A detailed introduction into fundamental transport models was presented, especially the quantum ballistic transport and the density matrix formulation. Special focus was laid on the semiclassical transport model and the stationary charge transport described by the PME.
Thereafter, we presented numerical solutions of the Schrödinger equation based on Robin boundary conditions in order to obtain current carrying states. Special attention has been turned to calculations of the fundamental quantities like the tunneling current density. The method has been applied to several heterostructure designs and the determined results have been compared to simulations using nonequilibrium Greens functions and the TsuEsaki model and also experimental measurements. Good numerical agreement has been obtained. In addition, we have investigated the asymptotic behavior of the wave functions. A proof that the solution of the Robin problem converges to the solution of the Dirichlet problem has been presented. This behavior has been also illustrated numerically by comparing simulated wave functions as well as the dipole matrix elements of the Robin and Dirichlet boundary value problems in situations with high energy levels. It has been shown that the tunneling current vanishes with decreasing electric field, which can be explained in terms of the transmission coefficient. The presented results indicate the necessity to treat QCLs as open quantum systems with non-selfadjoint boundary conditions.
In this thesis, I have described a semi-classical approach to model transport in QCLs. Over the course of this work, a simulator has been developed which solves the PME by means of a Monte Carlo method. As a prototypical example, we simulated a GaAs based QCL in the THz region. We have investigated the current density in dependence of the applied bias, and the electron distribution functions of the individual subbands have been computed. In general, it is demonstrated that the developed Monte Carlo simulator is an efficient approach for simulating stationary charge transport in quantum cascade structures governed by the PME. Special focus was laid on the study of intervalley scattering effects on the carrier dynamics. In particular, the simulation results indicate that the Γ-X electron transfer plays a considerable role and highlights the importance of intervalley charge transport for QCL design considerations. It has been shown that the modification of the Al content and the width of the collector barrier in a GaAs/AlGaAs QCL can yield a significant increase in current density when considering Γ-X intervalley scattering. This can be explained by an increase of the overlap between the upper X-state and the lower Γ-state of two adjacent stages, which is particularly important for QCL design considerations. Finally, a comparison of simulation results with measurements for a recently developed InGaAs/GaAsSb QCL has been presented. We were able to observe dominant impact due to optical phonon scattering and also a non negligible effect due to alloy scattering.
However, more remains to be done for future research. A large number of states are involved in transport, especially for THz QCLs. Subbands are close in energy and strongly coupled by Coulomb scattering which can play an important role [130]. For more precise simulations, a model for electron-electron scattering has to be added. Moreover, the semi-classical picture is accurate only for strong coupling and the Boltzmann-like formalism is sufficient for stationary states, but phase coherent phenomena are disregarded. The scattering induced phase coherence can be described by a density matrix formulation of the quantum transport theory. Hence, it will be important to incorporate the subband dependent dephasing which can yield a more accurate description of the electron transport. Furthermore, the parameters of InGaAs based material systems are not well characterized. This may be significant when adapting the Monte Carlo simulator to InGaAs based QCLs, which requires careful attention.