Modulated ellipsometric measurements and transfer-matrix calculation of the field-dependent dielectric function of a multiple quantum well.

Abstract

We perform a complete nondestructive optical characterization of a GaAs/${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As multiple-quantum-well (MQW) sample. Modulated ellipsometry, a combination of spectroscopic ellipsometry and photomodulation spectroscopy, is used as an experimental technique. The electric field is modulated via photoexcitation of electron-hole pairs in the p-i-n structure. The field-induced modification in the dielectric function of the multiple quantum well is derived in the spectral range from the GaAs to the ${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As band gap. We show that the modulated ellipsometry technique is particularly well suited to characterize higher quantized transitions. The experimental results are compared to an extensive theoretical model of the MQW dielectric function, based on a transfer-matrix solution of the Schr\"odinger equation. The effective-mass dispersion of electrons and light holes are treated by the Kane model, and the excitonic transitions are introduced by oscillator functions. A detailed comparison of the experimental spectra and this theory allows a very precise determination of the sample parameters, such as well thickness, barrier thickness, and aluminum content.