Hafnia-based ferroelectric films switch at room-temperature coercive fields of about $1\mathrm{MV}{\mathrm{cm}}^{-1}$, an order of magnitude below the fields predicted for homogeneous polarization reversal. The predicted switching fields can exceed the dielectric breakdown strength, although polarization reversal is experimentally observed before breakdown. In this work, we use first-principles calculations and symmetry-reduced atom mappings to compare switching in the polar $Pca{2}_1$ and $Pmn{2}_1$ structures of ${\mathrm{HfO}}_2$ and ${\mathrm{Hf}}_{0.5}{\mathrm{Zr}}_{0.5}{\mathrm{O}}_2$ (HZO). Systematic comparison of the matched pathways shows that Zr substitution lowers nearly all sampled $Pca{2}_1$ barriers and produces a particularly low-barrier $Pmn{2}_1 {180}^{\circ }$ reversal route. The comparison identifies a favorable $Pmn{2}_1 {180}^{\circ }$ reversal whose barrier is substantially reduced in HZO. For a field normal to the predominantly exposed $\left\{111\right\}$ facets, the estimated room-temperature critical field for this route, obtained using phenomenological fourth-order Landau-Ginzburg-Devonshire temperature scaling, decreases from 2.57 $\mathrm{MV}{\mathrm{cm}}^{-1}$ in ${\mathrm{HfO}}_2$ to 1.02 $\mathrm{MV}{\mathrm{cm}}^{-1}$ in HZO. These results clarify how Zr substitution lowers the switching field. The identified reversal pathway helps explain the discrepancy between conventional homogeneous-switching predictions and low-field measurements.

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• Critical field
• Doping effects
• Ferroelectricity
• Phase transitions