Very large scale integration is the forthcoming design in semiconductor technology. This implies that in modern integrated electron devices the scale length of individual components becomes comparable with the distance between successive carrier interactions with the crystal, and the well-established drift-diffusion models describing the carrier transport lose their accuracy [Markowich et al. (1990)]. Consequently, to cope with high-field and sub-micron phenomena, Boltzmann transport equations (BTEs) must be applied [Ferry (1991)]. In femtosecond laser experiments non-equilibrium longitudinal-optical (LO) phonons 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 [Vaissiere et al. (1992); Vaissiere et al. (1996)].

Deterministic as well as stochastic procedures can be considered as solution approaches to these extremely sophisticated equations, the so-called Bloch-Boltzmann-Peierls (BBP) equations. So far, mainly stochastic solution methods have been applied to solve the Boltzmann transport equations [Jacoboni and Lugli (1989); Jacoboni and Reggiani (1983); Fiscetti (1991); Jungemann and Meinzerhagen (2003)]. Although Monte Carlo (MC) methods combined with drift-diffusion or hydrodynamic models can be considered as an approved method for device simulation, purely deterministic procedures are characterized by some essential advantages. Standard Monte Carlo methods are unable to resolve almost empty regions of two-dimensional devices, e.g., areas close to the gate of a MESFET, while deterministic approaches can do. Hence, deterministic results should be used as benchmarks for Monte Carlo, hydrodynamic or drift-diffusion results, even though they are not competitive with Monte Carlo...