Direct Solution of the Boltzmann Transport Equation in Nanoscale Si Devices
Predictive semiconductor device simulation faces a challenge these days. As devices are scaled to nanoscale lengths, the collision-dominated transport equations used in current device simulators can no longer be applied. On the other hand, the use of a better, more accurate Boltzmann Transport Equation (BTE) is hampered by the fact that it is a complicated integro-differential 6-dimensional kinetic equation and is extremely difficult to solve. Previous works on solving the BTE have used either a stochastic method or an approximate method, both of which do not have the suitable properties for practical device simulation. Therefore, this work describes the first direct numerical solution of the BTE for semiconductors that can be used for practical device simulation. This is done by using powerful mathematical techniques to discretise the BTE in energy and angle without making any approximations about the angular shape of the distribution function or the collision integral. Such a direct discretisation results in a very large matrix equation,with N = 106–107 unknowns. In order to address the need for efficient and fast solutions, this work also reports the first application of a preconditioned iterative method (GMRES) to the BTE. This method is not only fast (on the order of N1.2) but also has low memory requirements because it does not require explicit storage of the matrix elements. The technique developed in this work is also highly suitable for self-consistent device simulations because it shows smooth and stable convergence when coupled to the Poisson equation. Finally, this method is applied to study transport in two representative nanoscale devices — a one-dimensional 50nm n+-p-n+ diode and a two-dimensional 50nm ultra-thin body dual-gate nMOSFET. The report ends with a summary and a discussion of possible future improvements in this field.
Purdue University, West Lafayette, IN
Kausar Banoo received his PhD from Purdue University in December 2000.
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