  At present, hydrogen-based compounds constitute one of the most promising
classes of materials for applications as a phonon-mediated high-temperature
superconductors. Herein, the behavior of the superconducting phase in tellurium
hydride (HTe) at high pressure ($p=300$ GPa) is analyzed in details, by using
the isotropic Migdal-Eliashberg equations. The chosen pressure conditions are
considered here as a case study which corresponds to the highest critical
temperature value ($T_{c}$) in the analyzed material, as determined within
recent density functional theory simulations. It is found that the
Migdal-Eliashberg formalism, which constitutes a strong-coupling generalization
of the Bardeen-Cooper-Schrieffer (BCS) theory, predicts that the critical
temperature value ($T_{c}=52.73$ K) is higher than previous estimates of the
McMillan formula. Further investigations show that the characteristic
dimensionless ratios for the the thermodynamic critical field, the specific
heat for the superconducting state, and the superconducting band gap exceeds
the limits of the BCS theory. In this context, also the effective electron mass
is not equal to the bare electron mass as provided by the BCS theory. On the
basis of these findings it is predicted that the strong-coupling and
retardation effects play pivotal role in the superconducting phase of HTe at
300 GPa, in agreement with similar theoretical estimates for the sibling
hydrogen and hydrogen-based compounds. Hence, it is suggested that the
superconducting state in HTe cannot be properly described within the mean-field
picture of the BCS theory.
