4,0)nanotube, there exists only one pi-pi* transition with significant magnitude in the perpendicularcase. This transition happens at ∼2 eV between the 3rd valence and 2nd conduction bands. For a(16,0) nanotube there exist two large peaks in the perpendicular polarization spectrum, happeningbetween the 1st and 2nd VHS points. The sharp peak at 0.9 eV is the result of a significant dipolemoment and also a small energy gap between the 1st valence and the 2nd conduction bands.In the UV range (3-11 eV), on the other hand, the strength and the number of peaks for paralleland perpendicular polarizations become comparable for (4,0) and (8,0) nanotubes, while for a (16,0)nanotube the peaks are considerably weaker for perpendicular polarization. Therefore, while theW‖W⊥ratio is significant for a (16,0) nanotube in the UV range, for (4,0) and (8,0) nanotubes theperpendicular light seems to be absorbed more compared to the parallel light. For a (4,0) nanotube,the overall absorption of the parallel light is almost half of that of the perpendicular light in thisregion. Transitions with considerable magnitude in the UV region are all non-pi-pi* (σ-pi*, pi-σ*, orσ-σ*) transitions for perpendicular polarization in a (4,0) nanotube while for parallel polarizationnon-pi-pi* transitions have small magnitudes. For a (16,0) nanotube, on the other hand, pi-pi* andnon-pi-pi* transitions are all of small magnitudes in the UV range for perpendicular polarization,whereas in the parallel polarization spectra, there are a few sharp peaks in the UV region, all as aresult of pi-pi* transitions.To study how much of the behaviour of the interband transition is due to the trends in the dipolemoment, we perform a comparison similar to that of Table 4.1 for the squared magnitude of thedipole moment in the IR/VIS and UV regions (Table 4.2). Notice that the trends are similar to thetrends in Table 4.1, implying that the dipole moment (and, therefore, the shape and interactions68of the wave functions with each other) is the component mostly responsible for the trends in thetransition rates in each region (as opposed to the other players in the transition rate, such as energyor JDOS).Table 4.2: Ratio of the dipole moments squared for parallel and perpendicular polarization oflight (D2‖D2⊥) in different energy ranges. For each polarization, D2 is the sum of the square of themagnitudes of the dipole moments in the specified range for the nanotube under study.Range (4,0) (8,0) (16,0)IR/VIS (0-3 eV) 11.69 5.82 2.98UV (3-11 eV) 0.86 0.96 15.18The values ofW‖W⊥combined with the depolarization effect can provide an estimate for overallweakening of the absorption spectra for perpendicular light compared to the one for parallel light.Studies of the static polarizabilities of carbon nanotubes have shown a screening factor of ∼5 fornanotubes under a transverse field [97,98], implying that the effective field becomes approximately5 times weaker than the applied field due to the depolarization effect. Therefore, for an (8,0)nanotube, for example, one might estimate the overall weakening of the transition spectra in theIR/VIS range to be approximately 5×3.79=∼19 times after including the antenna effect. Or forinstance, experimental results in [93] show an overall weakening of ∼10 times for the maximumpeak in the IR/VIS spectrum of (7,5) nanotubes. Hence, one might estimate the weakening due tothe interband transitions (W‖W⊥) for these peaks to be about 105 =∼2 times. Considering the diameterof these nanotubes (∼0.8 nm) and the values in Table 4.1, two times weakening of the band-to-band transition spectrum under the perpendicular light can be a reasonable prediction for (7,5)nanotubes. Obviously, these are all very rough estimations and many factors such as the frequencydependence of the screening factor or the chirality dependence of theW‖W⊥need to be taken intoaccount for better predictions of these values.4.4 SummaryIn summary, the interband optical transition rate spectra for (4,0), (8,0), and (16,0) nanotubeswere calculated for both parallel and perpendicular polarizations of light in the 0-11 eV range.69We observed that a modified angular quantum number, >m, should be used when considering theselection rules in order to explain all the allowed transitions for perpendicular polarization of light,similarly to what has been shown for parallel polarization in Chapter 2. Our results showed theoverall suppression of the transition rate spectra for perpendicular polarization compared to thosefor parallel polarization for all three nanotubes, although at certain photon energies the probabilityof absorption for perpendicular light turned out to be surprisingly high.We noticed that, in the IR/VIS range, theW‖W⊥ratio increases as the nanotube diameter de-creases. For UV light, not only does this trend becomes reversed, but also perpendicular light isabsorbed with higher probability compared to the parallel light for (4,0) and (8,0) nanotubes. Thiscan be important in UV application of small-diameter nanotubes in optoelectronic devices. Weobserved that non-pi-pi* transitions play a major role in the perpendicular light absorption of a(4,0) nanotube in the UV region.All these calculations were performed without including the depolarization effect in order toshow the polarization dependence of the band-to-band transition spectra. If, after inclusion of thedepolarization effect, some of the peaks in the perpendicular spectra carry a significant magnitudeso as to be detectable experimentally, such as those shown in [93], there will be new opportunitiesfor applications of nanotubes, such as in sensitive angle detection devices. Furthermore, lightpolarization can be used as an external parameter for tuning the optical properties of nanotube-based optoelectronic devices. This is of importance since, in this case, internal changes to thedevice will not be needed, and the orientation of the device with respect to the incident light canmodify the wavelength of absorption or emission. The results of this chapter have been publishedin [99].70Chapter 5CHAPTER 5: Effect of Spin andLength on the Optical Properties5.1 IntroductionDue to their confinement along the length, short carbon nanotube segments can exhibit interestingelectronic and optical properties different from those of long nanotubes. Understanding the effectof this confinement on the optical absorption properties can be useful in the study of devicesand structures made with short nanotube segments such as single-electron transistors [100] andoptoelectronic components, as well as selective tagging of nanotube-based biological transporters[101]. In this part of our research, we study the optical absorption behaviour of zigzag carbonnanotube segments that are terminated with hydrogen at both ends.It has been shown that the boundary conditions at the edges of nanotube segments can result indoubly degenerate half-filled states close to the Fermi level, and hence, the non-zero electron spin(S) in finite-length carbon nanotubes [102, 103]. Electron spin is the intrinsic angular momentumcharacterized by quantum number 1/2 and spin multiplicity is defined as 2S + 1. Okada et.al.showed that this spin multiplicity is dependent on the chirality of the nanotube [103]. Lengthdependence of the spin multiplicity has also been investigated [102]. In this work, we first calculatethe metastable configuration of the electron spin in the nanotube segments. We then calculate theoptical properties of the finite-length CNTs with correct spin configuration. The effect of spin onthe optical absorption spectra is investigated.In order to study the effect of length, we calculate and compare the absorption spectra ofnanotube segments with different numbers of unit cells, from 1 up to 20 unit cells (∼4 A˚ to ∼8571A˚), as well as an infinitely long nanotube (periodic structure). Figure 5.1 shows segments with1 up to 7 unit cells of an (8,0) nanotube. It is also of interest to find the minimum length of ananotube segment that can approximately represent an infinitely long nanotube in terms of opticalproperties.Figure 5.1: From top to bottom: 1 unit cell (32 toms and 4.26 A˚), 3 unit cells (96 toms and 12.78A˚), 5 unit cells (160 toms and 21.3 A˚) and 7 unit cells (224 atoms and 29.82 A˚) of an (8,0) nanotube.5.2 MethodologyWe perform the calculations of this section with SIESTA due to the capability of this software inhandling the calculations for thousands of atoms, as opposed to other programs such as Gaussian09 that can only deal with small numbers of atoms (∼100 atoms). We use (8,0) zigzag nanotubesegments that are terminated with hydrogen and perform the calculations for both cases where thenanotube ends are fixed in the space and when the ends are free. Our simulations are based onLDA calculations, norm-conserving pseudopotentials and vacuum size of ∼10 A˚. The polarizationof light is assumed to be parallel to the nanotube axis (z-direction).For periodic nanotubes, an energy shift of 50 meV is recommended [104]. The basis set used inthe literature for periodic nanotubes is generally in the form of singly polarized double-ζ (DZP) for722s and 2p orbitals [77,105]. It is also shown that the inclusion of the diffuse 3s orbital (single-ζ) isessential for correct description of the electronic structure of periodic nanotubes beyond 3 eV [106].The cut-off radii based on the 50 meV energy shift are: 5.12 bohr for 2s, 6.25 bohr for 2p and d,and 10 bohr for 3s [106].According to our tests, the basis set above also works well for energy calculations of non-periodic nanotubes. However, the optical calculations could not be achieved with this basis set andthe simulations did not converge. The problem turned out to lie in the value of the cut-off radiusfor the 3s orbital, which had to be reduced for the optical calculations to converge. The optimumvalue for this cut-off radius in our calculations was found to be 4.5 bohr (Figure 5.2). We also ransome tests in order to find the proper value of the vacuum size along the nanotube axis. Based onour calculations, 10 A˚ seemed to be a reasonable value (Figure 5.3).4 5 6 7 8 9 10−5200.5−5200−5199.5−5199−5198.5−5198Cutoff Radius (bohr)Total Energy (eV)Figure 5.2: Total energy of the system for different cut-off radii chosen for the 3s orbital of a 1-unit cell(8,0) nanotube segment terminated with hydrogen.5.3 Effect of SpinIn order to achieve the most stable configuration for the spin multiplicity, we run two sets ofseparate tests for each of the nanotube segments. One is to allow the structure to relax and reachthe metastable spin configuration. The other is to force a fixed number of spin in the structure and735 10 15 20 25 30−5400−5200−5000−4800−4600−4400−4200−4000−3800Lattice Constant (Ang)Total Energy (eV)Figure 5.3: Total energy of the system for different lengths of the vacuum along the nanotube axis for a1-unit cell (8,0) nanotube segment terminated with hydrogen.calculate the total energy. The minimum total energy happens at the correct spin multiplicity. Weshow that the results of both simulations are in agreement with each other.For 3 unit-cell of (8,0) and (7,0) nanotubes, the result of our spin polarized calculations lead toa total spin (S) of 1 and 2, respectively. That is in agreement with the results in [102]. Our forcedspin calculations reach the same conclusions (Figure 5.4 and Figure 5.5).To explore the effect of length on the spin multiplicity, we repeat the simulations for variousnumbers of unit cells of an (8,0) nanotube. Our spin polarized calculations predict a total spin of 0for the 1 unit-cell segment. Our spin-forced calculations also, lead to the exact same results for thisnanotube segment. Figure 5.6 shows the total energy of a 1-unit cell (8,0) nanotube versus the spin.We can see that the most stable configuration happens when there is no spin. Our calculations inGaussian 09 with HF, LSDA, and BLYP methods (6-31G basis set) confirm this (Figure 5.7).740 0.5 1 1.5 2−1.15−1.1−1.05−1−0.95−0.9−0.85−0.8−0.75SpinTotal Energy (eV)0 1 2 3SpinTotal Energy (eV)Figure 5.4: Total energy versus spin for a 3-unit cell (8,0) nanotube segment calculated withSIESTA. The inset shows our calculation with Gaussian 09 (BLYP/6-31G method).0 0.5 1 1.5 2 2.5 3−1.3238−1.3238−1.3238−1.3238−1.3237−1.3237−1.3237−1.3237−1.3237−1.3236 x 104SpinTotal Energy (eV)Figure 5.5: Total energy versus spin for a 3-unit cell (7,0) nanotube segment calculated withSIESTA.750 0.5 1 1.5 2 2.5−5208−5207.5−5207−5206.5−5206−5205.5−5205−5204.5−5204SpinTotal Energy (eV)Figure 5.6: Total energy versus spin for a 1-unit cell (8,0) nanotube segment calculated withSIESTA.0 1 2 3 4 5−1220.6−1220.5−1220.4−1220.3−1220.2−1220.1−1220SpinTotal Energy (eV)0 1 2 3 4 5−1221.5−1221.4−1221.3−1221.2−1221.1−1221−1220.9SpinTotal Energy (eV)0 1 2 3 4 5−1228−1227.95−1227.9−1227.85−1227.8−1227.75−1227.7−1227.65−1227.6−1227.55−1227.5SpinTotal Energy (eV)(c)(a) (b)Figure 5.7: Total energy versus spin for a 1-unit cell (8,0) nanotube segment calculated in Gaussian09 with HF (a), LSDA (b), and BLYP (c) methods and 6-31G basis set.760 0.5 1 1.5 2 2.5 3 3.5 4−2.5066−2.5065−2.5065−2.5065 x 104SpinTotal Energy (eV)Figure 5.8: Total energy versus spin for a 5-unit cell (8,0) nanotube segment calculated withSIESTA.When we increase the number of unit cells to 5 and more, however, the simulations result inthe total spin of 3. We confirmed this for up to 20 unit cells for an (8,0) nanotube segments. Theresults for a 5-unit cell segment are demonstrated in Figure 5.8. In summary, our spin polarizationcalculations lead to a total spin (S) of 0, 1, and 3 or the spin multiplicity (2S+1) of 1, 3, and 7 forone, three and ≥ five unit cells of an (8,0) nanotube, respectively.The length dependent calculations of spin multiplicity presented in [102] are in agreement withour results. The trend can be explained by applying the zone-folding method to the band structureof a nano-ribbon as shown in Figure 5.9. The nanotube segment can be considered as a rollednano-ribbon with a finite width. The doubly degenerate, flat dispersion bands in the nano-ribbonband structure are the results of the edge effects. In the 3-unit cell segment of the (8,0) nanotube,only one of the k-points (k = pia ) falls in the flat-band states region, leading to the total spin of1. As we increase the width of the nano-ribbon, the boundary of the flat-band region graduallyapproaches to k = 2pi3a [103]. This means that as we increase the nanotube length, two more statesat k = ±6pi8a become degenerate at the Fermi level, resulting in the spin of 3.77By looking at the Mulliken Population, we can see that most of the spin density is localized onthe outer carbon rings at both edges for the 3-unit cell segment. For the 5-unit cell segment, whilemost of the spin density is still on the outer carbon rings, there is also some spin density on theinner rings with the 3rd and 5th rings having the highest densities, respectively.Figure 5.9: Band structure of a zigzag nano-ribbon with the length equal to the 3-unit cell of azigzag nanotube, obtained by the LDA calculations (top) and the allowed k-points for the (7,0),(10,0), and (8,0) nanotube segments with 3-unit cell length (bottom). Copyright c© 2003 ThePhysical Society of Japan [103].78Figure 5.10 shows the comparison between the optical absorption spectra of a 3-unit cell (8,0)segment, before and after considering the spin multiplicity. As we can see, the difference between thetwo plots is negligible and the transition energies are almost identical in both cases. The differenceis, however, more noticeable in the case of a 5-unit cell segment (Figure 5.11). Particularly, the firstpeak is shifted to a higher energy and reduced significantly in relative magnitude after includingthe spin. Figure 5.12, which is the zoomed in version of Figure 5.11, shows that except for thispeak, the rest of the plot does not undergo a significant change. This is predictable since the spinmultiplicity mostly affects the states near the Fermi level. It results in widening the band gap,hence, explaining the shift of the first peak to the higher energy. Since the transition happens overa larger energy gap, weakening of the peak magnitude is expected.Figure 5.10: Absorption spectrum of a 3-unit cell (8,0) nanotube segment before (a) and after (b)considering the spin multiplicity.79Figure 5.11: Absorption spectrum of a 5-unit cell (8,0) nanotube segment before (a) and after (b)considering the spin multiplicity.Figure 5.12: Zoomed in version of the absorption spectrum of a 5-unit cell (8,0) nanotube segmentbefore (a) and after (b) considering the spin multiplicity.805.4 Effect of Length on the Absorption SpectraNow that we have obtained the correct spin configuration for each of the nanotube segments, we canproceed to study the effect of length on the absorption spectra of these structures. Our calculationsfor a 1-unit cell nanotube segment in the 0-6 eV range results in the absorption spectrum shownin Figure 5.13 (a). Four distinct peaks are observed. Looking at the absorption spectrum of a3-unit cell segment (Figure 5.13 (b)), we notice a shift of the peaks to the lower energy. Pleasenote that the first peak in the 1-unit cell spectrum (denoted by *) has disappeared. This peak thatis the result of HOMO to LUMO (lowest unoccupied molecular orbital) transition in the 1-unitcell segment is not allowed any longer for the 3 and higher number of unit cells. The same energyshifting trend exists for the peaks of 5 unit cells and 7 unit cells as we can see in Figures 5.13 (c)and (d), respectively. The only exception is for peak A, which shifts to a higher energy as we gofrom 3 to 5 unit cells. As explained before, that is due to the difference in the spin multiplicity ofthese two structures. Peak A continues to shift to lower energies as we go from 5 to 7 unit cellsand higher.As observed, except for what happens for peak A in the transition from 3 to 5 unit cells due tothe change in spin multiplicity, the rest of the shifting trend is in accordance with the “particle in abox” (POB) theory, where the energy levels decrease by increasing the dimensions of the box. Basedon our calculations, the transition energy for the corresponding peaks is inversely proportional tothe nanotube length (E ∝ 1L), as opposed to the squared of this length in the simple particle in abox theory (E ∝ 1L2 in each dimension). Figure 5.14 shows the transition energy versus the inverseof the segment length for peak B, as an example.81Figure 5.13: Optical absorption spectra of (8,0) nanotube segments with 1(a), 3 (b), 5 (c) and 7(d) unit cells.820 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.41.822.22.42.62.831/LTransition Energy (eV)Figure 5.14: Transition energy versus the inverse of length (normalized) for peak B.We were able to calculate the absorption spectra of the (8,0) nanotube segments up to 20 unitcells (640 atoms and ∼85A˚). In Figure 5.15, we compare the absorption spectrum of the 15- and20-unit cell nanotube segments. Although there are differences between the two spectra, the shapeand energy of the main peaks (D, B, and C) are very similar to each other. This suggests thatthese main peaks will most likely appear in the absorption spectrum of a nanotube with infinitelength. Figure 5.16 shows the absorption spectrum for an infinitely long (periodic) nanotube.83Figure 5.15: Optical absorption spectrum of a 15- (a) and 20- (b) unit cell (8,0) nanotube segment.We confirm this by extrapolating our data of transition energy versus inverse of length. Table5.1 shows the comparison between the extrapolated values and the energies resulting from thesimulations of a periodic nanotube. The values of peak B and C are in very good agreementwith the simulation data, confirming the fact that these two peaks appear in the spectra of a longnanotube. The predicted energy for peak D is smaller than the actual value. This may be againrelated to the spin multiplicity of states close to the Fermi level, which disappears in the case of ananotube with infinite length.84Figure 5.16: Optical absorption spectrum of an infinitely long (periodic) (8,0) nanotube.Table 5.1: Transition energies of peaks D, B, and C obtained by simulations and extrapolation ofthe data from nanotube segments.Calculated E(eV ) Extrapolated E(eV )Peak D 1.24 0.92Peak B 1.58 1.61Peak C 2.55 2.585.5 SummaryWe calculated the optical absorption spectra of (8,0) carbon nanotube segments for up to 20unit cells using DFT and the SIESTA software package. We showed that spin multiplicity existsfor zigzag nanotube segments, which is dependent both on the nanotube index and nanotubelength. We also showed that the spin multiplicity can drastically affect the energy and magnitudeof the optical transitions happening close to the Fermi level. As expected, the confinement of thenanotube segments in the longitudinal direction resulted in optical properties different from those85of an infinitely long nanotube. We observed that as the length of the segments increases, peaks ofthe absorption spectra shift to the lower energies, similar to the simple case of the particle in a box.The only exception is in the first peak of the absorption spectra when we go from 3 to 5 unit cells,due to the change in the spin value. The transition energies showed to be inversely proportionalto the nanotube length. By extrapolation and also simulation of an infinitely long nanotube, wepredicted that the three main absorption peaks of a 20-unit cell segment would appear in thespectrum of an infinitely long nanotube. Parts of this work were presented in [107].86Chapter 6CHAPTER 6: Conclusion6.1 SummaryThe optical absorption properties of carbon nanotubes have been widely studied through exper-iments and theoretical calculations [25, 26, 28, 29, 39, 40]. However, a systematic study of theoptical absorption mechanisms based on first-principles methods is needed in order to investigateand properly include the effects of structural and environmental elements on the optical absorptionproperties of these structures.In this thesis, we first performed a detailed study of the dipole moments, optical transitionmechanisms and the selection rules for nanotubes, using first-principles methods and quantummechanical software packages, such as Gaussian 09 and SIESTA. We then investigated the effect ofseveral structural and external factors on the absorption properties.In Chapter 2, we described a method based on Fermi’s golden rule to calculate the band-to-band transition dipole moment of carbon nanotubes. This method allowed for calculation ofthe transition rate in the wide range of IR-VIS-UV and a thorough investigation of the opticalabsorption mechanism and selection rules. Also, by employing first-principles approaches, thecurvature effects were automatically taken into account in our calculations. Based on the obtainedresults, we noted that the conventional selection rules based on the tight-binding calculations neededto be revised and a modified angular momentum number had to be used in order to include allthe possible optical absorptions. We also showed that besides the pi-pi* transitions, pi-σ*, σ-pi*,and σ-σ* transitions, that are neglected in the pi-based tight-binding method, contribute to theoptical absorption spectrum. In fact, σ-σ* transitions generally led to the highest value of thedipole moment magnitude and in some cases, resulted in strong absorption peaks in the low UV87region.The difference between our results and those from the tight-binding method is mostly related tothe effect of the curvature of the nanotube sidewall that is ignored in tight-binding. Especially fornanotubes with small diameters, including the effects of curvature in the calculations is essential.For this reason, we dedicated Chapter 3 of this thesis to a detailed investigation of several effectscaused by the curvature of the nanotube sidewall and the level of improvement that including eachof these effects can bring into the calculations. We showed that the effect of curvature is not onlylimited to the the σ-pi rehybridization, but other effects caused by this curvature, such as the effect ofbond length, need to be taken into account for accurate calculation of the optical absorption spectraof small-diameter nanotubes. The effect of σ-pi rehybridization on the transition energies in ourcalculations was predicted to be as high as∼ 64% and the effect of bond length resulted in changes aslarge as ∼46%. This also showed the importance of the geometry optimization for carbon nanotubeswith small diameter. We calculated the bond length of graphene using different methods, basedon which we recommended DFT calculations with HSEH1PBE functional for accurate calculationof the geometry of carbon nanotubes. The overall effect of σ-pi rehybridization and bond lengthwas shown to depend not only on the diameter, but also on the chirality of the nanotube, wherezigzag nanotubes with n(mod)3=1 showed a different behaviour than the ones with n(mod)3=2 asthe diameter of nanotube increased.The optical properties of 1-dimensional structures such as carbon nanotubes can be sensitive tothe polarization of the incident light. In Chapter 4, we studied the effect of light polarization on theband-to-band transition spectra of nanotubes in different frequency regions and investigated howthis effect could be changed with the nanotube diameter. Once again, we showed that a modifiedangular quantum number needed to be used in the selection rules in order to explain all the allowedtransitions for perpendicular polarization of light.Although the overall absorption spectra were suppressed for perpendicular polarization of lightcompared to the ones with parallel polarization, at certain energies, perpendicular polarizationresulted in peaks with significant magnitudes. Most interestingly, for small-diameter nanotubes inthe UV region, the perpendicular polarization could be absorbed with a higher probability compared88to the parallel polarization, mostly as a result of non-pi-pi* transitions. In the IR/VIS range, on theother hand, we had more probability of absorption for parallel polarization and the ratio of parallelto perpendicular absorption increased as the nanotube diameter decreased.It was also of interest to study the optical absorption spectra of short carbon nanotube segmentsand examine the effect of length on their optical properties. As expected, our calculations for shortnanotube segments resulted in optical absorption spectra different from those with long lengths.Although the energy and magnitude of the peaks changed by increasing the length, we predictedthat some of the absorption peaks of the short nanotube segments still appear in the spectrum ofa long nanotube, i.e. the one that is calculated as a periodic structure. The energy of transitionsshowed to be proportional to the inverse of the nanotube length. Spin multiplicity, which is one ofthe characteristics of the nanotube segments, also showed to affect the optical absorption spectra.6.2 Thesis ContributionsThe work in this thesis provides a fundamental study of the electronic properties, optical band-to-band transition mechanism and selection rules in carbon nanotubes. The absorption propertiesresulting from our first-principles calculations are shown to be in a good agreement with experi-mental data and the improvements over the previously existing theoretical methods are discussed.The effects of structural characteristics such as the diameter, chirality, carbon-carbon bond lengthand length of the nanotube on the absorption properties are studied in detail. It is discussed howand to what extent each of these factors come into play when calculating the transition energiesand transition probabilities. This information can be useful in predicting the optical behaviour ofnanotube-based optoelectronic devices or estimating their optimum wavelength of operation basedon the structural geometry of the nanotube. For example, for transmitter and receiver devicesthat communicate with each other, such as the ones proposed in Figure 1.13, these predictionscan be beneficial in choosing the right nanotubes with suitable geometrical properties prior to thefabrication of devices in order to assure the operation of these devices at the desired frequency.Upon increasing the length of the nanotube segment in our study of the length dependence of theabsorption spectra, we reach a point where the absorption properties become similar to those of the89so-called ”infinitely long” or ”periodic” nanotubes. Due to simplicity and lower computational costs,properties of carbon nanotubes are often calculated for ”infinitely long” nanotubes by applyingboundary conditions to one or a few unit cells of nanotubes. However, in practice, we always dealwith nanotubes with finite lengths. The information from this part of the thesis can be used tosee whether and how we can estimate the properties of nanotube segments based on the propertiescalculated for infinitely long nanotubes. The results of the length dependence of the absorptionspectra can also be used for tuning the working frequency of optoelectronic components made ofshort nanotube segments or in selective tagging of nanotube-based biological transporters [101], aswell as characterizing the single-electron transistors [100].The light polarization dependence of the absorption spectra that is discussed in this thesiscan also be useful in tuning the wavelength of absorption and emission for nanotube-based opticaldevices by using the polarization of light as an external controlling parameter without having tochange the internal characteristics of the device. Again, for devices in Figure 1.13, as an example,varying the orientation of the transmitter with respect to the receiver results in changing thepolarization of the light to be absorbed by the receiver device, hence changing the frequency ofabsorption. Therefore, by only changing the orientation of the two devices with respect to eachother, the operation frequency of the system can be tuned. The polarization dependence can alsobe used in other applications such as sensitive angle detection devices. Our results also show thepossibility of absorption of light in nanotubes beyond the infrared frequencies, which suggests theapplication of these structures for visible and UV absorption.The main contributions of this thesis are summarized as follows:• A first-principles study of the optical absorption in carbon nanotubes• Redefinition of selection rules for interband transitions in carbon nanotubes• Elucidation of the effect of the nanotube sidewall curvature on optical properties• Evaluation of the effects of some of the structural and environmental parameters on theoptical absorption properties of nanotubes• A broad-range study that goes beyond the infrared frequencies (infrared-visible-ultraviolet)906.3 Future DirectionsIn this work, we investigated the effects of some of the main structural and external elements onthe optical properties of carbon nanotubes. Other environmental parameters or changes in thenanotube structure can also affect the optical properties, which their studies fall beyond the scopeof this work. However, the calculation method described in this thesis can be used to evaluate theeffect of these parameters on the optical properties.One of such structural changes is the mechanical deformation in the structure of carbon nan-otubes, which can happen in the form of curvature caused by either local or distributed deformationover the length of the tube (Figure 6.1). Strong local bend strains (kinks) result in local σ-pi rehy-bridization [108]. On the other hand, uniform bends do not lead to significant σ-pi rehybridization[109,110]. Instead, they keep the states delocalized over the entire length of the tube. The changein the electronic structure due to this deformation has been attributed mostly to the C-C bondlength deformation [110]. Both σ-pi rehybridization and C-C bond length effects are extensivelystudied in our work and can be utilized to investigate the optical properties of deformed nanotubes.Figure 6.1: Kinked (a) and bent (b) nanotubes at several deformation angles. Copyright (2003) by theAmerican Physical Society [111].Introducing impurities such as boron or nitrogen in carbon nanotubes, or interactions of nan-otubes with molecules like oxygen or hydrogen can also affect their electronic structures [112–118]and consequently, their optical properties. Although the impact of impurities and defects on opti-cal transitions in nanotubes have not been widely investigated, the change in the band structureresulting from them suggests their potential in tuning the optical properties of nanotubes. Thedetailed study of the electronic structure and optical absorption presented in this work allows for91calculating the absorption spectra and understanding the band-to-band transition mechanism insuch structures.It has been shown that the quasiparticle corrections and electron-hole interactions affect theband gap and play a crucial role in the optical absorption spectra of semiconducting carbon nan-otubes [37, 38]. Although the results presented in this thesis are based on the single-electronband-to-band transitions in carbon nanotubes, the data obtained from our calculations, such as thewave functions and transition energies, can be used to calculate the binding energy and radiativelifetime of excitons within simplifying approximations. The details of these approximations andthe applicability of our method in calculating the binding energy and radiative lifetime of excitonsare explained in Appendix D. An alternative is to account for the excitonic effects by including theself-energy term (within the GW approximation) and solving the Bethe Salpeter equation (BSE),respectively [37, 38]. Implementing these corrections is time-consuming and computationally ex-pensive; the calculations for an (8,0) nanotube with only 32 atoms in its unit cell appear to be astruggle [37]. Our method, on the other hand, can be used for nanotubes with hundreds of atomsin their unit cells.92References[1] F. Kreup, A. Graham, M. Liebau, G. Duesberg, R. Seidel, and E. Unger, “Carbon nanotubesfor interconnect applications,” IEEE International Electron Devices Meeting, pp. 683–686,2004.[2] D. S. Bethune, C. H. Klang, M. S. de Vries, G. Gorman, R. Savoy, J. Vazquez, and R. Bey-ers, “Cobalt-catalysed growth of carbon nanotubes with single-atomic-layer walls,” Nature,vol. 363, no. 6430, pp. 605–607, 1993.[3] S. Iijima and T. 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Louie, “Theory and ab initio calcula-tion of radiative lifetime of excitons in semiconducting carbon nanotubes,” Physical ReviewLetters, vol. 95, no. 24, p. 247402, 2005.106Appendices107Appendix AGaussian 09 Simulation ParametersA.1 Sample Gaussian 09 Input FileBelow is a sample Gaussian input file for calculating the energy of a periodic zigzag (8,0) nanotubeusing the BLYP method and 6-31G basis set.—————————————-%Mem=90MW%CHK=output.chk#P BLYP/6-31G SCF(MaxCyc=1000) IOp(5/98=1) IOp(5/103=50) SCF=Fermi NoSymm GFIn-put formcheckzigzag (8,0) (Periodic, 1 unit cell)0 1C -2.045303 -1.969511 2.492103C -0.935808 -1.247094 3.085338C -2.843557 -1.24711 1.519926C -2.843557 0.18211 1.519926C -2.045303 0.904511 2.492103C -0.935808 0.182094 3.085338C -2.492104 -1.969511 -2.045303C -3.08534 -1.247094 -0.935806108C -3.209037 -1.969542 0.316164C -3.209037 0.904542 0.316164C -1.519927 -1.24711 -2.843557C -1.519927 0.18211 -2.843557C -2.492104 0.904511 -2.045303C -3.08534 0.182094 -0.935806C 2.045304 -1.969511 -2.492104C 0.935807 -1.247094 -3.08534C -0.316164 -1.969542 -3.209037C -0.316164 0.904542 -3.209037C 2.843559 -1.24711 -1.519926C 2.843559 0.18211 -1.519926C 2.045304 0.904511 -2.492104C 0.935807 0.182094 -3.08534C 2.492103 -1.969511 2.045304C 3.085339 -1.247094 0.935809C 3.209038 -1.969542 -0.316162C 3.209038 0.904542 -0.316162C 1.519925 -1.24711 2.843557C 1.519925 0.18211 2.843557C 2.492103 0.904511 2.045304C 3.085339 0.182094 0.935809C 0.316162 -1.969542 3.209036C 0.316162 0.904542 3.209036TV 0.000000 4.303239 0.000000109A.2 Sample Cubegen Input FileBelow is a sample Cubegen input file for calculating several molecular orbitals of a periodic zigzag(8,0) nanotube.—————————————-g09 < C80BLYP.comexport GAUSS MEMDEF=20000000cubegen 0 MO=48 C80BLYP.FChk C80BLYP48.cube -4 ncubegen 0 MO=47 C80BLYP.FChk C80BLYP47.cube -4 ncubegen 0 MO=46 C80BLYP.FChk C80BLYP46.cube -4 ncubegen 0 MO=45 C80BLYP.FChk C80BLYP45.cube -4 ncubegen 0 MO=44 C80BLYP.FChk C80BLYP44.cube -4 ncubegen 0 MO=43 C80BLYP.FChk C80BLYP43.cube -4 ncubegen 0 MO=42 C80BLYP.FChk C80BLYP42.cube -4 ncubegen 0 MO=41 C80BLYP.FChk C80BLYP41.cube -4 ncubegen 0 MO=40 C80BLYP.FChk C80BLYP40.cube -4 ncubegen 0 MO=39 C80BLYP.FChk C80BLYP39.cube -4 ncubegen 0 MO=38 C80BLYP.FChk C80BLYP38.cube -4 ncubegen 0 MO=37 C80BLYP.FChk C80BLYP37.cube -4 ncubegen 0 MO=36 C80BLYP.FChk C80BLYP36.cube -4 ncubegen 0 MO=35 C80BLYP.FChk C80BLYP35.cube -4 ncubegen 0 MO=34 C80BLYP.FChk C80BLYP34.cube -4 ncubegen 0 MO=33 C80BLYP.FChk C80BLYP33.cube -4 ncubegen 0 MO=32 C80BLYP.FChk C80BLYP32.cube -4 ncubegen 0 MO=31 C80BLYP.FChk C80BLYP31.cube -4 n110Appendix BSIESTA Simulation ParametersB.1 Sample SIESTA Input FileBelow is a sample SIESTA input file for calculating the energy of a 1-unit cell (8,0) nanotubesegment using LDA.—————————————-#−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−# FDF for (8,0) nanotube: 1U Optical Absorption#−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−SystemName 80NPHSystemLabel 80NPHNumberOfAtoms 48NumberOfSpecies 2MeshCutoff 270.0 Ry%block ChemicalSpeciesLabel1 6 C2 1 H%endblock ChemicalSpeciesLabel111PAO.EnergyShift 0.050 eVPAO.SplitNorm 0.15%block PAO.BasisC 3n=2 0 20.000 0.0001.000 1.000n=2 1 2 P0.000 0.0001.000 1.000n=3 0 1 P4.5001.000H 10 2 P0.000 0.0001.000 1.000%endblock PAO.BasisAtomicCoordinatesFormat NotScaledCartesianAngLatticeConstant 4.26 Ang112%block LatticeVectors # Lattice vectors in units of LatticeConstant3.521127 0.000000 0.0000000.000000 3.521127 0.0000000.000000 0.000000 3.521127%endblock LatticeVectors%block AtomicCoordinatesAndAtomicSpecies3.13067920 0.00000136 0.29349785 2 H2.92064381 1.21024676 2.08276816 1 C3.17198182 -0.00000125 1.40423392 1 C2.92083213 1.21020715 3.53601850 1 C3.17239469 0.00000086 4.21452565 1 C3.13119221 0.00000023 5.32555621 2 H2.21381807 2.21381494 0.29247760 2 H1.21024880 2.92063989 2.08276479 1 C2.24343905 2.24343736 1.40375415 1 C1.21020542 2.92083248 3.53601661 1 C2.24371826 2.24371824 4.21470034 1 C2.21391620 2.21391442 5.32586904 2 H0.00000102 3.13067428 0.29349630 2 H-1.21024556 2.92063982 2.08276551 1 C0.00000256 3.17197632 1.40423114 1 C-1.21020618 2.92083005 3.53601614 1 C0.00000040 3.17239205 4.21452467 1 C0.00000105 3.13118704 5.32555471 2 H-2.21381689 2.21381655 0.29247567 2 H-2.92064079 1.21024850 2.08276425 1 C-2.24343803 2.24343842 1.40375267 1 C113-2.92083264 1.21020334 3.53601507 1 C-2.24371752 2.24371583 4.21469811 1 C-2.21391475 2.21391214 5.32586646 2 H-3.13067692 0.00000037 0.29349634 2 H-2.92064239 -1.21024647 2.08276515 1 C-3.17197890 0.00000114 1.40423033 1 C-2.92083315 -1.21020599 3.53601492 1 C-3.17239370 -0.00000095 4.21452388 1 C-3.13118822 -0.00000183 5.32555368 2 H-2.21381775 -2.21381576 0.29247716 2 H-1.21024819 -2.92064056 2.08276637 1 C-2.24343959 -2.24343781 1.40375362 1 C-1.21020538 -2.92083082 3.53601798 1 C-2.24371660 -2.24371877 4.21469908 1 C-2.21391398 -2.21391548 5.32586761 2 H-0.00000147 -3.13067477 0.29349725 2 H1.21024618 -2.92064257 2.08276585 1 C-0.00000119 -3.17197773 1.40423187 1 C1.21021051 -2.92083298 3.53601719 1 C0.00000138 -3.17239328 4.21452626 1 C0.00000197 -3.13118946 5.32555587 2 H2.21381715 -2.21381704 0.29247677 2 H2.92064524 -1.21024649 2.08276345 1 C2.24343881 -2.24343766 1.40375349 1 C2.92083627 -1.21020483 3.53601782 1 C2.24371945 -2.24371883 4.21470071 1 C2.21391802 -2.21391638 5.32586961 2 H%endblock AtomicCoordinatesAndAtomicSpecies114##### DFT, Grid, SCFXC.functional LDAXC.authors CASpinPolarized .false.MaxSCFIterations 1000DM.MixingWeight 0.05DM.RequireEnergyConvergenceDM.Tolerance 1.d-5DM.EnergyTolerance 1.d-5 eVDM.NumberPulay 5#####Eigenvalue problem: order-N or diagonalizationSolutionMethod diagonElectronicTemperature 0 K##### Optical PropertiesOpticalCalculation .true.%block Optical.Mesh1 1 1%endblock Optical.MeshOptical.EnergyMinimum 0 eVOptical.EnergyMaximum 12 eVOptical.Scissor 0 RyOptical.Broaden 0.05 eVOptical.NumberOfBands 240115Optical.PolarizationType polarized%block Optical.Vector0.0 0.0 1.0%endblock Optical.Vector##### Output optionsWriteDenChar .true.WriteCoorInitialWriteCoorStepWriteForcesWriteKpointsWriteEigenvaluesWriteKbandsWriteBandsWriteMullikenPop 1WriteCoorXmol .false.WriteMDCoorXmol .false.WriteMDhistoryWriteCoorXmol .false.WriteWaveFunctions .true.COOP.Write .true.116B.2 Optical SimulationsIn order to obtain the optical properties of structures, we need two programs, input.f and optical.f.The following commands need to be executed:input < Filename.EPSIMGwhich results in a file named ”e2.dat”, followed byoptical < e2.datwhich produces several files (absorp coef.out, absorp index.out, conductivity.out, e2.interband.out,epsilon img.out, epsilon real.out, reflectance.out, refrac index.out, e1.interband.out), each contain-ing information about different optical properties. ”epsilon img.out” is used to calculate the opticalabsorption spectra.117Appendix CMATLAB Code for Calculating theDipole MomentBelow is an example of the MATLAB code for calculating the dipole moment between the 3rdvalence and the first 16 conduction bands of a (4,0) nanotube, with the nanotube axis being alongthe y direction.—————————————-close all;clear all;a=0.0441;Nx=227;Ny=222;Nz=227;N=Nx*Ny*Nz;fname=’Ev3’fid = fopen(’C40Ev3.txt’);A = textscan(fid,’%f’);PsiI= A{1}’;118DelPsi=PsiI;Del1=(1/(a))*((diag(sparse((ones(1,Ny))))-(diag(sparse(ones(1,Ny-1)),1))));Del1(Ny,1)=1/a;for ix=1:Nxfor iz=1:NzPsiY= PsiI((ix-1)*Ny*Nz+iz:Nz:(ix-1)*Ny*Nz+(Ny-1)*Nz+iz);DelPsi((ix-1)*Ny*Nz+iz:Nz:(ix-1)*Ny*Nz+(Ny-1)*Nz+iz)=Del1*PsiY’;Yrecord=Del1*PsiY’;endendFPsi={’C40Lumo.txt’,’C40Ec2.txt’,’C40Ec2-2.txt’,’C40Ec3.txt’,’C40Ec3-2.txt’,’C40Ec4.txt’,’C40Ec5.txt’,’C40Ec5-2.txt’,’C40Ec6.txt’,’C40Ec7.txt’,’C40Ec8.txt’,’C40Ec8-2.txt’,’C40Ec9.txt’,’C40Ec10.txt’,’C40Ec10-2.txt’, ’C40Ec11.txt’,’C40Ec11-2.txt’,’C40Ec12.txt’,’C40Ec12-2.txt’,’C40Ec13.txt’,’C40Ec13-2.txt’,’C40Ec14.txt’, ’C40Ec14-2.txt’,’C40Ec15.txt’,’C40Ec15-2.txt’,’C40Ec16.txt’,’C40Ec16-2.txt’};for i=1:27FileName=cell2mat(FPsi(i));fid = fopen(FileName);B = textscan(fid,’%f’);PsiF= B{1}’;I(i)=106 ∗ (a3) ∗ (PsiF ∗DelPsi′);end119Appendix DExcitonic EffectsD.1 Binding Energy of ExcitonsThe exciton wave function can be approximated according to the following:ψ(~re, ~rh) = C∑v,cAvcφc(~re)φ∗v(~rh)e(−ze−zh)2/2σ2 , (D.1)where φc and φv are conduction electron and valence hole states. The sum is performed over thefour band-edge states of c = ±m and v = ±m. The coefficients Avc are determined according tothe symmetry of the orbitals, as described in [119].The energy of the exciton is composed of three terms: direct, exchange, and kinetic energy. Thedirect and exchange energies can be calculated according to the following [119]:〈K〉 =∫ψ∗(~re, ~rh)V (~re − ~rh)ψ(~re, ~rh)d~red~rh= C2∑v,c,v′,c′AvcAv′c′∫φ∗c(~re)φ′c(~re)φv(~rh)φ′∗v (~rh)× e(−ze−zh)2/σ2V (~re − ~rh)d~red~rh,(D.2)where V is the screened and unscreened Coulomb interaction for the direct and exchange energies,respectively. These Coulomb energies can be parametrized according to the Ohno formula [120].Finally the kinetic part of the exciton energy can be obtained by:〈T 〉 =~24m∗σ2, (D.3)120where the exciton reduced mass, m∗, is defined by 1/m∗ = 1/me + 1/mh. The terms for the directand exchange energies have so far been calculated for the first two exciton binding energies andbased on the TB wave functions for simplification. The values calculated, however, deviate from theab-initio calculation of the exciton binding energies because of the approximation and simplificationsconsidered [119]. The direct and exchange energies can be calculated more accurately by performingthe integral in Equation D.2 based on the orbital coefficients obtained from our first-principlescalculations. Although these calculations are beyond the time and scope of this thesis, they can beused in future works for estimating the binding energy of excitons for nanotubes with hundreds ofatoms in their unit cells. The calculations can also be extended beyond the first two excitons sincethe information for other wave functions is available in our method.D.2 Radiative Lifetime of ExcitonsThe decay rate of the excitons in nanotubes has been shown to be dependent on the squaredtransition dipole moments per unit length (µ2aa ), according to the following [121]:γ(Q) =2pie2Ω(0)2~c2µ2aaΩ2(Q)− c2Q2Ω2(Q), (D.4)where Ω is the transition energy and Q is the momentum of the exciton. For a zigzag nanotube withzero momentum of exciton, the above expression for the intrinsic radiative decay rate reduces to2pie2Ω(0)2µ2a~ac2 . The squared dipole moments calculated in our work, along with the transition energiesat the Γ point, can be used to determine the radiative lifetime of the excitons not only for the firsttwo, but also for higher order transitions.121

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