Excitonic States in GaAs/AlxGa1−xAs Quantum Wells: Direct Coulomb Interaction Modeling via Finite Element Electrostatics and Parametric Analysis Under Impurity and Field Effects

This study presents a comprehensive numerical investigation of excitonic states in GaAs quantum wells embedded in AlxGa1-xAs barriers, incorporating the effects of donor and acceptor impurities, external electric and magnetic fields, and varying well widths. The electron and hole wavefunctions are computed by directly solving the Schrödinger equation using the finite element method in cylindrical coordinates, without assuming trial forms. To evaluate the exciton binding energy, the implementation and comparison of two independent approaches were performed: a numerical integration method based on elliptic function corrections, and a novel finite element electrostatic formulation using COMSOL Multiphysics v5.6. The latter computes the Coulomb interaction by solving Poisson's equation with the hole charge distribution and integrating the resulting potential over the electron density. Both methods agree within 1% and capture the spatial and field-induced modifications in excitonic properties. The results show that quantum confinement enhances binding in narrow wells, while donor impurities and electric fields reduce binding via spatial separation of carriers. Magnetic fields counteract this effect by providing radial confinement. The FEM-based electrostatic method demonstrates high spatial accuracy, computational efficiency, and flexibility for complex heterostructures, making it a promising tool for exciton modeling in low-dimensional systems.