Pseudo-gap and Vertex Correction of Electron-Phonon Interaction in High Transition-Temperature Superconductors

The strong-coupling Eliashberg theory plus vertex correction is used to calculate the maps of transition temperature (T (c)) in parameter-space characterizing superconductivity. Based on these T (c) maps, complex crossover behaviors are found when electron-phonon interaction increases from weak-coupling region to strong-coupling region. The doping-dependent T (c) of cuprate superconductors and most importantly the emergence of pseudo-gap region can be explained as the effects of vertex correction.