Tendencies in ABO 3 Perovskite and SrF 2 , BaF 2 and CaF 2 Bulk and Surface F -Center Ab Initio Computations at High Symmetry Cubic Structure

: We computed the atomic shift sizes of the closest adjacent atoms adjoining the (001) surface F -center at ABO 3 perovskites. They are signiﬁcantly larger than the atomic shift sizes of the closest adjacent atoms adjoining the bulk F -center. In the ABO 3 perovskite matrixes, the electron charge is signiﬁcantly stronger conﬁned in the interior of the bulk oxygen vacancy than in the interior of the (001) surface oxygen vacancy. The formation energy of the oxygen vacancy on the (001) surface is smaller than in the bulk. This microscopic energy distinction stimulates the oxygen vacancy segregation from the perovskite bulk to their (001) surfaces. The (001) surface F -center created defect level is nearer to the (001) surface conduction band (CB) bottom as the bulk F -center created defect level. On the contrary, the SrF 2 , BaF 2 and CaF 2 bulk and surface F -center charge is almost perfectly conﬁned to the interior of the ﬂuorine vacancy. The shift sizes of atoms adjoining the bulk and surface F -centers in SrF 2 , CaF 2 and BaF 2 matrixes are microscopic as compared to the case of ABO 3 perovskites.


Introduction
In the last 50 years, great attention has been paid to an in-depth understanding of the radiation formation of lattice defects in alkali and alkaline earth metal halides, as well as in simple and complex oxides.Several review articles were published, where both the mechanisms of their creation and the properties of the main defects were described in detail [1][2][3][4][5].Lattice-point defects in these materials are traditionally called color centers and are classified into several large groups, among which electron centers, hole centers and interstitial particles are the most noticeable primary point radiation defects.As for electronic centers, F-centers are among the most intensively studied point defects in alkali and alkaline-halides.An ordinary F-center consists of one electron captured by a halogen vacancy [1,2,5].Such F (Farbe) centers in alkali halides and alkaline-earth halides and oxides have been studied for many decades [1,2,[5][6][7][8][9][10][11][12].Thus, F-centers are defects in ionic crystals in which an anion is replaced by one or more trapped electrons.These vacancy-trapped electrons, confined and screened by the surrounding crystal lattice, thus form gap states with unique optical, electrical and magnetic properties [7][8][9][10][11][12][13][14][15][16] that are relevant to different optoelectronic devices.How efficient the formation is of point defects, including F-centers, is also the determining factor of the material radiation resistance in nuclear applications [17][18][19][20][21]. F-centers at surfaces, along with other anionic defects, are also relevant in catalysis [22].
The F-center in ABO 3 perovskites is the oxygen vacancy (V 0 ), which catches 2 electrons.The V 0 is the most frequent point defect in ABO 3 perovskites [23][24][25][26][27]. Bearing in mind the technological relevance of ABO 3 perovskite matrixes, as well as quantity of F-centers, which unavoidably exist in these materials and deteriorate their quality, the number of F-center studies is still insufficient [28][29][30][31][32][33][34][35].In order to save the computer time, we performed bulk and (001) surface F-center ab initio computations in ABO 3 perovskites in their high-symmetry cubic phase.The SrZrO 3 , PbTiO 3 , BaTiO 3 and SrTiO 3 cubic unit cells accommodate five atoms.The coordinates (0, 0, 0) have the A atom (A = Sr, Pb, Ba or Sr).It is placed at the corner position of the cube.The Ti atom is placed in the body center position of the cube, with the following coordinates ( 12 , 1 2 , 1 2 ).Finally, the three O atoms are placed in the face centered positions of the cube with the coordinates ( 12 , 1 2 , 0), ( 1 2 , 0, 1 2 ) and (0, 1 2 , 1 2 ).All our computed SrZrO 3 , PbTiO 3 , BaTiO 3 and SrTiO 3 perovskites at their high symmetry cubic structure belong to the space group Pm3m with the space group number 221.Our computed SrZrO 3 , PbTiO 3 , BaTiO 3 and SrTiO 3 perovskites exhibit a large variety of different structural phase transitions from the high temperature and symmetry cubic paraelectric phase as the temperature is reduced.For example, the SrZrO 3 exhibits a cubic perovskite-type structure.The space group (Pm3m: 221) incorporates 48 symmetry operations.The Wyckoff positions for the atoms are as follows: Sr 1a (0.0, 0.0, 0.0), Zr 1b (0.5, 0.5, 0.5) and O 3c (0.0, 0.5, 0.5).The SrZrO 3 perovskite undergoes three phase transitions: orthorhombic (Pnma), orthorhombic with another space group (Cmcm), tetragonal (I4/mcm) and cubic (Pm3m) at temperatures of approximately 970, 1100 and 1440 K, respectively.In contrast, the SrTiO 3 crystal has a polar soft mode, but it does not exhibit a ferroelectric phase transition and always remains in its high symmetry cubic phase.Finally, BaTiO 3 is a perovskite that undergoes a phase transition from a ferroelectric tetragonal phase to a paraelectric cubic phase due to heating above 403 K in temperature.
Taking into account the soaring industrial significance of SrF 2 , BaF 2 and CaF 2 matrixes, it is only logical that there exist a lot of studies devoted to those materials .The SrF 2 , BaF 2 and CaF 2 unit cells are equal, and they contain three ions.The cation is located in the coordinate origin (0, 0, 0).The two anions are located at ( 1 4 , 1 4 , 1  4 ) and ( 3 4 , 3 4 , 3  4 ).The fluorites SrF 2 , BaF 2 and CaF 2 are the cubic Fm3m large-band-gap insulators.Their space group number is 225.The F-center in CaF 2 has been observed experimentally by electron-spin-resonance techniques.The F-center resonance in CaF 2 has been detected by Arends [58] in additively colored crystals and associated with the optical absorption band at 3.3 eV.Nepomnyashchikh et al. [59] experimentally detected that X-ray irradiation at 77 K of undoped BaF 2 produces V k and F-centers.They have absorption bands at 3.4 and 2.3 eV, respectively [59]. Finlly, the experimentally measured F-center absorption energy at 4 K temperature in the SrF 2 crystal is equal to 2.85 eV [60].Present-day knowledge of defects in SrF 2 , BaF 2 and CaF 2 crystals has helped to generate the new area of high technology named defect engineering. SF 2 , BaF 2 and CaF 2 crystals could be the important optical materials if we can manage the photo-induced defect process [61,62].
The aim of this contribution is to report our comprehensive ab initio computation results dealing with F-centers in SrZrO 3 , PbTiO 3 , BaTiO 3 and SrTiO 3 matrixes, as well as SrF 2 , BaF 2 and CaF 2 fluorites.Next, we analyzed our ab initio computation results and created a unified theory, which describes systematic tendencies in ab initio computations dealing with F-centers in SrZrO 3 , PbTiO 3 , BaTiO 3 and SrTiO 3 perovskite matrixes, as well as SrF 2 , BaF 2 and CaF 2 fluorites.

Details of Ab Initio Computations of the F-Centers in ABO 3 Perovskites, as Well as in SrF 2 , BaF 2 and CaF 2 Fluorites
We computed the F-centers in SrZrO 3 , PbTiO 3 , BaTiO 3 and SrTiO 3 perovskites, as well as SrF 2 , BaF 2 and CaF 2 fluorites, by means of the CRYSTAL computer program package [63].For our ab initio F-center computations, we used the B3PW hybrid exchange-correlation functional [64].On the contrary to the plane-wave program packages [65,66], the CRYSTAL brings into play Gaussian-type functions as the basis [63].It is worth noting that Gaussiantype functions have been intensively investigated during the last years [67,68].For our numerical computations of F-centers in SrZrO 3 , PbTiO 3 , BaTiO 3 and SrTiO 3 perovskites, we used the basis sets created by Piskunov et al. [69].We bring into play the 8 × 8 × 8 extension of the Pack Monkhorst mesh [70].The ABO 3 perovskite F-center computations for bulk were performed employing 3 × 3 × 3 extended supercell with 134 atoms and one F-center (Figure 1).It is important to note that the generalized-gradient (GGA) and local-density (LDA) approximations to the density functional theory (DFT) strongly underestimate the band gap of insulators and complex oxide materials [46,71].From another side, the Hartree-Fock (HF) method considerably overestimate the band gap of insulators and complex oxide materials [46,71].In order to obtain as good as possible results for the band gaps of SrF 2 , BaF 2 and CaF 2 fluorites, as well as SrZrO 3 , PbTiO 3 , BaTiO 3 and SrTiO 3 perovskites, we performed our ab initio computations by means of the B3PW hybrid exchange-correlation functional.The B3PW functional contain a portion of exact exchange energy density from the HF method (20%).The rest of the exchange-correlation part consisted of different approaches, including both exchange and correlation.Therefore, the B3PW functional, since it is a superposition of HF and DFT methods, is much better suited for our band gap and optical property (defect level positions in the band gap) ab initio computations in SrF 2 , BaF 2 , CaF 2 , SrZrO 3 , PbTiO 3 , BaTiO 3 and SrTiO 3 matrixes.With aimed to compute the ABO 3 perovskite; for example, for the the SrZrO 3 (001) surface F-center, we selected a 3 × 3 × 1 times increase in the size of the surface supercells.Namely, with aim to compute the F-center situated on the ZrO 2 -terminated SrZrO 3 (001) surface, we took away the arbitrary surface oxygen atom (Figure 2).In order to maximally exact describe the F-center, we used an extra basis function [63].Namely, we fixed inside the oxygen vacancy auxiliary basis function [63], identical to the commonly named "ghost atom".The formation energy of the bulk oxygen vacancy in ABO 3 perovskites was computed by us the using subsequent equation: where E(oxygen) is our computed total energy for an oxygen atom, and E(F) and E(perfect) are our computed total energies for the F-center containing and perfect ABO 3 crystal, respectively.We discuss our computational methodology for F-centers in SrF 2 , BaF 2 and CaF 2 matrixes by using the CaF 2 case as an example.In our F-center computations in CaF 2 matrix, we used a supercell consisting of 48 atoms (Figure 3).In order to produce the F-center, we removed one fluorine atom and kept the supercell charge neutral (Figure 3).To compute the fluorine vacancy with electron (F-center) formation energy in CaF 2 matrix, we employed the subsequent expression: where E(fluorine) is our ab initio computed total energy for the isolated fluorine atom; and E(F) and E(perfect) are our ab initio computed total energies for the defective (F-center containing) CaF 2 crystal and the perfect CaF 2 crystal.We utilized the Mulliken representation for the elucidation of the bond populations and the effective atomic charges [72-74].As an opener of our F-center computations in SrZrO 3 , PbTiO 3 , BaTiO 3 , SrTiO 3 , SrF 2 , BaF 2 and CaF 2 matrixes [36,42,[75][76][77][78][79], we performed computations for the pristine crystal bulk lattice constants.Our B3PW approximation obtained bulk lattice constants for SrZrO 3 (4.163Å), PbTiO 3 (3.936Å), BaTiO 3 (4.008Å), SrTiO 3 (3.904Å), SrF 2 (5.845Å), BaF 2 (6.26 Å) and CaF 2 (5.50 Å), and these are in a fair correspondence with the literature's experimental information [80][81][82][83][84][85][86][87][88][89][90][91][92][93][94] (Table 1).For example, our computed SrTiO 3 bulk lattice constant (3.904Å) is almost in perfect agreement with the experimentally detected SrTiO 3 bulk lattice constant (3.89845Å).Moreover, our ab initio B3PW computed SrF 2 (5.845Å), BaF 2 (6.26 Å) and CaF 2 (5.50 Å) bulk lattice constants are in a fair agreement with the respective experimental information (5.799, 6.20 and 5.46 Å).Our computation data, dealing with the atomic shifts around the bulk and (001) surface F-centers in SrZrO 3 , PbTiO 3 , BaTiO 3 and SrTiO 3 matrixes, are listed in Table 2.
As we can see from Table 2, all B (Zr or Ti) atoms are repulsed from the bulk and (001) surface F-centers in all four our calculated perovskites.The repulsion magnitudes for the Zr atoms from the (001) surface F-center located on the ZrO 2 -terminated SrZrO 3 (001) surface (+9.17% of a 0 ), as well as for the Ti atoms from the F-centers located on the TiO 2 -terminated PbTiO 3 and SrTiO 3 (001) surfaces (+9.98 and +14.0% of a 0 ), are considerably larger than the B atom repulsion magnitudes from the F-centers located in the bulk of SrZrO 3 , PbTiO 3 and SrTiO 3 matrixes (+3.68, +1.63 and 7.76% of a 0 , respectively).On the contrary, the Ti atom repulsion from the F-center located on the BaO-terminated BaTiO 3 (001) surface (0.1% of a 0 ) is smaller than the relevant Ti atom repulsion from the F-center located in the BaTiO 3 bulk matrix (1.06% of a 0 ) (Table 2).The second nearest adjacent O atoms, both in the SrZrO 3 , PbTiO 3 , BaTiO 3 and SrTiO 3 bulk and on its (001) surfaces, always are shifted towards the F-center.Namely, the second nearest adjacent O atoms are attracted towards the bulk F-center in SrZrO 3 , PbTiO 3 , BaTiO 3 and SrTiO 3 matrixes, by (−2.63, −0.88, −0.71 and −7.79% of a 0 , respectively).Even more strongly than in the bulk, the second nearest adjacent O atoms are shifted towards the (001) surface F-center in SrZrO 3 , PbTiO 3 , BaTiO 3 and SrTiO 3 perovskites by (−4.16, −5.58, −1.4 and −8.0% of a 0 ) (Table 2).
Even less charge is confined in the (001) surface oxygen vacancies in SrZrO 3 (−1.10e) and BaTiO 3 (−1.052e) matrixes.The (001) surface oxygen vacancy formation energies in SrZrO 3 , PbTiO 3 , BaTiO 3 and SrTiO 3 matrixes are equivalent to 7.52, 5.99, 10.20 and 6.22 eV (Table 3).Finally, the (001) surface-oxygen-vacancy-induced defect level is situated at the Γ-point 0.93, 0.71, 0.07 and 0.25 eV below the CB at SrZrO 3 , PbTiO 3 , BaTiO 3 and SrTiO 3 matrixes (Table 3).In all three pristine SrF 2 , BaF 2 and CaF 2 fluorites, our B3PW computed fluorine atom effective charges (−0.954 e, −0.923 e and −0.902 e) are very close to the net ionic fluorine charge equivalent to −1.00 e (Table 4).According to our B3PW computations, even less charge is confined inside the bulk fluorine vacancy in the SrF 2 , BaF 2 and CaF 2 matrixes (−0.848 e, −0.801 e and −0.752 e) (Table 4).The nearest Sr atom is sifted towards the bulk fluorine vacancy in the SrF 2 crystal by a very small displacement size (−0.02% of a 0 ), whereas the nearest Ba and Ca atoms are repulsed from the bulk fluorine vacancy by (+0.03 and +0.15% of a 0 ) in the BaF 2 and CaF 2 matrixes (Table 4).The bulk fluorine vacancy formation energy in the SrF 2 , BaF 2 and CaF 2 matrixes is equal to 10.33, 7.82 and 7.87 eV, respectively (Table 4 and Figure 4).Our computed band-gap widths for the SrF 2 , BaF 2 and CaF 2 containing the single bulk fluorine vacancy at the Γ point are equal to 11.34, 11.28 and 10.99 eV, respectively (Table 4).The bulk-fluorinevacancy-induced defect levels are located inside the band gap at 3.67, 4.27 and 4.24 eV, respectively, below the conduction band (CB) bottom at the Γ-point in the SrF 2 , BaF 2 and CaF 2 matrixes (Table 4 and Figure 5).
Finally, for the fluorine vacancy located on the BaF 2 (111) surface, only the (−0.790 e) charge is localized inside it (Table 4).The nearest Ba atom is shifted towards the (111) surface fluorine vacancy in the BaF 2 matrix by (−0.13% of a 0 ), and the second nearest F atom is shifted towards the fluorine vacancy even more strongly, by (−0.37% of a 0 ).The fluorine vacancy formation energy on the BaF 2 (111) surface is equal to 7.48 eV (Table 4).The BaF 2 (111) surface-fluorine-vacancy-induced defect level is located inside the BaF 2 band gap, 4.11 eV below the conduction band (CB) bottom at the Γ-point (Table 4 and Figure 5).
It is worth noting that we observed in our ab initio B3PW computations strong increase of the chemical bond covalency between the F-center and its nearest neighbor cations in our calculated materials.For example, our B3PW computed chemical bond covalency between the Zr and O atoms in the pristine SrZrO 3 perovskite is equal to 0.100 e, but the chemical bond covalency between the Zr atoms and bulk F-center in SrZrO 3 is more than two times larger and equivalent to 0.244 e.The same situation, according to our B3PW computations, is also in the BaTiO 3 perovskite.Namely, the Ti-O chemical bond covalency in the pure BaTiO 3 bulk is equivalent to 0.100 e.We observed, according to our ab initio B3PW computations, the strong increase of the chemical bond covalency between the adjacent Ti atoms and the BaTiO 3 bulk F-center equivalent to 0.320 e.

Conclusions
The shift magnitudes of the nearest-neighbor atoms around the (001) surface oxygen vacancy in SrZrO 3 , PbTiO 3 , BaTiO 3 and SrTiO 3 perovskites (Table 3), in most cases, are larger than the respective shift magnitudes of the nearest atoms around the bulk oxygen vacancy.The atomic shift amplitudes around both (001) surface and bulk oxygen vacancies in SrZrO 3 , PbTiO 3 , BaTiO 3 and SrTiO 3 perovskites (Table 3) almost always are significantly larger than the atomic shift magnitudes around the fluorine vacancy in SrF 2 , BaF 2 and CaF 2 fluorites (Table 4).
In the SrZrO 3 , PbTiO 3 , BaTiO 3 and SrTiO 3 perovskites, the charge, as a rule, is much better localized inside the bulk oxygen vacancy than inside the (001) surface oxygen vacancy (Table 3).In ionic materials, such as, SrF 2 , BaF 2 and CaF 2 , the charge is considerably better localized (Table 4) inside the fluorine vacancy than inside the oxygen vacancy of SrZrO 3 , PbTiO 3 , BaTiO 3 and SrTiO 3 perovskites (Table 3).More than 75% of charge is localized inside the bulk and (111) surface fluorine vacancy in the SrF 2 , BaF 2 and CaF 2 matrixes (Table 4).We obtained the strong increase of a chemical bond covalency among the F-center and its nearest adjoining cations, in comparison to the pristine crystals.
In the SrZrO 3 , PbTiO 3 , BaTiO 3 and SrTiO 3 perovskites, as well as BaF 2 fluorite, the surface-oxygen-induce or fluorine-vacancy-induced defect levels are located closer to the CB band bottom than in the bulk cases (Tables 3 and 4).The B3PW computed energy difference between the SrZrO 3 , PbTiO 3 , BaTiO 3 , SrTiO 3 and CaF 2 bulk and surface oxygen or fluorine vacancies (Tables 3 and 4) triggers the vacancy segregation from the bulk towards the surface.
Our computation results for the F-center defect ground state levels in CaF 2 , BaF 2 and SrF 2 fluorites, located at 4.24, 4.27 and 3.67 eV under the CB bottom, suggest a possible explanation of the optical absorption observed experimentally in CaF 2 , BaF 2 and SrF 2 fluorites at 3.3, 2.3 and 2.85 eV.According to our computations, the experimentally observed optical absorption may be due to an electron transition from the F-center ground state, located at 4.24, 4.27 and 3.67 eV under the CB, to the CB.Namely, rhe F-center defect band at CaF 2 , BaF 2 and SrF 2 is located 4.24, 4.27 and 3.67 eV under the CB, which is very close to the experimentally detected optical absorption energy equal to 3.3, 2.3 and 2.85 eV, respectively.

Table 2 .
B3PW computed 3 closest adjacent atom shifts around the bulk and (001) surface F-center (in% of the a 0 ).

Table 4 .
B3PW computed atomic and electronic structure of the fluorine vacancy in SrF 2 , BaF 2 and CaF 2 matrixes.