Mathematical Sciences: Classical and Quantum Fluid Transport Models for Semiconductor Devices

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Jerome The investigator studies semiconductor devices, focusing on the simulation and mathematical analysis of the hydrodynamic and quantum hydrodynamic models for such devices. Microwave and multi-species models, and the decisive role of friction, are emphasized in the classical studies, and hysteresis, bistable-state analysis, are emphasized in the quantum studies. The rationale is to establish a way of dealing with various aspects of hot electron effects in classical structures, and the effects of tunneling and repulsion in heterogeneous structures, or compound semiconductors. The algorithms to be employed for the simulations are shock capturing algorithms of finite difference and finite element type. The multi-species models permit simulation of the circuit involving the Gunn diode, and its three valley electron carriers, by new hydrodynamic models. These are compared with the computationally intensive results obtained by the Urbana group, via Monte Carlo simulation of the Boltzmann equation. Mathematical and computer models of semiconductor devices have increasingly necessitated the incorporation of two types of effects: (1) energetics, often called the hot electron effect, and (2) small scale, or quantum mechanical, effects. This proposal describes two important models, the first based upon classical, and the second upon quantum, physics. Although the basic physical laws have been understood for at least seventy-five years, the efforts to apply these laws to semiconductor devices in a way that allows for reasonably efficient modeling and simulation represents an enormous challenge to scientists, engineers, and mathematicians. An example of classical modeling is the Gunn diode, which is acknowledged to be of critical importance in microwave manufacturing applications; such devices appear in the circuitry of the automobile, for example. Its function depends upon delicate oscillations of current and voltage, which must be captured by computer models if they are to be of any usefulness. The principal investigator is open to any collaboration in the manufacturing sector which might result from this work. A possible application of the quantum studies is the development (by others) of logic gates in circuits that do not operate according to binary "off-on" laws, but allow for multiple states of occupancy. The work described is highly collaborative and interdisciplinary, including electrical engineers from the University of Illinois at Urbana. Computational mathematicians from Arizona State and Brown Universities, and the University of Minnesota, and mathematical analysts from the Courant Institute and from Stanford University, are involved. The results of past work have been regularly presented at workshops on computational electronics, and published in journals of electrical engineering and computational physics. This practice is continued.