  HfO$_{2}$-based ferroelectric thin films are promising for their application
in ferroelectric devices. Predicting the ultimate magnitude of polarization and
understanding its switching mechanism are critical to realize the optimal
performance of these devices. Here, a generalized solid-state variable cell
nudged elastic band (VCNEB) method is employed to predict the switching pathway
associated with domain-wall motion in (Hf, Zr)O$_{2}$ ferroelectrics. It is
found that the polarization reversal pathway, where three-fold coordinated O
atoms pass across the nominal unit-cell boundaries defined by the Hf/Zr atomic
planes, is energetically more favorable than the conventional pathway where the
O atoms do not pass through these planes. This finding implies that the
polarization orientation in the orthorhombic Pca2$_{1}$ phase of HfO$_{2}$ nd
its derivatives is opposite to that normally assumed, predicts the spontaneous
polarization magnitude of about 70 ${\mu}$C/cm$^{2}$ that is nearly 50% larger
than the commonly accepted value, signifies a positive intrinsic longitudinal
piezoelectric coefficient, and suggests growth of ferroelectric domains, in
response to an applied electric field, structurally reversed to those usually
anticipated. These results provide important insights into the understanding of
ferroelectricity in HfO$_{2}$-based ferroelectrics.
