Design and Modeling of High Temperature Terahertz Quantum Cascade Lasers

The portion of the electromagnetic spectrum between roughly 300 GHz and 10 THz is nicknamed the "THz Gap" because of the enormous difficulty encountered by researchers to devise practical sources covering it. Still, the quantum cascade laser (QCL) has emerged over recent years as the most promising approach to a practical source in the 1-5 THz range. First developed in the higher-frequency mid-IR, where they are now widely available, QCLs were later extended to the THz where a host of greater design challenges awaited. Lasing in QCLs is based on intersubband optical transitions in semiconductor quantum wells, the energy of which can be chosen by design ("bandstructure engineering"). However, simply building a THz optical transition is insufficient; a good design must also produce significant population inversion by the applied cascading electron current, and this requires deep understanding of the transport physics. So far, no THz QCL has operated above the temperature of 200 K, even though the reasons prohibiting high temperature operation are well known.The goal of this Thesis is to put novel ideas for high-temperature operation of THz QCL active regions through rigorous theoretical testing. The central enabling development is a density-matrix-based model of transport and optical properties tailored for use in QCLs, which is general enough that widely varying design concepts can be tested using the same core principles. Importantly, by simulating QCLs more generally, fewer a priori assumptions are required on part of the researcher, allowing for the true physics to emerge on its own. It will be shown that this gives rise to new and useful insights that will help to guide the experimental efforts towards realization of these devices.One specific application is a quantum dot cascade laser (QDCL), a highly ambitious approach in which the electrons cascade through a series of quantum dots rather than wells. Benefits are expected due to the suppression of nonradiative scattering, brought about by the discrete spectrum of electronic states. However, this in turn leads to a highly different physics of transport and effects that are not well understood, even in the case of perfect materials. This work will show that while the benefits are clear, naive scaling of existing QCL designs to the quantum dot limit will not work. An alternative strategy is given based on a revised understanding of the nature of transport, and is put to a test of practicality in which the effects of quantum dot size inhomogeneity are estimated.Another application is to the already existing method of THz difference frequency generation in mid-IR QCLs, which occurs via a difference-frequency susceptibility $\chi^{(2)}$ in the active region itself. For this purpose, the model is extended to enable a coherent and nonperturbative calculation of optical nonlinearities. First, the generality of the method is displayed through the emergence of exotic nonlinear effects, including electromagnetically-induced transparency, in mock quantum-well systems. Then, the modeling concepts are applied to the real devices, where two new and important mechanisms contributing to $\chi^{(2)}$ are identified. Most importantly, it is predicted that the QCL acts as an extremely fast photodetector of itself, giving rise to a current response to the mid-IR beatnote that provides a better path forward to the generation of frequencies below ~2 THz.Finally, the fundamentals of density matrix transport theory for QCLs are revisited to develop a model for conventional THz QCL designs eliminating the usual phenomenological treatment of scattering. The new theory is fully developed from first principles, and in particular sheds light on the effects of scattering-induced electron localization. The versatility of the model is demonstrated by successful simulation of varying active region designs