Optical dispersion relations for Si and Ge

Abstract

A method is described for calculation of the optical constants (the refractive index, extinction coefficient, absorption coefficient, and normal-incidence reflectivity) of Si and Ge in the entire range of photon energies (0–0.6 eV). The imaginary part of the dielectric function [ε2(ω)] is derived first from the joint-density-of-states functions at energies of various critical points (CPs) in the Brillouin zone; then its real part [ε1(ω)] is obtained analytically using the Kramers–Kronig relation. The indirect-band-gap transitions are also assumed to provide a gradually increasing absorption spectrum expressed by a power law of (ℏω−EIDg)2, where ℏω is the photon energy and EIDg is the indirect-band-gap energy. The optical dispersion relations are expressed in terms of these model dielectric functions. The present model reveals distinct structures in the optical data at energies of the E0, E0+Δ0 [three-dimensional (3D) M0 critical point (CP)], E1, E1+Δ1 [3D M1 or two-dimensional (2D) M0 CP], E2 [a mixture of damped harmonic oscillator (DHO) and 2D M2 CP], E′0 triplet (DHO), and E1 (DHO). Calculated optical spectra are in satisfactory agreement with the experimental data over a wide range of the photon energies.