  $\mathrm{HfO_2}$-based dielectrics are promising for nanoscale ferroelectric
applications, and the most favorable material within the family is
Zr-substituted hafnia, i.e., $\mathrm{Hf_{1-x}Zr_xO_2}$ (HZO). The extent of Zr
substitution can be great, and x is commonly set to 0.5. However, the band gap
of $\mathrm{ZrO_2}$ is lower than $\mathrm{HfO_2}$, thus it is uncertain how
the Zr content should influence the electronic band structure of HZO. A reduced
band gap is detrimental to the cycling endurance as charge injection and
dielectric breakdown would become easier. Another issue is regarding the
comparison on the band gaps between $\mathrm{HfO_2}$/$\mathrm{ZrO_2}$
superlattices and HZO solid-state solutions. In this work we systematically
investigated the electronic structures of $\mathrm{HfO_2}$, $\mathrm{ZrO_2}$
and HZO using self-energy corrected density functional theory. In particular,
the conduction band minimum of $Pca2_1$-$\mathrm{HfO_2}$ is found to lie at an
ordinary k-point on the Brillouin zone border, not related to any interlines
between high-symmetry k-points. Moreover, the rule of HZO band gap variation
with respect to x has been extracted. The physical mechanisms for the
exponential reduction regime and linear decay regime have been revealed. The
band gaps of $\mathrm{HfO_2}$/$\mathrm{ZrO_2}$ ferroelectric superlattices are
investigated in a systematic manner, and the reason why the superlattice could
possess a band gap lower than that of $\mathrm{ZrO_2}$ is revealed through
comprehensive analysis.
