  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.
