A heuristic approach to precisely represent optical absorption and refractive index data for photon energies below, at, and above the band gap of semiconductors: The case of high-purity GaAs. Part I

摘要

The existence of band tails even in nominally undoped and uncompensated GaAs, and the nonparabolic absorption shape above the band gap are the prime examples for the discrepancy between theory and experiments. To overcome this difficulty, we propose a heuristic approach that is guided by the fundamental theoretical aspects and the experimental facts. For this reason, we fit the available absorption data with exponential functions over a photon energy interval from below the band gap to 3 eV. This analytical representation remains well within the known experimental uncertainties over a temperature range from cryogenic to room temperature and beyond. The fitting functions are interpreted to represent the absorption contributions by the band tails, the continuum, and the excitons. This descriptive absorption function implicitly results from a perturbation of the imaginary part of the dielectric function that takes a host of unspecified contributions into account. The real part of the dielectric function due to the high-energy critical points is represented by Lorentzian functions with critical energies taken at 3 and 5 eV. Its square root defines the refractive index due to these critical points, n_h. The refractive index, n, is represented by the sum of n_h and the truncated Kramers-Kroenig transformation of the absorption function. The determination of n_h is made by fitting n to the precise refractive index data of Marple [J. Appl. Phys. 35, 1241 (1964)]. This procedure yields a very precise description ( < 0.1% below and within experimental uncertainty at and above the band gap) of the published refractive index data from 0 to 2.2 eV (except for the narrow reststrahl band near 0.033 eV) over a temperature range similar to that mentioned for the absorption function. We feel confident to predict the refractive index for temperatures as low as 4 K. An analytical expression for the refractive index describes the temperature and energy dependences very precisely below the band gap. The analytic expression is also very precise in the band-gap region for a temperature range estimated from 80 to 400 K. We also compare the form of the absorption function derived from photoluminescence spectra. We find that low quasiequilibrium carrier densities lead to important modifications of the absorption function. These experimental findings together with the fitting procedures serve as a basis for a heuristic theory to calculate the electronic and optical properties under injection.