Temporal response of the Franz-Keldysh effect

Abstract

External perturbations such as temperature, stress, or electric fields, which can modulate the band gap of a semiconductor, play a major role in the study of energy band structures. In electroabsorption, in particular, the modulation of the band gap of a semiconductor by an externally applied electric field, referred to as the Franz-Keldysh effect, can be detected as a change in the sample's reflectivity. By measuring the relative change in the reflectivity due to a modulating external perturbation as a function of wavelength, one can study the band structure of semiconductors. The relative change in the reflectivity as a function of wavelength is given by the third derivative of the unperturbed dielectric constant.1 Yacoby2 discussed the case where the duration of the perturbing electric field is comparable to or shorter than the characteristic time of the system. Here we show, using the time gauge and the Fourier time transform of the time-dependent current operator, that if the duration of the perturbing electric field is shorter than the characteristic time of the system, the relative change in the reflectivity due to the Franz-Keldysh effect will be proportional to the first-order rather than the third-order derivative and will therefore yield results similar to those obtained in piezoreflectance, thermoreflectance, and wavelength modulation.1