The energy level spectrum of the yellow excitons in cuprous oxide

Abstract

This article discusses the experimental status achieved in the assessment of the hydrogen-like series of Wannier–Mott excitons, using the semiconductor cuprous oxide, Cu${}_2$O, as material platform. While for other crystals the observed exciton series are limited to low principal quantum numbers $n$ and typically a particular orbital angular momentum $L$, recently a major extension of the number of detected states has been achieved for the so-called yellow exciton series in cuprous oxide. About 60 quantum number combinations $(n,L)$, defining different shells of possible exciton states, were detected in high-resolution one-photon absorption and second harmonic generation spectroscopy, also complemented with application of external electric or magnetic fields. The extension concerns not only the optically active states (the orthoexcitons) that are allowed in different orders of light–matter coupling, but also the states that are optically forbidden due to spin conservation in optical transitions (the paraexcitons). The hydrogen model provides a good overall description of the exciton level spectrum. However, an analysis with sufficient energy resolution reveals significant deviations evidenced by shell splittings, which arise from breaking of the rotational into discrete symmetries in the cubic crystal environment. The resulting fine structure splitting between different shells and within a shell $(n,L)$ is mainly determined by the valence band dispersion in Cu${}_2$O showing pronounced band mixing effects. The corresponding extensions in the exciton Hamiltonian bear similarity to those causing the fine structure splitting in hydrogen, namely a higher order kinetic energy term and a spin–orbit coupling term. In addition, the electron–hole exchange interaction arising for the orthoexcitons and corrections to the dielectric screening provide further contributions to the fine structure splitting. As a consequence, the hydrogen wavefunctions are valid only approximately for describing excitons, being in fact coupled in the exciton envelopes. Despite the broken $L$-degeneracy of the exciton levels, further symmetry protected degeneracies remain, which can be removed by applying external fields. We describe the evolution of the fine structure spectrum in electric and magnetic fields towards Stark ladders and Landau fans, respectively. The optical spectra depend on the crystal orientation relative to the external field in addition to their dependence on the chosen optical axis. Also, the deviations from an isotropic medium become obvious, as the symmetry reduction and the resulting state coupling lead to avoided crossings when levels come into field-induced resonance. Applying the field such that all symmetries are broken, quantum chaos is observed in the state range with principal quantum numbers exceeding $n=5$, as evidenced by observing exclusively anticrossings at state resonances. The synopsis of the data shows that high resolution exciton spectroscopy has reached a level of accuracy that approaches that of atomic systems.