  Free electrons can screen out long-range Coulomb interaction and destroy the
polar distortion in some ferroelectric materials, whereas the coexistence of
polar distortion and metallicity were found in several non-central-symmetric
metals (NCSMs). Therefore, the mechanisms and designing of NCSMs have attracted
great interests. In this work, by first-principles calculation, we found the
polar distortion in the lone-pair driven ferroelectric material PbTiO$_3$ can
not only persist, but also increase with electron doping. We further analyzed
the mechanisms of the persisting of the polar distortion. We found that the Ti
site polar instability is suppressed but the Pb site polar instability is
intact with the electron doping. The Pb-site instability is due to the
lone-pair mechanism which can be viewed as a pseudo-Jahn-Teller effect, a mix
of the ground state and the excited state by ion displacement from the central
symmetric position. The lone-pair mechanism is not strongly affected by the
electron doping because neither the ground state or the excited state involved
is at the Fermi energy. The enhancement of the polar distortion is related to
the increasing of the Ti ion size by doping. These results show the lone-pair
stereoactive ions can be used in designing NCSMs.
