Accuracy of transfer matrix approaches for solving the effective mass Schr\"{o}dinger equation

TL;DR

This paper evaluates various transfer matrix methods for solving the effective mass Schrödinger equation in one dimension, comparing their accuracy and computational efficiency, especially highlighting the advantages of a symmetrized approach over traditional methods.

Contribution

It introduces and assesses a symmetrized transfer matrix approach, demonstrating its comparable accuracy to the Airy function method with lower computational cost and improved numerical stability.

Findings

Symmetrized transfer matrix approach matches Airy function accuracy.

Symmetrized method reduces numerical issues and computational cost.

Validated with analytical models and self-consistent simulations.

Abstract

The accuracy of different transfer matrix approaches, widely used to solve the stationary effective mass Schr\"{o}dinger equation for arbitrary one-dimensional potentials, is investigated analytically and numerically. Both the case of a constant and a position dependent effective mass are considered. Comparisons with a finite difference method are also performed. Based on analytical model potentials as well as self-consistent Schr\"{o}dinger-Poisson simulations of a heterostructure device, it is shown that a symmetrized transfer matrix approach yields a similar accuracy as the Airy function method at a significantly reduced numerical cost, moreover avoiding the numerical problems associated with Airy functions.