Prediction of crystalline Ta4O9 phase using first principles-based cluster expansion calculations

Tantalum is the only element of Group 5 in the periodic table that lacks any experimental reports on the existence of reduced crystalline oxide between the pentoxide (Ta2O5) and the dioxide (TaO2). We computationally predict the existence of a novel tantalum oxide with Ta4O9 stoichiometry, which lies at the midpoint between Ta2O5 and TaO2. The ground-state Ta4O9 structure was found through simulated annealing based on a cluster expansion model, which is trained using 186 density functional theory calculations. The newfound Ta4O9 material has space group number 10 (P2/m), and it can be viewed as an oxygen-deficient λ-Ta2O5 structure in which oxygen vacancies aggregate pairwise in nearest-neighbor sites. Tad–Tad bonds fill the spatial void of the oxygen vacancies, keeping the system non-magnetic and non-metallic. The synthesis of the new Ta4O9 crystal is deemed feasible through a controlled reduction of λ-Ta2O5. The reported Ta4O9 has the potential to open new avenues in catalysis and resistive switching device applications where the reduced tantalum oxides are broadly employed. © 2020 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0027018., s


I. INTRODUCTION
Great attention has been focused on tantalum oxide since the turn of the millennium. Various phases of tantalum oxide have been investigated for use in photocatalysis, 1 in electrocatalysis, 2 as coating in biomedical applications, 3 and as the resistive layer in Resistive Random Access Memory (ReRAM) devices. 4 The most stable stoichiometry of the tantalum-oxide system is Ta 2 O5. Although Ta 2 O5 fabricated at room temperature is amorphous, [5][6][7] samples are known to crystallize when fabricated at temperatures above ∼400 ○ C. 8 Likewise, annealing samples at temperatures between 500 ○ C and 700 ○ C also leads to crystallization. 5,[9][10][11] Recently, Ta 2 O5 crystallites were observed in an electroformed nanofilament of a ReRAM device. 12 This finding raises a need for further investigation of crystalline tantalum oxide even in resistive switching applications where devices are usually reported as nonstoichiometric. 4,13,14 The atomic coordinates of low-temperature orthorhombic Ta 2 O5, or β-Ta 2 O5, were established in 2002 through Rietveld refinement of existing x-ray spectra. 15 However, a density functional theory (DFT) study later revealed imaginary phonon frequencies for β-Ta 2 O5, 16 which indicate a structural instability. 17 Lee et al. instead proposed the orthorhombic λ phase and used DFT calculations to support their proposition. 17 The λ phase yields a lower DFT energy than the β phase, as well as a more accurate prediction of the experimental bandgap of 4.0 eV. 18 Although Ta 2 O5 has received the most attention of all tantalum-oxide phases, reduced TaOx phases constitute an important area of study as well. The work of Awaludin, Okajima, and Ohsaka, 2 for instance, characterized reduced Ta 2 O5-x for catalysis applications. The switching process in TaOx ReRAM devices relies on the valence change mechanism, 19 in which regions of the device are reduced until a conductive filament is formed. 4,20,21 The chemistry of reduced Ta 2 O5-x, though, has received far less attention than Ta 2 O5, with the exception of crystalline TaO 2 . Despite being a metastable phase of the tantalum oxide system, TaO 2 has been observed both in the rutile phase [22][23][24][25]  On a theoretical level, oxygen vacancies (vO) in λ-phase Ta 2 O5 are well studied in terms of formation energy and defect orbitals 27 as well as migration barriers. 28,29 However, the cited works have focused on the effects of singular vO defects exclusively. In contrast, little is known about the behavior of larger concentrations of vO in crystalline Ta 2 O5, which is an important topic for ReRAM devices. 19 In this article, we report on a large-scale investigation of oxygen vacancy-defective Ta 2 O5 in the λ phase, for vO concentrations from 0% to 25%. Using DFT calculations combined with cluster expansion (CE), we have found a new ground state of the TaOx system with the chemical formula Ta 4 O 9 , which is stable relative to phase separation of Ta 2 O5 and TaO 2 . Exploring the atomic structure reveals that threefold coordinated oxygen vacancies aggregate in pairs at high (∼10%) vO concentrations, with no decrease in coordination numbers anywhere in the system. Neighboring Ta ions are sufficiently close that their d orbitals overlap, restoring their coordination number and stabilizing the pair defect. We believe that the Ta 4 O 9 phase may be synthesized through reduction of orthorhombic Ta 2 O5. Furthermore, the discovery of the pair defect is an important contribution to ReRAM research, where Ta 2 O5 crystallites are known to form and vO are created in large enough quantities to allow for pair-wise aggregation.

II. COMPUTATIONAL METHODS
We created in total 186 structures with vO concentrations ranging from 0% to 25%, i.e., stoichiometric Ta 2 O5 to stoichiometric TaO 2 using the Cluster Expansion in Atomic Simulation Environment (CLEASE) software. 30 Working from the 14-atom unit cell of λ-Ta 2 O5 (see Ref. 17), vO distributions with various cell sizes and shapes were used to train the CE model, with the largest training structures containing 168 atoms. For all distributions of vO, we used DFT to relax the atomic forces. The correlation between the input atomic structures and the final formation energies calculated from DFT were then determined using the CLEASE code. The trained CE model was used to suggest the optimal arrangements of vO for different concentrations, which were calculated using DFT with the same relaxation scheme.
All DFT calculations were performed using the Vienna Ab Initio Simulation Package (VASP). [31][32][33][34] O and Ta atoms were described by PAW pseudopotentials 35 with valence electron configuration 5p 6 5d 3 6s 2 and 2s 2 2p 4 , respectively. Unless noted otherwise, our calculations employed the Perdew-Burke-Ernzerhof (PBE) Generalized Gradient Approximation (GGA) to describe the electronic exchange and correlation. 36 Additional high-precision calculations were performed incorporating 25% exact exchange using the Heyd-Scuseria-Ernzerhof functional with a range-separation parameter of 2.0 × 10 −1 (i.e., HSE06). [37][38][39] Our setup used a plane wave basis for the electronic wave functions with a cutoff energy of 520 eV. To sample the first Brillouin zone, we used Monkhorst-Pack grids 40 with a density of at least 3.50 per Å −1 , shifted to the Γ point. For relaxations, the atomic coordinates were considered converged when interatomic forces were lower than 2.5 × 10 −2 eV Å −1 , while for the electronic density, the convergence threshold was 1.0 × 10 −5 eV. The final CE model had a formation energy prediction accuracy of 16.5 meV/atom based on the 10-fold cross-validation score. The optimal vO arrangements were predicted using the CE model in conjunction with the simulated annealing technique. The structures with the 2 × 2 × 3 supercell containing 168 atoms (vacancies were counted as atoms in this context) were used for generating the structures with the optimal vO arrangements. The minimumenergy structure for each vO concentration was found by gradually lowering the temperature from 5000 K to 1 K with 50 000 attempted atomic swaps per temperature.

A. Convex hull and structural features
The results of our CE-enhanced structure search are summarized in the convex hull of Fig. 1. Each investigated structure is represented with its energy relative to phase separation of λ-Ta 2 O5 and TaO 2 . For TaO 2 , we use the 96-atom tetragonal cell of Khitrova, Klechkovskaya, and Pinsker 26 (labeled "K1967"). The ICSD lists two similar tetragonal structures from Khitrova, Klechkovskaya, and Pinsker's work, with Coll. codes 60 267 and 60 268. We chose the latter because our calculations found it to have a slightly lower total energy per formula unit (f.u.). All of the energies reported in Fig. 1(a) are based on DFT calculations, while Fig. 1(b) shows the accuracy of the CE model used to predict the minimum-energy structure for each stoichiometry. Most remarkably, the hull shows preference for a solid solution of vO, with a wide range of structures falling just below zero relative energy. The lowest point on the hull lies at x = 0.5 and has a stoichiometry of Ta 4 O 9 (Ta 2 O5-0.5 in axis coordinates).
Apart from the K1967 model, Fig. 1(a) depicts two additional calculations with stoichiometries corresponding to x = 1 (i.e., TaO 2 ). The first, and overall most stable, is a phase separation into Ta and Ta 2 O5 (labeled "Ta + Ta 2 O5"). We include this to show that TaO 2 itself is metastable with respect to decomposition into Ta and Ta 2 O5. Second, we include the rutile TaO 2 -structure documented by Schönberg et al., Niebuhr, and Terao 22-24 (labeled "Rutile TaO 2 ").
The comparison to Ta + Ta 2 O5 shows that our newfound Ta 4 O 9 phase is unstable compared with that reference. However, the two TaO 2 phases marked with triangles in Fig. 1(a) have been synthesized experimentally despite lying ∼358 meV/f.u. (rutile) and ∼216 meV/f.u. (K1967) above phase-separated Ta + Ta 2 O5 according to our calculations. Since the Ta 4 O 9 structure reported here is stable with respect to λ-Ta 2 O5 and K1967's TaO 2 -structure, it should be synthesizable. In Sec. S1 of the supplementary material, we corroborate the stability of Ta 4 O 9 with a phonon analysis using Phonopy. 41 We further confirmed the stability of our Ta 4 O 9 structure by recreating it in the Ta 2 O5 structure proposed by Ramprasad. 42 The vO-defective λ phase was found to be 0.24 eV/f.u. more stable than the vO-defective model of Ramprasad. The crystalline Ta 4 O 9 structure described here is a form of vO-defective λ-Ta 2 O5 by construction. Hence, the newfound Ta 4 O 9 structure can be reached by reduction of crystalline Ta 2 O5. In addition, based on the observation of Ta 2 O5 nanocrystallites by Ma et al., 12 we hypothesize that Ta 4 O 9 may be formed during switching operations of ReRAM devices due to the large quantities of vO generated during said operations.
Having established a new metastable state in the TaOx system, we set to work characterizing its atomic structure. The crystal structure of λ-Ta 2 O5 is depicted in Fig. 2(a)   As seen in (f), this raises the coordination number of all involved atoms by 1; the fourfold coordinated Ta atoms become fivefold coordinated, the fivefold coordinated Ta atoms become sixfold coordinated, and the shifted twofold coordinated O atoms become threefold coordinated. The rectangular arrangement of Ta ions surrounding the pair site is intact, albeit slightly compressed. The inter-planar distance has increased by 0.04 Å, from 3.83 Å to 3.87 Å, which is a relatively minor expansion of ∼1%, likely owing to the fact that the Ta-O chains in the out-of-plane direction are intact.
The newfound Ta 4 O 9 structure can be reduced to a primitive cell with 26 atoms, which is depicted in Fig. 3. This primitive structure contains two neighboring vO and belongs to space group number 10 (P2/m). The lattice vectors and Wyckoff positions for this structure are listed in Table I.

B. Electronic properties of Ta 4 O 9
We now proceed to characterize the electronic properties of the newfound primitive cell, beginning with the band structure and orbital-Projected Density of States (PDOS). Since PBE is known to underestimate bandgaps, we performed a single-point calculation with the Heyd-Scuseria-Ernzerhof (HSE) functional on the PBE-relaxed structure using a k grid reduced to (2 × 5 × 2) from (3 × 6 × 3). The Ta 4 O 9 band structure visualized in Fig. 4(a) exhibits two bands localized in the middle of the gap, resulting from the two removed O atoms. The PDOS spectrum in Fig. 4(b) shows that the orbital character of the defect state is primarily Ta d, secondarily O p, and tertiarily Ta s (see the inset). The band edges of Ta 4 O 9 are otherwise well aligned with those of Ta 2 O5. The defect state is filled according to the location of the Fermi energy for this state, with an energy gap of 1.12 eV between the defect state and the conduction band minimum. Since Ta 4 O 9 has previously been investigated in the amorphous TaOx system, 44,45 we believe that the ∼1.1 eV bandgap of crystalline Ta 4 O 9 can be used to distinguish it from its amorphous counterpart. To further characterize the Ta 4 O 9 phase, we present its optical absorption spectrum in Sec. S3 of the supplementary material. We find that the edge of the optical absorption lies at ∼1.1 eV, which may make it suitable for photovoltaic applications. We compare the above band structure to that of λ-Ta 2 O5 in Sec. S2 of the supplementary material. A thorough analysis of the nature of these two mid-gap bands may help to understand the particular stability of the oxygen divacancy center that gives rise to the Ta 4 O 9 structure. In the following, we analyze the wave function at the Γ point of the two mid-gap states, which we will refer to as the Highest Valence Band (HVB) and second-highest Valence Band (SHVB).
From an electrostatic point of view, it is expected that the four electrons needed to compensate the two O 2− vacancies will localize in the two TaO5 complexes (the only Ta left fivefold coordinated after the reconstruction). This localization would imply a change in the nominal oxidation state of these two Ta ions from +5 to +3, thus filling some of their d orbitals. Another important consideration is that the two TaO5 complexes, which adopt a distorted square pyramidal shape, are very close to each other with just a void region between the bases of the two pyramids. In this scenario, the distance between the two pentacoordinated Ta ions is only 2.94 Å, and substantial overlap between their corresponding d orbitals may occur. A significant overlap of the d orbitals is corroborated by the wave functions of the SHVB and HVB shown in Fig. 5.
HVB has a character of a bonding orbital between the dyz orbitals of the two pentacoordinate Ta ions [the local axis of the distorted square pyramids is shown in Fig. 5(c)]. SHVB has a character of a bonding orbital between the d 3z 2 −r 2 orbitals of the two pentacoordinated Ta ions. The equatorial part of the d 3z 2 −r 2 orbitals has disappeared due to the slight hybridization with the Ta s orbitals [see the inset of Fig. 4(b)]. The bonding interaction between the Ta d orbitals in SHVB is so strong that it makes the state resemble an F center, i.e., a state fully localized in the void left by an anionic vacancy. This intense interaction between the d orbitals of the two TaO5 complexes restores the missing bonds to complete full sixfold coordination for every Ta ion in the system, conferring the unique stability of the Ta 4 O 9 structure.

IV. CONCLUSIONS
We have combined DFT calculations and CE to perform a structure search for vO-defective λ-Ta 2 O5. We found a phase with stoichiometry Ta 4 O 9 at the midpoint between Ta 2 O5 and TaO 2 , which is stable with respect to phase separation of the latter two structures. The Ta 4 O 9 structure has a highly ordered defect distribution with five distinct features: (i) vacancies form exclusively on threefold coordinated sites, (ii) vacancies aggregate pair-wise on neighboring sites, (iii) vacancy pairs are arranged with maximal inplane spacing, (iv) vacancy pairs form on equivalent sites in successive planes, mimicking the chain-like atomic order in the out-ofplane direction, and (v) nearby twofold coordinated O atoms shift closer to undercoordinated Ta atoms, which are thus compensated for their loss in the coordination number due to the defects. The high degree of order in the defect structure allowed us to reduce the ground state supercell to a monoclinic primitive cell consisting of 26 atoms (Ta 8 O 18 ).
We further characterized the ground-state structure with respect to PDOS and frontier orbital wave functions. The pair defect results in a localized and filled mid-gap defect state with predominantly Ta d orbital character and lesser contributions from O p and Ta s. Spatially, the wave function associated with the defect pair is localized between the individual vO sites and projected primarily onto the d orbitals of the nearest Ta neighbors. The two Ta atoms closest to the defect site are sufficiently close that these d orbitals overlap, forming a bonding orbital. This metal bonding restores the Ta atoms to full sixfold coordination, explaining the extraordinary stability of the pair defect. Finally, the newfound ground state is non-magnetic and non-metallic.
The most immediate study concerning the Ta 4 O 9 phase is to find experimental pathways to synthesize it. If successful, it is worthwhile to investigate the Ta 4 O 9 surface for catalysis applications since the nature and concentration of the defect states may have a profound impact on the catalytic properties of the surface. Finally, it APL Mater. 8, 121101 (2020); doi: 10.1063/5.0027018 8, 121101-5 scitation.org/journal/apm is worth investigating whether the pair-defects found for Ta 4 O 9 may linger in ReRAM device crystallites when switching to the OFF state, and if so, how they might influence crucial parameters such as ON/OFF current ratio and device longevity.

SUPPLEMENTARY MATERIAL
See the supplementary material for information about the phonon spectra and band structures of λ-Ta 2 O5 and Ta 4 O 9 and dielectric function of Ta 4 O 9 .