Derivative transfer matrix method: Machine precision calculation of electron structure and interface phonon dispersion in semiconductor heterostructures

Abstract

We develop a machine precision transfer matrix method that can be used for a wide range of ordinary differential equations and eigenvalue problems. One of the major drawbacks of transfer matrix approaches is the requirement to sweep parameters in a shooting-like manner, thus lacking in precision in comparison to finite difference methods. We resolve this by finding the zero of the analytically calculated first derivative of the transfer matrix. This allows us to outperform the finite difference approach and compute eigenvalues with high precision and linear numerical complexity. We test the developed model in the following scenarios in semiconductor quantum heterostructures: standard Schrödinger equation under effective mass approximation with parabolic subbands, with two-band nonparabolicity, a $4^{\mathrm{th}}$ order Schrödigner equation that accounts for nonparabolic subbands using the 14 k⋅p approach and calculation of the interface phonon modes dispersion relations and the mode profiles. We show that the developed derivative transfer matrix method outperforms the finite difference method by being able to handle higher spatial resolution and having better time performance. The numerical implementation of our models is available as an open-source package in MATLAB version that can be found on https://github.com/AcaDemicNanoLab/dTMM_Schrodinger.