Modeling and Simulation of Electrostatics of Ge$_{\text{1-x}}$Sn$_{\text{x}}$ Layers Grown on Ge Substrates

This work introduces a comprehensive simulation tool that provides a robust 1D Schrödinger <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">–</b> Poisson solver for modeling the electrostatics of heterostructures with an arbitrary number of layers, and non-uniform doping profiles along with the treatment of partial ionization of dopants at low temperatures. The effective masses are derived from the first-principles calculations. The solver is used to characterize three Ge<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$_{\text{1-x}}$</tex-math></inline-formula>Sn<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$_{\text{x}}$</tex-math></inline-formula>/Ge heterostructures with non-uniform doping profiles and determine the subband structure at various temperatures. The simulation results of the sheet carrier densities show excellent agreement with the experimentally extracted data, thus demonstrating the capabilities of the solver.