  We use first-principles density functional theory to investigate how the
polar distortion is affected by doping in multiferroic hexagonal yttrium
manganite, h-YMnO$_3$. While the introduction of charge carriers tends to
suppress the polar distortion in conventional ferroelectrics, the behavior in
improper geometric ferroelectrics, of which h-YMnO$_3$ is the prototype, has
not been studied to date. Using both background charge doping and atomic
substitution, we find an approximately linear dependence of the polar
distortion on doping concentration, with hole doping suppressing and electron
doping enhancing it. We show that this behavior is a direct consequence of the
improper geometric nature of the ferroelectricity. In addition to its doping
effect, atomic substitution can further suppress or enhance the polar
distortion through changes in the local chemistry and geometry.
