Mathematical Models for Nanoscale Semiconductor Device Engineering

The simulation of manufacturing processes and the electromagnetic activities in today`s semiconductor devices is one of the most demanding subjects in applied mathematics and electronics and of substantial importance in industry. The simulation of the electrical behavior of the devices enables semiconductor manufacturers to estimate the properties of future devices prior to the beginning ofthe production cycle. Highly expensive test runs can be eliminated by deepening the understanding of the physical processes occurring during the operation of the devices. Using this knowledge devices can be optimized in an early phase and the manufacturing processes can be improved with respect to the quality of the resulting devices and manufacturing throughput. As the scaling of semiconductor devices continued into the nano scale regime, quantum effects became increasingly important and are now indispensable for correct simulations and for understanding device behavior. The goal of this research project is to establish a mathematically and physically correct formalism to extend particle based device simulation methods down to the nano scale level, i.e., to 25nm and below. This includes deriving a proper quantum potential framework for effective potential calculation and developing algorithms that capture the physical effects occurring in MOSFET devices scaled down to 1nm gate length. These effects include quantization of motion in the channel, tunneling through the gate oxide, source to drain tunneling, fluctuations associated with inhomogeneities, and carrier-carrier interactions treated quantum mechanically. Finally partial differential equations arising in semiconductor device simulation and especially from transport models will be studied by applying and extending the ideas of symmetry analysis and geometric integration.