Phononen-induzierte Elektronenstreuung und elektrische Leitfähigkeit in verdrehtem zweilagigen Graphen

The isolation of the two-dimensional carbon allotrope graphene in 2004 and the subsequent experimental proof of its remarkable electronic properties have refueled the research interest in low dimensional materials significantly. By now various two-dimensional materials have been discovered, and more recently the possibility of stacking different two-dimensional allotropes to create low dimensional few-layer systems has been discussed. These van-der-Waals coupled multilayers are very promising for future technological applications since (i) the weak interlayer interaction implies rich stacking possibilities for the few layer system, e.g., rotation or translation of layers and (ii) the electronic and transport properties of the multilayer depend sensitively on the stacking order. However, the theoretical treatment of arbitrarily stacked van-der-Waals coupled multilayers presents a significant theoretical challenge. A small rotation of one layer or a network of partial dislocations can lead to systems of essentially mesoscopic size, that fall outside the standard methods of electronic structure theory. Furthermore, such stacking faults can fundamentally change the physical properties of the material, and thus carefully accounting for their presence is essential to the correct treatment of low dimensional systems. In this thesis we describe an effective electronic structure theory, based on a general -- and exact -- mapping of the Slater-Koster tight-binding Hamiltonian to an effective continuum Hamiltonian H(r,p), that we show can successfully treat such systems. While our electronic structure theory can be applied to arbitrary multilayer or single layer structures, in this work we focus on one particular prototype: the twisted bilayer graphene. This material, consisting of two mutually rotated graphene monolayers, is simple in structure, yet captures the rich physics introduced by the stacking degree of freedom to full extent. We deploy an effective Hamiltonian approach that enables us to investigate the electronic structure for, essentially, any twist angles. Above a critical angle of approximately 0.02 degree the low energy electronic spectrum changes significantly as a function of the twist angle, while below this angle it is largely unchanged and resembles a superposition of all possible stacking types of the bilayer. Fermi surface calculations as a function of angle reveal that from a large angle limit in which the two Dirac cones of each layer are preserved there emerges a strong coupling low angle regime possessing a remarkably baroque Fermiology. At intermediate angles a number of isolated topological changes in the Fermi surface can be identified as a function of doping. In order to investigate the impact of this rich electronic structure on the transport properties of twisted bilayer graphene, we consider the scattering of electrons due to phonons within the Boltzmann transport formalism and calculate the resulting in-plane electric conductivity as a function of rotation angle, doping, and temperature. The low-energy (quasi-)acoustic phonon spectrum is derived by treating the two layers of the twisted bilayer as weakly coupled elastic planes. We find that the conductivity depends very sensitively on the twist angle of the layers, as well as on doping of the bilayer system. At large angles (more than 8 degree), the conductivity is -- within a certain energy range -- almost independent of doping, however the electronic decoupling of the layers that is found in the ground state only persists at low enough temperatures T < 10 K. At higher temperatures the layers are re-coupled due to the excitation of phonons. In the small angle regime (less than 2 degree) the conductivity is reduced by two orders of magnitude and varies significantly with doping, reflecting the complexity of the electronic ground state. Finally, we investigate the effect of a layer-perpendicular electric field and find that it strongly affects the transport properties at low temperatures whereas this effect is washed out at room temperature