Quantum theory of the dispersion of the refractive index near the fundamental absorption edge in compound semiconductors

Abstract

A calculation of the real part of the refractive index near the fundamental absorption edge is given, which is based on a quantum mechanical calculation of the complex dielectric constant using the quantum density matrix method. An analytical expression is obtained in terms of experimentally known quantities for a given semiconductor and compared with available experimental data. The band structure of the Kane theory, which applies to direct gap III-V and II-VI compounds, is assumed. The expression obtained is a function of the band-gap energy, the effective electron and heavy hole masses at the bandedge, the spin orbit splitting energy, the carrier concentration for n-type or p-type materials, the temperature, and the frequency of the incident radiation. The temperature dependence occurs through the dependence of the bandgap energy and the effective mass on temperature for degenerate n-type or p-type materials, and there is an additional temperature-dependent factor for nondegenerate materials. The expression also involves the value of n at the absorption edge which is not accessible to measurement. However, an equation for n at the absorption edge can be found in terms of experimentally obtainable values of n near the absorption edge and solved to give the desired quantity. This can then be used to predict the refractive index to a high degree of accuracy over the entire frequency spectrum up to the bandedge. Within limits of the above statement regarding n at the absorption edge, there are no adjustable parameters involved, and this constitutes a significant improvement over previous theories of the refractive index of a semiconductor. In particular, the theory predicts the dispersion near the fundamental absorption edge which has been observed experimentally for a number of III-V and II-VI compounds and enables its precise calculations as a function of frequency. This fact is expected to be of considerable importance in technological applications involving integrated optics. Theory is compared with experimental results for a number of III-V and II-VI compounds.