Efficient 2D k-Space Discretization and Non-Linear Interpolation Schemes

Abstract

Within this chapter the primed coordinate system is denoted as (x, y, z) for convenience. The 2D k-space is discretized in polar coordinates, which are denoted as k, ϕ. The foci of this section are the efficient 2D k-space discretization and non-linear interpolation schemes. The k ⋅p solver consumes typically the largest part of the CPU time during transport simulations including the k ⋅p SE [1, 2, 3]. An efficient discretization of the 2D k-space combined with an efficient interpolation is therefore required in order to reduce the number of grid points in the 2D k-space and thus the CPU time spent on the k ⋅p solver. The k ⋅p SE is solved in this work on an orthogonal grid in the 2D (k, ϕ)-space. The discretization of the 2D k-space with polar coordinates is actually not new and has been used by Fischetti et al. [3] before. However, in [3] a linear interpolation of the subband energy based on the coordinates k, ϕ has been used, which requires to solve the k ⋅p SE for many (k, ϕ) grid points in order to get an accurate energy interpolation. In this work, the subband energy and the probability density function for an arbitrary in-plane wave vector are determined by an efficient interpolation which requires a much smaller number of (k, ϕ) grid points. Firstly, the dispersion relation for each radial angle ϕ _n is interpolated with monotonic cubic splines. Secondly, the resultant interpolation coefficients for constant ϕ are interpolated for a constant modulus of the wave vector k _l with harmonics in ϕ.