  Ferroelectricity observed in thin film $\mathrm{HfO_2}$, either doped with
Si, Al, etc. or in the $\mathrm{Hf_{0.5}Zr_{0.5}O_2}$ form, has gained great
technical significance. However, the soft mode theory faces a difficulty in
explaining the origin of such ferroelectricity. In this work, we propose that
the 7 cation coordination number of $\mathrm{HfO_2/ZrO_2}$ lies at the heart of
this ferroelectricity, which stems from the proper ionic radii of Hf/Zr
compared with O. Among the numerous compounds with non-centrosymmetric nature,
e.g., $mm2$ point group, $\mathrm{HfO_2}$ and $\mathrm{ZrO_2}$ are special in
that they are close to the border of 7 and 8 cation coordination, such that the
8-coordination tetragonal intermediate phase could greatly reduce the switching
barrier. Other 7-coordination candidates, including $\mathrm{SrI_2}$, TaON,
YSBr and YOF are also studied in comparison to
$\mathrm{HfO_2}$/$\mathrm{ZrO_2}$, and six switching paths are analyzed in
detail for the $Pca2_1$ phase. A rule of preferred switching path in terms of
ionic radii ratio and coordination number has been established. We also show
the possible route from ferroelectric $Pca2_1$ phase to monoclinic $P2_1/c$
phase in $\mathrm{HfO_2}$, which is relevant to the fatigue phenomenon.
