{'fraction_non_alphanumeric': 0.02831261231525287, 'fraction_numerical': 0.01371621373821836, 'mean_word_length': 4.951113051069402, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Crossover in the pressure evolution of elementary distortions in RFeO3 perovskites and its impact on their phase transition

This work reports on the pressure dependence of the octahedra tilts and mean Fe-O bond lengths in RFeO3 (R=Nd, Sm, Eu, Gd, Tb and Dy), determined through synchrotron X-ray diffraction and Raman scattering, and their role on the pressure induced phase transition displayed by all of these compounds. For larger rare-earth cations (Nd-Sm), both anti- and in-phase octahedra tilting decrease as pressure increases, whereas the reverse behavior is observed for smaller ones (Gd-Dy). EuFeO3 stands at the borderline, as the tilts are pressure independent. For the compounds where the tilts increase with pressure, the FeO6 octahedra are compressed at lower rates than for those ones exhibiting opposite pressure tilt dependence. The crossover between the two opposite pressure behaviors is discussed and faced with the rules grounded on the current theoretical approaches. The similarity of the pressure-induced isostructural insulator-to-metal phase transition, observed in the whole series, point out that the tilts play a minor role in its driving mechanisms. A clear relationship between octahedra compressibility and critical pressure is ascertained.


Introduction
Hydrostatic pressure has been increasingly considered in the study of critical phenomena, since it allows to modify the interatomic distances and, consequently, the interactions to a greater extent than any other external parameter, like temperature or magnetic field. Thus, the number of published experimental [1][2][3][4][5] and theoretical [6][7][8][9] reports concerning the pressure evolution of elementary distortions and phase transitions in many materials has steadily increased over the last few years. In this regard, the effect of pressure on ABO3 perovskites has been the focus of intense research, because of their remarkable pressure-induced phase transitions sequences, many of them undergoing many structural transformations accompanied by changes on the magnetic, transport and ferroelectric properties. [1][2][3][4][5] The ABO3 perovskites exhibit a rather simple crystallographic structure that can be described as a corner-sharing BO6 octahedra network, with the A-cations placed between them, forming AO12 dodecahedra. 10 The structure of ABO3 perovskites can be obtained from basic distortions of the ideal Pm3 ̅ m cubic phase. 11,12 For perovskites with a tolerance factor less than unity, the most important distortion is characterized by short-period BO6 octahedral rotations. 13 In the case of Pnma orthorhombic perovskites, these distortions are the in-phase and anti-phase tilts about the [010]pc and [101]pc pseudocubic directions, 13 which transform according to the M3 + and R4 + irreducible representations of the Pm3 ̅ m space group, respectively. 11,12 These two distortions are the primary order parameters associated with the symmetry lowering, and present the largest amplitudes, which increase as the A-cation size decreases. 11 Other non-symmetry breaking distortions occur together with octahedra tilting, where the anti-parallel motion of the A-cations along the z pseudocubic direction with symmetry X5 + is the most relevant, as it bares the largest amplitude among the secondary distortions. Moreover, it couples to both tilts via a specific trilinear coupling term that provides an energy gain crucial to the stabilization of the Pnma phase. 11,14 In order to predict and explain the structural behavior of perovskites under high-pressure, several rules have been proposed, grounded on theoretical models and DFT calculations. [6][7][8] Based on the bond-valence concept, Zhao et al. 6 predicted that the AO12 dodecahedra are expected to be significantly less compressible than the BO6 octahedra in orthorhombic perovskites with both A and B cations having the formal charge +3 (3:3 perovskites), as it is the case of rare-earth orthoferrites (RFeO3). Thus, the octahedral tilts should decrease with increasing pressure. 6 This model also evidences the correlation between the ratio of the BO6 and AO12 compressibilities and the rate of change of the octahedra tilting. 6 Moreover, the decrease of the octahedra tilting with pressure should yield a structural phase transition from the orthorhombic into a higher symmetric structure at some critical pressure. 6 This is the case of NdNiO3 and LaGaO3 which undergo a high-pressure structural phase transition into a rhombohedral symmetry, but with a different tilt system. As a matter of fact, these rules are not followed by many of 3:3 perovskites. For instance, new results in rare-earth chromites (RCrO3), with small R-cations, show that octahedra tilting increase with pressure and some rare-earth orthoferrites undergo a pressure induced isostructural phase transition. 15,16 In order to unravel the mechanisms underlying the distinct pressure dependences of octahedra tilting, Xiang et al. 7 conducted first-principles calculations on representative perovskites and proposed a set of rules governing the tilt evolution with pressure. In this framework, they simulated the pressure dependence of both in-phase and anti-phase octahedra tilts of LaFeO3 and LuFeO3, as border cases in the rare-earth orthoferrites series. 7 For LaFeO3, they found that both octahedra tilts are suppressed as pressure increases in agreement with Zhao's prevision. 7 However, pressure suppresses the anti-phase but enhances the in-phase tilting in LuFeO3. 7 This pressure behavior was interpreted by taking into account the contribution of a trilinear coupling between these two rotations and the anti-polar mode involving the A-cation. 7 Thus, a new rule emerges for the orthorhombic Pnma perovskites, which simultaneously exhibit the in-phase and anti-phase octahedra tilts, stating that they are not inevitably both suppressed or enhanced by pressure. 7 Unfortunately, the authors did not present results concerning intermediate rareearth orthoferrites in order to check where the crossover between the two aforementioned distinct pressure behaviors occurs.
Despite the intensive research already done, [16][17][18][19][20][21] a systematic study of the pressure evolution of the elementary structural distortions in orthorhombic perovskites is still missing, as well as, the careful search for crossover events, including the experimental determination of the borderline compound. In this regard, the study of the RFeO3 series under high pressure is particularly interesting, as it is expected to show different pressure dependences of the elementary distortions across the series. 7,8 Moreover, this system bears a distinct advantage, as it does not show distortions other than those with direct origin in octahedra tilting, like Jahn-Teller distortion, which can influence the mean B-O bond length. 2 In this work, we present an experimental study of the structure of the RFeO3 (R = Nd, Sm, Eu, Gd, Tb and Dy) as a function of pressure, by means of synchrotron X-ray diffraction and Raman scattering. We first analyze the pressure evolution of the octahedra tilt angles across the series in order to reach an overall picture of its dependence on the rare-earth cation size. Then, the pressure variation of the mean Fe-O bond length obtained by Raman scattering is explored in order to correlate the pressure tilt behavior and the mechanisms to accommodate pressure. We also discuss the experimental results within the scope of the theoretical models' predictions.
Finally, the dependence of the critical pressure on the rare-earth cation size is examined in terms of the pressure behavior of the elementary distortions.

Experiment aspects
SmFeO3 single crystals were grown in an optical-floating-zone furnace, 22 and NdFeO3 and TbFeO3 powder was prepared using single crystals growth by floating zone method in a FZ-T-4000 (Crystal Systems Corporation) mirror furnace. EuFeO3 powder was obtained by conventional solid-state reactions, while GdFeO3 and DyFeO3 powder was prepared using the urea sol-gel combustion method. 23 The quality of the samples was previously characterized by means of X-ray diffraction, Fourier transform infrared spectroscopy, X-ray photoemission spectroscopy and scanning electron microscopy. The ceramics were manually grinded to acquire a homogeneous powder and loaded in a diamond anvil cell (DAC) with diamond culets of 300 μm diameter, and with helium as a pressure-transmitting medium. The pressure was monitored through the standard fluorescence method of a ruby loaded next to the sample. High-pressure synchrotron X-ray diffraction (XRD) experiments on SmFeO3 and TbFeO3 were performed at the European Synchrotron Radiation Facility (ESRF, Proposal Ref. HC-2153) on the ID27 highpressure beamline (λ=0.3738 Å). The diffraction data were analyzed by Le Bail and Amplimodes refinements using FullProf and Jana2006 softwares. The Raman spectra were recorded on a Horiba LabRam at MINATEC (Grenoble, France) using a He-Ne laser at 633 nm for TbFeO3 and on a Horiba T64000 spectrometer at Institut Néel (Grenoble, France) using an Ar+ laser at 514.5nm for RFeO3 (R=Nd, Sm, Eu, Gd and Dy). In both cases, the laser power was kept below 8 mW on the DAC to avoid sample heating. Raman spectra were fitted by a sum of independent damped oscillators using IgorPro® software.

Results
a. High pressure XRD: SmFeO3 and TbFeO3 Figure 1 shows the pressure dependence of the pseudocubic lattice parameters = /√2, = /2, and = /√2 (values can be found in Table I   As the pressure increases, the pseudocubic lattice parameters monotonously decrease and suddenly down shift at around 41 GPa for SmFeO3, and at 46 GPa for TbFeO3; as a consequence, the pseudocubic volume is reduced 4.3% and 6.6%, respectively. The later result gives clear evidence for a pressure-induced structural phase transition, which we will address in detail in the last section of this work. On further pressure increase, a smooth pressure evolution of the pseudocubic lattice parameters, and volume is again observed. The symmetry of both low-and high-pressure phases of SmFeO3 and TbFeO3 is found to be Pnma. This symmetry was reported for the high pressure phase of other rare-earth orthoferrites. 16,19 In the low pressure phase, the pressure dependence of the pseudocubic cell volume of SmFeO3 where B0 the bulk modulus and B'0 its pressure derivative, all taken at room pressure, and is given by: where is the pseudo-cubic volume at the pressure P, and (0) its room pressure value.   Figure S3 of Supplemental Material. They exhibit simultaneously all Raman-active modes due to the unpolarized recording condition. A detailed mode assignment of the Raman bands and the corresponding atomic motions for rare-earth orthoferrites is presented elsewhere. 3 The Raman spectra for all the studied materials exhibit similar trends with increasing pressure: Raman bands shift towards higher wavenumbers, due to an overall pressure-induced bond shortening and volume reduction, and become broader; their intensity reduces and disappear above a certain pressure that depends on the compound. The later result corroborates the existence of a structural phase transition at high pressures for all the studied compounds. The nature and structure of the high-pressure phase will be discussed in the last section.
In the following, we focus our attention on the Raman-active modes assigned to the FeO6 octahedra rotations, mirroring the anti-phase and in-phase octahedra tilts. Figure 2 presents the pressure dependence of the wavenumber of the aforementioned Raman modes of the studied compounds. The pressure dependence of the wavenumber of the Raman modes does not exhibit anomalous behavior up to the critical pressure, corroborating that the Pnma structure is preserved in the low-pressure range. A linear pressure dependence of the wavenumber of the Raman bands is observed below to 20 GPa. Concerning NdFeO3, our results agree with those already reported below 11 GPa. 21 From the best fit of a linear function to the experimental data below to 20 GPa, shown in Figure 2, we have determined the corresponding slopes and wavenumbers at room conditions which are presented in Table 2. The slopes of the linear pressure dependence of the wavenumber of both in-phase and anti-phase tilt modes tend to increase as the rare-earth cation size decreases. We can see this result in another way: for larger tilt angles measured at ambient pressure, greater slopes are observed, as it is predicted by theoretical models. 7 Table 2. Wavenumbers at room pressure, and slopes of the linear pressure dependence of the inphase and anti-phase tilt modes, involving the data recorded below to 20 GPa.

a. Compilation and analysis of XRD data
The quality of the XRD patterns obtained at high pressures hinders the full Rietveld refinement of the atomic positions and, so, the calculation of the tilt angles. This is however possible with the Amplimodes analysis. 11,26 During the structure refinement, instead of allowing the atomic positions to vary in the three dimensional space without restriction to find the global minimum, Amplimodes are used to improve the structure refinement by describing the displacement of the atoms, relatively to their positions in the high symmetry Pm3 ̅ m structure, as the superposition of symmetry-adapted distortions. 26 The refinements show that the internal octahedral distortions have small amplitudes (less than 0.05 Å), whereas the octahedral tilts distortions and the rare-earth shifts are apparently larger (between 1 and 2 Å value of global amplitude). However, the refined mode amplitudes when plotted versus pressure show dispersion, especially in the modes mainly involving oxygen motions (that is, the tilting modessee Figure S4 of Supplemental Material). This is not surprising since high-pressure powder diffraction presents peak overlapping and the contribution of the oxygen atoms to the diffraction peaks intensity is small compared to the rest of the atoms (that is Sm or Tb, and Fe).
The Amplimodes refinement results, especially regarding the octahedral tilts, should be contrasted with other observations.
Thus, we estimated the tilt angles from the lattice parameters, using the equations of Megaw et al. 27,28 This formula assumes that the change of the unit cell volume is originated by the octahedral tilting, with no significant additional octahedra distortion. 27,28 The formula is most suitable for larger tilts than for smaller tilts, as for the latter other contributions to the change of the unit cell volume may be comparable to the contribution from the tilts. 27,28 Therefore, for our compounds, the estimation using Megaw's formula is better for the anti-phase tilt than for the in-phase tilt, and for smaller tolerance factors than for larger ones. The reliability of this approach was assessed by comparing the values of the tilt angles for these compounds, obtained at room conditions using the Megaw's formula, with their reference ones, obtained from the refinement of the atomic positions, as shown in Figure 3. Furthermore, their pressure behaviors follow similar trends as the ones found by the Amplimodes analysis, though with much less dispersion (see Figure S4 in Supplemental Material). Moreover, as we shall discuss in the following, this approach is also consistent with the pressure dependence of the spontaneous e4 strain, calculated using the refined lattice parameters (see Figure S5 in Supplemental Material), which is inherently connected to the two tilts. 13 The good agreement between the literature and our experimental observations ensures the validity of this method for other compounds where similar conditions are expected, such as LuFeO3, EuFeO3 and NdFeO3. For the these compounds, we have calculated the tilt angles from the lattice parameters measured at different pressures already published, 16,18 as these values are relevant for the discussion of how their pressure behavior depends on the rare-earth ionic radius (see following section).  As stated by Xiang et al., 7 the stabilization of the Pnma phase of perovskites is predicted to be due to an energy gain coming from a trilinear coupling of both R4 + and M3 + distortions with the X5 + distortion, associated with the displacement of the rare-earth cation. 8,14 In this theoretical framework, the magnitude of the X5 + distortion mode is predicted to be proportional to the product of both tilt distortions. 8 If this is the case, then this distortion should follow the same trend as the tilts to which it is coupled with. In fact, the pressure evolution of the X5 + distortion amplitude, which we have experimentally obtained by the Amplimodes analysis, is consistent with this model, since it mimics the behavior of the tilts for the respective rare-earth, as seen in  The slopes of the linear pressure dependence of the Raman rotation modes, presented in Figure   5(a), are always positive. The case of EuFeO3 is straightforward because the tilt angles are pressure independent, and thus, m1 = 0 (see Eq. 3). So, the linear pressure dependence of both Raman rotation modes in EuFeO3, observed in the 0 -20 GPa range, comes from the reduction of the <Fe-O> value, and m2 = mtotal, which is represented by the horizontal dashed lines in Figure   5(a). In this case, the wavenumber of the rotational modes mirrors the reduction of the unit cell volume, due to the decrease of the mean B-O bond lengths with pressure, and a linear relation between the mode wavenumber and volume is experimentally evidenced, as shown in Figure  5(c) (similar plots for TbFeO3 and SmFeO3 are presented in Figure S6 of Supplemental Material for which the linear relation does not hold, as the tilts are not constant for these compounds).
In order to unravel the effect of the tilt angles on the pressure dependence of the Raman tilt mode wavenumber for the other compounds, the m2 = mtotal value calculated for EuFeO3 might not the best reference, since the reduction of the FeO6 octahedra volume may have different values for each compound, as it will be shown. According to Eq. 3, the pressure derivative of the wavenumber of the Raman tilt modes is: In  Table 3. The estimated < − > values varied only around 0.001 Å/GPa when using the two different rotation modes. It is worth to note that, Weber et al. 3 have found different proportionality constants ( ) between  and  for the anti-and in-phase tilts for the RFeO3 system. The presented value agrees with that obtained by Todorov et al. 31 for the anti-phase tilt, but not for the in-phase tilt. Therefore, the most suitable value for the actual < − > should be the one obtained with the anti-phase tilt. c. Crossover for the behavior of tilts in the rare-earth series: experiment and theory We now focus on the evolution of the tilt angles in the vicinity of zero (atmospheric) pressure across the series. From Figure 5 We now compare this compilation of experimental results with predictions made by theoretical approaches for the pressure evolution of tilt angles. A first approach is based on the compressibility of the polyhedra forming the perovskite structure. In this approach, the ratio of compressibilities ßA/ßB between the AO12 and BO6 polyhedra is used as a predictor of the behavior of tilt angles under pressure, and is calculated by a valence bond sum model: tilt angles are expected to increase when this ratio is larger than unity, and decrease otherwise. The case of the rare-earth orthoferrites has been treated in Ref. 6, and the corresponding data are reported in Figure 5(b) for comparison. The ratio is always positive, but decreases as the ionic radius increases and reaches values very close to unity for the smallest cations. 6 According to this criterion, all compounds in the series should see their tilt angles reduced under pressure, with the exception of TmFeO3. 6 This is not what is found experimentally, but the overall evolution bears a striking resemblance with the experimental one, up to an overall shift.
More recently, in a DFT-based approach, a set of rules for the evolution of tilts under pressure was proposed. 7 In this paper, the compounds with the smallest A-cations (Lu, Tm), are predicted to exhibit an unusual behavior whereby the two tilt angles behave in the opposite way with pressure: the antiphase tilts are reduced whereas the in-phase tilts are enhanced. 7 For larger cations, both tilts behave the same way and are reduced as pressure increases. We find here no evidence for this behavior, but instead, both tilt angles behave the same in all investigate compounds. Also, Ref. 7 does not predict any case where both tilts increase under pressure.
The discrepancies between theoretical and experimental results seemingly point to a specific difficulty in predicting the tilt behavior for the smallest cations in the series. While we cannot be conclusive at this point about the precise reasons for this discrepancy, we observe that, in both approaches, the predicted tilt changes under pressure reach extremely small values, typically below 0.01°/GPa for LuFeO3 in Ref. 7. For EuFeO3, which in fact has absolute pressure derivatives below 0.01°/GPa, the slopes for in-phase and anti-phase tilts are of opposite sign (0.006 and -0.01°/GPa, respectively), but with values negligibly small that fall within the experimental uncertainty. In that context, it is conceivable that some additional parameters, legitimately neglected for large cations, become relevant and suffice to change the trend from slightly positive to negative or vice versa. In particular, RFeO3 with very small cations (Er-Lu) are known to exhibit some distortions of the AO12 polyhedra, so that the classical separation of the 12 A-O bonds into 4 longer bonds and 8 shorter bonds becomes less satisfactory. 33 We hypothesize that this has an influence of the bond valence sum and compressibilities calculated in Ref. 7. In the DFT approach, where calculations are performed at 0 K, and the behavior of tilts is the result of a delicate balance between the pressure evolution of the Landau parameters, one might question the role of temperature effects. Differences in tilt angles of the order of 0.1° between 0 and 300 K -which is reasonable following Ref. 34 -would be equivalent to several tens of GPa and cause significant shifts in the predicted behavior. Altogether, further work will be needed to clarify the picture for small cations, which calls for detailed experimental studies and reexamination of theoretical models.

d. High-pressure phase transition
Despite the crossover between enhancement and suppression of the tilt angles on the lowpressure regime, for all the studied compounds a similar isostructural pressure-induced structural phase transition occurs. This implies that the tilts play no major role on triggering this phase transition, nor on the symmetry of the high-pressure phase. This phase transition is clearly evidenced by the changes on the XRD patterns, mirroring the sudden changes of the pseudocubic lattice parameters. The structural phase transition reveals itself by the disappearance of the Raman signal above a certain critical pressure, hereafter designated by PIM.
The crystallographic structure of the high-pressure phase of SmFeO3 and TbFeO3 is Pnma, as it has been reported for other rare-earth orthoferrites. 16 This structure allows for Raman signal.
Therefore, the disappearance of the Raman spectra above the critical pressure suggests the metallic character of the high-pressure structural phase.
The pressure hysteresis evidenced by the XRD patterns of SmFeO3 and TbFeO3, and Raman spectra, for all studied compounds, recorded on increasing and decreasing pressure runs (3 to 6 GPa depending on the rare-earth cation), evidences for the first-order character of the highpressure structural phase. Taking into account both the XRD and Raman data, we can estimate the critical pressure PIM for the studied compounds, which is depicted in Figure 6. As it can be observed, the critical pressure linearly increases as the rare-earth cation size decreases. High-pressure Mössbauer experiments give evidence for a transition from high-spin to low-spin configurations occurring at PIM. 19 As the tilts have little impact on this transition, it is sensible to assume that is depends mostly on the chemistry found inside the FeO6 octahedra. We suggest that, as pressure increases and the FeO6 octahedra become smaller, a critical volume of the octahedron is reached where the electronic repulsion between the oxygens p-electrons and the iron eg-electrons is such, that it triggers this electronic reconfiguration, as it is energetically favorable for the eg -electrons to pair with the t2g-electrons, avoiding to come closer to the oxygen p-electrons. The critical volume can be estimated for EuFeO3. As the tilt angles do not change with pressure, there is an isotropic volume reduction of the unit cell, already evidenced in Figure 5(c), and thus the ratio of the FeO6 octahedron with the pseudocubic unit cell volume (VFeO6/Vpc) can be assumed pressure-independent. Considering a regular octahedron at room pressure, the ratio VFeO6/Vpc = 0.189 is obtained. Using the value of Vpc before the critical pressure, the critical volume for EuFeO3 is estimated to be VFeO6 = 9.5±0.2 Å 3 .
It is expected that for larger R-cations, where from Table 3 we know the FeO6 octahedra reduce their volume at a larger rate, the aforementioned phenomenon is promoted at a lower critical pressure. Conversely, for the smaller R-cations, as the FeO6 octahedra reduce their volume at a lower rate, they present higher values of critical pressure. This prediction explains the behavior of PIM with the rare-earth size, shown in Figure 6. Moreover, one can infer that a similar role is played by the octahedra tilting in the similar pressure-driven phase transition observed in RMnO3. 2 The critical pressures for RMnO3 have similar dependence on the rare-earth ionic size, being always slightly higher by a constant value of 2 GPa. This difference can be assigned to the Jahn-Teller distortion present in RMnO3, as it provides an additional pressure accommodation mechanism than their respective RFeO3 compounds. It is worth to note that the tilt angles are almost the same, thus allowing for this comparison. 32 This fact also supports the importance of the chemistry inside the octahedra in determining the symmetry of the high-pressure phase. This is because, unlike the isostructural phase transition found for the RFeO3, the RMnO3 present many different symmetries of the high-pressure phase. 2

Conclusions
In this work, we present an experimental study of the pressure dependence of the main structural distortions and lattice dynamics in the RFeO3, by XRD and Raman spectroscopy. Firstly, we have ascertained the crossover of the pressure dependence of both anti-and in-phase octahedra tilting in RFeO3 series. EuFeO3 stands distinctly at the borderline, where both tilts are pressure independent. For larger rare-earth cations, we have found that the octahedra tilting decreases as pressure increases, whereas the reverse behavior is observed for smaller ones.