Hierarchical quantum master equation approaches to nonequilibrium charge transport through single-molecule junctions

In the field of molecular electronics, single-molecule junctions, which comprise a molecule contacted by electrodes, represent the ultimate miniaturization of electronic components in an electric circuit. In this context, experiments on single-molecule junctions have revealed interesting transport phenomena, including transistor- and diode-like behavior as well as negative differential resistance. These transport effects mimic the basic functions of conventional semiconductor devices. Additionally, molecules provide intrinsic functionalities which may result in applications like molecular sensors and machines, memory and spintronic devices, as well as optoelectronics. In contrast to rigid solid-state components, the transport characteristics in molecular junctions are strongly influenced by the intricate interplay of electronic and nuclear (vibrational) degrees of freedom due to their small size and flexible structure. A central task in the field of molecular electronics is to attribute the transport phenomena to the structure and the properties of the molecule in the contact. To this end, theoretical model studies are performed, which facilitate the understanding of this complex nonequilibrium transport problem on the nanoscale. In these studies, the analysis of the average current is complemented by the investigation of current fluctuations in order to obtain detailed knowledge on the underlying transport processes and mechanisms. In this thesis, we investigate electron transport through single-molecule junctions on the basis of numerically exact results. In particular, we focus on transport phenomena induced by electronic-vibrational coupling. To this end, we develop two approaches within the hierarchical quantum master equation (HQME) framework, which differ by the treatment of electronic-vibrational coupling, and thus cover a large spectrum of parameters ranging from the nonadiabatic to the adiabatic regime and including both resonant and nonresonant transport. In particular, we show that the nonequilibrium vibrational excitation significantly influences the transport characteristics of a single-molecule contact in all these regimes by comparison with results where the vibrational mode is equilibrated. In addition, we extend the HQME approach to the evaluation of the full counting statistics, which allows the calculation of higher-order current cumulants beyond the average current. This extension is not limited to the description of vibrationally coupled transport but can be applied to all transport setups which can be treated within the HQME formalism. The numerically exact HQME results are used to benchmark approximate master equation and nonequilibrium Green's function methods. Our findings demonstrate that vibrational nonequilibrium effects play an important role in a wide range of parameters, and thus cannot be neglected in the description of this transport problem. In particular in the nonresonant transport regime, the inelastic cotunneling signal is analyzed for a vibrational mode in full nonequilibrium, revealing a complex interplay of different transport processes and deviations from the commonly used G${}_0$/2-rule of thumb. The inelastic correction to noise also exhibits strong deviations from the prediction for a thermally equilibrated vibration. Additionally, we investigate how the phenomenon of vibrational instability, that is, the increase of current-induced vibrational excitation for decreasing electronic-vibrational coupling, is influenced by level broadening due to molecule-lead coupling as well as broadening of the Fermi distribution caused by temperature. Results obtained for the first two moments suggest that the vibrational excitation is always described by a geometric distribution in the weak electronic-vibrational coupling limit, which confirms our approximate analytic description. Moreover, we study the influence of cotunneling on avalanche-like transport in the regime of strong electronic-vibrational coupling. We find signatures in the current noise reflecting the complex interplay of inelastic cotunneling processes and resonant avalanches.