Emergence of the logarithmic average phonon frequency in the superconducting critical temperature formula

We analytically demonstrate the essential role of the logarithmic average phonon frequency in describing the superconducting critical temperature, directly from a predictive function. The current study assumes that the Eliashberg spectral function follows the Debye model in the low frequency spectrum, whereas contributions from optical phonons dominate outside this range. Our findings confirm that, under a specific condition, we obtained a formula for superconducting transition temperature. Furthermore, we compared our formula with the Allen-Dynes formula and its modified version, and the exact solutions. They reveal notable correlations.