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Interplay of lattice distortion and bands near the Fermi level in $A$TiO$_3$ ($A$=Ca, Sr, Ba)

The structural and electronic properties of $A$TiO$_3$ ($A$=Ca, Sr, Ba) have been investigated under strain-free situation and with realistic constraints using first-principles calculations. We endeavored to find out the interplay between mild lattice distortions and bandgap in three $A$TiO$_3$ family members that has remained skeptical to date. We found out that the electronic structure was particularly sensitive to strains (compressive or tensile) as expected in most materials science studies. Our results indicate that under mild strains; the bandgap ($E_{gap}$), increased under compression and decreased under tension. In all the three materials, the bandgap and the lattice parameter ($a$) were found to relate as $E_{gap}\propto \frac{1}{a^x}$ for mild distortions with $2.19<x<3.1$. All these changes are attributed to the interplay of electrostatics and covalency in these crystals. This work acts as a yardstick on bandgap engineering to achieve desired properties in these titanates for feasible future applications.