Group Theory Analysis to Study Phase Transitions of Quasi-2D Sr$_{3}$Hf$_{2}$O$_{7}$

We present an $ab-initio$ study performed in the framework of density functional theory, group-subgroup symmetry analysis and lattice dynamics, to probe the octahedral distortions, which occur during the structural phase transitions of the quasi-2D layered perovskite Sr$_{3}$Hf$_{2}$O$_{7}$ compound. Such a system is characterized by a high-temperature I4/mmm centrosymmetric structure and a ground-state Cmc21 ferroelectric phase. We have probed potential candidate polymorphs that may form the $I4/mmm$ → $Cmc2$$_{1}$ transition pathways, namely $Fmm2$, $Ccce$, $Cmca$ and $Cmcm$. We found that the band gap widths increase as the symmetry decreases, with the ground-state structure presenting the largest gap width (∼5.95 eV). By probing the Partial Density of States, we observe a direct relation regarding the tilts and rotations of the oxygen perovskite cages as the transition occurs; these show large variations mostly of the O $p$-states which contribute mostly to the valence band maximum. Moreover, by analyzing the hyperfine parameters, namely the Electric Field Gradients and asymmetric parameters, we observe variations as the transition occurs, from which it is possible to identify the most plausible intermediate phases. We have also computed the macroscopic polarization and confirm that the $Cmc2$$_{1}$ phase is ferroelectric with a value of spontaneous polarization of 0.0478 C/m$^{2}$. The ferroelectricity of the ground-state $Cmc2$$_{1}$1 system arises due to a second order parameter related to the coupling of the rotation and tilts of the O perovskite cages together with the Sr displacements.