Two-dimensional helium-like atom in a homogeneous magnetic field: Numerically exact solutions

A two-dimensional helium atom (2D-helium) is a real subject for current studies, particularly regarding a hot topic of negatively charged excitons (trions) in semiconducting monolayers. The present study considers a 2D-helium-like atom in a homogeneous magnetic field. We are able to rewrite its Schrödinger equation into a polynomial form concerning dynamic variables. This form is useful for utilizing the algebraic calculation by annihilation and creation operators, enabling the successful application of the Feranchuk-Komarov (FK) operator method to obtain numerically exact solutions (energies and wave functions) for this system. The polynomialization of the equation allows obtaining analytical expressions of all matrix elements, which saves the computational resources significantly. Numerical results for the case without a magnetic field are comparable to other calculations. Moreover, the precise separation of the center-of-mass motion, as provided in this study, leads to an equation for the relative motion of the electrons in a magnetic field, incorporating all previously neglected terms. This result is useful for further study of trions where the electron effective mass is comparable with the hole effective mass. Additionally, we provide a FORTRAN program designed to solve the problems above.