In this work, the structural, electronic, and optical properties of CdS nanoparticles with sizes up to 4nm have been calculated using density-functional theory (DFT). Inaccuracies in the description of the unoccupied states of the applied density-functional based tight-binding method (DFTB) are overcome by a new SCF-DFTB method. Density-functional-based calculations employing linear-response theory have been performed on cadmium sulfide nanoparticles considering different stoichiometries, underlying crystal structures (zincblende, wurtzite, rocksalt), particle shapes (spherical, cuboctahedral, tetrahedral), and saturations (unsaturated, partly saturated, completely saturated). For saturated particles, the calculated onset excitations are strong excitonic. The quantum-confinement effect in the lowest excitation is visible as the excitation energy decreases towards the bulk band gap with increasing particle size. Dangling bonds at unsaturated surface atoms introduce trapped surface states which lie below the lowest excitations of the completely saturated particles. The molecular orbitals (MOs), that are participating in the excitonic excitations, show the shape of the angular momenta of a hydrogen atom (s, p). Zincblende- and wurtzite-derived particles show very similar spectra, whereas the spectra of rocksalt-derived particles are rather featureless. Particle shapes that confine the orbital wavefunctions strongly (tetrahedron) give rise to less pronounced spectra with lower oscillator strengths. Finally, a very good agreement of the calculated data to experimentally available spectra and excitation energies is found.

In this work, the structural, electronic, and optical properties of CdS nanoparticles with sizes up to 4nm have been calculated using density-functional theory (DFT). Inaccuracies in the description of the unoccupied states of the applied density-functional based tight-binding method (DFTB) are overcome by a new SCF-DFTB method. Density-functional-based calculations employing linear-response theory have been performed on cadmium sulfide nanoparticles considering different stoichiometries, underlying crystal structures (zincblende, wurtzite, rocksalt), particle shapes (spherical, cuboctahedral, tetrahedral), and saturations (unsaturated, partly saturated, completely saturated). For saturated particles, the calculated onset excitations are strong excitonic. The quantum-confinement effect in the lowest excitation is visible as the excitation energy decreases towards the bulk band gap with increasing particle size. Dangling bonds at unsaturated surface atoms introduce trapped surface states which lie below the lowest excitations of the completely saturated particles. The molecular orbitals (MOs), that are participating in the excitonic excitations, show the shape of the angular momenta of a hydrogen atom (s, p). Zincblende- and wurtzite-derived particles show very similar spectra, whereas the spectra of rocksalt-derived particles are rather featureless. Particle shapes that confine the orbital wavefunctions strongly (tetrahedron) give rise to less pronounced spectra with lower oscillator strengths. Finally, a very good agreement of the calculated data to experimentally available spectra and excitation energies is found.

Die vorliegende Arbeit beschäftigt sich mit den Transporteigenschaften defektbehafteter Kohlenstoffnanoröhrchen (CNTs). Als Beispiel werden dabei einfache und doppelte Fehlstellen betrachtet. Der Fokus liegt auf der Berechnung des Transmissionsspektrums und der Leitfähigkeit mit einem schnellen, linear skalierenden rekursiven Greenfunktions-Formalismus, mit dem große Systeme quantenmechanisch behandelt werden können. Als Grundlage wird ein dichtefunktionalbasiertes Tight-Binding-Modell verwendet. Der Einfluss der Defektdichte und des CNT-Durchmessers wird im Rahmen einer statistischen Analyse untersucht. Es wird gezeigt, dass im Grenzfall kleiner Transmission die Leitfähigkeit exponentiell mit der Defektanzahl skaliert. Das System befindet sich im Regime starker Lokalisierung, wobei die Lokalisierungslänge von der Defektdichte und dem CNT-Durchmesser abhängt.

We theoretically investigate the influence of defect-induced long-range deformations in carbon nanotubes on their electronic transport properties. To this end we perform numerical ab-initio calculations using a density-functional-based tight-binding model for various tubes with vacancies. The geometry optimization leads to a change of the atomic positions. There is a strong reconstruction of the atoms near the defect (called 'distortion') and there is an additional long-range deformation. The impact of both structural features on the conductance is systematically investigated. We compare short and long CNTs of different kinds with and without long-range deformation. We find for the very thin (9, 0)-CNT that the long-range deformation additionally affects the transmission spectrum and the conductance compared to the short-range lattice distortion. The conductance of the larger (11, 0)-or the (14, 0)-CNT is overall less affected implying that the influence of the long-range deformation decreases with increasing tube diameter. Furthermore, the effect can be either positive or negative depending on the CNT type and the defect type. Our results indicate that the long-range deformation must be included in order to reliably describe the electronic structure of defective, small-diameter zigzag tubes.

We investigate the electronic transport properties of semiconducting (m, n) carbon nanotubes (CNTs) on the mesoscopic length scale with arbitrarily distributed realistic defects. The study is done by performing quantum transport calculations based on recursive Green's function techniques and an underlying density-functional-based tight-binding model for the description of the electronic structure. Zigzag CNTs as well as chiral CNTs of different diameter are considered. Different defects are exemplarily represented by monovacancies and divacancies. We show the energy-dependent transmission and the temperature-dependent conductance as a function of the number of defects. In the limit of many defetcs, the transport is described by strong localization. Corresponding localization lengths are calculated (energy dependent and temperature dependent) and systematically compared for a large number of CNTs. It is shown, that a distinction by (m − n)mod 3 has to be drawn in order to classify CNTs with different bandgaps. Besides this, the localization length for a given defect probability per unit cell depends linearly on the CNT diameter, but not on the CNT chirality. Finally, elastic mean free paths in the diffusive regime are computed for the limit of few defects, yielding qualitatively same statements.

This thesis theoretically investigates the electronic transport properties of defective carbon nanotubes (CNTs). For the defects the focus is set to vacancy types. The calculations are performed using quantum transport theory and an underlying density-functional-based tight-binding method. Two algorithmic improvements are derived, which accelerate the common methods for quasi one-dimensional systems for the specific case of (i) randomly distributed defects and (ii) long unit cells. With this, the transmission spectrum and the conductance is calculated as a function of the CNT length, diameter, chiral angle, defect type, defect density, defect fraction, and temperature. The diffusive and the localized transport regime are described by extracting elastic mean free paths and localization lengths for metallic and semiconducting CNTs. Simple analytic models for estimating or even predicting the conductance dependence on the mentioned parameters are derived. Finally, the formation of defect-induced long-range deformations and its influence on the conductance are studied.:1 Introduction 2 Fundamentals 2.1 Carbon nanotubes 2.1.1 Structure 2.1.2 Properties 2.1.3 Defects 2.1.4 Synthesis 2.1.5 Characterization 2.1.6 Applications 2.2 Electron structure theory 2.2.1 Introduction 2.2.2 Density functional theory 2.2.3 Density-functional-based tight binding 2.2.3.1 First-order expansion 2.2.3.2 Creation of the parameter set 2.2.3.3 Second-order expansion 2.2.3.4 Usage 2.3 Electron transport 2.3.1 Equilibrium Green’s-function-based quantum transport theory 2.3.2 Transport regimes 2.3.3 Classical derivation: drift-diffusion equation with a sink 2.3.4 Quantum derivation: Dorokhov-Mello-Pereyra-Kumar theory A Improved recursive Green’s function formalism for quasi one-dimensional systems with realistic defects (J. Comput. Phys. 334 (2017), 607–619) A.1 Introduction A.2 Quantum transport theory A.3 Recursive Green’s function formalisms A.3.1 Forward iteration scheme A.3.2 Recursive decimation scheme A.3.3 Renormalization decimation algorithm A.4 Improved RGF+RDA A.5 Performance test A.5.1 Random test matrix A.5.2 Transport through carbon nanotubes A.6 Summary and conclusions B Strong localization in defective carbon nanotubes: a recursive Green’s function study (New J. Phys. 16 (2014), 123026) B.1 Introduction B.2 Theoretical framework B.2.1 Transport formalism B.2.2 Recursive Green’s function formalism B.2.3 Electronic structure B.2.4 Strong localization B.3 Modeling details of the defective system B.4 Results and discussion B.4.1 Single defects B.4.2 Randomly distributed defects B.4.3 Localization exponent B.4.4 Diameter dependence and temperature dependence of the localization exponent B.5 Summary and conclusions Supplementary material C Electronic transport in metallic carbon nanotubes with mixed defects within the strong localization regime (Comput. Mater. Sci. 138 (2017), 49–57) C.1 Introduction C.2 Theoretical framework C.3 Modeling details C.4 Results and discussion C.4.1 Conductance C.4.2 Localization exponent C.4.3 Influence of temperature C.4.4 Conductance estimation C.5 Summary and conclusions D An improved Green’s function algorithm applied to quantum transport in carbon nanotubes (arXiv: 1806.02039) D.1 Introduction D.2 Electronic transport D.3 Decimation technique and renormalization-decimation algorithm D.4 Renormalization-decimation algorithm for electrodes with long unit cells D.4.1 Surface Green’s functions D.4.2 Bulk Green’s functions and electrode density of states D.5 Complexity measure and performance test D.6 Exemplary results D.7 Summary and conclusions E Electronic transport through defective semiconducting carbon nanotubes (J. Phys. Commun. 2 (2018), 105012) E.1 Introduction E.2 Theoretical framework E.3 Modeling details E.4 Results and discussion E.4.1 Transmission and transport regimes E.4.2 Energy dependent localization exponent and elastic mean free path E.4.3 Conductance, effective localization exponent and effective elastic mean free path E.5 Summary and conclusions Supplementary material F Influence of defect-induced deformations on electron transport in carbon nanotubes (J. Phys. Commun. 2 (2018), 115023) F.1 Introduction F.2 Theory F.3 Results F.4 Summary and conclusions 3 Ongoing work 4 Summary and outlook 4.1 Summary 4.2 Outlook 5 Appendix 5.1 Bandstructure of graphene 5.2 Quantum transport theory and Landauer-Büttiker formula References List of figures List of tables Acknowledgement Selbstständigkeitserklärung Curriculum vitae List of publications / Diese Dissertation untersucht mittels theoretischer Methoden die elektronischen Transporteigenschaften von defektbehafteten Kohlenstoffnanoröhren (englisch: carbon nanotubes, CNTs). Dabei werden Vakanzen als Defekte fokussiert behandelt. Die Berechnungen werden mittels Quantentransporttheorie und einer zugrunde liegenden dichtefunktionalbasierten Tight-Binding-Methode durchgeführt. Zwei algorithmische Verbesserungen werden hergeleitet, welche die üblichen Methoden für quasi-eindimensionale Systeme für zwei spezifische Fälle beschleunigen: (i) zufällig verteilte Defekte und (ii) lange Einheitszellen. Damit werden das Transmissionsspektrum und der Leitwert als Funktion von CNT-Länge, Durchmesser, chiralem Winkel, Defekttyp, Defektdichte, Defektanteil und Temperatur berechnet. Das Diffusions- und das Lokalisierungstransportregime werden beschrieben, indem die elastische freie Weglänge und die Lokalisierungslänge für metallische und halbleitende CNTs extrahiert werden. Einfache analytische Modelle zur Abschätzung bis hin zur Vorhersage des Leitwertes in Abhängigkeit besagter Parameter werden abgeleitet. Schlussendlich werden die Bildung einer defektinduzierten, langreichweitigen Deformation und deren Einfluss auf den Leitwert studiert.:1 Introduction 2 Fundamentals 2.1 Carbon nanotubes 2.1.1 Structure 2.1.2 Properties 2.1.3 Defects 2.1.4 Synthesis 2.1.5 Characterization 2.1.6 Applications 2.2 Electron structure theory 2.2.1 Introduction 2.2.2 Density functional theory 2.2.3 Density-functional-based tight binding 2.2.3.1 First-order expansion 2.2.3.2 Creation of the parameter set 2.2.3.3 Second-order expansion 2.2.3.4 Usage 2.3 Electron transport 2.3.1 Equilibrium Green’s-function-based quantum transport theory 2.3.2 Transport regimes 2.3.3 Classical derivation: drift-diffusion equation with a sink 2.3.4 Quantum derivation: Dorokhov-Mello-Pereyra-Kumar theory A Improved recursive Green’s function formalism for quasi one-dimensional systems with realistic defects (J. Comput. Phys. 334 (2017), 607–619) A.1 Introduction A.2 Quantum transport theory A.3 Recursive Green’s function formalisms A.3.1 Forward iteration scheme A.3.2 Recursive decimation scheme A.3.3 Renormalization decimation algorithm A.4 Improved RGF+RDA A.5 Performance test A.5.1 Random test matrix A.5.2 Transport through carbon nanotubes A.6 Summary and conclusions B Strong localization in defective carbon nanotubes: a recursive Green’s function study (New J. Phys. 16 (2014), 123026) B.1 Introduction B.2 Theoretical framework B.2.1 Transport formalism B.2.2 Recursive Green’s function formalism B.2.3 Electronic structure B.2.4 Strong localization B.3 Modeling details of the defective system B.4 Results and discussion B.4.1 Single defects B.4.2 Randomly distributed defects B.4.3 Localization exponent B.4.4 Diameter dependence and temperature dependence of the localization exponent B.5 Summary and conclusions Supplementary material C Electronic transport in metallic carbon nanotubes with mixed defects within the strong localization regime (Comput. Mater. Sci. 138 (2017), 49–57) C.1 Introduction C.2 Theoretical framework C.3 Modeling details C.4 Results and discussion C.4.1 Conductance C.4.2 Localization exponent C.4.3 Influence of temperature C.4.4 Conductance estimation C.5 Summary and conclusions D An improved Green’s function algorithm applied to quantum transport in carbon nanotubes (arXiv: 1806.02039) D.1 Introduction D.2 Electronic transport D.3 Decimation technique and renormalization-decimation algorithm D.4 Renormalization-decimation algorithm for electrodes with long unit cells D.4.1 Surface Green’s functions D.4.2 Bulk Green’s functions and electrode density of states D.5 Complexity measure and performance test D.6 Exemplary results D.7 Summary and conclusions E Electronic transport through defective semiconducting carbon nanotubes (J. Phys. Commun. 2 (2018), 105012) E.1 Introduction E.2 Theoretical framework E.3 Modeling details E.4 Results and discussion E.4.1 Transmission and transport regimes E.4.2 Energy dependent localization exponent and elastic mean free path E.4.3 Conductance, effective localization exponent and effective elastic mean free path E.5 Summary and conclusions Supplementary material F Influence of defect-induced deformations on electron transport in carbon nanotubes (J. Phys. Commun. 2 (2018), 115023) F.1 Introduction F.2 Theory F.3 Results F.4 Summary and conclusions 3 Ongoing work 4 Summary and outlook 4.1 Summary 4.2 Outlook 5 Appendix 5.1 Bandstructure of graphene 5.2 Quantum transport theory and Landauer-Büttiker formula References List of figures List of tables Acknowledgement Selbstständigkeitserklärung Curriculum vitae List of publications

Die vorliegende Arbeit beschäftigt sich mit den Transporteigenschaften defektbehafteter Kohlenstoffnanoröhrchen (CNTs). Als Beispiel werden dabei einfache und doppelte Fehlstellen betrachtet. Der Fokus liegt auf der Berechnung des Transmissionsspektrums und der Leitfähigkeit mit einem schnellen, linear skalierenden rekursiven Greenfunktions-Formalismus, mit dem große Systeme quantenmechanisch behandelt werden können. Als Grundlage wird ein dichtefunktionalbasiertes Tight-Binding-Modell verwendet. Der Einfluss der Defektdichte und des CNT-Durchmessers wird im Rahmen einer statistischen Analyse untersucht. Es wird gezeigt, dass im Grenzfall kleiner Transmission die Leitfähigkeit exponentiell mit der Defektanzahl skaliert. Das System befindet sich im Regime starker Lokalisierung, wobei die Lokalisierungslänge von der Defektdichte und dem CNT-Durchmesser abhängt.:1 Einleitung 2 Physikalische Grundlagen 2.1 Vom Graphen zum Kohlenstoffnanoröhrchen 2.1.1 Geometrische Struktur 2.1.2 Elektronische Eigenschaften 2.2 Schrödingergleichung 2.3 Dichtefunktionaltheorie 2.4 Tight-Binding-Verfahren 2.5 Dichtefunktionalbasiertes Tight-Binding-Verfahren 2.6 Fermienergie, Zustandsdichte und Bandstruktur 2.7 Landauer-Formalismus 2.8 Transportmechanismen und Lokalisierungseffekte 3 Greenfunktions-Formalismus 3.1 Definition der Greenfunktion 3.2 Greenfunktion für die Schrödingergleichung 3.3 Dezimierungstechnik 3.4 Einfacher Algorithmus für periodische Matrizen 3.5 Renormierungs-Dezimierungs-Algorithmus 3.6 Erste Nebendiagonalgreenfunktionsblöcke für periodische Matrizen 3.7 Rekursiver Greenfunktions-Formalismus für endliche Matrizen 4 Elektronische Struktur und quantenmechanischer Transport 4.1 Quantenmechanische Systeme mit Elektrodenkopplung 4.1.1 Reduktion und Lösung der Schrödingergleichung 4.1.2 Elektronische Struktur: Spektralfunktion und Zustandsdichte 4.1.3 Elektronischer Transport: Transmissionsspektrum und Strom 4.2 Quasi-eindimensionale Systeme 4.2.1 Zustandsdichte 4.2.2 Transmissionsspektrum 4.3 Numerischer Aufwand 5 Simulation: Software und Algorithmen 5.1 Atomistix ToolKit 5.2 DFTB-Parametersätze 5.3 LAPACK, BLAS 5.4 Überblick über selbst implementierte Programme 6 Ergebnisse 6.1 Testrechnungen 6.1.1 Genauigkeitstest 6.1.2 Geschwindigkeitstest 6.1.3 Parametersatz 6.1.4 Konsistenztest 6.2 Darstellung der Strukturen 6.3 Transmissionsspektren für einen Defekt 6.4 Transmissionsspektren für zwei Defekte 6.5 Transmissionsspektren für zufällig verteilte Defekte 6.6 Abhängigkeit der Leitfähigkeit von der Defektanzahl 6.7 Abhängigkeit der Leitfähigkeit vom CNT-Durchmesser 6.8 Abschließende Bemerkungen und Vergleich zu anderen Arbeiten 7 Zusammenfassung und Ausblick A Anhänge A.1 Orthogonale Transformation der p-Orbitale A.2 Operatordarstellung der Greenfunktion A.3 Berechnung der Greenfunktionsblöcke A.4 Transmission durch die doppelte Potentialbarriere Tabellenverzeichnis Abbildungsverzeichnis Literaturverzeichnis Danksagung Selbstständigkeitserklärung