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.

二维氦原子（2D-helium）是当前物理学研究的热点课题之一，尤其聚焦于半导体单层材料中带负电激子（trions，三激子）这一前沿研究方向。本研究以均匀磁场中的类二维氦原子体系为研究对象：我们可将该体系的薛定谔方程转化为关于动力学变量的多项式形式，该形式可借助产生算符与湮灭算符开展代数运算，从而成功应用费兰丘克-科马罗夫（Feranchuk-Komarov，FK）算符方法，获得该体系的数值精确解（包括能量与波函数）。对该方程的多项式化处理可推导出所有矩阵元的解析表达式，大幅节省计算资源。无磁场情形下的数值计算结果与同类已有研究的计算结果吻合度良好。此外，本研究通过精准分离质心运动分量，推导出磁场中电子相对运动的控制方程，囊括了此前研究中被忽略的全部项；该结果可用于电子有效质量与空穴有效质量相当的三激子体系的后续研究。最后，本文还提供了用于求解上述问题的FORTRAN程序。