Adaptive variable-order spherical harmonics expansion of the Boltzmann Transport Equation

The spherical harmonics expansion method provides a deterministic solution method for the Boltzmann Transport Equation for semiconductors. While first-order expansions have been used in early works, higher-order expansions are required for modern scaled-down devices. The drawback of higher-order expansion is that the number of unknowns in the resulting system of equations increases quadratically with the expansion order, leading to high memory consumptions and long simulation times. In this work we show that a considerable number of unknowns can be saved by increasing the expansion order only locally in the simulation domain. Moreover, we propose a scheme that adaptively increases the order starting from a uniform first-order expansion. For the considered n⁺nn⁺-diode, savings in the number of unknowns of up to a factor of five are obtained without sacrificing any accuracy of the numerical solution.