Complex Scorer Function Model for the Franz–Keldysh Dielectric Function of Single-Electron, Indirect, and Phonon-Assisted Transitions

This work presents a complex-valued analytical model for the Franz–Keldysh effect (FKE) on the dielectric function (DF) of single-electron, indirect, phonon-assisted transitions in the vicinity of the energy-band edge. We employ the Airy transform of the quadratic energy relation for the imaginary part of the DF to derive its electric-field-induced change. The derived DF is consistent with the Kramers–Kronig relation and accommodates Lorentzian broadening by a shift in the complex energy. Relative to the electro-optic energy, we investigate theoretically the impact of Lorentzian, Gaussian, and Faddeeva broadening. An empirical model for the electro-optic coefficients that accounts for the indirect transverse acoustic and transverse optic phonon-assisted interactions of electrons with light and heavy holes is extracted from published experimental data for the optical parameters of Si. The electro-optic-induced changes in the absorption coefficient (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Delta \alpha $ </tex-math></inline-formula>), the index of refraction (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Delta {\mathbf {n}}$ </tex-math></inline-formula>), and the extinction coefficient <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(\Delta {\mathbf {k}})$ </tex-math></inline-formula> are calculated for electric fields from 100 to 500 kV/cm. The extrema of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(\Delta \alpha)$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(\Delta {\mathbf {n}})$ </tex-math></inline-formula> are computed as a function of the electric fields for different Lorentzian broadening strengths. The broadening strength plays a significant role in fitting previously published experimental data. We suggest the use of a modified transparent-gate recessed-channel metal–oxide–semiconductor structure to characterize the FKE better for indirect optical transitions in Si. The proposed structure provides a uniform and higher-field alternative to the fields generated in the space-charge region of pn-junction diodes. The proposed structure enables an enhanced assessment of the characterization of Si using modulation spectroscopy, yielding an improved empirical electro-optic model for practical computer-aided design. This analytical framework provides a pathway for further exploration of the electro-optical properties of indirect-gap semiconductor systems under strong electric fields.