  We clarify the nature of hafnia as a proper ferroelectric and show that there
is a shallow double well involving a single soft polar mode as in well-known
classic ferroelectrics. Using symmetry analysis, density-functional theory
(DFT) structural optimizations with and without epitaxial strain, and density
functional perturbation theory (DFPT), we examine several important possible
hafnia structures derived ultimately from the cubic fluorite structure,
including baddeleyite ($P2_{1}/c$) tetragonal antiferroelectric $P4_{2}nmc$,
$Pbca$ (nonpolar and brookite), ferroelectric rhombohedral ($R3m$ and $R3$),
$Pmn2_{1}$, and $Pca2_{1}$ structures. The latter is considered to be the most
likely ferroelectric phase seen experimentally, and has an antiferroelectric
parent with space group $Pbcn$, with a single unstable polar mode and a shallow
double well with a well depth of 24 meV/atom. Strain is not required for
switching or other ferroelectric properties, nor is coupling of the soft-mode
with any other modes within the ferroelectric $Pca2_{1}$, $Pmn2_{1}$, $R3m$ or
$R3$ phases.
