  HfO$_2$- and ZrO$_2$-based ferroelectric thin films have emerged as promising
candidates for the gate oxides of next generation electronic devices. Recent
work has experimentally demonstrated that a tetragonal/orthorhombic (t/o-)
phase mixture with partially in-plane polarization can lead to negative
capacitance (NC) stabilization. However, there is a discrepancy between
experiments and the theoretical understanding of domain formation and domain
wall motion in these multi-phase, polycrystalline materials. Furthermore, the
effect of anisotropic domain wall coupling on NC has not been studied so far.
Here we apply 3D phase field simulations of HfO$_2$- and ZrO$_2$-based
mixed-phase ultra-thin films on silicon to understand the necessary and
beneficial conditions for NC stabilization. We find that smaller ferroelectric
grains and a larger angle of the polar axis with respect to the out-of-plane
direction enhances the NC effect. Furthermore, we show that theoretically
predicted negative domain wall coupling even along only one axis prevents NC
stabilization. Therefore, we conclude that topological domain walls play a
critical role in experimentally observed NC phenomena in HfO$_2$- and
ZrO$_2$-based ferroelectrics.
