Optical representation of thermal nuclear fluctuation effect on band-gap renormalization

The bandgap of insulating materials is renormalized in various ways by the electron-phonon interaction owing to the dynamical and quantum fluctuations of nuclei. These fluctuation effects are considered in the perturbative Allen-Heine-Cardona theory using the formulae for the Fan-Migdal and Debye-Waller terms. However, the material dependence is not clear in the formulae. Thus, in this study, we focus on the analytical form of the Debye-Waller term and find that the term can be reformulated using the momentum matrix. In addition, the optical selection rule is found to play a role. For diamond-type materials, the Debye-Waller term can be approximately decomposed into a product of the optical transition energy, the mean square displacement of nuclei, and the dipole transition probability. The decomposition can also be applied with an additional approximation to zinc-blende-type materials, as revealed by our first-principles calculation. The magnitudes of the Debye-Waller term of several materials can thus be estimated using basic physical quantities prior to performing the calculation of the electron-phonon interaction.