Band-Structure Analysis from Electro-Reflectance Studies

This paper provides the analytical tools for an interpretation of the electro-reflectance effect in semiconductors, which is observed as a change in the reflectance in the presence of an electric field at approximately the photon energies of interband transitions. Previously only applicable to ${\epsilon }_2$ at an $M_0$-type absorption edge, the theory of Franz and Keldysh is expanded for the most general case of field-induced changes of both ${\epsilon }_1$ and ${\epsilon }_2$ at both $M_0$- and $M_1$-type edges. Every member of this group of four components displays a highly individual line shape and symmetry character, facilitating the recognition of basic patterns in the experimentally observed structure. By comparing such structure with the calculated line shapes, an assignment to interband transitions of a certain type and at certain photon energies will be suggested. The coefficients which determine the fractional contributions of $\Delta {\epsilon }_1$ and $\Delta {\epsilon }_2$ to the change in reflectance in different spectral regions are calculated for Ge, Si, and GaAs. The analysis is applied to the 3.4-eV region in Si, and satisfactory agreement with experiment is obtained by placing a parabolic $M_0$-type edge at 3.33 eV and a saddle-point $M_1$-type edge at 3.41 eV. The calculation produces a shift of the effect with increasing average field toward shorter wavelengths for the parabolic and toward longer wavelengths for the saddle-point edge. This is in agreement with experimental observations for Si and GaAs and confirms a valuable experimental criterion which discriminates between the two different types of interband transitions.