In this thesis, the electronic transport properties of mesoscopic condensed matter systems based on graphene are investigated by means of numerical as well as analytical methods. In particular, it is analyzed how the concepts of quantum interference and disorder, which are essential to mesoscopic devices in general, are affected by the unique electronic and transport properties of the graphene material system. We consider the famous Aharonov–Bohm effect in ring-shaped transport geometries, and, besides providing an overview over the recent developments on the subject, we study the signatures of fundamental phenomena such as Klein tunneling and specular Andreev reflection, which are specific to graphene, in the magnetoconductance oscillations. To this end, we introduce and utilize a variant of the well-known recursive Green’s function technique, which is an efficient numerical method for the calculation of transport observables in effectively non-interacting open quantum systems in the framework of a tight binding model. This technique is also applied to study the effects of a specific kind of disorder, namely short-range resonant scatterers, such as strongly bound adatoms or molecules, that can be modeled as vacancies in the graphene lattice. This numerical analysis of the conductance in the presence of resonant scatterers in graphene leads to a non-trivial classification of impurity sites in the graphene lattice and is further substantiated by an independent analytical treatment in the framework of the Dirac equation. The present thesis further contains a formal introduction to the topic of non-equilibrium quantum transport as appropriate for the development of the numerical technique mentioned above, a general introduction to the physics of graphene with a focus on the particular phenomena investigated in this work, and a conclusion where the obtained results are summarized and open questions as well as potential future developments are highlighted. / In dieser Arbeit werden die elektronischen Transporteigenschaften von Graphen-basierten mesoskopischen Festkörpersystemen mittels numerischer und analytischer Methoden untersucht. Im Besonderen wird analysiert, wie Konzepte von Quanteninterferenz und Unordnung, die eine wesentliche Rolle für mesoskopische Systeme spielen, durch die einzigartigen elektronischen und Transporteigenschaften von Graphen beeinflusst werden. Wir betrachten den berühmten Aharonov-Bohm-Effekt in ringförmigen Transportgeometrien, geben einen Überblick über die Entwicklung dieses Themas in den letzten Jahren und befassen uns mit den charakteristischen Merkmalen, die fundamentale Phänomene wie Klein-Tunneln und gerichtete Andreev-Reflexion, welche spezifisch für Graphen sind, in den Magnetooszillationen der elektrischen Leitfähigkeit aufweisen. Dazu führen wir eine Variante der Methode der rekursiven Greenschen Funktionen ein, die ein effizientes numerisches Verfahren zur Berechnung von Transportobservablen in effektiv nicht-wechselwirkenden, offenen Quantensystemen im Rahmen eines „tight binding“-Modells darstellt. Diese Methode wird desweiteren zur Erforschung eines speziellen Typs von Unordnung herangezogen, nämlich kurzreichweitiger, resonanter Streuzentren wie stark gebundene Adatome oder Moleküle, die als Fehlstellen in der Graphen-Gitterstruktur modelliert werden können. Diese numerische Analyse der elektrischen Leitfähigkeit bei Anwesenheit resonanter Streuzentren in Graphen führt zu einer nicht-trivialen Klassifizierung von Fremdatom-Gitterplätzen innerhalb des Graphen-Gitters und wird durch eine unabhängige analytische Behandlung im Rahmen der Dirac-Gleichung bekräftigt. Die vorliegende Arbeit enthält weiterhin eine formale Einführung in das Thema des Nichtgleichgewichts-Quantentransports, wie es für die Entwicklung der genannten numerischen Methode dienlich ist, eine allgemeine Einführung in die Physik von Graphen mit Fokus auf die speziellen Aspekte, die in dieser Arbeit untersucht werden, sowie eine abschließende Darstellung, in der die erhaltenen Ergebnisse zusammengefasst und offene Fragen sowie mögliche zukünftige Entwicklungen hervorgehoben werden.

In this thesis, we investigate the quantum transport properties of disordered three dimensional topological insulator (3DTI) nanostructures of BiSe and BiTe in detail. Despite their intrinsic bulk conductivity, we show the possibility to study the specific transport properties of the topological surface states (TSS), either with or without quantum confinement. Importantly, we demonstrate that unusual transport properties not only come from the Dirac nature of the quasi-particles, but also from their spin texture. Without quantum confinement (wide ribbons), the transport properties of diffusive 2D spin-helical Dirac fermions are investigated. Using high magnetic fields allows us to measure and separate all contributions to charge transport. Band bending is investigated in BiSe nanostructures, revealing an inversion from upward to downward bending when decreasing the bulk doping. This result points out the need to control simultaneously both the bulk and surface residual doping in order to produce bulk-depleted nanostructures and to study TSS only. Moreover, Shubnikov-de-Haas oscillations and transconductance measurements are used to measure the ratio of the transport length to the electronic mean free path ltr/le. This ratio is measured to be close to one for bulk states, whereas it is close to 8 for TSS, which is a hallmark of the anisotropic scattering of spin-helical Dirac fermions. With transverse quantum confinement (narrow wires or ribbons), the ballistic transport of quasi-1D surface modes is evidenced by mesoscopic transport measurements, and specific properties due to their topological nature are revealed at very low temperatures. The metallic surface states are directly evidenced by the measure of periodic Aharonov-Bohm oscillations (ABO) in 3DTI nanowires. Their exponential temperature dependence gives an unusual power-law temperature dependence of the phase coherence length, which is interpreted in terms of quasi-ballistic transport and decoherence in the weak-coupling regime. This remarkable finding is a consequence of the enhanced transport length, which is comparable to the perimeter. Besides, the ballistic transport of quasi-1D surface modes is further evidenced by the observation of non-universal conductance fluctuations in a BiSe nanowire, despite the long-length limit (L > ltr) and a high metallicity (many modes). We show that such an unusual property for a mesoscopic conductor is related to the limited mixing of the transverse modes by disorder, as confirmed by numerical calculations. Importantly, a model based on the modes' transmissions allows us to describe our experimental results, including the full temperature dependence of the ABO amplitude.

In this thesis, we investigate the quantum transport properties of disordered three dimensional topological insulator (3DTI) nanostructures of BiSe and BiTe in detail. Despite their intrinsic bulk conductivity, we show the possibility to study the specific transport properties of the topological surface states (TSS), either with or without quantum confinement. Importantly, we demonstrate that unusual transport properties not only come from the Dirac nature of the quasi-particles, but also from their spin texture. Without quantum confinement (wide ribbons), the transport properties of diffusive 2D spin-helical Dirac fermions are investigated. Using high magnetic fields allows us to measure and separate all contributions to charge transport. Band bending is investigated in BiSe nanostructures, revealing an inversion from upward to downward bending when decreasing the bulk doping. This result points out the need to control simultaneously both the bulk and surface residual doping in order to produce bulk-depleted nanostructures and to study TSS only. Moreover, Shubnikov-de-Haas oscillations and transconductance measurements are used to measure the ratio of the transport length to the electronic mean free path ltr/le. This ratio is measured to be close to one for bulk states, whereas it is close to 8 for TSS, which is a hallmark of the anisotropic scattering of spin-helical Dirac fermions. With transverse quantum confinement (narrow wires or ribbons), the ballistic transport of quasi-1D surface modes is evidenced by mesoscopic transport measurements, and specific properties due to their topological nature are revealed at very low temperatures. The metallic surface states are directly evidenced by the measure of periodic Aharonov-Bohm oscillations (ABO) in 3DTI nanowires. Their exponential temperature dependence gives an unusual power-law temperature dependence of the phase coherence length, which is interpreted in terms of quasi-ballistic transport and decoherence in the weak-coupling regime. This remarkable finding is a consequence of the enhanced transport length, which is comparable to the perimeter. Besides, the ballistic transport of quasi-1D surface modes is further evidenced by the observation of non-universal conductance fluctuations in a BiSe nanowire, despite the long-length limit (L > ltr) and a high metallicity (many modes). We show that such an unusual property for a mesoscopic conductor is related to the limited mixing of the transverse modes by disorder, as confirmed by numerical calculations. Importantly, a model based on the modes' transmissions allows us to describe our experimental results, including the full temperature dependence of the ABO amplitude.