Impact of the valence band structure of Cu$_{2}$O on excitonic spectra

Abstract

We present a method to calculate the exciton spectra of all direct semiconductors with a complex valence band structure. The Schrödinger equation is solved using a complete basis set with Coulomb Sturmian functions. This method also allows for the computation of oscillator strengths. Here we apply this method to investigate the impact of the valence band structure of cuprous oxide $\left(\mathrm{Cu_{2}O}\right)$ on the yellow exciton spectrum. Results differ from those in [Phys. Rev. Lett. 115, 027402 (2015)]; the differences are discussed and explained. The difference between the second and third Luttinger parameter can be determined by comparisons with experiments, however, the evaluation of all three Luttinger parameters is not uniquely possible. Our results are consistent with band structure calculations. Considering also a finite momentum $\hbar K$ of the center of mass, we show that the large $K$-dependent line splitting observed for the $1S$ exciton state [Phys. Rev. Lett. 91, 107401 (2003)] is not related to an exchange interaction but rather to the complex valence band structure of $\mathrm{Cu_{2}O}$.