Effects of Homogeneous Doping on Electron-Phonon Coupling in SrTiO₃

Bulk n-type SrTiO₃ (STO) has long been known to possess a superconducting ground state at an exceptionally dilute carrier density. This has raised questions about the applicability of the BCS-Eliashberg paradigm with its underlying adiabatic assumption. However, recent experimental reports have set the pairing gap to the critical temperature (Tc) ratio at the BCS value for superconductivity in Nb-doped STO, even though the adiabaticity condition the BCS pairing requires is satisfied over the entire superconducting dome only by the lowest branch of optical phonons. In spite of the strong implications these reports have on specifying the pairing glue, they have not proved sufficient in explaining the magnitude of the optimal doping. This motivated us to apply density functional theory to Nb-doped STO to analyze how the phonon band structures and the electron-phonon coupling evolve with doping. To describe the very low doping concentration, we tuned the homogeneous background charge, from which we obtained a first-principles result on the doping-dependent phonon frequency that is in good agreement with experimental data for Nb-doped STO. Using the EPW code, we obtain the doping-dependent phonon dispersion and the electron-phonon coupling strength. Within the framework of our calculation, we found that the electron-phonon coupling forms a dome in a doping range lower than the experimentally observed superconducting dome of the Nb-doped STO. Additionally, we examined the doping dependence of both the orbital angular momentum quenching in the electron-phonon coupling and the phonon displacement correlation length and found the former to have a strong correlation with our electron-phonon coupling in the overdoped region.