Complex Airy analysis of photoreflectance spectra for III-V semiconductors.

Abstract

We present a detailed analysis of photoreflectance (PR) spectra of semiconductors using complex Airy functions and their derivatives. We demonstrate that photoreflectance spectra can be treated in terms of a single complex Airy function with an energy-dependent broadening parameter. We show analytically and numerically that this complex Airy PR treatment is functionally equivalent within field conditions appropriate for PR to the model of R. N. Bhattacharya, H. Shen, P. Parayanthal, F. H. Pollak, T. Coutts, and A. Aharoni [Phys. Rev. B 37, 4044 (1988)], where the effects of gradient electric field and non-flat-band modulation are treated explicitly. The equivalence occurs because the field gradient and non-flat-band modulation effects are included in our model in the energy dependence of the phenomenological broadening parameter ${\mathrm{\ensuremath{\Gamma}}}^{\mathrm{*}}$=(${\mathrm{\ensuremath{\Gamma}}}_{0}$/\ensuremath{\Elzxh}\ensuremath{\theta})exp[\ensuremath{\delta}(\ensuremath{\Elzxh}\ensuremath{\omega}-${\mathit{E}}_{\mathit{g}}$)], where \ensuremath{\Elzxh}\ensuremath{\omega} is the photon energy, ${\mathit{E}}_{\mathit{g}}$ is the band-gap energy, ${\mathrm{\ensuremath{\Gamma}}}_{0}$ is the nominal broadening at the band-gap energy, and \ensuremath{\delta} is a parameter directly proportional to the electric-field gradient and modulation between two finite fields. The major utility of our model is that a single effective layer can be treated instead of a more computationally intensive laminar model.We apply our complex Airy model to bulk semiconductors such as GaAs, InP, and ${\mathrm{In}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As. In the photoreflectance spectra of these semiconductors, our model considers three distinct but convolved features at ${\mathit{E}}_{0}$ which represent the light- and heavy-hole valence bands and an exciton-impurity feature below ${\mathit{E}}_{0}$. At ${\mathit{E}}_{0}$+${\mathrm{\ensuremath{\Delta}}}_{0}$ our model considers two features which are related to the spin-orbit-split valence band and a second state just below this critical point. For GaAs, we determined from our PR modeling that the band-gap energy for these films was 1.422\ifmmode\pm\else\textpm\fi{}0.003 eV, which agreed, within experimental error, with the band-gap energy measured by room-temperature photoluminescence. A feature was found below the ${\mathit{E}}_{0}$ gap in the GaAs samples with energies ranging from 3 to 4 meV below the band gap, which is similar to excitonic binding energies in this material system.A below-critical-point feature was evident in one GaAs sample at 11 meV below the ${\mathit{E}}_{0}$+${\mathrm{\ensuremath{\Delta}}}_{0}$ transition. Also, in GaAs we determined the ratio of light and heavy interband effective masses, ${\mathrm{\ensuremath{\mu}}}_{\mathrm{LH}}$/${\mathrm{\ensuremath{\mu}}}_{\mathrm{HH}}$, to be 0.6865, which is in good agreement with values determined in previous studies. For bulk InP the band-gap energy was determined to be 1.340\ifmmode\pm\else\textpm\fi{}0.003 eV, which agreed, within experimental error, with the band-gap energy determined from PL. Below-critical-point features were also found for InP with energies of 4.0 and 3.5 meV below the ${\mathit{E}}_{0}$ and ${\mathit{E}}_{0}$+${\mathrm{\ensuremath{\Delta}}}_{0}$ transitions, respectively. For both GaAs and InP, the surface electric fields determined from the ${\mathit{E}}_{0}$ and ${\mathit{E}}_{0}$+${\mathrm{\ensuremath{\Delta}}}_{0}$ critical points were in agreement within experimental error. For ${\mathrm{In}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As on InP films near lattice-matched conditions, the complex Airy line shape was applied to both intermediate electric field (Franz-Keldysh oscillations) and low-field-like PR spectra illustrating the utility of the complex Airy functional analysis in fitting both simple and complicated PR responses. We found that the band-gap energy from the PR spectral fits to those determined from photoluminescence measurements agreed within experimental error for these ${\mathrm{In}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As films.