A direct measurement of the details of charge transport in semiconductor devices is impossible due to the small characteristic lengths of semiconductor devices. Therefore, accurate simulations are essential for understanding the physical effects inside the device and for further improvements of device performance. While the drift-diffusion model has long been the workhorse of semiconductor device simulation, the ongoing miniaturization implies that carrier transport through the device is no longer dominated by scattering with the crystal lattice or other carriers, which invalidates the model.

The loss in accuracy can be compensated by a numerical solution of the Boltzmann transport equation instead of simplified transport models derived from its moments, provided that quantum mechanical effects are sufficiently small. However, the high dimensionality of the Boltzmann transport equation makes direct numerical solutions very difficult. Consequently, the most commonly used method is the stochastic Monte Carlo method, which allows for the inclusion of many details. However, the method leads to excessive execution times and other problems, which are absent in macroscopic transport models. The deterministic numerical solution approach based on spherical harmonics expansions considered throughout this thesis does not suffer from the disadvantages of the Monte Carlo method, yet it provides virtually the same accuracy.

Further improvements of the spherical harmonics expansion methods are proposed in this work. First, a method for the inclusion of carrier-carrier scattering at arbitrary expansion orders is proposed. Then, the structure of the resulting equations is analyzed and a scheme for the efficient storage of the resulting system matrix is proposed. The method is then extended to unstructured grids at arbitrary expansion order. Variable-order expansions and adaptive schemes are proposed in order to keep computational costs under control. Moreover, the trend towards parallel computing architectures is addressed by the development of a parallel preconditioner scheme. With the combination of the presented schemes it is shown for the first time that the simulation of truly three-dimensional field-effect transistors using the spherical harmonics expansion method is feasible. Overall, the improvements suggested in this thesis lead to a reduction of memory requirements by one order of magnitude and execution times by two orders of magnitude.