  The polar metal is a material that hosts both polar distortion and
metallicity. Such a material is expected to show exotic magneto-electric
phenomena if superconducts. Here, we theoretically explore ferroelectric and
superconducting properties in a series of perovskite-type oxyhydrides ATiO$_2$H
(A=K, Rb, Cs) under hole-doping conditions using the first-principles
calculations based on the density functional theory. Our simulation shows that
these compounds host spontaneous polarization and superconductivity at optimal
doping concentration. The unusual coexistence of superconductivity and large
polarization (~100 ${\mu}$C/cm$^2$) originates from weak coupling of the polar
distortion and superconducting states, the reason of which is separation of the
displaced atoms and spacially confined metallic carriers. Besides, the
superconductivity is enhanced by the unique electronic properties near the
valence band maximum: quartic band dispersion with a sizable contribution of
hydrogen 1s states. Our study thus feature the oxyhydrides as possible model
polar superconducting systems, which may be utilizable for future
magneto-electric devices.
