Pressure-Induced Phase Transformations of Quasi-2D Sr3Hf2O7

We present an abinitio study of the quasi-2D layered perovskite Sr3Hf2O7 compound, performed within the framework of the density functional theory and lattice dynamics analysis. At high temperatures, this compound takes a I4/mmm centrosymmetric structure (S.G. n. 139); as the temperature is lowered, the symmetry is broken into other intermediate polymorphs before reaching the ground-state structure, which is the Cmc21 ferroelectric phase (S.G. n. 36). One of these intermediate polymorphs is the Ccce structural phase (S.G. n. 68). Additionally, we have probed the C2/c system (S.G n. 15), which was obtained by following the atomic displacements corresponding to the eigenvectors of the imaginary frequency mode localized at the Γ-point of the Ccce phase. By observing the enthalpies at low pressures, we found that the Cmc21 phase is thermodynamically the most stable. Our results show that the I4/mmm and C2/c phases never stabilize in the 0–20 GPa range of pressure values. On the other hand, the Ccce phase becomes energetically more stable at around 17 GPa, surpassing the Cmc21 structure. By considering the effect of entropy and the constant-volume free energies, we observe that the Cmc21 polymorph is energetically the most stable phase at low temperature; however, at 350 K, the Ccce system becomes the most stable. By probing the volume-dependent free energies at 19 GPa, we see that Ccce is always the most stable phase between the two structures and also throughout the studied temperature range. When analyzing the phonon dispersion frequencies, we conclude that the Ccce system becomes dynamically stable only around 19–20 GPa and that the Cmc21 phase is metastable up to 30 GPa.


INTRODUCTION
Rational design and the discovery of new materials are essential toward the advancement of future technological innovations.Perovskite structures (with general chemical formula ABX 3 , where A and B are typically large and small cations, respectively, and X is an anion) are one of the most versatile classes of materials, with a variety of fascinating characteristics such as magnetism, superconductivity, thermoelectricity, electrocatalysis, ferroelectricity, and optical properties. 1,2hile the greater part of the known perovskite oxides are centrosymmetric, a small fraction of these crystallizes as noncentrosymmetric (polar perovskites). 3−6 Among perovskites, naturally layered perovskite oxides, such as quasi-2D Ruddlesden−Popper (RP) and Dion−Jacobson (DJ) structures, have particular interest because of their recently discovered mechanism for ferroelectricity, termed as "hybrid improper ferroelectricity" (HIF). 7,8Structures with HIF exhibit two non-polar structural distortion modes, which lead to the third polar distortion mode, and are known as the trilinear coupling, resulting in a macroscopic spontaneous polarization. 7,9For perovskite structures which undergo distortions toward the Pnma space group (i.e., doubleperovskite structures), the HIF phenomenon leads to an antiferro-distortive displacement of the A-site cations and the adjacent layer displacement in opposite directions. 10,11The displacements of the adjacent layers cancel each other for ABO 3 single perovskites, leading to non-polar phases.However, naturally layered quasi-2D perovskites, with disconnected octahedral layers and having two inequivalent crystallographic A-sites, exhibit macroscopic layer polarization, 12,13 which are oriented oppositely with slightly different intensities.In the RP structured naturally layered oxides, with the (AO)(ABO 3 ) n stoichiometry form, the nlayers of the ABO 3 perovskite slabs are stacked between the AO rock-salt layers, with a relative shift of the neighboring perovskite slabs by a ( 1 2 , 1 2 ) translation.By increasing the value of n from 1 to ∞, it is possible to tune the structural dimensionality of these perovskites from quasi-2D to 3D.At high temperatures, each of the n layers consists of vertex sharing oxygen octahedra with a B-ion sitting at its center. 7,9As temperature decreases, these systems can undergo structural phase transitions due to reorientation of oxygen octahedra, leading to HIF.The HIF is also predicted to occur in oxide superlattices. 14In addition to HIF, the octahedral reorientations in layered perovskites are of continuing interest because these effects also trigger many interesting electronic properties, such as high-T c superconductivity, colossal magnetoresistance, metal-insulator transitions, and magnetic ordering. 15ecent theoretical works from Liu et al. 16 and da Silva et al. 17 confirmed that Sr 3 Hf 2 O 7 (SHO), a n = 2 quasi-2D RP layered perovskite system, with a ferroelectric ground state Cmc2 1 (S.G. n. 36), is similar to the previously reported HIF systems, such as Ca 2 Mn 3 O 7 and Ca 2 Ti 3 O 7 , in ref 15.SHO is also characterized by a high-temperature I4/mmm (S.G. n. 139) centrosymmetric paraelectric structure.Previous studies showed that by applying group theoretical analysis, it was possible to probe potential intermediate polymorphs that form as the temperature is lowered, enabling a transition pathway as I4/mmm → Cmcm (S.G. n. 63) → Cmca (S.G. n. 64) → Cmc2 1 (S.G. n. 36). 7,17The Ccce (S.G. n. 68) structural phase is a possible polymorph; however, it is not evidenced as being an intermediate structure, occurring for these structural RP symmetries, according to group theoretical analysis�a condensation of the X 1 − zone boundary mode would have to occur upon symmetry breaking of the centrosymmetric system.It was also found by computing the phonon dispersion curves of the Ccce phase 17 that the system is dynamically unstable at 0 K and 0 GPa, evidencing negative phonon modes localized at two of the high-symmetry points of the Brillouin zone (BZ): Γand Y-points.Therefore, the Ccce polymorph does not have the potential to crystallize at room conditions.We must note that such a phase has been experimentally evidenced as an intermediate phase for increasing temperatures in systems with similar stoichiometry, i.e., Ca 3 (Mn/Ti) 2 O 7 , 18,19 thus making it plausible for this polymorph to form through the first-order transition for the SHO system as well.
Only recently have high-pressure (HP) studies been devoted to understand the properties of the perovskite systems and the respective transition pathways.Pressure is an important thermodynamic variable, for it enables a precise control over the interatomic distances and hence the atomic interactions, in turn driving phase-transitions.Almost all materials will undergo several phase-transitions under compression, thus generating new polymorphs with promising properties, different to those properties of materials under room conditions.It has been found that factors which are known to control the ABO 3 perovskites pressure response, such as the A-and B-site cation formal charges, the tolerance factor, and the B-site chemical environments, also affect the pressure response of the layered perovskites. 20Therefore, the interest in applying pressure to probe the structural, electronic, and vibrational properties to such structures.
A recent experimental study on Sr 3 Sn 2 O 7 compound evidences a sequence of pressure-induced phase transitions as Cmc2 1 → Pbcn (S.G. n. 60) → Ccce → I4/mmm at room temperature. 21Such transition pathway would match the sequence of temperature-dependent structural transitions observed for this compound between 77 and 1000 K.For the Ca 3 M 2 O 7 -based oxides (M = Ti, Mn), 22 ferroelectric switching is inhibited by the irregular domains induced by the intermediate Ccce phase, and the switching may be realized when the intermediate phase is suppressed by applying chemical pressure.Moreover, it has been evidenced that the O octahedra tilt and rotation modes of Ca 3 Ti 2 O 7 undergo softening when hydrostatic pressure or heating is considered. 23,24This behavior under temperature and pressure reveals the softness of the antiphase tilt, which indicates the importance of the partially occupied d-orbitals of the transition metal ions, since these determine the stability of the oxygen octahedra distortion. 23,24Moreover, through Raman measurements, it was observed that under isotropic pressure, the polyhedra tilts of the Ca 3 Mn 2 O 7 compound can be suppressed within the low-pressure regime (1.4−2.3GPa). 25 On the other hand, through DFT calculations (0 K) performed on the Ca 3 Ti 2 O 7 compound, it was recently found by computing the enthalpies for several pressure values, up to 20 GPa, that in fact the Ccce phase can become energetically more favorable than Cmc2 1 , with the transition occurring slightly below 15 GPa. 20he goal of this work is to inspect, through first-principles calculations, the evolution of the structural properties and the energetic and dynamic stabilities of the Cmc2 1 and Ccce structural phases of the SHO compound for increasing values of applied hydrostatic pressure.Furthermore, enthalpy results on the I4/mmm and C2/c structural phases have also been computed.
To probe the energetic (thermodynamic) stability, we evaluate the enthalpy difference curves of the Ccce, C2/c, and I4/mmm structural phases, relative to the ground-state Cmc2 1 phase at 0 GPa, for hydrostatic pressures up to 20 GPa.We also analyze the free energies, where the zero-point energy is considered, in order to infer the energetic stability trend between the Ccce and Cmc2 1 structures.It is worth mentioning that we have considered not only the harmonic approximation (HA) but also the quasi-HA (QHA) at constant pressure of 19 GPa.Upon these results, the dynamic stability was probed by analyzing the phonon dispersion curves of the two energetically lowest structural phases�Ccce and Cmc2 1 �at relevant pressure values.
First, variable-cell relaxation calculations were performed for each structural phase (Cmc2 1 , Ccce, C2/c, and I4/mmm) of the SHO compound and for each applied hydrostatic pressure value, ranging from 0 GPa up to 20 GPa.In these structural relaxations, both the atomic positions and the unit-cell parameters were allowed to change by constraining the system to the targeted pressure value.The projector augmented wave (PAW) 33 data sets were used to treat semi-core electronic states, with the Sr[4s 2 4p 6 5s 2 ], Hf[5s 2 5p 6 5d 2 6s 2 ], and O[2s 2 2p 4 ] electrons being included in the valence shell.Additionally, the variable cell-shape relaxation, considering the damped Beeman ionic dynamics and the Wentzcovitch extended Lagrangian for the cell dynamics, 34 was performed with a The Journal of Physical Chemistry C plane-wave kinetic-cutoff of 70 Ry.The electronic BZ was sampled with a Γ-centered Monkhorst−Pack mesh 35 and defined with 6 × 6 × 12 subdivisions for the Ccce and C2/c phases; 12 × 12 × 6 subdivisions for the ground-state Cmc2 1 system, and 14 × 14 × 14 subdivisions for the I4/mmm phase.
The enthalpy curves, H, were then calculated by interpolating the V − p and E − V curves with the thirdorder Birch−Murnagham equation 36,37 to obtain the relation H = E + pV, where E is the total electronic energy of the system, p is the hydrostatic pressure to which the system is subjected to, and V is the volume per formula unit.The calculation of the relative enthalpy curves, with respect to the lowest enthalpy at 0 GPa of the Cmc2 1 phase, allows the analysis of the evolution of the thermodynamic stability of the several structural phases, relative to one another, with increasing values of applied pressure (up to 20 GPa).
For the (harmonic) lattice-dynamics calculations, the supercell finite-displacement method was considered, with the Phonopy software package, 38 where QE is used as the second-order force-constant calculator.The supercells employed to compute the phonon dispersion spectra were a 2 × 2 × 2 expansion of the primitive-cell.Phonon calculations were performed for the Ccce structure at several pressure values above which the system is energetically more favorable than the Cmc2 1 phase.Respective calculations were considered in order to probe the dynamical stabilities of the Ccce phase.Phonon frequency calculations were also considered for the Cmc2 1 system at pressure values between 20 and 30 GPa.
In the harmonic model, the equilibrium distance among atoms is not temperature dependent.The anharmonic effects needed to account for thermal expansion can be introduced by the QHA, in which the thermal expansion of the crystal lattice is obtained from the volume dependence of the phonon frequencies. 39,40To perform a QHA calculation, the phonon frequencies are computed for a range of expansions and compressions about the 0 K equilibrium volume, and the constant-volume free energy, for each configuration, is then evaluated as a function of temperature.From this approach, the equilibrium volume, bulk modulus, and Gibbs free energy can be obtained at several temperature values by fitting the free energy as a function of volume to the Vinet equation of state (EoS). 38,39,41We have performed QHA calculations for the two structural phases of interest: Cmc2 1 and Ccce.The EoS fit was computed for a constant pressure of 19 GPa and for a temperature range up to 500 K (more details on the EoS fitting is discussed in the Supporting Information).

RESULTS AND DISCUSSION
3.1.Sr 3 Hf 2 O 7 Polymorphs.Since the environment around the Sr atoms is inequivalent, the general formula of the SHO system can be better written as (AHfO 3 ) 2 A′O, where A′ = Sr 1 and A = Sr 2 .Just as other RP structures, the system results from the inter-growth of rock-salt (R) and perovskite (P) blocks.The P blocks are composed of two layers of HfO 6 octahedra along the a-axis, sharing the O corner ions.The A′site cations occupy a ninefold coordination site (R block), while the A-site cations have a coordination number of 12 (cuboctahedral in P block).
The high-symmetry structure is centrosymmetric and belongs to the I4/mmm space group (S.G. n. 139). 17By decreasing the temperature, lower symmetry structural phases may be generated by inducing tiltings and/or rotations of the O octahedra cages.In ref 17 it was theoretically shown that second-order phase transitions may occur, toward lower structural phases, to the Cmcm or Cmca space groups, from which the phonon instability modes of these two phases may direct the final transition to Cmc2 1 space group.Interestingly enough, it was experimentally observed that other related RP systems, such as Ca 3 Mn 2 O 7 and Ca 3 Ti 2 O 7 compounds, show a structural transition path from low-temperature Cmc2 1 , intermediate-temperature Ccce (S.G. n. 68), and high-temperature I4/mmm. 18Since the Ccce phase is not related by groupsymmetry analysis to the I4/mmm → Cmc2 1 transition and there is no single mode connecting the phases, we infer that The Journal of Physical Chemistry C this structural phase may occur as a first-order phase transition.Such a transition can occur by applying an external perturbation, such as pressure, in order to induce the condensation of the X 1 − zone boundary mode from the centrosymmetric system. 17e show in Figure 1 the four structural phases we have studied in this investigation.They are: 1.The ground-state structural phase at room conditions, evidencing a polar symmetry, Cmc2 1 (S.G. n. 36).This system phase is related to the centrosymmetric I4/mmm structure through group-subgroup relations, by the condensation of the X 2 + and the X 3 − modes, which respectively lower the high-symmetry to the Cmcm and Cmca phases. 17.The Ccce structural phase (S.G. n. 68), which has experimentally been observed in other similar RP compounds; 18 however, such a phase has not been evidenced in SHO by analyzing the group-subgroup relations from the I4/mmm toward the polar Cmc2 1 symmetry and as described in ref 17.Respective polymorph is induced through the condensation of the X 1 − mode, which does not occur spontaneously for the SHO system.Such a distortion mode would have to be induced by applying an external perturbation (i.e., hydrostatic pressure) to break the symmetry from I4/ mmm towards Ccce.We must note that there is no group−subgroup relation (by considering single mode analysis) between the Ccce phase and the Cmc2 1 phase. 173.The C2/c system (S.G. n. 15) was obtained by mapping out the anharmonic potential energy surfaces by following the eigenvectors associated with the soft (imaginary) phonon mode, localized at the Γ-point, of the Ccce structure (a more detailed description has been added in Supporting Information).From this analysis, it is possible to obtain a lower energy structure, corresponding to the minima of the potential energy surface.4.And finally, the high-symmetry I4/mmm structural phase.All three structural polymorphs are related to the highsymmetry, high-temperature, I4/mmm phase through tiltings and/or rotations of the O octahedra.When external perturbation is applied, symmetry-breaking occurs, thus lowering the energy of the system.

Energetic Stability.
To probe whether the structural phases of Figure 1 can become energetically competitive as a function of hydrostatic pressure, the enthalpy energies, for pressures up to 20 GPa, have been calculated.These results are presented in Figure 2, where the relative enthalpy energies, Δ H, with respect to that of the structure with lowest enthalpy at 0 GPa (the Cmc2 1 phase) are defined.
By analyzing Figure 2, we observe that at 0 GPa, the orthorhombic Cmc2 1 phase is energetically the most stable as already discussed in the literature. 17The C2/c is slightly higher in energy than Cmc2 1 (Δ H = ∼ 0.02 eV) at 0 GPa; however, it starts increasing the relative enthalpy as a function of pressure up until ∼10 GPa.After 12 GPa, the energetics of this system starts decreasing and, however, never becomes energetically stable throughout the studied pressure range.It is noteworthy of mentioning that the energetic tendency to decrease for increasing pressures evidences the possibility of the C2/c system stabilizing for pressures higher than 20 GPa.At 20 GPa, we observe a enthalpy difference of the C2/c system with respect to Cmc2 1 of Δ H = 0.15 eV.
At room conditions, the Ccce phase is, in terms of enthalpy, the second highest system with respect to the ground-state structure.However, for increasing pressure, we observe a considerable decrease in energy.At ∼5 GPa, this system will be competing energetically with the C2/c structure, thus becoming more stable than the latter, and above 17.12 GPa, the Ccce phase becomes energetically the most favorable among the four studied systems.
The high-symmetry I4/mmm system shows quite elevated energy at 0 GPa, which is expected since it is a hightemperature phase.At around 10 GPa, we observe that the I4/ mmm structural phase starts decreasing the enthalpy and, at 20 GPa, the energetic difference with respect to the Cmc2 1 phase is around 0.52 eV.In comparison to the most stable phase at 20 GPa, the Ccce structure, the energy difference between the two phases is of ∼0.60 eV.Once again, we envisage the possibility of the high-symmetry I4/mmm system to stabilize at much higher pressures (>30 GPa), than those considered in the present work.
The enthalpy calculations were performed at 0 K, without taking into consideration the contributions to the free energy from the vibrations of the solids.In order to probe whether these effects could alter the energy ordering between the Cmc2 1 and Ccce phases of Sr 3 Hf 2 O 7 , we have performed latticedynamics calculations on the equilibrium and compressed

The Journal of Physical Chemistry C
structures to evaluate the constant-volume Hemholtz (F) and constant-pressure Gibbs (G) free energies.
Within the HA, the Helmholtz free energy as a function of temperature can be obtained from the phonon frequencies and the lattice energy of the equilibrium structure.Temperature effects can be included through the Helmholtz free energy, which by introducing a transformation from the constant volume function, we obtain the thermal properties at constant pressure, and thus the Gibbs free energy. 40,43By increasing temperature, the volume dependence of the phonon free energy changes, which in turn results in different equilibrium volumes for different temperatures.This is regarded as thermal expansion in the QHA.
The temperature dependence of the relative Helmholtz, Δ F (0 GPa), and Gibbs, Δ G (19 GPa), free energies of the Cmc2 1 and Ccce phases are shown in Figure 3.
We observe from the Helmoltz free energy (Figure 3; red dashed line) that the Ccce structure becomes energetically more stable than the Cmc2 1 system (Figure 3; black solid line), surpassing the latter at 370 K.This observation leads to the conclusion that the Ccce may also be induced through temperature effects, with the transition temperature being quite close to room temperature.At 0 K and 0 GPa, the freeenergy difference between both phases is 38 meV, which is significantly lower than the relative enthalpy obtained without considering the entropy effects (∼220 meV).
Since the Ccce structure becomes dynamically stable around 19−20 GPa (more detailed information in Section 3.3) and that the Cmc2 1 structure is still stable up until 30 GPa (Section 3.3), we have calculated the Gibbs free energy at 19 GPa (Figure 3; blue dashed-dotted line) to probe the evolution of the free energies of the two phases of interest.We observe that Ccce is thermodynamically the most stable phase throughout the studied temperature range (up to 500 K).At 0 K, the free energy of the Ccce phase differs from the Cmc2 1 system with a relative value of Δ E ∼ 25 meV; as temperature increases, this energy difference also grows further apart (Δ E = 54 meV at 500 K).When comparing Δ G with Δ H (Figure 2) for the two systems at 0 K and 19 GPa, we observe that the enthalpy difference is ∼50 meV, which is in much closer agreement to when comparing the relative enthalpy with Δ F (for 0 GPa and 0 K).
We must note that the (quasi-)HA will not account for the influence of the soft modes of the Ccce system on the free energy, a deficiency which is amplified by the sensitivity of the volume to the free energy.Therefore, the transition temperature might be slightly underestimated than expected. 40Since at 19 GPa, neither the two analyzed phases evidence imaginary modes, we conclude that the QHA free energies would provide a much reasonable description of the free energy behavior between the two systems.Overall, and by comparing the free energies of Figure 3 to what was obtained for the relative enthalpy plots of Figure 2, we observe comparable behaviors between the energetics of both polymorphs: at 0 GPa and low temperatures, the Cmc2 1 system is more stable than Ccce; whereas with pressure (19−20 GPa), the Ccce is more stable than Cmc2 1 , even for finite temperatures.

Dynamical Stability.
Energetic stability is a necessary, but not a sufficient, condition for a structural phase to be synthetically accessible.Another condition that should be analyzed is the dynamical stability of the system, which requires the study of the phonon spectra.−49 Thus being, we have proceeded to calculating the phonon structure for the Ccce phase at several pressure values.These values were considered at: room pressure (and taken from ref 17 for comparison) and at values above which the energetical stability of the Ccce phase occurs − 12, 14, 16, 18, and 20 GPa, as shown in Figure 4.
As illustrated in Figure 4, the phonon dispersion curves of the Ccce structural phase at 0 GPa (retrieved from ref 17) show negative phonon modes localized at two of the high-symmetry points of the BZ: Γand Y-points.We can therefore infer that such a structural phase is not kinetically stable at ambient pressure.However, we observe that as the applied pressure increases, the frequencies of the negative modes shift toward positive values.The Γ-point soft-mode hardens more rapidly than the Y-point: at 18 GPa, only a mild instability is observed at the latter point since at Γ, the soft optical mode already evidences positive frequencies.However, it is only at 20 GPa that we arrive at a structure that is both thermodynamically and dynamically stable, with the Y phonon mode evidencing positive values as well.
We also show, in Figure 5, the phonon dispersion curves of the Cmc2 1 system at several pressure values, and we observe that the structure is dynamically stable still at 20 GPa, thus coexisting with Ccce phase, as a metastable compound.It is noteworthy of mentioning that around the high-symmetry Spoint, the lowest frequency branch starts to decrease toward a soft mode, which may be an indication that for higher pressure values, a structural transition will occur due to the instability at this q-point.Indeed, by increasing the pressure to 30 GPa, it is observed that this soft-mode becomes unstable, thus rendering this structure kinetically unviable above this pressure value.

CONCLUSIONS
In the present work, we have performed ab-initio calculations to study the thermodynamical stability of the room temperature Cmc2 1 phase and the high-temperature I4/mmm structure; as well as different structural phases of Sr 3 Hf 2 O 7 , which are not related to the aristotype polymorph through It was observed that, at room conditions, the Cmc2 1 phase is the most energetically stable, as expected; the C2/c phase being 0.05 eV higher in energy than the ground-state; followed by the Ccce system with an energy difference of ∼0.18 eV with respect to Cmc2 1 ; and finally, the I4/mmm polymorph with the highest energy difference of ∼0.41 eV in relation to the ground-state phase.We observe that the enthalpy of Ccce decreases as pressure increases, and at ∼17 GPa, it surpasses that of the Cmc2 1 structure, becoming energetically the most stable system of all four studied compounds.
To inspect in detail the two lowest energetic compounds, namely the Ccce and Cmc2 1 phases, we have calculated the free energies (where entropy is accounted for) and the phonon dispersion curves to analyze their respective dynamical stability.From the constant-volume free energies, we observe that the Cmc2 1 system is energetically the most stable compound below 370 K. Above this temperature, the Ccce polymorph has its free energy lowered, thus becoming more stable than Cmc2 1 phase.By probing the volume-temperature

The Journal of Physical Chemistry C
dependent free energies at 19 GPa, we observe that the Ccce is always the most stable phase when compared to Cmc2 1 , throughout the studied temperature range, up to 500 K.When observing the phonon dispersion spectra, we conclude that the Ccce structure only becomes dynamically stable at 20 GPa, which is at a higher pressure range than when the energetic transition occurs from Cmc2 1 .The Cmc2 1 phase, on the other hand, is dynamically stable up to 30 GPa when an imaginary mode at the high-symmetry S-point emerges.These results suggest that both phases may coexist energetically at pressure values from ∼19 GPa up to ∼30 GPa; at this pressure interval, the polar Cmc2 1 structure is a metastable system.
From the present calculations, we conclude that the quasi-2D layered RP perovskite Sr 3 Hf 2 O 7 compound has a stable phase at room conditions which is a ferroelectric ground-state system with Cmc2 1 symmetry.By applying hydrostatic pressure, this phase may undergo a transition to the Ccce structure, for it becomes thermodynamically (at ∼17 GPa) and dynamically stable (at 20 GPa).
Evolution of the lattice parameters of the four studied structural phases with applied pressure, potential-energy surface for the phonon instabilities associated with the high-symmetry Γ-point negative mode of the Ccce phase, band structure of the Ccce and Cmc2 1 structural phases, electronic band structure of Cmc2 1 with spin-orbit coupling effects, and QHA: Helmoltz free-energy EoS fits, temperature-dependent volume, and the volumetric thermal-expansion coefficient (ZIP)

Figure 1 .
Figure 1.Representation of the Cmc2 1 , Ccce, C2/c, and I4/mmm structural phases of Sr 3 Hf 2 O 7 .The red, green, and gold spheres represent, respectively, the oxygen, strontium, and hafnium ions.The VESTA visualization software was used to plot the figures.42

Figure 2 .
Figure 2. (a) Relative enthalpy curves of the I4/mmm, Ccce, C2/c, and Cmc2 1 structural phases as a function of pressure with respect to the structure with the lowest energy value at ambient pressure (Cmc2 1 ).The cross marker indicates the pressure value at which the Ccce structural phase becomes energetically stable.(b) Schematics of the most plausible transition pathways that Sr 3 Hf 2 O 7 can undergo and based also on ref 17.