Analytical and Computational Advances for Hydrodynamic Models of Classical and Quantum Charge Transport

In recent years, substantial advances have been made in understanding hydrodynamic models, both from the standpoint of analytical infrastructure, as well as the parameters which play a decisive effect in the behavior of such models. Both classical and quantum hydrodynamic models have been studied in depth. In this survey paper, we describe several results of this type. We include, for example, well-posedness for both classical and quantum reduced models, and the relaxation drift-diffusion limit as examples of analytical results. As examples of computational results, we include some discussion of effective algorithms, but most importantly, some information gleaned from extensive simulation. In particular, we present our findings of the prominent role played by the mobilities in the classical models, and the role of hysteresis in the quantum models. All models are self-consistent. Included is discussion of recent analytical results on the use of Maxwell’s equations. Benchmark devices are utilized: the MESFET transistor and the n+/n/n+ diode for classical transport, and the resonant tunneling diode for quantum transport. Some comparison with the linear Boltzmann transport equation is included. Acknowledgements: The author is supported by the National Science Foundation under grant DMS-9704458. ∗Department of Mathematics, Northwestern University, Evanston, Il 60208 1