Spherical Harmonics Expansion and Multi-Scale Modeling

Abstract

AbstractBy far, most numerical device simulations are still based on the drift-diffusion (DD) model, owing to its versatility, numerical efficiency, and robustness. However, the DD model is of limited physical accuracy for nanoscale devices and hot carrier effects. Advanced physical modeling in the regime of semi-classical transport must therefore be based on the Boltzmann transport equation (BTE), and the typical solution method for this equation is a Monte-Carlo (MC) algorithm owing to its small memory requirements. Unfortunately, this popular solution method makes the BTE model much less versatile than the DD approach, as, for example, no AC and noise analysis in the small-signal framework based on the Jacobian of the numerical model is available if the MC method is used. Moreover, evaluating the complete transport of carriers in a device by the BTE model is always several orders of magnitude less efficient in terms of CPU time than the application of the DD model. Thus, there is a clear demand to increase the versatility of the BTE model and to enhance the physical accuracy of the DD model without compromising its numerical efficiency and robustness. Solving the BTE model with the spherical harmonic expansion (SHE) method allows the calculation of the full Jacobian of the discrete system and to establish AC and noise analysis in the small-signal framework, thus making the BTE model much more versatile. Moreover, sequential and concurrent multi-scale methods using microscopic models such as the BTE model where necessary and macroscopic models such as the DD model otherwise are well suited to enhancing the physical accuracy of the DD model without compromising its efficiency and robustness. In addition, using the SHE method allows many of the discretization methods and principles previously developed for the DD model to be applied for the discretization of the BTE model. Therefore, a short overview of the historic development of discretization methods and principles, finally making the DD model as efficient and robust as it is today, is added as well.KeywordsMulti-scale simulationBoltzmann transport equationSpherical harmonics expansionMonotonicity-preserving discretizationHierarchical drift-diffusionHydrodynamic equations