  Polar metals are rare because free carriers in metals screen electrostatic
potential and eliminate internal dipoles. Degenerate doped ferroelectrics may
create an approximate polar metallic phase. We use first-principles
calculations to investigate $n$-doped LiNbO$_3$-type oxides (LiNbO$_3$ as the
prototype) and compare to widely studied perovskite oxides (BaTiO$_3$ as the
prototype). In the rigid-band approximation, substantial polar displacements in
$n$-doped LiNbO$_3$ persist even at 0.3 $e$/f.u. ($\simeq$ 10$^{21}$
cm$^{-3}$), while polar displacements in $n$-doped BaTiO$_3$ quickly get
suppressed and completely vanish at 0.1 $e$/f.u. Furthermore, in $n$-doped
LiNbO$_3$, Li-O displacements decay more slowly than Nb-O displacements, while
in $n$-doped BaTiO$_3$, Ba-O and Ti-O displacements decay approximately at the
same rate. Supercell calculations that use oxygen vacancies as electron donors
support the main results from the rigid-band approximation and provide more
detailed charge distributions. Substantial cation displacements are observed
throughout LiNbO$_{3-\delta}$($\delta = 4.2\%$), while cation displacements in
BaTiO$_{3-\delta}$($\delta = 4.2\%$) are almost completely suppressed. We find
that conduction electrons in LiNbO$_{3-\delta}$ are not as uniformly
distributed as in BaTiO$_{3-\delta}$, implying that the rigid-band
approximation should be used with caution in simulating electron doped
LiNbO$_3$-type oxides. Our work shows that polar distortions and conduction can
coexist in a wide range of electron concentration in $n$-doped LiNbO$_3$, which
is a practical approach to generating an approximate polar metallic phase.
Combining doped ferroelectrics and doped semiconductors may create new
functions for devices.
