Optical representation of thermal nuclear fluctuation effect on bandgap 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 focused on the analytical form of the Debye-Waller term and found that the term can be reformulated using the optical transition matrix. In addition, the optical selection rule is found to play a role. For the 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. Further, the decomposition can also be applied with an additional approximation to the zinc-blende-type materials, as revealed by our first-principles calculation. The magnitude of the Debye-Waller term of a number of materials can thus be estimated using the basic physical quantities prior to performing the calculation of the electron-phonon interaction.