Advanced Multigroup Methods for Electron Phonon Kinetics

Research project P 14669 Advanced Multigroup Methods for Electron-Phonon Kinetics Ferdinand SCHÜRRER 09.10.2000 Very large scale integration is the forthcoming design in semiconductor technology. In many cases the well established approach of drift-diffusion models describes very efficiently the carrier transport in semiconductors. However, in modem enhanced integrated electron devices these models lose their accuracy because the scale length of individual components becomes comparable with the distance between two successive carrier interactions with the crystal. In order to cope with high-field and submicron phenomena, Boltzmann like transport equations have to be applied. Moreover, in ferntosecond laser experiments non-equilibrium longitudinal-optical phonon distributions have been found to affect strongly the electron distribution function. Thus, for a unified treatment one has also to include kinetic equations for the evolution of phonons in a realistic description. Instead of a purely microscopic approach by means of the costly Monte Carlo methods, we intend to develop a mesoscopic strategy to deal with the dynamics of electrons and phonons. Our approach is based on the Bloch- Boltzinann-Peierls (BBP) kinetic equation, which is the most complete description of such systems at this level. The only needed parameters to be adjusted are the cross sections, the energy band structure of the electrons, the dispersion relation of the phonons and the impurity profile. This approach requires us to characterise rigorously the equilibria of the BBP equations and to investigate their uniqueness and stability in terms of a suitable Lyapunov functional. However, our main aim is to find methods to solve self-consistently the BBP equations coupled with the Poisson equation which governs the electric potential. We will treat both space homogeneous and space dependent regimes. To this end, we develop an overlapping multigroup formalism for phonons and electrons. This method has the advantage of taking into account external forces in a natural manner. However, instead of working with rigid shape functions, we resort to a smooth approximation of the energy dependence of the distribution function. This multigroup formalism combined with a spherical harmonics expansion with respect to the angle variables results in a system- of hyperbolic PDE. Since high electric fields can cause spatial discontinuities of the distribution function, we plan to work with adaptive local stencils (ENO) or convex combinations of contributions from local stencils (WENO). Time discretization can be performed via total variation diminishing Runge Kutta schemes. The advanced treatment of the energy dependence of the distribution functions for electrons and phonons (in contrast to the full moment method) plays a key role in our procedure. It allows for an accurate description of far- from-equilibrium. phenomena.