COMPLEX AIRY ANALYSIS OF PHOTOREFLECTANCE SPECTRA FOR III-V SEMICONDUCTORS

We present a detailed analysis of photoreflectance (PR) spectra of sem iconductors using complex Airy functions and their derivatives. We dem onstrate that photoreflectance spectra can be treated in terms of a si ngle complex Airy function with an energy-dependent broadening paramet er. We show analytically and numerically that this complex Airy PR tre atment 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 modulat ion are treated explicitly. The equivalence occurs because the field g radient and non-flat-band modulation effects are included in our model in the energy dependence of the phenomenological broadening parameter GAMMA = (GAMMA0/HthetaBAR)exp[delta(HomegaBAR - E(g)) ], where Homeg aBAR is the photon energy, E(g) is the band-gap energy, GAMMA0 is the nominal broadening at the band-pp energy, and delta is a parameter dir ectly proportional to the electric-field gradient and modulation betwe en two finite fields. The major utility of our model is that a single effective layer can be treated instead of a more computationally inten sive laminar model. We apply our complex Airy model to bulk semiconduc tors such as GaAs, InP, and InxGa1-xAs. In the photoreflectance spectr a of these semiconductors, our model considers three distinct but conv olved features at E0 which represent the light- and heavy-hole valence bands and an exciton-impurity feature below E0. At E0 + DELTA0 our mo del considers two features which are related to the spin-orbit-split v alence band and a second state just below this critical point. For GaA s, we determined from our PR modeling that the band-pp energy for thes e films was 1.422+/-0.003 eV, which agreed, within experimental error, with the band-gap energy measured by room-temperature photoluminescen ce. A feature was found below the E0 gap in the GaAs samples with ener gies ranging from 3 to 4 meV below the band pp, which is similar to ex citonic binding energies in this material system. A below-critical-poi nt feature was evident in one GaAs sample at 11 meV below the E0 + DEL TA0 transition. Also, in GaAs we determined the ratio of light and hea vy interband effective masseS, mu(LH)/mu(HH), to be 0.6865, which is i n good agreement with values determined in previous studies. For bulk InP the bandgap energy was determined to be 1.340+/-0.003 eV, which ag reed, within experimental error, with the band-gap energy determined f rom PL. Below-critical-point features were also found for InP with ene rgies of 4.0 and 3.5 meV below the E0 and E0 + DELTA0 transitions, res pectively. For both GaAs and InP, the surface electric fields determin ed from the E0 and E0 + DELTA0 critical points were in agreement withi n experimental error. For InxGa1-xAs on InP films near lattice-matched conditions, the complex Airy line shape was applied to both intermedi ate electric field (Franz-Keldysh oscillations) and low-field-like PR spectra illustrating the utility of the complex Airy functional analys is in fitting both simple and complicated PR responses. We found that the band-pp energy from the PR spectral fits to those determined from photoluminescence measurements agreed within experimental error for th ese InxGa1-xAs films.