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Polar metals are rare because free carriers in metals screen electrostaticpotential and eliminate internal dipoles. Degenerate doped ferroelectrics maycreate an approximate polar metallic phase. We use first-principlescalculations to investigate $n$-doped LiNbO$_3$-type oxides (LiNbO$_3$ as theprototype) and compare to widely studied perovskite oxides (BaTiO$_3$ as theprototype). 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 getsuppressed and completely vanish at 0.1 $e$/f.u. Furthermore, in $n$-dopedLiNbO$_3$, Li-O displacements decay more slowly than Nb-O displacements, whilein $n$-doped BaTiO$_3$, Ba-O and Ti-O displacements decay approximately at thesame rate. Supercell calculations that use oxygen vacancies as electron donorssupport the main results from the rigid-band approximation and provide moredetailed charge distributions. Substantial cation displacements are observedthroughout LiNbO$_{3-\delta}$($\delta = 4.2\%$), while cation displacements inBaTiO$_{3-\delta}$($\delta = 4.2\%$) are almost completely suppressed. We findthat conduction electrons in LiNbO$_{3-\delta}$ are not as uniformlydistributed as in BaTiO$_{3-\delta}$, implying that the rigid-bandapproximation should be used with caution in simulating electron dopedLiNbO$_3$-type oxides. Our work shows that polar distortions and conduction cancoexist in a wide range of electron concentration in $n$-doped LiNbO$_3$, whichis a practical approach to generating an approximate polar metallic phase.Combining doped ferroelectrics and doped semiconductors may create newfunctions for devices.

Coexistence of polar displacements and conduction in doped ferroelectrics: an ab initio comparative study