Computation of scaled spectra for excitons in a quantum well in the region from weak to strong confinement

Abstract

When an electron in the semiconductor is excited from the valence band to the conduction band, a quasi-particle is formed, consisting of an excited electron and the remaining positively charged hole. This quasi-particle is called an exciton. Excitons were first introduced in 1930 by J.A. Frenkel and experimentally verified in 1952 by Gross and Karryjew in Cu2O . This thesis focuses on the Wannier-Mott exciton model. Excitons are held together by the Coulomb interaction between the electron and hole and can therefore be described in a hydrogen like model. Recently, extensive research has been devoted to excitons in low-dimensional materials. In particular, there is growing interest in excitons in cuprous oxide. In this context, the confinement can be described by a quantum well in the z-direction with length L. An algorithm originally written by P. Belov computes exciton spectra in a quantum well for a certain width L using a B-spline basis. In this bachelor thesis, the Hamiltonian is scaled with the quantum well width L. The algorithm is then slightly adjusted in order to calculate the quantum well widths for a specified scaled energy E = EL. Furthermore, we will examine the parity of the states, using properties of the B-spline functions. Moreover, the spectra are divided into their even and odd parity subspaces. Using level statistics, the system will be further investigated for chaotic behavior, and it will be examined whether the system, which is not integrable on paper, exhibits integrable behavior for low scaled energy ranges and can therefore be approximated by its integrable analog in the region of weak confinement. Lastly, the scaled spectra are Fourier transformed, and the recurrence spectra will be investigated. First, there will be a general introduction to the quantum mechanical description of the exciton in the potential well and the properties of the scaling. In addition, a brief introduction to the semiclassical treatment of the system is provided and Gutzwiller's trace formula is introduced. Additionally, the method of numerical calculation of spectra, as well as the use of the B-spline basis, is also discussed. Then, universal level statistics will be introduced and discussed. Finally, the system's level statistics will be evaluated and the results discussed, and we will also Fourier transform and analyze recurrence spectra.