  The discovery of ferroelectricity in both pure and doped HfO$_2$-based thin
films have revitalized the interest in using ferroelectrics for nanoscale
device applications. To take advantage of this silicon-compatible
ferroelectric, fundamental questions such as the origin of ferroelectricity and
better approach to controlled realization of ferroelectricity at the nanoscale
need to be addressed. The emergence of robust polarization in HfO$_2$-based
thin films is considered as the cumulative effect of various extrinsic factors
such as finite-size effects and surface/interface effects of small grains,
compressive stress, dopants, oxygen vacancies, and electric fields. The kinetic
effects of phase transitions and their potential impacts on the emergence of
ferroelectricity in HfO$_2$ at the nanoscale are not well understood. In this
work, we construct the transition paths between different polymorphs of hafnia
with density functional theory calculations and variable-cell nudged elastic
band technique. We find that the transition barriers depend strongly on the
mechanical boundary conditions and the transition from the tetragonal phase to
the polar orthorhombic phase is a fast process kinetically under clamping. The
effects of growth orientations and epitaxial strains on the relative stability
of different phases of HfO$_2$ are investigated. The two orthorhombic phases,
polar $Pca2_1$ and non-polar $Pbca$, become thermodynamically stable in
(111)-oriented thin films over a wide range of epitaxial strain conditions.
This work suggests a potential avenue to better stabilize the ferroelectric
phase in HfO$_2$ thin films through substrate orientation engineering.
