Electronic Structures of Oxygen-deficient Ta2O5

We provide a first-principles description of the crystalline and oxygen-deficient Ta2O5 using refined computational methods and models. By performing calculations on a number of candidate structures, we determined the low-temperature phase and several stable oxygen vacancy configurations, which are notably different from the previous results. The most stable charge-neutral vacancy site induces a shallow level near the bottom of conduction band. Stability of different charge states is studied. Based on the results, we discuss the implications of the level structures on experiments, including the leakage current in Ta2O5-based electronic devices and catalysts.


I. Introduction
Tantalum pentoxide (Ta 2 O 5 ) has attracted considerable attention in recent years due to its potential applications in the electronics industry [1,2] and catalysis [3,4].
The high dielectric constant of Ta 2 O 5 [5][6][7][8][9][10][11][12] puts it a candidate to substitute SiO 2 in the conventional complementary metal-oxide-semiconductor (CMOS) devices. The high activity of Ta 2 O 5 in reducing organic molecules under the UV irradiation makes it a candidate for an effective photocatalyst [3]. Moreover, oxygen-deficient Ta 2 O 5 has attracted interests as a new non-noble metal cathode material for the polymer electrolyte fuel cell (PEFC) [4]. In these applications, the oxygen vacancy plays a key role: It is considered a source of the leakage current in CMOS and a major reaction center in PEFC. The role of oxygen vacancies in Ta 2 O 5 is contrasting in CMOS and PEFC applications: The vacancy is the less the better for the former while the more the better for the latter.
Despite its importance in industrial applications, the basic structural and electronic properties of Ta 2 O 5 are not well understood. Stephenson and Roth [13] proposed a crystal structure containing 11 formula units (22 Ta and 55 O atoms) for the low-temperature phase (referred to as L-Ta 2 O 5 ), where some oxygen sites are partially occupied to satisfy the stoichiometric ratio. It is yet unknown how the oxygen atoms occupy those sites to form the most stable crystalline structure. Later, the structures of simpler crystal models, such as the β-Ta 2 O 5 [14] and δ-Ta 2 O 5 [15], were also suggested.
Density functional theory (DFT) calculations were applied to Ta 2 O 5 on its 3 structural, electronic and dielectric properties within the local density approximation (LDA) or generalized gradient approximation (GGA) [16][17][18][19]. The works on oxygen-deficient Ta 2 O 5 used some simplified crystalline models for the L- Ta 2 O 5 phase to study the energy levels induced by the oxygen vacancies [16,17]. Recently, both β-Ta 2 O 5 [14] and δ-Ta 2 O 5 [15] were found meta-stable by phonon calculations [18], and the L-Ta 2 O 5 phase was found the most stable. This also cast doubt on the reliability of the simplified models [16,17]. Moreover, DFT-LDA/GGA severely underestimates the value of band gap [16,17,19].
In this work, we revisit the oxygen-deficient Ta 2 O 5 by providing refined theoretical data on its structural and electronic properties. We overcome the band gap problem encountered in the conventional DFT-LDA/GGA calculations by using hybrid functional for the exchange-correlation interactions, and the instability problem by using a crystal model without simplifying the unit cell of L-Ta 2 O 5 [13].
We have investigated a number of vacancy sites and the associated level structures, which are found distinct from the previous reports.

II. Computational & Modeling Methods
The calculations are carried out by the Vienna ab initio simulation package (VASP) [20,21], using a plane wave basis set and the projector-augmented-wave (PAW) potentials [22,23]. The exchange-correlation interactions of electrons are described by two types of functional: The PBE functional [24] and the PBE0 hybrid functional [25]. The energy cutoff for plane waves is 600 eV. For the calculation of unit cell, we 4 use a 4×2×4 k-mesh. For the (2×1×2) supercell (with and without oxygen vacancies), a 1×1×2 k-mesh is employed for structural relaxation and the calculation of electronic density of states (DOS). The k-meshes are generated using the Monkhorst-Pack scheme [26]. The parameters describing the Ta 2 O 5 unit cell are: a = 6.332 Ǻ, b = 40.921 Ǻ, c = 3.846 Ǻ; α = 90º, β = 90º, γ = 89.16º. In the experimental L-Ta 2 O 5 structure model [13], there are two types of partially occupied oxygen sites: four sites with 75% occupation and eight sites with 25% occupation. We have investigated where the first three terms are the total energies of the oxygen-deficient Ta [27].
Here, the oxygen vacancy at the charge-neutral, singly and doubly charged states will be investigated. Using the fact that the vacancy has several meta-stable structures that are dependent on the charge state, it is found effective to change the charge state during the geometry optimization to search the stable structure more globally.
Therefore, for the charge-neutral systems, we take two procedures to make a comparison. In procedure 1, we relax the structures while keeping the systems charge-neutral. In procedure 2, they are optimized at the doubly positively charged state followed by re-optimization at the charge-neutral state.

III. Results and Discussion
The building blocks of the L- Ta Table I, where only the charge-neutral state is tabulated because of space limitation. From Table I, one sees that most of the oxygen vacancies at the in-plane sites have lower formation energies and consequently more stable than the ones at the cap sites. The first significant difference from the previous works is that the magnitude of vacancy formation energies is only about one half of the values obtained by the work [17] mentioned above. This indicates that the simplified models in the previous works [16,17] provide quantitatively different structural properties of the oxygen-deficient system. The configuration D at the in-plane site prepared by procedure 2 turns out to be the most stable. Configurations prepared by procedures 1 and 2 are of almost the same stability except for D and E, where the two configurations prepared by procedure 2 are much more stable than the ones obtained using procedure 1.
The characteristics of the vacancy levels are also displayed in Table I To understand the difference resulted from the relaxation procedure 1 and procedure 2, we studied the connection between the total energy of the defective system and the lattice relaxation. The total energy of the relaxed structure can be written as follows where E [Ta 2 O 5-δ ] unrelaxed is the total energy of the corresponding unrelaxed defective structure, and ΔE is the amount of total energy lowering due to the relaxation of bond lengths and bond angles. The amount of the lattice relaxation is shown in Table II.
Here we list the variation of Ta  This is consistent with the fact that the formation energy (Table I)  semiconductor [30]. Considering the fact that most oxygen vacancy configurations under consideration (Table I)  capacitor persists and depends critically on the local atomic structures around Ta [31].
We then study the stability of Ta 2 O 5-δ at different charge states. By neglecting the entropy term [32], the grand potential of the system may be written as where [Ta 2 O 5−δ ] δq for δq = 0 is the total energy of most stable charge-neutral configuration, and for δq = +1 and +2 are the total energies of the relaxed configurations at the corresponding charge states; µ is the electron chemical potential (i.e., the Fermi level); δV is the potential to align the averaged electrostatic potential of the defect system to that of a perfect crystal [33]. In our case we find 0 < δV < 0.23 eV. The calculated grand potential is shown in Fig. 4 . S3). So, we select Type 1 as a representative of p2mm-Ta 2 O 5 . X-ray Diffraction (XRD): In order to compare the experimental XRD data with present identified crystal structure, we calculated the XRD by using a RETAIN-FP program included in VESTA program (see Fig. S4)

FIG. S4
The XRD data of Type 1 Ta 2 O 5 .