Semiclassical analysis and interpretation of quantum mechanically computed Cu2O exciton spectra

Abstract

In solid state physics the energy dispersion of the electrons is described by the band structure of the solid. When an electron is excited from a valence band into a conduction band it creates an unoccupied state in the valence band which is commonly known as a hole and can be treated as a positively charged quasi particle. Instead of considering all the interactions of the excited electron with the electrons remaining in the valence bands one can equivalently consider only the interaction between the excited electron and the hole. With the Coulomb attraction between those two they can form bound states which are called excitons. This suggests a simple description in analogy to the hydrogen atom which is a good approximation under certain conditions, for example a sufficiently large extension of the exciton such that the crystal background can be treated as a continuum. In this thesis we consider excitons in cuprous oxide which can be described as a hydrogen-like system in a first approximation. Depending on the valence bands and conduction bands involved one can distinguish between different exciton series which are named after the corresponding color of light needed for their excitation. The two series with the lowest excitation energies are therefore called yellow and green series. The yellow series with lowest excitation energy has been investigated intensely in experiments and a hydrogen-like exciton spectrum could be observed. However also deviations from the hydrogen-like behavior have been found which are visible as a fine-structure splitting in the spectrum. Theoretical investigations could attribute those deviations mainly to the complex band structure of cuprous oxide. For a more complete theoretical description of the yellow excitons a present coupling of the yellow and green series has to be taken into account. Therefore the valence bands involved in the green series have to be included. This can be achieved by the introduction of a quasi-spin which couples with the hole spin. The coupling strength of the two series is controlled by the spin-orbit coupling constant. Its value is given by the separation between the valence bands involved in the two series at the Gamma point. In 2014 highly excited yellow exciton states with a principal quantum number of up to n = 25 could be observe by T. Kazimierczuk et al. in experiments. For those large quantum numbers the correspondence principle becomes applicable and a classical or semiclassical treatment should be possible. In this thesis we want to investigate connections between the quantum mechanical spectrum and the associated classical dynamics of the yellow excitons. We include the complex valence band structure involved in the yellow and green series and therefore account for the deviations from a hydrogen-like behavior. Numerical calculation of the quantum mechanical exciton spectrum requires the diagonalization of a Hamiltonian using a large but truncated basis set. Although the agreement to experiments is very good, those calculations do not provide direct information about the associated classical exciton dynamics. For hydrogen-like systems classical orbits forming Kepler ellipses are connected to the Rydberg spectrum by the Bohr-Sommerfeld model. The classical phase space structure is not changing with energy as all bound states can be connected to classical elliptic Kepler orbits. This is not the case for excitons in cuprous oxide. The associated classical dynamics is different for every state in the spectrum as the ratio between the corresponding energy and the spin-orbit coupling constant varies. It is possible to avoid this energy dependence by scaling the coupling constant with the energy such that the ratio between energy and the resulting scaled coupling constant remains constant over the whole spectrum. This leads to a scaled quantum spectrum. For every bound classical dynamics characterized by a given energy there exists a corresponding scaled quantum spectrum. A connection between the scaled quantum mechanical exciton spectra and classical exciton dynamics is established by semiclassical trace formulas. They relate fluctuations of the quantum density of states to a superposition of oscillations with frequencies determined by the period or action of classical periodic orbits. Their amplitudes are related to stability properties of the orbits. We therefore apply a Fourier transform and a technique for high-resolution spectral analysis called harmonic inversion to numerically calculated scaled quantum exciton spectra. The resulting quantum recurrence spectra exhibit peaks at positions given by the action of classical periodic orbits of the associated classical dynamics. Their contribution to the quantum spectra is given by the amplitudes of the peaks. By appropriately including the band structure of cuprous oxide it is possible to treat the excitons classically. With the application of semiclassical theories a semiclassical recurrence spectrum can be obtained from parameters of numerically integrated classical periodic orbits. A comparison of the quantum and semiclassical recurrence spectra shows very good agreement. This allows us to obtain information about the periodic orbits (e.g. shape, action, stability, etc.) contributing to the scaled quantum spectra. This thesis thus provides a deeper insight into the classical exciton dynamics in cuprous oxide and the relation between the fine-structure splitting present in quantum spectra and the associated classical dynamics.