Group Theory Analysis to Study Phase Transitions of Quasi-2D Sr3Hf2O7

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 Sr3Hf2O7 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 → Cmc21 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 Cmc21 phase is ferroelectric with a value of spontaneous polarization of 0.0478 C/m2. The ferroelectricity of the ground-state Cmc21 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.


Introduction
Ruddlesden Popper (RP) layered perovskites are formed by a rock-salt layer positioned between perovskite (ABO 3 )n blocks-(AO)(ABO 3 )n, either within double-layered compounds-A 3 B 2 O 7 [1][2][3] or in A-site ordered single-layered compounds-AA'BO 4 [4,5]. These quasi-2D structures present advantages of possessing similar properties to that of conventional 2D materials [6], without the need to scale down the system to the atomic-layer thickness. Such properties include quantum confinement effects which lead to large exciton binding energies, high quantum yields and other photophysics properties, making these systems also desirable candidates for next-generation energy-efficient optoelectronics [6,7]. experimentally observed the Cmcm and Ccca phases, respectively, as being intermediate structures that constitute the pathway between the high-temperature and ground-state systems of the CMO compound. Moreover, with respect to the Sr-based systems, experimental studies by Yoshida et al. [2,20] confirmed the existence of an intermediate nonpolar Cmcm phase for Sr 3 Zr 2 O 7 and Ccce for Sr 3 Sn 2 O 7 along the structural transition from I4/mmm to Cmc2 1 . Moreover, it is found that for both the Sr 3 Zr 2 O 7 and the Sr 3 Sn 2 O 7 systems an antipolar phase, the Pbcn (S.G.60), competes with the equilibrium polar Cmc2 1 phase [20]. On the other hand, Wang et al. [19] predicts that it is the Cmca phase which will result from the orthorhombic structural distortions together with Cmc2 1 , therefore resulting in orthorhombic twins.
We intend to demonstrate from the present work that by employing first-principles calculations on SHO, it is possible to predict possible transition pathways based on groupsubgroup relations, as shown by Benedek and Fennie [13,15] on the CMO systems. We show an energetic stability trend of the potential intermediate phases between the I4/mmm towards the Cmc2 1 system, with the electronic energy lowering as the symmetry decreases. On the other hand, we observe an opposite trend of the electronic band gaps, since the widths increase as the symmetry of the structure decreases. By analyzing the lattice dynamics of all the potential systems we observe imaginary modes for all the phases (consistent of being transition states for the given conditions). Exception is observed for the Cmc2 1 polymorph, which evidences dynamical stability and therefore results confirm that Cmc2 1 is in fact the ground-state system for SHO. By computing the spontaneous polarization, we show that the Cmc2 1 phase is ferroelectric; these results are consistent to what has been reported in literature for the same family of compounds, i.e., CMO, CTO [3,13,18] and also for SHO [21].
Nuclear hyperfine techniques are particularly important to study local magnetic and electric interactions, such as the electric field gradient (EFG), at a given nuclear site. The measurement of the EFG, through perturbed angular correlations (PAC) experiments, combined with ab-initio density functional theory (DFT) calculations, has shown to be a valuable tool to probe the octahedral rotations of the cages during phase transition [18]. Therefore, we have computed the EFG for each structural phase and observe variations of respective parameters which allow the possibility to relate the intermediate phases with the most plausible structural transition pathway.

Materials and Methods
The calculations have been performed within the framework of the density functional theory (DFT) [24,25] in the Kohn-Sham scheme, as implemented in the Quantum Espresso (QE) code [26][27][28] and in WIEN2k [29]. The exchange-correlation functional was described by the semi-local generalized-gradient approximation with the Perdew-Burke-Ernzerhof parametrization revised for solids (PBEsol) [30,31].
The theoretical background regarding the harmonic lattice-dynamics calculations is detailed in References [35,36]. Lattice-dynamics calculations were performed using the supercell finite-displacement method implemented in the Phonopy software package [37], with QE used as the 2nd order force-constant calculator. Calculations to obtain the phonon dispersion curves were carried out on 2 × 2 × 2 expansions of the primitive-cell.
The Full Potential Linearized Augmented Plane Wave (FP-LAPW) method as implemented in the WIEN2k code [29] was employed to obtain the hyperfine parameters including electric field gradients at the nuclear sites (EFG) (largest diagonal component in the principal axis system, V zz , and the asymmetry parameter, η), have been calculated for the different phases of SHO, namely I4/mmm, Cmcm, Cmca, Ccce, Fmm2, Cmc2 1 (structures represented in Figure 1). As an all-electron method, WIEN2k has proven to be a benchmark to compute hyperfine parameters; the macroscopic polarization, by employing the Berry's phase expressions [38][39][40], have also been computed with the WIEN2k code and for the above mentioned structural phases. The Sr, Hf, and O radii of the muffin-tin atomic spheres were respectively set to 2.25, 2.07, and 1.78 a.u., the energy boundary between core and valence electronic states was -6 Ry, the cut-off parameter RMT × KMAX (which controls the size of the basis set) was 7.0 and G max , the Fourier expansion of the charge density, was restricted to 16 Ry 1/2 . The geometry optimizations of different phases were obtained through the total energy minimization with respect to the unit cell volume using the 3rd order Birch-Murnaghan equation of state (EoS) [41,42], and internal atomic positions were minimized to a force limit below 1 mRy/a.u. A mesh of (6 × 6 × 6) k-points in the irreducible part of the first BZ was applied to the self-consistent total energy calculation. We have also employed the modified Becke-Johnson (mBJ) exchange-correlation functional [43] to compute the band gap energies. The mBJ functional provides an efficient framework for band gap prediction and is an appealing alternative to hybrid functionals and the many-body perturbation GW method for large systems and has been successfully applied to different perovskites structures [44][45][46][47]. Representation of the different structural phases of Sr 3 Hf 2 O 7 , namely the parent I4/mmm structure, followed by the Ccce, Fmm2, Cmcm, Cmca and the ground-state Cmc2 1 structural phases. The Sr 1 and the Sr 2 located at the A' and at the A sites, respectively, are labeled accordingly on the I4/mmm phase, whose label can be extrapolated to the remaining phases. O ions are shown in red, Sr in green and Hf in gold. The figures were plotted using the VESTA visualization software [48].
The calculation of the EFG tensor can provide the signature of observable pathways on the structural phase transition from the I4/mmm system towards the ground-state Cmc2 1 structure. The EFG tensor components are defined as the second order spatial derivatives of the self-consistent potential V evaluated at a given nuclear site [49] where i = x, y, z. Since V ij is a symmetric (traceless) second rank tensor, it can be diagonalized (the principal system of axis) and the largest diagonal component V zz is commonly referred as the EFG and defined such that |V zz | |V yy | |V xx |. Due to the traceless property, only two parameters are enough to characterize the EFG tensor and these are usually chosen as V zz and the asymmetry parameter

Results
The general formula of SHO can be written as (AHfO 3 ) 2 A'O with A' = Sr 1 and A = Sr 2 . The environment around the two Sr atoms are inequivalent. As shown in Figure 1, different structures are formed by rock-salt (R) and perovskite (P) blocks which include the Sr 2 located at the A site and the Sr 1 on the A' site, respectively. The P block consists of two layers of corner-sharing HfO 6 octahedra along the c-axis. While the A'-site cations occupy a twelve-fold coordination environment (cuboctahedral), the A cation is at the nine-coordination site. As observed in Figure 1, the I4/mmm structure exhibits higher symmetry being the paraelectric high-temperature phase of the system, analogously to other RP oxides, such as Ca 3 Ti 2 O 7 and Ca 3 Mn 2 O 7 [13]. By decreasing the temperature, lower symmetry structural phases may be generated by inducing tiltings and/or rotations of the O octahedral cages, namely the Ccce, Fmm2, Cmcm, Cmca, and  Table 1. Small differences, in the usual range (3-4% in volume), can be observed due to the different methods used in these codes.

Symmetry Mode Analysis
As mentioned above regarding the effect of the temperature lowering, most of the low-symmetry perovskite phases can be derived from the high-symmetry structure by rigid rotations and tiltings of the octahedral units around respective symmetry axes. The structural distortions relating the two phases can be denoted as a symmetry-breaking distortion [50,51]. The equilibrium position of the cation can therefore be determined by the positions and rotations/tiltings of the anions forming the octahedral cage for a given set of bond angles. By employing symmetry-mode analysis, it is possible to fully describe the distorted ground-state Cmc2 1 structure (S.G. 36) by decomposing the structural distortion into contributions from lattice modes with different symmetries. These are characterized by the irreducible representations (IR) of the centrosymmetric tetragonal I4/mmm (S.G. 139) structure. The analysis is performed using the software package AMPLIMODES [52] of the Bilbao Crystallographic Server, being useful to determine the driving mechanisms of the structural phase transitions and the fundamental instabilities at the origin of the distorted phases. By providing the high-and low-symmetry structures, the use of AMPLIMODES [52] allows to perform atomic mappings (pairings) by identifying which atoms in the low symmetry structure correspond to the asymmetry unit of the reference structure and therefore computing the atomic displacements, which relate both structures.
From this approach, it is then possible to define a basis of symmetry-adapted modes and calculate the amplitudes and polarization vectors from the high-symmetry to the distorted structure [50,51,53]. Table 2 presents a summary of the basis of symmetry modes and respective IR which describe the atomic displacements in each Wyckoff orbit of the high symmetry phase. There are a total number of nineteen basis modes involving the displacements of the different atomic sites in the system. Eleven modes correspond to displacements of the three different O sites, namely with two referring to the sites at the 2a WP (Γ − 5 and X − 3 ), six to 8g (Γ + 1 , Γ − 5 , X + 2 , X − 3 ) and three to the 4e WP (Γ + 1 , Γ − 5 and X − 3 ). Three modes refer to displacements of the Hf sites, namely Γ + 1 , Γ − 5 and X − 3 . Regarding the Sr sites, two modes are related to the Sr 1 displacements (Γ − 5 and X − 3 ) and three to the Sr 2 site (Γ + 1 , Γ − 5 and X − 3 ). The atomic displacements of all atoms, multiplied by a given amplitude, yields the actual distortion of the low-symmetry structure. All O cages of the low-symmetry ferroelectric phase are rotated with respect to the I4/mmm phase; these rotations force the Sr atoms to deviate from the centrosymmetric positions, resulting in the antiferroelectric displacement of Sr atoms at the A-site. Noncanceled displacements then induce ferroelectric polarization.  Table 3 summarizes the mode decomposition, which lists the IR involved in the I4/mmm → Cmc2 1 distortion and the absolute amplitudes of the symmetry components of the global distortion. By analyzing Table 3, it can be observed that the modes with highest amplitudes are X + 2 and X − 3 , with values of Q X + 2 = 0.5487 Å and Q X − 3 = 0.7838 Å, respectively, indicating that these modes are the major contributors to the distortion towards the low symmetry phase. By analyzing Figure 2, it is possible to deduce that the X + 2 and X − 3 IR correspond to in-plane rotations and the tilting of the perovskite O cages (HfO 6 ), respectively, whereas IR Γ − 5 corresponds mostly to the ferroelectric displacement of the Sr ions. It is noteworthy to mention that based on Table 2, we can verify that the Γ − 5 mode also corresponds to displacements of the O atoms; however, these are basically negligible in comparison to the Sr movement, as can be observed in Figure 2. The IR modes obtained from respective analysis is compatible with the global structural distortion between the high-and low-symmetry phases relating both phases and these are compatible with symmetry-breakings observed in the same family of systems [3,13,15,18].

Atoms WP Modes
Cmcm (63) 0.7838 Figure 2. Decomposition of the structural distortion from I4/mmm → Cmc2 1 into contributions from lattice modes with different symmetries. The distorted structure derives from the high-symmetry structure through four frozen distortions, Γ + 1 , Γ − 5 , X + 2 and X − 3 . Oxygen ions are shown in red, Sr in green and Hf in gold. The figures were plotted by using the VESTA visualization software [48]. Figure 3 presents the potential energy surface along each distortion mode (frozen mode). The end-point u = 0 corresponds to the high-symmetry I4/mmm structure, and u = 1 represents the distortion corresponding to the isotropy subgroup of the respective frozen mode. The different frozen-mode distortions would thus correspond to the subgroups referenced in Table 3: Γ + 1 distortion to the symmetry breaking towards the I4/mmm symmetry; Γ − 5 distortion would correspond to the symmetry breaking towards the Fmm2 phase; and X + 2 and X − 3 distortions would correspond to the symmetry breaking towards the Cmca and Cmcm symmetry, respectively. The sum of the four different u = 1 end-points will agree with the ground-state Cmc2 1 symmetry. It can be observed that the energy lowers considerably along the X + 2 and X − 3 modes, which means that these distortions are mainly responsible for the symmetry-lowering to the Cmc2 1 phase. This corresponds to what has been discussed above (Table 3) and to these modes contributing with more weight to the distortion (Q X + 2 = 0.5487 Å and Q X − 3 = 0.7838 Å). These two modes are mostly due to the O cages, which evidence in-plane rotations and tiltings, respectively. This feature is consistent with what is observed from the electronic PDoS (Figure 4), for which the O-states are those which present major variations for the different studied structural phases. The Γ + 1 will be omitted from the discussion since the amplitude distortion is practically negligible when compared to the other distortion amplitudes. The Γ − 5 mode refers to motions of the Sr ions and produces an increase in energy when the frozen distortion occurs. In combination with other distortions that occur for the surrounding cage, the Sr movement leads to an average reduction in the overall energy of the SHO system.  In the following sections, we also include discussions of the analysis performed for another structural phase, i.e., Ccce (S.G. 68), since this is an experimentally known phase of the CMO system. Through space group relations, the Ccce phase is obtained through the out-of-plane rotations of the O cages, which corresponds to the condensation of a zone boundary X − 1 [18].

The Relative Electronic Energy
The enthalpy of formation of a crystal can be calculated by using the following expression: where E SHO is the total electronic energy (obtained from DFT calculations) and µ Sr , µ Hf and µ O are the chemical potentials of the constituent Sr, Hf and O elements, respectively. We must note that when comparing the enthalpies of formation among the different phases with the same composition, the chemical potentials cancel out, and therefore only the E SHO term survives. We have computed the total energies of the representative structures that constitute the pathway between the paraelectric reference I4/mmm phase and the ferroelectric groundstates Cmc2 1 structure (Table 4). From Table 4 we observe that the polar Cmc2 1 phase presents the lowest energy, which is in good agreement with Ref. [21], establishing that the SHO ground-state is therefore ferroelectric. The system with higher energy is the tetragonal I4/mmm, followed by the Ccce, Fmm2, Cmcm, and Cmca. As mentioned before, the Ccce phase does not constitute the pathway between I4/mmm → Cmc2 1 according to group theoretical analysis; however, for the sake of completeness we have computed the structural parameters for this structure as well. From this analysis, we concluded that the energy ordering agrees with the discussion performed in Section 3.1, for which the most plausible transition pathway from the high-to the low-symmetry phase would be: I4/mmm < Fmm2 < Cmcm < Cmca < Cmc2 1 . The energy of the Ccce phase is higher than that of the Fmm2 system. However, to induce the Ccce structure the condensation of the X − 1 mode would be required, which, and according to group theory analysis, does not occur naturally for the system. However, such a phase has been experimentally observed in Ca 3 Mn 2 O 7 [18], which raises the possibility of occurrence of the Ccce phase also in SHO but as a first-order phase transition. Moreover, it has been observed that the intermediate Ccce structural phase may also occur in the Sr 3 Sn 2 O 7 system [2], therefore providing further foundations for the theoretically investigation of this structural polymorph.  (36) 0.000

Partial Density of States
In order to investigate the effect of octahedral tilts/rotations on the electronic properties of SHO along the plausible path of the phase transition, the partial density of states (PDoS) have been computed ( Figure 4). As observed in Figure 4, the conduction band minimum (CBM) is mostly defined by the Hf d-states, whereas the fully occupied O p-states form the valence band maximum (VBM). There is considerable hybridization at the higher energy levels of the conduction band, among the different Hf and Sr d-states. The overall PDoS features of the studied SHO structural phases are similar and all evidence an insulator behavior. Major differences are observed for the O p-states, which can be related to rotations of the octahedral cages, as mentioned in Section 3.1, and evidence largest contribution to the symmetry-breaking in the path towards the ground-state.
Moreover, the Ccce and Cmca structures evidence differences at the CBM. While for the remaining phases the CBM is mostly defined by the Hf d-states, these two mentioned Ccce and Cmca structures show hybridized Hf and Sr d-states and O p-states at the CBM. These features evidence dissimilarities with the remaining phases, probably caused by the out-of-plane and in-plane rotations of the octahedral cages (Sr d-states and O p-states), respectively, which lead to variations of the Hf d-states.
The width of the energy band gaps by employing the PBEsol and mBJ exchangecorrelation functionals are summarized in Table 5. The mBJ approximation provides higher values than the PBEsol functional, as expected, due to the nature of the meta-GGA functional. To the best of our knowledge no experimental results regarding the band gaps of SHO have been reported, nevertheless the presently obtained PBEsol gap width for the Cmc2 1 phase is in good agreement with the 4.2 eV value obtained in Reference [21]. We must also note that the gap we have obtained is larger than in other RP systems, namely, for the heavier element system Ba 3 Ce 2 O 7 [54] (∼2.4 eV using the PBEsol functional) and the Ca 3 Mn 2 O 7 (∼0.8 eV using the PBE). We note that the higher symmetry structures present lowest gap widths, a feature observed for both employed functionals (Table 5). This may be due to the fact that the top of the valence band is primarily formed by O-p states which, and as can be evidenced from Figure 4, exhibit a larger splitting in the lower symmetry systems. such a feature is consistent with the changes occurring in the O perovskite cages.

Phonon Dispersion Curves
We have performed lattice dynamics calculations in order to analyse the dynamical (in)stability of the different studied phases of SHO.
By inspecting Figure 5, we can observe that the I4/mmm phase shows imaginary modes (represented by negative phonon frequencies) at the zone boundaries, more specifically at the high-symmetry points X, P and N. The negative modes at the X-point are consistent with the symmetry mode analysis and related to the rotations and tilts of the O cages, namely the X + 2 and the X − 3 modes. As the distortion lowers the symmetry to Fmm2, the negative modes delocalize throughout the whole BZ (with exception of the Y-and Γ-points), characterizing the respective space group as being unstable, and thus a transition state to yet another phase.
According to Table 3, Fmm2 is related to a mode that transforms as Γ − 5 IR and this mode is shown to be stable in the phonon dispersion of the I4/mmm phase. The phonon dispersion further shows that these antidisplacements of the Sr atoms do not occur alone, since they are delocalized in the whole BZ. The results suggest that the octahedral instabilities drive the transition from the I4/mmm either to the Cmcm or Cmca phase, where imaginary modes occur at the Y-and Γ-points. We must note that the Cmca system shows quite localized imaginary frequencies located at the high-symmetry points of Y and the zonecenter Γ. Interestingly enough the Ccce shows similarities in the phonon dispersions when compared to the Cmca system, with negative phonon branches located at the highsymmetry Y-and Γ-points. The distortions related to the tilting of the octahedral cages show similar phonon branches, with negative modes also localized at Y-and Γ-points. The only phonon dispersion spectra for which we cannot find any imaginary frequencies is for the Cmc2 1 phase, indicating that this phase is dynamically stable at 0 K for SHO, and in accordance to that obtained in Reference [21]. Moreover, as explained in Reference [13], the coupling between the oxygen octahedron rotation mode (X + 2 ) and an oxygen octahedron tilt mode (X − 3 ) establishes the polar Cmc2 1 space group, and for which a zone-center polar instability is not required (Γ − 5 ).

Spontaneous Polarization
The spontaneous polarization of SHO for the different phases (end-points and intermediate) have been calculated using the Berry phase approach. The macroscopic polarization resulted in finite values for the Fmm2 and Cmc2 1 phases, with P = 0.319 C/m 2 and P = 0.0478 C/m 2 , respectively. The latter value is consistent with that computed in Reference [21], where it has been found that P = 0.043 C/m 2 . The value of Fmm2 is quite large, which is expected since the movement of the Sr ions out of the equilibrium positions generate polarization. Since the Γ − 5 mode is polar, proper ferroelectricity is induced from the I4/mmm to the Fmm2 phase. However, and based on the phonon dispersion curves (mentioned above in Section 3.4), this transition would probably not occur since the system is dynamically unstable, evidencing negative frequencies throughout the whole BZ. For the I4/mmm, Cmcm, Ccce, and Cmca phases the polarization is null, which is consistent with the antiferrodistortive displacements of the O perovskite cages and also since these structures are centrosymmetric the polarization must be zero. We must also note that the value of polarization of the Cmc2 1 phase is comparable with those found for other compounds, such as of Sr 3 Sn 2 O 7 with P = 0 .039 C/m 2 and Sr 3 Zr 2 O 7 with P = 0.072 C/m 2 [4]. The polarization of the Sr 3 B 2 O 7 (B = Hf, Sn, Zr) compounds are found to be lower than the value of the Ca 3 Ti 2 O 7 (1.0 C/m 2 [1]) and Ca 3 Mn 2 O 7 (0.5 C/m 2 ) [13] systems, what can be related to the restricted atomic displacement of Sr (atomic radii: 215 pm) from the ideal position when compared to that of Ca (atomic radii: 197 pm), due to the cations radii size.

The Hyperfine Parameters of SHO
The calculated hyperfine parameters, namely the EFG principal component V zz and the asymmetry parameter η, at Sr 1 , Sr 2 and Hf atoms in the different phases, are shown in Figure 6. From our results the different pathways from the high symmetric phase, I4/mmm, and the polar Cmc2 1 structure, can be followed through the V zz and η parameters. At the Hf site, we observe that the lowest values for V zz are found in the high-symmetry I4/mmm phase, with V zz = 16.38 V/Å 2 and η = 0.00 due to the center-symmetric positions of the nondistorted octahedral HfO 6 cages. On the other hand, the largest magnitude of η, at the Hf site, (η = 0.70) was obtained for the Cmcm system, which corresponds to the tilting of the cages and the largest contribution to the distortion towards the polar ground-state phase (as observed in Table 3). However, the V zz = 48.87 V/Å 2 , obtained for the Cmcm phase is lower than that of the Cmca phase (associated with the in-plane rotations of the HfO 6 perovskite cages as shown in Figure 6) corresponding to the largest magnitude of the principal EFG component, with V zz = −89.63 V/Å 2 and η = 0.13. The characterization of the Sr local environments may also allow one to probe the HfO 6 octahedral rotations that underlie the structural phase transitions of the system. Therefore, with respect to the rock-salt Sr 2 sites, for Fmm2 phase, we find that the value of V zz and η are very close to that of the high symmetric I4/mmm phase, with V zz = −76.44 V/Å 2 , η = 0.00 and V zz = −77.91 V/Å 2 , η = 0.00 respectively corresponding to the absence of the octahedral rotations and/or tilts of the rock-salt environment. This similarity is understandable as the Fmm2 phase results from a very small amplitude displacement of the Sr ions. Upon octahedral rotations we observe that the value of η increases significantly in accordance to what was recently observed in Ca 2 MnO 4 where η increases from 0 to 0.8 as the Mn-O-Mn bond angle decreases from 180º to 160º [55]. Moreover, we observe that the asymmetry parameter evidences the largest values with η = 0.77 and η = 0.96, for the Cmcm and the ground-state Cmc2 1 , respectively. In these two mentioned phases the lowest values for V zz are found with V zz = −54.48 V/Å 2 and V zz = −58.31 V/Å 2 . Interestingly, we observe that the highest values of V zz at the rock-salt site are observed for the Cmca and Ccce, which are the phases which respectively correspond to the in-plane and out-of-plane rotations of the cages ( Figure 6). The asymmetry parameters are very close to each other (η = 0.606 and η = 0.672).
The EFG values found here for the I4/mmm, Ccce, and Cmc2 1 phases at the Sr 2 rock-salt site and Sr 1 perovskite site show similar trends as those calculated for Ca 3 Mn 2 O 7 system at analogous sites [18]. In addition, we observe that the largest η values at the perovskite sites, Sr 1 , are for the ground-state Cmc2 1 , with η = 0.73, and followed by the Cmcm phase with η = 0.72. On the other hand, the Cmca phase (X + 2 mode) evidences largest V zz magnitudes, not only for the Hf sites, as already mentioned, but also for both Sr 1 and Sr 2 sites, with V zz = 74.21 V/Å 2 and V zz = −119.55 V/Å 2 , respectively.
As mentioned before, the Ccce phase is not considered to be in the transition pathway from I4/mmm to Cmc2 1 , when assuming group theory analysis, as they are not in a group-subgroup relation. Nevertheless, this phase might manifest as an intermediate one, displaying a discontinuous first-order phase transition given the considerable gain of total energy with respect to I4/mmm. The EFGs values of the Ccce system (V zz = −58.21 V/Å 2 , V zz = −90.77 V/Å 2 and V zz = 72.37 V/Å 2 for the Sr 1 , Sr 2 and Hf sites, respectively) are larger than the ones found for Cmc2 1 , Cmcm, Fmm2, and I4/mmm phases, although with lower value when compared to the Cmca phase. Moreover, the asymmetry parameter, η, for the rock-salt Sr 2 -site is also quite high, with η = 0.67; however, for the remaining two sites η approximates to zero. These results are in agreement with the experimental data in Reference [18] for the Ca 3 Mn 2 O 7 compound, in which the highest magnitudes of V zz were observed for the Ccce phase when compared to the ground-state structural or I4/mmm phase.
In a more comprehensive point of view, and considering a close inspection of the EFG behavior at each site (perovskite Sr 1 , rock-salt Sr 2 and Hf sites) across the potential phases, which constitutes the transition pathway from I4/mmm → Cmc2 1 , clear EFG signatures can be observed. For example, for the rock-salt site the I4/mmm → Cmca → Cmc2 1 pathway has the following EFG signature: V zz = −77.91 V/Å 2 η = 0.00 → V zz = −119.55 V/Å 2 η = 0.61 → V zz = −58.31 V/Å 2 η = 0.96, as depicted in Figure 6. In addition, the hypothetical I4/mmm → Ccce → Cmc2 1 path shows the difference in the intermediate phase with V zz = −90.77 V/Å 2 , η = 0.67, with magnitudes of both parameters being quite close magnitudes of the asymmetric parameters. A similar situation is observed for the Sr 1 and Hf sites where distinguishable EFG (V zz , η pairs) within each EFG paths are evident (see Figure 6). These results show that an experimental EFG measurement, by using hyperfine techniques such as nuclear quadrupole resonance (NQR) or perturbed angular correlation (PAC), might allow to ascertain for the correct phases/pathways, namely the ones connected by group-subgroup relations. In such a way, these techniques can inform about the subtle octahedral tilting and rotations which are typically not easily accessible by long-range crystallographic techniques. The potential transition pathway, viewed from the rock-salt (Sr 2 ) site, may be correctly studied using PAC, as demonstrated in [18,55]. Moreover, the Hf site may be studied using the 181 Hf PAC probe. In addition, and as recently mentioned by Zao et al. [56], a local description of oxides is fundamental in the understanding of their properties, since polymorphs should be present in a given sample. In this sense, hyperfine quantities are especially well suited to probe very local environments.

Discussion and Conclusions
From symmetry mode analysis and energetic stability we infer that the most probable pathway transition from the aristotype towards the polar ground-state system of SHO is: I4/mmm > Cmcm > Cmca > Cmc2 1 . The energetic trend is here confirmed. Although the energetics of the Ccce phase is higher than that of Fmm2, we do not consider it to constitute the pathway between I4/mmm → Cmc2 1 according to group theory analysis, since the condensation of the X − 1 mode does not naturally occur as being a continuous second order phase transition for the SHO system. However, there is the possibility of such a mode being induced as a first-order transition (for example, through application of an external perturbation such as pressure), as observed in the literature for other systems. Moreover, the band gap widths also increase in a similar trend according to the energetic stability and shows slight variations along the potential phases that constitute the transition pathway. We must, however, note that the symmetry-breaking is mostly driven by the O octahedral rotations with different symmetry modes, namely the X + 2 and the X − 3 distortions, which respectively lower the high-symmetry to the Cmcm and Cmca phases. These O rotations and tilts are the primary order parameter of the phase transition for which a secondary order parameter will induce the spontaneous polarization of the Cmc2 1 system. The combination of both the Q X + 2 and Q X − 3 modes lower the overall energy of the system and the polarization arises due to the coupling of a hybrid order parameter (tilts and rotations of the oxygen octahedral cages and the Sr displacements). Based on the phonon dispersions, we may infer that the Fmm2 may not exist, since we observe that the phonon branches are stable at the Γ-point for the I4/mmm structure, implying that the system will most likely not undergo a transition to this phase. In addition, by observing Figure 3 it is here shown that the Γ − 5 mode alone increases the total energy, further evidencing that the I4/mmm → Fmm2 transition will most likely not occur. The I4/mmm instability at the X-point will drive the transition directly to the Cmcm or Cmca space groups, from which the Γ-point imaginary modes of either these two phases will probably direct the final transition to Cmc2 1 . We observe that the magnitudes of V zz and η are extremely sensitive to the octahedral rotations and tilting distortions across the phase transition path. Clearly distinct EFG signatures are observed at each (perovskite Sr 1 , rock-salt Sr 2 and Hf) site for the potential phase that constitutes the transition pathway from I4/mmm → Cmc2 1 . These EFG results are particularly relevant to establish experimentally the theoretical predicted phases/pathways using hyperfine measurements, especially when crystallographic long-range methods fail. We hope that the results presented here will stimulate further measurements into this challenging system.