The effect of critical coupling constants on superconductivity enhancement

In this study, we propose a phenomenological model to extend McMillan's results on a coupling strength equal to 2. We investigate possible strategies to enhance superconductivity by tuning the phonon frequency, carrier number, or pressure. In particular, we show that the critical coupling constants corresponding to the phonon frequency, carrier number, or pressure determine whether the variation of the critical temperature is positive or negative. These observations explain the contrasting behavior between weak and strong coupling superconductors and are consistent with experimental observations. We also demonstrate the dome observed in the carrier number effect and pressure effect. Additionally, these critical coupling constants systematically separate superconductivity into three regions: weak, intermediate, and strong coupling. We find that the enhancement strategies for weak and strong coupling regions are opposite, but both inevitably bring superconductivity into the intermediate coupling region. Finally, we propose general zigzag methods for intermediate coupling superconductors to further enhance the critical temperature.