Numerical Methods for the Solution of the Boltzmann Transport Equation in Semiconductors

The subject of this research is the development of an original numerical method for the approximate solution of the Boltzmann Transport Equation in silicon devices. The main motivation for this is the necessity to reach a trade-off between minimizing cpu-time consumption in the simulation of two or three-dimensional devices, and the necessity of knowing the carrier distribution function at high energies. First, a technique based on the solution of the spherical-harmonics expansion of the Boltzmann Transport Equation has been studied. This method has been applied to the spatially-uniform and one-dimensional case with satisfactory results. The simulations were able to reproduce the measured collector-current multiplication factor due to impact ionization in a BJT. Then the model has been extended to the two-dimensional case and it has been applied to the simulation of {\sl n}-MOS transistors. The results have pointed out aspects of the electron transport which cannot be detected with other kind of analyses. The model has been extended to incorporate the semiconductor full-band structure. The information about the bands is contained inside the density of states and group velocity. This description allows to represent more accurately the dynamics of high-energy electrons. The same description has been adopted for the valence band. An impact-ionization model consistent with the silicon full-band structure has been developed. It is a multi-threshold model which better describes the available experimental data. A new numerical scheme which allows to compute the solution in the spatially 2-dimensional case has been developed. This scheme has a computational cost 5 times lower than the original one and guarantees the same accuracy of the solution.

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