Gaussian and Faddeeva broadening of the Franz-Keldysh oscillations at three-dimensional critical-points

Abstract

Abstract Faddeeva broadening (FB) couples the physical characteristics of Gaussian and Lorentzian broadening into a single profile shaping the spectral response of dielectric functions. The electric field induces Franz-Keldysh oscillations (FKOs) as deviations in the dielectric function of semiconductors. We investigated the impact of Gaussian and FB on FKOs at the isotropic M0 of three-dimensional critical points. The twenty-pole Martin–Donoso–Zamudio quadrature was applied to compute the FB of FKOs in differential reflectivity based on the electro-optic function, H(z), with the complex Airy function, Ai(z). H(z) was simplified into a compact product of K ( z ) = A i ′ ( z ) + z A i ( z ) to reduce the truncation error and determine its asymptotics. We derived a general asymptotic expansion of order N, accounting for Stokes’ phenomenon as the arg(z) crosses 2 π / 3 . The computation of H(z) is detailed for twelve digits of accuracy. The effects of higher Gaussian and FB widths on the damping of FKOs are assessed using FKO peak-to-peak attenuations. For the low-field regime, this work H(z) smoothly evolves into Aspnes’ third derivative functional form (TDFF). Between low- to intermediate-field regimes, we investigated the boundaries for the relevant use of the TDFF by assessing parameters extracted from simulated lineshapes, and the average root mean-squared error. We fitted of the Faddeeva broadened single Airy model Δ R / R to published experimental photoreflectance (PR) lineshapes of InGaAsP. To model degenerate light-and heavy-holes transitions, we fitted of the Faddeeva broadened double Airy model Δ R / R to published experimental contactless electroreflectance (CER) lineshapes of GaAs. In addition to legacy mean-squared error ( χ 2 ) and coefficient of determination ( R 2 ), the goodness of fit was evaluated using the unbiased Akaike information criteria. The effectiveness of Faddeeva versus Lorentzian broadening in sample PR and CER spectra analysis was assessed using Airy function FKO models. Concludingly, the FB of the differential reflectivity Δ R / R model outperformed the Lorentzian broadening even with the empirical energy-dependent width.