  Ferroelectricity in hafnia is often regarded as a breakthrough discovery in
ferroelectrics, potentially able to revolutionize the whole field. Despite
increasing interests, a comprehensive understanding of the many factors driving
the ferroelectric stabilization is still lacking. We here address the phase
transition in terms of a Landau-theory-based approach, by analyzing
symmetry-allowed distortions connecting the high-symmetry paraelectric
tetragonal phase to the low-symmetry polar orthorhombic phase. By means of
first-principles simulations, we find that the $\Gamma_{3-}$ polar mode is only
weakly unstable, whereas the other two symmetry-allowed distortions, non-polar
Y$_{2+}$ and anti-polar Y$_{4-}$ are hard modes. None of the modes, taken alone
or combined with one other mode, is able to drive the transition: the key
factor in stabilizing the polar phase is identified as the strong trilinear
coupling among the three modes. Furthermore, the experimentally acknowledged
importance of substrate-induced effects in the growth of HfO$_2$ ferroelectric
thin films, along with the lack of a clear order parameter in the transition,
suggested the extension of our analysis to strain effects. Our findings suggest
a complex behaviour of the Y$_{2+}$ mode, which become unstable under certain
strain conditions and an overall unstable behaviour for the $\Gamma_{3-}$ polar
mode for all the strain states. A robust result emerges from our analysis:
independently of the different applied strain (compressive or tensile, applied
along orthorhombic axes), the need of a simultaneous excitation of the three
coupled modes remain unaltered. Finally, when applied to mimic experimental
growth conditions under strain, our analysis show a further stabilization of
the ferroelectric phase with respect to the unstrained case, in agreeement with
experimental findings.
