In the present study, we investigate the thermodynamic properties of the Ba$_{x}$K$_{1-x}$BiO$_{3}$ (BKBO) superconductor in the under- ($x=0.5$) and over-doped ($x=0.7$) regime, within the framework of the Migdal-Eliashberg formalism. The analysis is conducted to verify that the electron-phonon pairing mechanism is responsible for the induction of the superconducting phase in the mentioned compound. In particular, we show that BKBO is characterized by the relatively high critical value of the Coulomb pseudopotential, which changes with doping level and does not follow the Morel-Anderson model. In what follows, the corresponding superconducting band gap size and related dimensionless ratio are estimated to increase with the doping, in agreement with the experimental predictions. Moreover the effective mass of electrons is found to take on high values in the entire doping and temperature region. Finally, the characteristic dimensionless ratios for the superconducting band gap, the critical magnetic field and the specific heat for the superconducting state are predicted to exceed the limits set within the Bardeen-Cooper-Schrieffer theory, suggesting pivotal role of the strong-coupling and retardation effects in the analyzed compound. Presented results supplement our previous investigations and account for the strong-coupling phonon-mediated character of the superconducting phase in BKBO at any doping level.
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