  The structural, thermal, and dielectric properties of the ferroelectric phase
of HfO$_2$, ZrO$_2$ and Hf$_{0.5}$ Zr$_{0.5}$ O$_2$ (HZO) are investigated with
carefully validated density functional computations. We find, that the free
bulk energy of the ferroelectric orthorhombic Pca2$_{1}$ phase is unfavorable
compared to the monoclinic P2$_{1}$/c and the orthorhombic Pbca phase for all
investigated stoichiometries in the Hf$_{\chi}$Zr$_{1-\chi}$O$_2$ system. To
explain the existence of the ferroelectric phase in nanoscale thin films we
explore the Gibbs / Helmholtz free energies as a function of stress and film
strain and find them unlikely to become minimal in HZO films for technological
relevant conditions. To assess the contribution of surface energy to the phase
stability we parameterize a model, interpolating between existing data, and
find the Helmholtz free energy of ferroelectric grains minimal for a range of
size and stoichiometry. From the model we predict undoped HfO$_2$ to be
ferroelectric for a grain size of about 4 nm and epitaxial HZO below 5 nm.
Furthermore we calculate the strength of an applied electric field necessary to
cause the antiferroelectric phase transformation in ZrO$_2$ from the P4$_2$/nmc
phase as 1 MV/cm in agreement with experimental data, explaining the mechanism
of field induced phase transformation.
