While in the early years of the semiconductor industry, macroscopic models such as the drift-diffusion model or the hydrodynamic model were sufficient for device simulation, accurate simulations of modern nanoscale devices require the use of more precise models. As long as quantum mechanical effects are not dominant, the microscopic electron transport may be described by the Boltzmann Transport Equation (BTE), which may be considered to be the most appropriate semi-classical description of electrons in a semiconductor.
The most commonly used technique for the numerical solution of the BTE is the Monte Carlo method, primarily because it is very flexible and allows the incorporation of modeling details such as complicated band structures and scattering processes. The main disadvantage of the Monte Carlo method is its computational cost, especially when attempting to reduce the statistical noise in the low density tails of the distribution function.
As an alternative to the stochastic Monte Carlo method, the deterministic Spherical Harmonics Expansion (SHE) method of first order was introduced in the early 1990s. The major challenge for SHE is the huge memory requirement on state-of-the-art computers, already reported for two-dimensional devices. In order to apply SHE to real three-dimensional devices, we develop algorithmic improvements that reduce run times and memory requirements of SHE.
In current implementations, most of the required memory is used for the storage of the global system matrix. We developed a method to reduce the memory required by the system matrix such that most of the memory is actually consumed by the unknowns of the system. For a given amount of memory, our new method allows handling of systems that are two orders of magnitude larger compared to existing approaches. With these savings at hand, our method paves the way for an application of SHE to three-dimensional simulations.