ITERATIVE SPECTRAL SOLUTION OF THE POISSON-BOLTZMANN EQUATION IN SEMICONDUCTOR-DEVICES

We present an iterative spectral method for solving the coupled Poisson-Boltzmann equation in semiconductor devices. Both theory and preliminary numerical results, in one spatial dimension (three phase-space dimensions), are reviewed. The method relies on a multidimensional Hermite-Gaussian-product expansion of the carrier distribution function's velocity dependence. Pseudotemporal and spatial variables are discretized using finite differences. Stability of the resulting numerical spectral equations is achieved using an added pseudoviscous term. The present formulation allows full inclusion of realistic phonon-scattering probability rates by means of a "collision matrix." Numerical solutions have been obtained for a variety of one-dimensional semiconductor diode problems. These include ballistic, collisional, equilibrium, and nonequilibrium cases. Comparison, where possible, with analytical solutions has confirmed the validity and accuracy of this spectral approach. © 1994 American Institute of Physics.