First Principles Study on Intrinsic Vacancies in Cubic and Orthorhombic CaTiO 3

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1 Materials Transactions, Vol. 5, No. 5 (29) pp. 977 to 983 Special Issue on Nano-Materials Science for Atomic Scale Modification #29 The Japan Institute of Metals First Principles Stud on Intrinsic Vacancies in Cubic and rthorhombic 3 Haksung Lee 1, Teruasu Mizoguchi 1; *, Takahisa Yamamoto 1;2 and Yuichi Ikuhara 1;2;3 1 Institute of Engineering Innovation, The Universit of Toko, Toko , Japan 2 Nanostructures Research Laborator, Japan Fine Ceramics Center, Nagoa , Japan 3 WPI, Advanced Institute for Materials Research, Tohoku Universit, Sendai , Japan The structural relaxations and the formation energies of intrinsic defects in cubic and orthorhombic 3 were investigated b a first principles projector-augmented wave method. It was found that cations and oxgen vacancies in both phases cause extra levels near the valence band maximum and the conduction band minimum, respectivel, and the -vacanc induced level in orthorhombic 3 is closer to the valence band maximum than that in cubic 3. Among the neutral defect species, including neutral isolated vacanc, partial Schottk, and full Schottk, it was found that the 2 þ V 2þ and V are the most preferable defect species for orthorhombic 3 under reduction and oxidization conditions, respectivel, whereas the 2 þ V 2þ partial Schottk is alwas stable in an atmosphere in cubic 3.As compared to cubic 3, it was found that orthorhombic 3 shows higher defect formation energies. [doi:1.232/matertrans.mc2813] (Received December 15, 28; Accepted March 11, 29; Published April 15, 29) Kewords: calcium titanate, defect energetics, calculation 1. Introduction 3 is an important material for a radioactive waste and electric materials with doping. 1,2) Due to its similarit in the crstal structure, 3 is also utilized as a dopant and a member of the superlattice together with other perovskite materials, such as Ba 3 and Sr ) It has been known that the electric properties and microstructures of those perovskite materials were strongl dependent on the oxgen partial pressure, indicating that defects pla an essential role to bring about their properties. 6 12) Thus, an understanding of defect energetics is indispensable for further developing and applications of these perovskite oxides. Although the defect energetics of Sr 3 and Ba 3 has been studied so far, 13 16) there are onl a few reports on the defect energetics in 3. Recentl Mather et al. reported the energetics of defects, dopant-vacanc association, and the nanoscale clusters formation in orthorhombic 3 based on a static lattice calculation with empirical pair potentials. 17) First principles calculations of 3 were also performed b Cockane et al. and Wang et al., but the were mainl focused on the dielectric constant, optical properties, and the surface structures. 18,19) In this stud, the intrinsic defect formation energies in cubic and orthorhombic 3 were calculated b a firstprinciples projector-augmented wave (PAW) method. The formation energies and atomic relaxations around vacancies were investigated. 2. Methodolog *Corresponding author, Mizoguchi@sigma.t.u-toko.ac.jp 2.1 First principles calculations of cubic and orthorhombic 3 A first-principles projector augmented wave (PAW) method within the generalized gradient approximation (GGA) implemented in VASP code was used to calculate the defect energetics. 2 22) The wave functions were expanded in a plane-wave basis set with a plane-wave cutoff energ of 4 ev. 3 shows three tpes of polmorphs depending on temperature: orthorhombic, tetragonal and cubic. 23) In this stud, defect energetics in cubic and orthorhombic 3 were calculated, because tetragonal phase is a transient compound in ver limited temperature (1523 K < T < 1622 K), and the cubic and orthorhombic 3 are stable over wide temperature range T > 1622 K and T < 1486 K, respectivel. First, the unitcell calculations were performed to find optimized structures for two polmorphs. The primitive cell of cubic 3 was calculated using k-point mesh generated b the Monkhors-Pack scheme (35 irreducible k-points). From the energ-volume curve, the lattice constant and the bulk modulus were estimated to be 3.89 Å and 173 GPa, respectivel, which are good agreement with previousl reported experimental value. 18) For orthorhombic 3, the structure was also calculated using k-point mesh (125 irreducible k-points) allowing relaxation until its residual forces were less than.5 ev/å. Figure 1 shows the optimized atomic structures of the two polmorphs. It can be seen that the calculations are satisfactoril reproduce the experimental structures. Figure 2 shows the calculated band structures of cubic and orthorhombic 3. Similar to Ba 3 and Sr 3, the valence-band maximum (VBM) composed of 2p orbitals and the conduction-band minimum (CBM) composed of 3d orbital respectivel. In cubic 3, VBM at M is.5 ev higher than that at and the indirect gap between M and is 1.97 ev and the direct band gap at is 2.5 ev. For orthorhombic, it shows the direct band gap at, 2.54 ev. The difference of the direct band gap between cubic and orthorhombic 3,.4 ev, is small. Although a feature of the band structure is consistent with the previous GGA report 19) for cubic 3, the calculated band gap is below the experimental value, 3.5 ev. 24) The difference between experimental and calculated band gap could underestimate the formation energies of oxgen vacancies, which will be discussed later.

2 978 H. Lee, T. Mizoguchi, T. Yamamoto and Y. Ikuhara (a) Cubic 3 (b) rthorhombic 3 Pm3m Pbnm 1 2 l. Exp. l. Exp. a(å) a (Å) Bulk modulus(gpa) b (Å) c (Å) x z x x x z z z Fig. 1 The crstal structures of cubic and orthorhombic 3. The calculated and experimental lattice parameters and bulk modulus are shown in the table, and the optimized atomic coordinates are also shown for the orthorhombic structure. The experimental data were obtained from Ref. 23). (a) Cubic 3 5 (b) rthorhombic Γ X M Γ R X Wave Vector -5 Y Γ X M Γ R Wave Vector Fig. 2 lculated energ band structure for (a) cubic and (b) orthorhombic 3. The VBM at M and was set at ev for cubic and orthorhombic, respectivel. The optimized primitive cells of cubic and orthorhombic 3 were expanded b and 2 2 2, and the supercells with 135-atoms and 16-atoms were respectivel obtained. B using these supercells, a vacanc can be located apart from the neighboring vacancies in the adjacent supercells b 11.6 Å for cubic 3 and more than 1.7 Å for orthorhombic 3. To introduce an isolated vacanc of,, or, an interior atom was removed from the supercell. The third nearest neighbor sites from the vacanc for cubic 3 were allowed to relax until their residual forces were less than.5 ev/å. In orthorhombic 3,it is difficult to determine the third nearest neighbor exactl. Therefore, atoms inside a sphere of 3.9 Å, which is slightl bigger than the relaxation radius of cubic 3, centered at the defect were relaxed. For the large supercell calculations, the k-points mesh generated b the Monkhorst- Pack scheme was used for the numerical integrations over the Brillouin zone.

3 First Principles Stud on Intrinsic Vacancies in Cubic and rthorhombic Defect Formation Energ The formation energies of vacancies in 3 were calculated b the similar procedure as the previous reports. 15,16) The formation energ (E f ) is evaluated as following equation: E f ¼ E T ðdefect : qþ E T ðperfectþ þfn þ n þ n gþqð" F þ E VBM Þ ð1þ Here, E T (defect:q) and E T (perfect) are the total energies of the supercells containing a defect in a charge state q and that of the perfect supercell without an defect, respectivel. n, n and n are the numbers of,, and atoms removed from the large supercells to introduce a vacanc.,, and are the atomic chemical potentials, and " F is the Fermi energ measured from the VBM. In this stud, chargeneutral (V,V, and V ) and ionized (V 2,V 2, V 4, and V 2þ ) vacancies were considered. As can be seen eq. (1), E VBM needs to be determined. For this purpose, it was assumed that the electrostatic potentials in the supercell without vacancies were similar to those far from a defect in the supercell with a vacanc. This treatment can compensate distortion to the band structure around the band gap, which is caused b a vacanc ) Since the theoretical band gap was lower than the experimental value, the energ of the electrons at the donor state is underestimated. The band gap energ was calculated from the total energies of the supercell as following equation: E g ¼ E perfect CBM Eperfect VBM ¼fðE 1 T ET Þ ðe T Eþ1 T Þg ð2þ F where E q T indicates the total energ of a perfect supercell with the charge state: q. The calculated band gap of cubic 3 and that of orthorhombic 3 is 2.7 ev and 2.15 ev smaller than experimental value, respectivel. In the previous report, the formation energ of an oxgen vacanc was adjusted to be increased the donor level b the band gap difference between the calculation and experiment. 28) Adjusting the defect formation energ of V is added 2 Eg and for V þ1, 1 Eg is added for calculating the defect formation energ. The formation energies also depend on the atomic chemical potentials. The chemical potentials of the constituent atoms var with the equilibrium conditions among their related phases. Figure 3 shows a schematic phase diagram of the ternar -- sstem, and the vertices of three phase regions were named AG as for the previous reports. 15,16) The values of the atomic chemical potential of three elements were evaluated at those points. Here, it was assumed that 3 was stable at ever point, so that the chemical potentials of the three elements were obtained from the combinations of the chemical potentials of,,,, 2 3, 2 and 2. The chemical potential of 2 was obtained from the supercell calculation at the point for an isolated 2 molecule using a large vacuum cell Å 3. Spin-polarized calculation was performed for the chemical species because in the previous report spinpolarized calculations were closer to the experimental values. 15,16) Especiall, for calculating the chemical potential of 2 3, antiferromagnetic state was considered because antiferromagnetism in 2 3 has been reported in literature. 3) The heats of formation obtained from the calculations are summarized in Table 1. It is seen that spinpolarized calculations satisfactoril reproduce the experimental values. As shown in Fig. 1, orthorhombic 3 has two oxgen sites, 1 and 2. Their formation energies were separatel calculated. As a result, it was found that their formation energies are ver similar and those of V 1 q(,1,2) is slightl lower compared to those of V 2 q. Thus, the defect formation energies of V 1 and V 1 q were emploed for calculating the formation energies of the neutral vacanc and the Schottk reactions. 3. Results and Discussion 3.1 Atomic and electronic structures of isolated vacancies Figure 4 shows the one-electron energ levels around the band gap and the number of electrons occuping these states. Since the existence of vacancies causes significant distortion to the band structures around the band gap, the alignment of the band structures of the perfect and defective supercells is necessar. For this purpose, it was assumed that the potentials in the perfect supercell are similar to those far from a defect in a defective supercell. The VBM positions of defective supercells were determined b the following equation: 31,32) E VBM ¼ E perfect VBM E D G C þ Vdefect av A B 23 2 Fig. 3 Schematic phase diagram of the ternar sstem --. A hatched polgon shows the region of 3. Corners of the polgon are indicated b A-G. Table 1 lculated and experimental formation enthalpies for the reference materials. The experimental data were obtained from Ref. 29). Formation enthalpies (ev/atom) Experiment lculation F m3m 3:3 3:3 F m3m 2:82 2:29 2 P 42/mmm 3:25 3:3 2 3 R3C 3:16 2:67 3 P m3m 3:11 3:17 P bnm V perfect av ð3þ

4 98 H. Lee, T. Mizoguchi, T. Yamamoto and Y. Ikuhara (a) Cubic CBM (a) V VBM (b) (b) rthorhombic CBM V1 V2 (c) VBM 1+ Fig. 4 ne-electron energ levels for,v and V of 3 in various charge states for (a) cubic and (b) orthorhombic 3. Electron energ levels are given with respect to the VBM in the case of and V, while those from the CBM in the case of V. V Vacanc The VBM of perfect cubic and orthorhombic 3 was set at ev for each case. In the case of the cation vacancies ( and V ), the positions of the states in the band measured from the VBM are displaed in this figure. In contrast, the positions of the V induced states are measured from the CBM. As can be seen in Fig. 4, each vacanc induces an extra level in the band gap. In the case of and V, the extra levels are located near the VBM and the energ levels become larger with increasing negative charge for both cubic and orthorhombic 3. Figures 5(a) and 5(b) shows the contour maps of the squares of the wave functions for the extra states induced b or V in cubic 3 on the {1} plane, respectivel. It can be seen that these states are mainl composed of 2p orbitals of ions surrounding the vacancies. In cubic 3, the wave function of the V induced level shows more localized than that of the V induced level, indicating that the electrons occupied in the V induced level would suffer more electronic repulsions around the vacanc site as compared with the V induced level. It could explain that the energ levels for V with negative charges tend to be higher than those for, as shown in Fig. 4(a). In contrast, the extra levels b V are located close to the CBM and thus this level is mainl composed of 3d orbitals. Those characteristics in the defect-induced levels are ver similar to the previousl reported results for Sr 3. 15) Fig. 5 Contour maps of the square of the wave functions of vacancinduced levels of (a)v, (b)v and (c)v on the {1} plane of cubic 3. The contour lines are drawn from.5 to.2 with an interval.1 in the unit of electrons/å 3. Figure 4(b) and Figure 6 show the energ levels of the vacanc induced states and the corresponding contour maps for orthorhombic 3. It is interestingl mentioned that the V induced level in orthorhombic 3 is closer to the VBM than that in cubic 3. This can be ascribed to the fact that the V induced level and the V induced level in orthorhombic 3 is more delocalized and localized, respectivel, as compared with that in cubic 3. (Figs. 5 and 6) The introduction of vacancies induces not onl extra levels in the band gap but also structural relaxations of the surrounding ions. Table 2(a) lists the calculated interatomic distances between each vacanc and the first and second NN ions before and after structural relaxations in cubic 3. For all vacanc species, the first NN ions exhibit outward relaxations b more than 3%, irrespective of the charge states. The outward relaxations of the first NN ions can be ascribed to that the chemical bonds between the removed ion and surrounding ions vanishes b introducing the vacanc. In contrast, the second NN cations of and V showed

5 First Principles Stud on Intrinsic Vacancies in Cubic and rthorhombic (a) (b) (c) V1 (d) V2 Vacanc Fig. 6 Contour maps of the square of the wave functions of vacanc-induced levels of (a) V, (b) V and (d) V 2 on the (1) plane and that of (c)v 1 on the (1) plane of orthorhombic 3. The (1) plane was selected onl for V 1 to show the site on the plane. The contour lines are drawn from.5 to.2 with an interval.1 in the unit of electrons/å 3. Table 2 The interatomic distance of the neighboring ions from a vacanc (Å) after the structural relaxation in (a) cubic and (b) orthorhombic 3. (a) Cubic 3 (b) rthorhombic 3 First NN Second NN First NN Second Species 2.75(12) 3.37(8) 2.34(4) 2.62(4) 3.31(8) 2.86(3.83%) 3.32( 1:57%) 2.45(4.89%) 2.71(3.57%) 3.26( 1:66%) (3.83%) 3.32( 1:57%) 2.45(4.71%) 2.72(3.52%) 3.25( 1:69%) (3.83%) 3.32( 1:57%) 2.44(4.4%) 2.72(3.77%) 3.25( 1:88%) 1.94(6) 3.37(8) 1.96(6) 3.31(8) V 2.1(3.54%) 2.98( 11:7%) 2.7(5.34%) 3.19( 3:79%) 1 V 2.1(3.54%) 2.98( 11:7%) 2.7(5.27%) 3.19( 3:83%) 2 V 2.1(3.59%) 2.97( 11:7%) 2.7(5.45%) 3.18( 3:91%) 3 V 2.1(3.59%) 2.97( 11:9%) 2.7(5.64%) 3.17( 4:3%) 4 V 2.1(3.61%) 2.94( 12:6%) 2.8(5.89%) 3.16( 4:7%) 1.94(2) 2.75(8) (2) 2.37(2) V 2.3(4.49%) 2.57( 6:25%) 2.12(8.34%) 2.51(6.5%) 1þ V 2.3(4.62%) 2.56( 6:2%) 2.12(8.62%) 2.51(6.7%) 2þ V 2.4(4.84%) 2.56( 6:27%) 2.13(8.73%) 2.51(5.86%) 2.75(4) (2) 2.53(3) 2.92(6.69%) 2.14(8.87%) 2.66(5.6%) 2.92(6.74%) 2.14(9.2%) 2.66(5.1%) 2.92(6.93%) 2.14(9.18%) 2.66(5.1%) inward relaxations because electrostatic repulsions between second NN cations and the removed cation are reduced b the vacancies. V in cubic 3 induces around 11% inward relaxations of the second NN while in cubic 3 induces less than 2% inward relaxations of. In the case of V, there are two kinds of atoms, and which are the second NN before relaxations. After relaxations, moves outward and moves inward due to electrostatic repulsions and attraction, respectivel. Table 2(b) shows structural relaxations b defects in orthorhombic 3. Since orthorhombic 3 is a lower smmetr phase than cubic 3, the coordination number and the distances of surrounding ions are more complex. Although it is difficult to define exact nearest neighbor (NN) atoms, the difference of the bond length within.1 Å is regarded as the same coordination members for the first NN and their average distances were evaluated. For example, although 2 in Fig. 1 is coordinated b two ions with Å and Å bond lengths, the are considered as the same coordination. The second NN bond length was calculated b the average of the distance between a vacanc and the second NN atomic species. For example, second NN atomic species for V are ions, which locate 3.13 Å, 3.26 Å, 3.33 Å and 3.52 Å apart from V. The

6 982 H. Lee, T. Mizoguchi, T. Yamamoto and Y. Ikuhara average distance becomes 3.31 Å before relaxations and 3.19 Å after relaxations. For V in orthorhombic 3, there are two oxgen vacanc positions (1 and 2) which have almost identical surroundings. 1 and 2 were commonl coordinated b two with Å and 1.96 Å bond length, respectivel. The first NN relaxation due to V 1, 8.34%, is close to that due to V 2, 8.87%. Comparing the first NN structural relaxations due to V 1 in cubic and orthorhombic 3, the relaxations in orthorhombic, 8.34%, is approximatel twice of that in cubic 3, 4.49%. In addition, the relaxation around V in cubic 3 is 11:7%, which is much larger than those in orthorhombic 3, 3:79%. (a) Cubic 3 18 Defect Formation Energ per defects, Ef / ev V V V 2- +V V 2- +V 3-2V +3V 4- V +2V 3.2 Defect formation energies of neutral isolated vacancies and Schottk reactions The formation energies of the isolated neutral vacancies were evaluated. As mentioned above, the formation energ for the 1 site in orthorhombic 3 is slightl lower than those for the 2 site. Therefore, the for orthorhombic 3 hereafter corresponds to 1 site. The formation energ of the isolated vacancies,, and, were shown in Fig. 7. It is commonl found that the vacanc formation energ of is relativel higher than that of in both phases. This trend is the same as Sr 3 and cubic Ba 3, in which the defect formation energ of V is also higher than that of V A(Sr, Ba). 15,16) In order to compare the formation energies in cubic and orthorhombic 3, the average values of the formation energies over all equilibrium points were evaluated. The average values for V in cubic and orthorhombic 3 are 11.8 and 13.1 ev and those for V in cubic and orthorhombic 3 are 5. and 6.33 ev, respectivel. In addition, the average values for V in cubic 3, 4.5 ev, are lower than those in orthorhombic 3, 4.47 ev. Namel, it can be concluded that all intrinsic vacancies are difficult to form in the orthorhombic phase as compared with those in the cubic phase. In additional to those isolated vacancies, the following Schottk reactions were also considered: V 2 2þ þ V ( partial Schottk), V 2 2þ þ V ( partial Schottk), 2V 3 þ 3V 2þ ( 2 3 partial Schottk), V 4 þ 2V 2þ ( 2 partial Schottk), V 2 þ V 4 þ 3V 2þ ( 3 full Schottk). In those Schottk reactions, each vacanc was separatel calculated and the association between vacancies was not considered. The defect formation energies at each point are plotted in Fig. 7. The calculated formation energies of the full Schottk, 2.19 ev for cubic 3 and 2.66 ev for orthorhombic 3, are close to the previousl reported value b G. C. Mather et al., of 2.49 ev, which was obtained for orthorhombic 3 b the static lattice calculation. 17) As compared with the full Schottk, the partial Schottk defect formation energ changed with equilibrium point. In the -rich conditions (B, C, D and E), the partial Schottk reaction for cubic 3 was 1.21 ev, which is smaller than that in -rich condition (A), 1.44 ev, and the partial Schottk reaction is the most dominant defect species for the cubic phase in an equilibrium condition. In the case of orthorhombic 3, partial Schottk energ is 1.73 ev in -rich (B, C and A B C D E F G Point in Phase Diagram (b) rthorhombic 3 Defect Formation Energ per defects, Ef / ev D) and 2.11 ev in -rich condition (A), and is the most stable defect species except for point G. n the other hand, it is found that both, 2 3 and 2 partial Schottk reactions are alwas higher formation energ as compared with other Schottk reactions. This higher formation energ of the, 2 3 and 2 Schottk reactions can be ascribed to the higher formation energ of the vacancies. B comparing cubic 3 with orthorhombic 3, it was found that all Schottk reactions in orthorhombic 3 show the higher formation energies due to the higher formation energies of the isolated vacancies. 4. Summar V V V 2- +V V 2- +V 2V 3- +3V V 4- +2V A B C D E F G Point in Phase Diagram Fig. 7 Defect formation energies calculated for (a)cubic and (b)orthorhombic 3 at points in schematic phase diagram. As described in text, in the orthorhombic case corresponds to the site 1 in Fig. 1. First-principles plane-wave-based PAW calculations were performed for the defect energetic in cubic and orthorhombic 3. (1) It was found that cations and oxgen vacancies in both phases cause extra levels near the valence band maximum and the conduction band minimum, respectivel, and the -vacanc induced level in orthorhombic 3 is closer to the valence band maximum than that in cubic 3.

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