Phase comparison and equation of state for Ta2O5

Tantalum pentoxide (Ta2O5) is among the most technologically useful heavy transition metal oxides. Unfortunately, its crystal structure is the subject of long-standing and unresolved disagreement. Among other consequences, this uncertainty has made it impossible to formulate a robust high pressure equation of state for Ta2O5. Here, we report the results of high pressure x-ray diffraction experiments indexed against a comprehensive list of proposed Ta2O5 phases. Five of the proposed phases produce good matches to experimental observations, and the compressibility parameters for these phases are all consistent within uncertainty. This means that regardless of the particular phase chosen, the Ta2O5 equation of state can be described with bulk modulus K0=138±3.68  GPa and pressure derivative K0′=1.82±0.45 . Combining these experimental results with new density-functional theory calculations allows us to identify the λ phase as the best structural model of Ta2O5 at ambient conditions.

Unusually for such a widely used material, the crystal structure of Ta 2 O 5 at ambient conditions remains controversial, with numerous candidate structures appearing in the literature and disagreement between studies that attempt to adjudicate between them.Synthesized bulk Ta 2 O 5 is often amorphous [9,10], while crystals tend to have high concentrations of defects that impede structural determinations [11].Resolving the ambient crystal structure is important because predictions of various properties, including the band structure [12,13] and mechanical strength [7,14] depend on it.Additionally, the crystal structure of Ta 2 O 5 is necessary for determining its phase diagram and equation of state (EoS).These properties are important for interpreting material behavior at nonambient pressures (P) and temperatures (T), such as the results of shock experiments (which rely on EoS parameters to determine bulk sound speeds) [15] or synthesis procedures [16].There have been two experimental crystalline Ta 2 O 5 EoS studies, but each indexed their data to a different crystal structure.Additionally, one of these studies did not determine the P-derivative of the bulk modulus (K ′ 0 ) [17], and the other reported an unrealistically low K ′ 0 value (≈ 0.1) [18], making a direct comparison of their results difficult.
Most studies that have systematically searched for the best candidate Ta 2 O 5 structure did so from an ab initio perspective, using density-functional theory (DFT) to determine the structure with the minimum total energy (E) [19][20][21][22][23][24].In this work, we combine DFT simulations and new high-P powder x-ray diffraction (XRD) experiments to evaluate a variety of proposed Ta 2 O 5 phases.This comparison allows us to identify the most realistic of the proposed Ta 2 O 5 phases and determine their EoS parameters.

Proposed Ta 2 O 5 structures
The literature on Ta 2 O 5 crystallography dates back 70 years and includes more than a dozen crystalline phases, as well as many studies that use the same name for different structures.Several previous publications provide overviews and comparisons of Ta 2 O 5 structure proposals [10,12,19,25], which we briefly summarize here.Early studies on Ta 2 O 5 recognized two forms that crystallized at varying T [26][27][28].Up to T ≈ 1300 • C, a low-T phase (originally called β, L, or T) was understood to be orthorhombic, while the structure of the high-T form (α or H) was disputed [29].Later studies proposed new phases and alternate descriptions of the original two, eventually leading to a complex body of literature.Table 1 lists the names, space groups, and published ambient unit cell volumes (V 0 ) of the phases considered in this study.This list is based on similar compilations in the literature [10,19,21,39].The supplementary material provides a list of other proposed phases.

Experimental
Experiments consisted of high-P angle-dispersive synchrotron XRD in a membrane-driven diamond anvil cell [40].Data were collected at the Advanced Photon Source Sector 16, managed by the High Pressure Collaborative Access Team (HPCAT).Samples were prepared from a commercial crystalline Ta 2 O 5 powder (Cerac lot #607010-1) that has been previously characterized and used for high P experiments [15].Samples were loaded into 60 µm diameter sample chambers drilled into pre-indented steel gaskets along with an internal P standard and quasi-hydrostatic P-transmitting medium.Three experiments were performed.Experiments 1 and 2 took place at HPCAT beamline 16-ID-B (λ = 0.4246 Å) with Cu as the P standard and mineral oil as the P medium.Experiment 1 reached a peak pressure of ∼60 GPa, experiment 2 reached ∼20 GPa, and data were collected on both compression and decompression.Experiment 3 took place at HPCAT beamline 16-BM-D (λ = 0.4133 Å) with Au as the pressure standard and gaseous Ne as the pressure-transmitting medium; this experiment reached ∼12 GPa and data were collected on compression only.The sample-to-detector distance (determined with a CeO 2 calibrant) was 211.9 mm for experiments 1 and 2 and 200.6 mm for experiment 3. The x-ray spot size for all experiments was 4 × 5 µm, and a 50 µm diameter pinhole was used to reduce beam tails.

Analysis and computation
The two dimensional diffraction images were integrated into one dimensional patterns with Dioptas [43] over the wavevector range 0.9 ⩽ Q ⩽ 5.3 Å −1 and compared to calculated powder profiles for the structures listed in table 1.Each integrated pattern was analyzed with full-profile Rietveld refinement for each of the proposed phases (plus the P standard) using GSAS-II [44].The P of each pattern was calculated from the published EoS [45] and the refined unit cell volume (V) of the P standard.
DFT calculations were performed using the Vienna Ab initio Simulation Package (VASP) version 5.4.4.[46][47][48][49] using the Perdew-Burke-Ernzerhof exchange-correlation functional for solids [50] with projector augmented wave (PAW) potentials [51,52] in which Ta has five valence electrons (isolated atom configuration 6s 2 5d 3 ) and O has six valence electrons (isolated atom configuration 2s 2 2p 4 ).We used the standard VASP PAW potentials, where the minimum plane-wave energy cutoffs required are 223 eV (2.9 Bohr cutoff radius) for Ta and 400 eV (1.52 Bohr cutoff radius) for O. Plane-wave energy cutoffs were 520 eV for all calculations.For some structures (β R , β A , δ, and λ), the PAW potentials overlapped slightly at the equilibrium V. We therefore also tested non-overlapping Ta potentials which included the 5p states (minimum 223 eV plane wave cutoff energy and 2.5 Bohr cutoff radius) and 'hard' O potentials (minimum 700 eV plane wave cutoff energy and 1.1 Bohr cutoff radius).These results did not change the cold curves (i.e.static lattice energies) significantly (see the supplementary material for details), so we used the standard PAW potentials for all calculations.
To calculate cold curves (i.e.static lattice energies at 0 K), we first tested convergence with respect to k-mesh size for each structure.At least 40 points/Å were needed to adequately converge E, corresponding to k-mesh sizes of 5 × 8 × 1 for L G , 8 × 1 × 5 for L SR , 8 × 1 × 5 for T, 11 × 6 × 5 for β A , 6 × 10 × 7 for β R , 6 × 6 × 10 for δ, and 8 × 7 × 13 for λ.For each structure, we performed a full relaxation in which lattice constants and atomic positions were allowed to change without any volumetric constraints; a minimum of three successive relaxations were performed in order to update the plane-wave basis.These relaxed equilibrium structures were used to isotropically scale the lattice constants to produce an initial structure over a range of compressions corresponding to lattice constant ratios a i /a 0 i = 0.93, 0.94, . . ., 1.01, where a i is the compressed lattice vector (i = 1, 2, 3) and a 0 i is the fully relaxed equilibrium lattice vector.At each compression, three successive relaxations were performed in which both the atomic positions and lattice constants were allowed to adjust at fixed V. Gaussian smearing was used for all relaxations.a L SR was originally reported in plane group pm.Pmm2 is the space group of the standardized crystal structure [41].b L G was originally reported in non-standard space group C112/m.C2/m is the space group of the standardized crystal structure [41], which has a 50% smaller V 0 .c βR was originally reported without a space group.We determined the space group from the original publication's reported atomic positions with AFLOW [42].
The final relaxed structure at each V was used to perform a more accurate E calculation using tetrahedral integration with Blochl's correction [53].Several proposed Ta 2 O 5 structures use partially occupied atomic sites (or partial vacancies) to maintain the stoichiometric 2:5 atomic ratio of Ta to O. DFT cannot account for partial occupancy, so calculations with these structures required treating partially occupied sites as either filled or vacant.Site occupancy treatments have been discussed in previous studies [19,54] and details of individual sites can be found in the supplementary material.

Experimental
Our observed powder XRD patterns closely match observations from previous studies on bulk Ta 2 O 5 [17,18,55] and Ta 2 O 5 thin films [39,56].Figure 1 compares an integrated pattern collected at ambient conditions with calculated profiles of the proposed Ta 2 O 5 phases.As expected, agreement with ambient-condition phases was better than ones proposed for high P or high T, but not all proposed ambient phases were good matches.Six of the phases evaluated (B, Z, H L , H SR , γ 1 , γ) did not match the major features of the XRD pattern and therefore did not converge during Rietveld refinement.The other seven phases (δ, λ, β A , β R , L SR , L G , T) all readily converged on a Rietveld solution (figure 2), though β R required adjustment of its unit cell volume upward by ∼8% in order to achieve convergence.We will refer to these seven phases as the 'plausible' Ta 2 O 5 phases.
Upon compression (figure 3), we observed the previously reported merging of the Ta 2 O 5 doublet peaks (i.e. the features which begin at Q ≈ 2.0 and 2.6 Å −1 ) at ≈3 GPa, followed by a transition to an amorphous phase starting at ≈25 GPa [17,18].We did not observe any phase changes upon further compression to 60 GPa, and decompressing the amorphous phase did not result in Ta 2 O 5 recrystallization.Additionally, decompressing the crystalline phase to ambient P did not result in reappearance of the doublet peaks.
Refined unit cell V for the plausible phases as a function of P are shown in figure 4. Uncertainties in V are calculated from the standard deviations of refined Ta 2 O 5 unit cell parameters, and uncertainties in P are calculated from those of the refined P standard unit cell parameters as well as the uncertainties on their published EoS parameters.Both sets of uncertainties were included in an orthogonal distance regression fit to a third-order Birch-Murnaghan EoS, the results of which are shown in table 2.

DFT
Relaxations were performed and cold curves calculated as described in section 2.2 for all plausible structures.All structures except β R retained their original space group symmetry during relaxation.The β R unit cell relaxed to an orthorhombic Pmm2 structure (the same space group as T and L SR ), so the DFT compression curve for β R reflects that structure, rather than the published one.By allowing the lattice constants and atomic positions to relax at fixed V, the cold curves allow for the lattice constants to move away from the initial, isotropically scaled, structures.This results in a softer EoS than would be obtained from isotropic scaling.The same EoS fitting method described above was applied to the DFT calculations for each phase, with EoS parameters listed in table 2.

Discussion
While any of the seven plausible phases could be used to model the experimental data, β R and δ are less plausible than the others.As noted above, the published β R V 0 was 8% too small to match the x-ray observations (all other refined V 0 Comparison between proposed Ta 2 O 5 structures.Red lines show simulated powder XRD patterns for each structure that were calculated from their published unit cells.Black lines show an experimental XRD pattern taken at ambient conditions with the background subtracted.The 'plausible' phases (δ, λ, β A , β R , L SR , L G , T) are clearly better matches to the experimental pattern than the other phases.The peak marked with an asterisk is a reflection from the Cu P standard.
were within 1% of corresponding published values).This, combined with the instability of the β R structure during DFT calculations implies that β R may not be the equilibrium structure for Ta 2 O 5 .Similarly, δ is a significantly worse match to the observed XRD patterns than any of the other phases because it predicts single peaks instead of the observed doublets for some of the highest intensity XRD features.The observed merging of the doublet peaks around 3 GPa could be interpreted as a transition to the δ phase from a lower-symmetry structure.However, considering the asymmetry of the merged Figure 2. Sample Rietveld refinement results for each of seven plausible Ta 2 O 5 phases (i.e.those phases for which refinement converged) at ambient conditions.No Le Bail terms were used during refinement.Reflections from the Cu P standard are indicated with asterisks.Refinements were performed with GSAS-II [44].peaks after the 'transition' and the broadening visible in other peaks, we consider it more likely that the merging is due to P-induced broadening of adjacent peaks.This phenomenon has been previously attributed to local disorder among competing Ta coordination polyhedra [18,57] and may be the source of the residual stress buildup interpreted to lead to the amorphous transition [17,58].We tried fitting a δ phase EoS to only those patterns at P above the peak merger, but the results did not differ from the δ EoS parameters fit over the full range of patterns.Thus, our data do not support a Ta 2 O 5 phase transformation at high P and ambient T until the amorphous transition at 25 GPa, despite suggestions from shock experiments [15,21].
Figure 5 shows a comparison between the experimentally determined EoS parameters for each of the plausible phases.We find that the bulk modulus (K 0 ) and its P derivative (K ′ 0 ) are consistent for all plausible phases except δ.Since δ has a much higher symmetry than the other plausible phases, and is the worst match to the experimental data, this discrepancy is not surprising.The other plausible phases (whether or not β R is included) have a combined parameter range of K 0 = 138.03± 3.68 GPa and K ′ 0 = 1.82 ± 0.45.These can be viewed as the most likely room-temperature EoS parameters for Ta 2 O 5 in the absence of definitive evidence to support one structure over the others, even if the analyzed material is a mix of phases, as may be the case for certain synthesis methods [10,59].Assuming the Ta 2 O 5 phase is L SR , we find it to be more compressible than determined by Stavrou et al (K 0 = 199 GPa, K ′ 0 = 0.1) [18].That study's low K ′ 0 (the error bars of which extend into negative values) was likely an artifact of fitting with a low density of data (16 points compared to the 80 points used in this study).Nonetheless, we do find that Ta 2 O 5 has a K ′ 0 value substantially lower than the standard value of 4.0, meaning that its compressibility is relatively insensitive to P. Assuming the Ta 2 O 5 phase is T, our K 0 is consistent with Li et al (K 0 = 139 GPa) [17].
There is substantial disagreement between the EoS parameters fit to experimental data and those fit to DFT calculations, with the DFT fits resulting in K 0 and K ′ 0 that are larger and more widely varied between phases (table 2).Our DFT calculations reproduce those values of V 0 reported by Perez-Walton et al [19] and come within 4% of that study's K 0 (except for β R and L G , which each disagree by ∼30%).The DFT-based K 0 of Wu et al [14] closely match our experimental values for δ and L SR , but both our experimental and computational values disagree with that study's β A values, as well as the calculated λ EoS parameters of Yang et al [22].Experiment-based EoS parameters should be more compressible than DFT predictions since experiments are performed well above 0 K, but the magnitude of the difference suggests other effects may also be contributing.In particular, DFT simulations at the scale of a unit cell do not capture the disorder effects mentioned above and therefore do not predict the trend toward amorphization.Larger scale DFT-based molecular dynamics simulations would be computationally expensive but might be able to more accurately reproduce these effects.
Our calculated E-V curves (figure 6) agree with previous studies [10,19,31] in finding that λ is the most energetically favorable of the plausible phases, followed by L SR .Phase B and several variations of γ have been calculated with even lower E [19,22], as well as good matches to the experimentally observed band structure [23,38].These low-energy phases may be important at non-ambient temperatures or as building blocks for more complex structures.However, as noted above, these phases cannot account for the observed XRD signatures of bulk Ta 2 O 5 or Ta 2 O 5 thin films at room temperature.

Discriminating between plausible structures
While none of the plausible phases provide an obviously superior match to experimental data, visual inspection suggests that L SR reproduces the observed XRD features (figure 1), especially low intensity ones, slightly better than the other structures.Indeed, Stavrou et al [18] observed XRD patterns almost identical to ours and concluded that L SR was the correct structure based on this correspondence.However, our Rietveld refinements do not quantitatively support this interpretation.For example, an equally valid interpretation might be that a minor coexisting phase is responsible for the low-intensity peaks.L SR does not produce substantially lower residuals than the other plausible phases (figure 2 and supplementary material) and it is not statistically valid to discriminate on the basis of small differences in refinement quality [60].Overall, it is unlikely that the correct Ta 2 O 5 phase can be conclusively identified with powder XRD alone.Since xrays are sensitive to electron density, the high atomic number of Ta gives it an x-ray atomic form factor >30 times larger than that of O.As a result, XRD patterns are insensitive to the O positions that comprise most of each structure.Neutron scattering power, on the other hand, depends less on atomic number.Neutron diffraction has been employed for structural refinement of Ta-Al oxides [61] and should be considered for future investigations of pure Ta 2 O 5 .
Since several phases are approximately equal in refinement quality and experimental EoS parameters, we used our DFT results to discriminate between them.Among the plausible structures with small unit cells (V 0 ∼ 177 Å 3 ), λ is the most compressible (smallest calculated K 0 ), and the same is true for L SR among those with large unit cells (V 0 > 800 Å 3 ).Since these 'softer' values more closely approach the experimental K 0 , this may be taken as support for λ and L SR as candidates for the best Ta 2 O 5 structure.Overall, we consider λ to be a slightly better candidate due to its lower E, consistency in the relative compressibilites of the unit cell directions, and simplicity for use in simulations (see the supplementary material).The difference in minimum E between λ and L SR is small, so it is possible that using different exchange correlation functionals or incorporating room temperature thermal expansion effects via phonon calculations could potentially change the relative ordering of the phases.Similarly, subtle differences in XRD patterns may reasonably be interpreted as supporting L SR (or perhaps L G ) as the single best candidate.Nonetheless, in the absence of definitive evidence, λ should provide a satisfactory model of Ta 2 O 5 for more applications than the other proposed structures.

Conclusions
We evaluated a suite of proposed Ta 2 O 5 structures to determine which ones could produce the XRD patterns observed in experiments.We found several proposed phases (B, Z, H L , H SR , γ 1 , γ) could not reasonably reproduce the observations.Seven proposed phases could, however, and of these plausible structures, five (λ, β A , L SR , L G , T) cannot be conclusively discriminated between.Indexing high-P experiments against these plausible phases reveals that, no matter which is the correct structure, it is stable at T up to the amorphous transition at 25 GPa, and its compression can be described with the EoS parameters K 0 = 138 ± 3.68 GPa and K ′ 0 = 1.82 ± 0.45.Significant discrepancies remain between experimental and DFT-calculated compression curves, probably due to the effects of yet-unmodeled microstructural disorder, but the best agreement is obtained for λ and L SR .λ is also the most energetically favorable of the plausible phases, and thus the strongest single candidate for the Ta 2 O 5 structure at ambient conditions.

Figure 1 .
Figure 1.Comparison between proposed Ta 2 O 5 structures.Red lines show simulated powder XRD patterns for each structure that were calculated from their published unit cells.Black lines show an experimental XRD pattern taken at ambient conditions with the background subtracted.The 'plausible' phases (δ, λ, β A , β R , L SR , L G , T) are clearly better matches to the experimental pattern than the other phases.The peak marked with an asterisk is a reflection from the Cu P standard.

Figure 3 .
Figure 3. Integrated XRD patterns collected during compression of experiment 1.The four peaks remaining after Ta 2 O 5 becomes amorphous are reflections from the gasket (cross) and the Cu P standard (asterisks).

Figure 4 .
Figure 4. Unit cell V for the plausible Ta 2 O 5 phases as a function of P. Solid lines are fits to the experimental data and dashed lines are cold curves calculated with DFT (table 2).

Figure 6 .
Figure 6.Total energy versus volume curves of each of the plausible phases calculated with DFT.λ has the lowest energy at all V and is thus the most thermodynamically favorable structure.Energies have been normalized to the E 0 of λ, which is −71.04 eV per formula unit.

Table 1 .
Proposed Ta 2 O 5 phases evaluated in this study.

Table 2 .
Birch-Murnaghan EoS parameters for the plausible Ta 2 O 5 phases.Numbers in parentheses show 1σ uncertainties on the last digit.Covariance parameters are listed in the supplementary material.EoS parameters K 0 and K ′ 0 fit to experimental data for each of the plausible phases (table2).