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A first-principles investigation of the stabilities and electronic properties of SrZrO3 (1 1 0) (1  ×   1) polar terminations

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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 395002 (14pp)

doi:10.1088/0953-8984/26/39/395002

A first-principles investigation of the stabilities and electronic properties of SrZrO3 (1 1 0) (1  ×  1) polar terminations Hong Chen1, Ying Xie1,2, Guo-xu Zhang1,3 and Hai-tao Yu1,2 1

  School of Chemistry and Materials Science, Heilongjiang University, Harbin 150080, People’s Republic of China 2   Key Laboratory of Functional Inorganic Material Chemistry, Ministry of Education, Heilongjiang University, Harbin 150080, People’s Republic of China 3   Present address: Fritz-Haber-Institut der MPG, Theory Department, Faradayweg 4–6, 14195, Berlin, Germany E-mail: [email protected] Received 9 May 2014, revised 3 July 2014 Accepted for publication 16 July 2014 Published 20 August 2014 Abstract

The stabilities and electronic properties of SrZrO3 (1 1 0) (1  ×  1) polar terminations were investigated systematically by the first-principles density functional theory method. Five possible polar surfaces, i.e. O-deficient, O-rich, stoichiometric, SrO-rich and SrO-deficient ones, were considered. The calculated results indicated that the charge neutralization and polarity compensation condition could be achieved by charge redistributions of surface atoms. For the O-deficient (1 1 0) termination, some filled electronic states were separated from the original conduction bands, while a surface reconstruction was found for the O-rich (1 1 0) surface. The remaining three (1 1 0) terminations remained insulated. Furthermore, a stability diagram involving seven different terminations was constructed using the surface grand potential technique, in which the effect of the chemical environment was included. The calculated results indicated that three (1 1 0) (O-rich, SrO-rich and stoichiometric) and 2 (0 0 1) (ZrO2 and SrO) terminations could be stabilized in distinct areas, whereas the O-deficient surface was unstable within the whole region. Finally, we drew a comparison of stability behaviors between SrZrO3 (1 1 0) (1  ×  1) polar surfaces and the counterparts of ATiO3 (A = Ba, Pb, Sr) and BaZrO3 materials. Keywords: polarity compensation, SrZrO3 (1 1 0) polar surface, thermodynamical stability, surface relaxation, first-principles calculation, stability diagram, polar termination S Online supplementary data available from stacks.iop.org/JPhysCM/26/395002/mmedia (Some figures may appear in colour only in the online journal)

1. Introduction

because of its suitable melting point [5, 6], proton conductivity [7, 8], breakdown strength [9] and dielectric constant [10–12]. Its excellent proton conductivity at high temperatures makes it a potential candidate as an electrolyte in novel electrochemical devices such as solid-oxide fuel cells and hydrogen or steam sensors [13, 14]. Furthermore, the SrZrO3 film deposited on SrTiO3 substrates exhibits a large ferroelectric anisotropy [15]. Its surface structures, properties and stabilities are of primary importance for all of the above applications.

Over the past few decades the perovskite family has been identified as an important group of functional materials [1–3]. The pace of development has increased markedly recently because of an emerging realization of the actual and potential advantages they possess [4]. Among the reported perovskite members, SrZrO3 is one of the most promising candidates in high-temperature, high-voltage and high-reliability capacitor applications 0953-8984/14/395002+14$33.00

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© 2014 IOP Publishing Ltd  Printed in the UK

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different distances R1 and R2. The condition for cancellation of the macroscopic dipole moment can be written as [51]

According to the orientations and classical electrostatic criteria [16], ABO3-type perovskite surfaces can be classified into nonpolar and polar terminations. In the past several decades, the nonpolar surfaces have attracted many experimental [17–23] and theoretical [24–33] investigations because of their favorable physical nature. It has been reported that these surfaces can act as bottom electrodes, allowing an epitaxial regrowth to develop new functionalities, such as dielectricity [23], magnetism [18], ferroelectricity [19–21], superconductivity [22] and optical transmittance [23]. Also, the relevant theoretical investigations of band structure, density of states, magnetic moment, surface energy, and thermodynamic stability provide us with new insights into the nature of ABO3type perovskite surfaces [24–33]. The (0 0 1) surface of cubic SrZrO3 is a nonpolar surface because the stack sequence of neutral layers cannot result in a non-zero dipole moment along the surface normal and therefore it was considered to be very stable. This was strongly supported by a number of theoretical studies ­[34–38]. Evarestov and Bandura performed a hybrid Hartree–Fock/density functional theory (HF/DFT) study to evaluate the surface relaxation and surface energy of the SrZrO3 (0 0 1) surface and found that the SrO-terminated surface was slightly lower in surface energy than the ZrO2 termination [38]. They also confirmed that SrZrO3 had a larger ionicity than SrTiO3. Sambrano and coworkers reported a first-principles investigation into the structural and electronic properties of the cubic SrZrO3 (0 0 1) surface, based on a DFT calculation [36]. Their results indicated that the ZrO2-terminated surface was more stable than the SrO-terminated surface. Relative to nonpolar surfaces, polar surfaces appear to be a more challenging prospect for experimental and theoretical investigations. The stack sequence of charged layers can lead to a non-zero dipole moment perpendicular to the surface, resulting in a polar surface with a diverging electrostatic surface energy [16]. Therefore, a non-zero dipole moment can be considered the origin of the instability of a polar surface. Relevant investigations into polar surfaces have been hindered in the past because of their characteristic instability. The difficulties usually lay in 2 aspects: the availability of experimental techniques for preparing stable polar thin films and that of appropriate characterization methods for exploring the properties of the prepared polar surfaces. With developments in preparation and characterization techniques over recent years, some polar surfaces, including ATiO3 (A = Ba, Sr) (1 1 0) and (1 1 1), MgO (1 1 1) and LaAlO3 (0 0 1), have been successfully produced [39–42]. Also, their structures and properties have been characterized by experimental techniques such as low-energy electron diffraction, Auger electron spectroscopy, scanning tunneling microscopy and x-ray photoelectron spectroscopy [39–49]. These experiments predicted that some polar surfaces could be stabilized in certain conditions. The extremely significant experiments with polar surfaces mentioned above strongly motivated investigations into the physical nature of polar surface stabilization. In one innovative report, Noguera gave a very reasonable interpretation of how and why a polar surface could be stabilized [50, 51]. Suppose a metal oxide is composed of alternating layers with opposite charges, +σ and −σ and the alternating layers are separated by

m



∑σj = − j=1

σm + 1 ⎡ R2 − R1 ⎤ ⎢(−1)m − ⎥, R2 + R1 ⎦ 2 ⎣

(1)

where m corresponds to the number of modified outer layers, σj represents the charge of layer j, which has been changed and no longer equals the bulk value and σm + 1 is the charge of the (m + 1)th layer and equals the bulk value. Typically for cubic ABO3 compounds, R1 is equal to R2 and equation (1) can be further simplified. Motivated by equation (1), some theoretical investigations on the (1 1 1) and/or (1 1 0) polar surfaces of ABO3 (A = Sr, Ba or Pb, B = Ti or Zr [52–59]) have been carried out. These theoretical investigations proved the existence of stable polar surfaces, in agreement with experimental data [39–42]. From those relevant theoretical studies we can determine that the corresponding polarity compensation mechanisms and stability domains are distinct, to some extent, because of a change in the atoms located at the A and B positions [60]. In an ab initio calculation [57], Bottin and coworkers proposed that three terminations (Sr, TiO and O) of the SrTiO3 (1 1 0) polar surface, whose polar compensations are actually achieved through surface stoichiometry change, possessed high stability over a large domain. However, for SrTiO and O2 terminations with the compensation mechanism fulfilled by surface electronic structure change, only the SrTiO termination is stable over a small domain corresponding to O-poor and Sr-rich conditions. A recent theoretical investigation by our group also confirmed that the polarity compensation mechanism and stability domains of a BaTiO3 (1 1 0) polar surface were similar to that of SrTiO3 [58]; however, for PbTiO3 (1 1 0) polar terminations, despite the compensation mechanism being similar to that of BaTiO3 and SrTiO3, no stable PbTiO and O2 terminations were found [59]. It should be stressed that besides the above achievements on the electronic properties and stabilization mechanism, some significant advances in ab initio and first-principles thermodynamics for oxide surfaces and interfaces have been established [61–68]. When investigating the reconstruction of a (2  ×  2) GaAs (1 1 1) polar semiconductor surface, Kaxiras and coworkers confirmed that the relative chemical potential played a crucial role in determining the lowest-energy geometry of the surface [61]. To investigate the equilibrium and adhesion of a metal/oxide interface, Batyrev et al have derived a formula under ambient temperature and partial pressure of oxygen, which is useful for first-principles calculation and relates the free energy of a metal/oxide interface to the free energies of surfaces and the work of separation of the interface. And then the works of separation, surface and interfacial energies and the equilibrium work of adhesion can be evaluated [62]. Zhang et al also established a powerful link between the standard states and thermodynamical variables and those of ab initio methods and the connection is valuable for extending the predictability of ab initio techniques [63]. Relying on the equilibrium thermodynamic methods [62–65], Rohrer and coworkers considered the growth of alumina thin 2

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Figure 1.  Side views of optimized eleven-layer slab models for the O-deficient (left) and O-rich (right) terminations.

2.  Computational and theoretical details

film on TiC surfaces. They predicted that a non-bonding TiC/ alumina interface was favorably stable, which is seemingly in conflict with experimental results [65]. To solve this issue, Rohrer et al then proposed a non-equilibrium method of ab initio thermodynamics of deposition growth [66, 67] and with the aid of the method, obtained a consistent result with available experiments [67]. The results suggested that the nonequilibrium methods [66, 67] would gain wider usefulness in predicting surface or interface configurations in the future. For polar surfaces, it is no doubt that the environmental parameters, such as temperature and partial pressure, have a significant impact on stable surface configurations. Typically, for the perovskites, the ionic or covalent characteristics of the A-O bonds and the antiferrodistortive (AFD) or antiferroelectric (AFE) distortions of bulks can also lead to different surface stability domains [69, 70]. SrZrO3 can undergo a series of phase transitions with decreasing temperature from cubic (Pm3m) to tetragonal (I4/mcm) and further to orthorhombic (Cmcm and Pnma) at 1400 K, 1100 K and 970 K, respectively [5, 70–74]. Therefore, SrZrO3 should be an illustrative example for investigating whether the AFD or AFE characteristics of bulk materials can result in different stability domains from materials without AFD phase transitions. To gain insight into this issue, a detailed investigation of SrZrO3 polar surfaces is crucial. Based on these considerations, herein, five (1 1 0) terminated surfaces of SrZrO3 were computed by the firstprinciples DFT method. We investigated not only their electronic properties and polarity compensation mechanisms, but also the stability intervals of different terminations using the surface grand potential (SGP) method. The resultant stability interval diagram can directly provide valuable pointers for further experimental studies.

2.1.  Computational methods and models

In this study, we used the pseudo potential method based on DFT [75] with the aid of the CASTEP code [76, 77]. The Vanderbilt ultrasoft pseudo potentials (UPS [78]) and the generalized gradient approximation (GGA [79]) exchangecorrelation functionals using the Perdew-Wang parameterization (PW91 [80]) were applied. The energy cut-off was tested and set to 380 eV, while the Monkhorst-Pack meshes (k-point [81, 82]) of 6   ×   6  ×  6 and 6   ×   4  ×  1 were used for sampling the Brillouin zones of bulks and surfaces, respectively. In the present case, the original valence configurations were 4s24p65s2 for Sr, 4s24p64d25s2 for Zr and 2s22p4 for O. Stable surface configurations were obtained by geometry optimization for ideal unrelaxed surfaces. The optimization procedure was repeated until the force on each atom was less than 0.01 eV/Å and the energy change was less than 5.0  ×  10–6 eV/ atom. To treat the partially filled d-orbital of Zr, the energies and electronic properties of bulks and surfaces were calculated using the GGA+U method, in which the Hubbard U parameter (5.79 eV) was obtained according to [83]. The effect of the Hubbard U parameter and functionals on the results is discussed in figures S1–S3 and tables S1–S3 in the supplementary information (stacks.iop.org/JPhysCM/26/395002/ mmedia). For all optimized structures, the atomic charges were evaluated by the Mulliken scheme. Before starting surface calculations, we optimized the bulk structure of SrZrO3. By the optimized cubic SrZrO3 lattice, we constructed O-deficient, O-rich, stoichiometric, SrO-rich and SrO-deficient surfaces, as depicted in figures  1 and 2. 3

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Figure 2.  Side views of optimized eleven-layer slab models for the SrO-rich (left), stoichiometric (middle) and SrO-deficient (right) terminations.

To accurately describe surface charge distribution, a slab model was used in this calculation. A sufficient slab thickness is necessary to maintain the bulk electronic properties of the central layer of a slab because of the large penetration length of surface perturbations. The test results suggested that slabs should contain not less than 11 atomic layers. To avoid pseudo-interactions between periodic images along the z axis, a vacuum region of 11 Å was required, based on the preliminary test results. Furthermore, the energy changes of the surfaces with respect to the same stoichiometry models with a 30 Å vacuum region were less than 0.7 meV, which led to a negligible uncertainty in the calculation of the SGP. Moreover, it should be pointed out that an ABO3 perovskite crystal is comprised of alternating charged planes, and the corresponding slab model contains either 2N + 1 or 2N + 2 atomic planes. The surface energy of a slab with even atomic layers (2N + 2) depends on the slab thickness and is divergent due to the presence of a non-zero dipole moment [51] and thus this kind of slab cannot be used. Therefore, the slab with odd atomic layers (2N+1) was applied in the present calculation. Because two outermost layers in a slab are symmetrically equivalent, the dipole moment can be suppressed effectively. However, in the ionic limit, the O-deficient (SrZrO) and O-rich (O2) (1 1 0) surfaces possess charges of +4e and −4e. It is only when the neutralization is achieved and the dipole moment is cancelled that the polar surfaces are likely to be stabilized. The actual charges of the O-deficient and O-rich surfaces are impossible to determine due to the following reasons: (1) SrZrO3 is not a purely ionic crystal and the charges of Sr, Zr and O atoms depend on the definition of population and are different from the nominal charges (i.e. +2,+4 and −2), to a large extent. It is difficult to assign suitable charges to these surfaces; (2) The charge of a surface atom generally does not equal the charge of the corresponding atom in bulk, due to the reduction in surface coordination number; (3) The two surfaces may become neutral when the electron redistribution

occurs in response to the polar electrostatic filed. Therefore, it is reasonable to assign the O-deficient and O-rich surfaces as neutral and then go back to examine the energetics and the changes in electronic structure to clarify the origin of charge redistribution. The assignment also allows us to compare the energies of different surfaces, since the stoichiometric, ­SrO-rich and SrO-deficient surfaces are neutral in principle. 2.2.  Theoretical details of SGP

The SGP (Ω) of SrZrO3 is defined as 1 Ω= (2) [Eslab − NZrμZr − NSrμSr − NOμO], 2S

where N is the number of the corresponding atoms in the slab; μSr , μZr and μO represent the chemical potentials of Sr, Zr and O atomic species; Eslab is the total energy of a relaxed termination and S represents the surface area per unit cell. Note that the PV term and vibrational contributions to Gibbs free energy were neglected for the condensed state materials investigated. In this case, to achieve a charge and relaxation balance, the central layer should be in equilibrium with the bulk crystal. Therefore, the bulk chemical potential can be regarded as approximately equal to the total energy of the bulk crystal ( μSrZrO3 ≈ Ebulk), and the following relation must be fulfilled: bulk E SrZrO ≈ μSrZrO3 = μSr + μZr + 3μO . (3) 3

To simplify the relation and reduce the number of variables, we substitute μSr and μO for μZr in equation (2) and the SGP can be rewritten as Ω=



1 ⎡ bulk − μ (N − 3N ) − μ (N − N )⎤. ⎣E slab − NZrE SrZrO O Zr Sr Zr ⎦ O Sr 3 2S

(4) Because the absolute chemical potential of each species cannot be calculated in specific conditions, it is necessary to 4

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introduce the relative value of chemical potential with regard to the most stable phase of each atomic species, i.e.

3.  Results and discussion

bulk, ΔμSr = μSr − ESr (5)

3.1.  Surface relaxations

The computed lattice constant of the cubic SrZrO3 is 4.17 Å, which is similar to the experimental value of 4.15 Å [72], as well as other theoretical results [35, 36]. The atomic relaxations and rumplings of the fully optimized considered terminations are listed in tables 1 and 2, respectively. Note that only data for the upper layers was listed because the model was mirror symmetrical with respect to the center layer. As shown in table  1, for the O-deficient (1 1 0) surface, larger relaxations occur in three outer layers. For instance, the O atoms in the first layer are displaced outward (toward the vacuum) by 0.30 Å, whereas this displacement decreases to 0.10 Å in the 3 layer and almost disappears in the inner layers. However, the Sr atoms move inward and their relaxations are somewhat smaller than the O atoms, as can be noted from the corresponding value being 0.19 Å in the first layer and decreasing to 0.11 Å in the 3 layer. The rumpling between oxygen and strontium atoms is 0.49 Å in the outmost layer and becomes smaller in the 3 layer (0.21 Å), but is still non-­negligible. With an increasing layer number, relaxation equilibrium is gradually established. According to previous reports on SrTiO [57] and BaTiO [58] (1 1 0) terminations, both the surface Sr and Ba atoms were found to move inward with displacements of 0.35 Å and 0.07 Å, respectively, which were similar to that of the O-deficient (SrZrO) surface. For the O-rich (1 1 0) surface, some important modifications can be observed from table 1 and figure 1. First, the 2 O atoms in the 1st layer move close to each other, with a length of 1.51 Å. Second, the displacements of the 2 O atoms are no longer equivalent, and the surface permits a distortion that propagates into the slab. The relaxation thus displays a clear alternating rotational characteristic along the [1 0 0] direction. Furthermore, the computed inward displacement of the Zr atoms in the second layer is also very large (0.31 Å) and disappears gradually with an increasing layer number. However, for the counterpart terminations in SrTiO3, BaTiO3 and PbTiO3, the rotational displacements along the y axis are very small [57–59]. For the SrO-rich and SrO-deficient (1 1 0) surfaces, the atomic relaxations are relatively moderate. In the first layer of the SrO-rich termination, Sr atoms move outward by 0.56 Å, while the displacements of the O atoms in the second and third layers are only 0.11 Å and 0.02 Å, respectively. For the SrOdeficient termination, the observed largest relaxations, −0.19 Å and 0.31 Å for Zr and O atoms, respectively, are located in the first layer. As the layer number increases, the atomic relaxations of the two terminations become quickly damped. Furthermore, a non-equivalent relaxation can be noted for the O atoms in the topmost layer of the stoichiometric (1 10 ) surface, which was also observed in the relevant terminations in the PbTiO3 [59], BaTiO3 [54, 58] and SrTiO3 [57] compounds. It is reasonable to attribute this special behavior to the following two aspects. The first aspect is the broken mirror symmetry along the [1 1 0] direction. Heifets and coworkers reported that in the O terminations of the

ΔμZr = μZr − E bulk (6) Zr , EOmol 2 (7) ΔμO = μO − , 2

in which Sr and Zr atoms are in their respective bulk metal phases and O atoms are in the gas phase O2 molecule. Finally, the SGP in equation (4) can be re-expressed as 1 Ω=ϕ− (8) [ΔμO (NO − 3NZr) + ΔμSr (NSr − NZr)] 2S

with ϕ=



⎤ E Omol 1 ⎡ 2 bulk(N − N )⎥. ⎢E slab − NZrEbulk (NO − 3NZr) − ESr Sr Zr SrZrO3 − 2S ⎣ 2 ⎦

(9) Moreover, because the surface Sr, Zr and O atoms cannot precipitate outside the slabs, the chemical potential of each species must be lower than the energy of the corresponding atom in its stable phase. Therefore, the upper limit of chemical potential can be determined as E Omol 2 (10) ΔμO = μO − ≤ 0, 2 bulk ≤ 0, ΔμSr = μSr − ESr (11) bulk ≤ 0. ΔμZr = μZr − E Zr (12)

By combining equations  (3), (10), (11) and (12), the lower limit can be determined as f ΔμSr + 3ΔμO > − ESrZrO , (13) 3 f where ESrZrO is the formation energy of SrZrO3 with respect 3 to Sr and Zr atoms in the bulk phase and O atoms in the gas phase O2 molecule and can be expressed as bulk bulk − E bulk − 3 E mol. f −E SrZrO = E SrZrO − ESr (14) Zr O 3 3 2 2

For each allowed μSr and μO , the most stable termination  should possess the smallest SGP. Thereby, the relative stabilities of different SrZrO3 (1 1 0) terminations can be compared with one another according to the calculated SGPs. 2.3.  Relaxed cleavage energy

The cleavage energy can be considered as the energy requirement for cutting a crystal. This can be defined as 1 ⎡ rel rel (B) − nE bulk ⎤, E clrel = (15) ⎣E slab(A) + E slab SrZrO3⎦ 4S rel and E bulk are the total energies of the relaxed Where Εslab SrZrO3 slab and the bulk crystal, respectively and S represents the surface area per unit cell. A and B denote two complementary terminations.

5

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Table 1.  Relaxations (Å) and rumplings (Å) of the O-deficient and O-rich terminations of the SrZrO3 (1 1 0) surface. Positive (negative) displacement represents the outward (inward) direction from (to) the surface.

Layer

Relaxation

Termination

No.

Atom

X axis

Y axis

Z axis

O-deficient

1

Sr Zr O O1 O2 Sr Zr O O1 O2 Sr Zr O O1 O2 O1 O2 Sr Zr O O1 O2 Sr Zr O O1 O2 Sr Zr O

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −0.03 −0.09 −0.08 −0.08 −0.08 −0.07 −0.08 −0.07 −0.06 −0.06 −0.05 −0.03 0 0 0

0 0 0 0.07 0.07 0 0 0 0.01 −0.01 0 0 0 0 0 −1.06 −0.90 0.30 0.16 0.20 0.39 0.64 −0.09 0.01 0.04 −0.30 −0.30 0 0 0.01

−0.19 0.03 0.30 0.03 0.03 −0.11 0.02 0.10 0.01 0.01 0.00 0.01 0.03 −0.01 −0.01 −0.33 −1.09 −0.11 −0.31 −0.55 −0.48 0.42 −0.18 −0.03 −0.07 0.31 −0.33 0.01 0.01 0.02

2 3 4 5 Center O-rich

1 2 3 4 5 Center

Rumpling 0.49 0 0.21 0 0.03 0 0.76 0.44 0.90 0.15 −0.64 0.01

compensation condition (equation (1)) of the O-deficient termination is fulfilled. When the neutralization is achieved and the surface polarity is suppressed, the O-deficient termination is likely to be stabilized. To further elucidate the origin of the charge redistribution, the electronic structures were calculated. From figure  3(c) we can determine that the Fermi energy level, taken as the zero point (E‒Ef = 0.0 eV), is located across the conduction bands (CBs) and some CB states are thus filled. The filling states mainly consist of the contributions of Sr and Zr atoms. Such a behavior can be expected because the O-deficient (1 1 0) surface is free from surface defects and reconstructions. To achieve the charge neutralization, the filling states are required. As a result, the charges of surface atoms are reduced, as observed in table 3. Furthermore, the projection of the bulk band structure into the (1 1 0) surface is calculated and shown in figure 4. It can be readily noted that some newly formed states appear in the gap of the projected bulk band structure. As already confirmed above, these new surface states mainly come from the contributions of Zr4d, Zr4p and Sr5s orbitals. Similarly, for the counterpart (1 1 0) terminations of SrTiO3 [57] and BaTiO3 [58] compounds, the filling states were also found to derive mainly from Ti3d orbital. However, for the counterpart (1 1 0) termination of PbTiO3 crystal, the filling states are more complex and were reported to be dominated by Pb6p orbits [59]. Considering the covalency between Pb and O atoms in Pb-based perovskites and the fact that the Sr-O and Ba-O

BaTiO3 (1 1 0) surface, with and without mirror symmetry along the [1 1 0] direction [54], the symmetric configuration was higher in energy than the asymmetric one and only the asymmetric O termination revealed an in-plane displacement. The second aspect is that such behavior corresponds to an antiferrodistortive distortion, which can also be found in the corresponding (1 1 0) termination of SrTiO3 [57]. For the reasons described above, it can be expected that the rumpling of the inner O2 plane of the stoichiometric SrZrO3 termination is larger than those of SrTiO3, BaTiO3 and PbTiO3 [57–59]. 3.2.  Charge redistributions and electronic properties

For bulk SrZrO3, the calculated Sr, Zr and O atomic charges are 1.39 e, 1.15 e and  −0.85 e, respectively. Therefore, the total charges of SrZrO and O2 planes are 1.69 e and −1.70 e, respectively. Table  3 gives the computed atomic and layer excess charges of the O-deficient termination. As expected, the largest changes in atomic charge, 0.44 e and 0.30 e for Sr and Zr atoms, respectively, were observed in the first layer. Furthermore, significant decreases in atomic charge of Sr and Zr atoms can be noted, i.e. they are only 0.05 e and 0.01 e in the third layer and can almost be neglected in the fifth layer. However, the charges of O atoms in different layers vary slightly. By summing the layer charges, the total charge of five outermost layers was predicted to be 0.86 e, which is close to half the bulk value (1.69 e). Therefore, the polarity 6

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Table 2.  Relaxations (Å) and rumplings (Å) of the SrO-deficient, SrO-rich, and stoichiometric terminations of the SrZrO3 (110) surface. Positive (negative) displacement represents the outward (inward) direction from (to) the surface.

Layer

Relaxation

Termination

No.

Atom

X axis

Y axis

Z axis

SrO-deficient

1

Zr O O1 O2 Sr Zr O O1 O2 Sr Zr O O1 O2 Sr O1 O2 Sr Zr O O1 O2 Sr Zr O O1 O2 O Sr Zr O O1 O2 Sr Zr O O1 O2 Sr Zr O

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0.1 −0.1 0 0 0 0 0 0 0 0 0.01 0.01 0 0.11 −0.11 0 0 0 0.02 −0.02 0 0 0 0.01 −0.01 −0.41 −0.16 −0.22 0.26 0.32 0.37 −0.18 −0.09 0.11 −0.25 −0.28 −0.07 −0.01 0.04

−0.19 0.31 −0.06 −0.06 −0.22 0.05 −0.21 0.01 0.01 0.16 −0.03 0.10 −0.01 −0.01 −0.56 0.12 0.11 0.03 0.02 −0.02 0.02 0.02 −0.02 0 −0.02 0 0 −0.67 0.04 −0.16 0.13 0.25 −0.44 −0.25 0.01 −0.11 −0.30 0.25 0.06 −0.02 0.02

2 3 4 5 Center SrO-rich

1 2 3 4 5 Center

stoichiometric

1 2 3 4 5 Center

Rumpling 0.50 0 0.27 0 0.19 0 0 0.01 0.05 0 0.02 0 0 0.28 0.69 0.26 0.55 0.08

(0.14)a (0.14)b (0.15)c (0.14)a (0.38)b(0.66)c (0.03)a (0.12)b (0.13)c (0.24) b (0.46)c (0.02)b

a

  The O termination of BaTiO3 (1 1 0) surface [64].   The O termination of PbTiO3 (1 1 0) surface [65]. c   The O termination of SrTiO3 (1 1 0) surface [63]. b

and form a chemical bond with a length of 1.51 Å, which falls in between the O-O bond length (1.446 Å) of a H2O2 gas molecule and that (1.621 Å) of an O22 − isolated gas ion at the MP2/6‒311 G(d,p) level calculated by Gaussian03 code [84]. Therefore, a peroxo group denoted as O22 − is formed on the surface. As shown in figures 3(b) and 5, we can observe a distinct change in band structure with respect to the bulk crystal. In bulk SrZrO3, the indirect gap between R and Γ points is 3.79 eV, while the contribution to the valence band (VB) and CB mainly originates from O2p and Zr4d states. The VB possesses a threefold degenerate characteristic at Г point of the Brillouin zone (figure 6). In the ionic limit, Zr and Sr atoms can form six- and twelve-coordination, respectively, with O atoms. Therefore, the O2p orbits are fully occupied, being consistent with the calculated bulk band structure. However, when the Zr-O and Sr-O bonds are cleaved to form an O-rich

bonds are highly ionic in corresponding compounds, the different filling behaviors of the O-deficient (1 1 0) terminations can perhaps be distinguished by A-O interaction characteristics [59]. For the O-rich (1 1 0) termination, based on the computed Mulliken charges, a significant charge redistribution can also be identified. The total charge of five outermost layers was calculated to be  −0.86 e, which is close to half of the bulk value (−1.70 e). Thus, the polarity compensation condition in equation (1) is also achieved. It should be noted that surface defects are also absent in this slab and a remarkable change in surface electronic structure is expected to respond to the charge redistribution. From the excess charges listed in table 3, we can readily determine that the key atoms with considerable charge change, i.e. the O atoms, are located in the first layer. As depicted in figure 1, the 2 O atoms in the first layer of the O-rich termination move close to each other 7

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J. Phys.: Condens. Matter 26 (2014) 395002 a

b

Table 3.  Atomic charges (in e) and layer excess charges (Δe, in e) of the stoichiometric.

O-deficient termination

O-rich termination

Layer

Sr

Zr

O

Δe

1 2 3 4 5 6 Sum c

0.95

0.85

1.34

1.16

1.39

1.14

−0.90 −0.83 −0.86 −0.85 −0.85 −0.85

−0.79 0.04 −0.05 0.00 −0.01 0.00

0.86

Sr

Zr

1.35

1.43

1.28

1.22

1.34 −0.86

1.2

O −0.46 −0.87 −0.83 −0.86 −0.85 −0.86

Δe −0.55 −0.87 −0.85

0.69 0.22 0.00 −0.05 0.00 −0.01

a

  Calculated charges of the Sr, Zr and O atoms in bulk crystal are 1.39, 1.15 and −0.85 e, respectively.   Δe, layer excess charge (elayer ‒ ebulk‒plane), which are calculated by layer and bulk plane charges. c   The sum is the total charge of five outer layers. b

Figure 3.  Calculated SDOSs and partial density of states (PDOSs) of the SrZrO3 bulk phase (a), O-rich termination (b) and O-deficient termination (c). The smearing width is 0.05 eV.

termination, the surface coordination number decreases by half, leading to the reduction of electrons captured by the O atoms in the first layer, as illustrated in figure  5. Supposing that the surface O atoms do not form peroxo groups, every O atom can accept only one electron in the ionic limit from the nearby Sr and Zr atoms. Therefore, the surface should show a metallic property. However, the calculated surface band structure (figure 5) and the projected bulk band structure (figure 4(c)) clearly confirm that the O-rich (1 1 0) termination still retains insulating characteristics. Moreover, the results shown in figures 3(b) and 4(c) further substantiate the formation of a peroxo group. The new peaks of density of state (DOS) at about  −5.1,  −3.9,  −0.5 and 5.0 eV are assigned to σ, π π, π* and σ* states, respectively, mainly formed by O2p orbits. Accompanied by the surface reconstruction, the electron redistribution thus results in an empty σ* anti bonding state, while the termination remains insulated.

the surface dipole moments of the SrO-rich, SrO-deficient and stoichiometric (1 1 0) terminations are suppressed, which is guaranteed by keeping the symmetric equivalence between two faces of a slab. Furthermore, with the absorption (desorption) of Sr and O atoms to (from) the ideal surfaces, the requirement for neutralization is also achieved. As surface vacancies may provide an additional degree of freedom, it can be expected that the charge redistributions of the three terminations will not significantly affect their surface electronic structures. To check the deduction, we computed the summation of DOSs (SDOS) of the first and second layers of the SrO-rich, SrO-deficient, and stoichiometric (1 1 0) terminations. The results shown in figure  7 illustrate that the insulating nature of the 3 terminations is reserved and no filling states were found. Moreover, as can be seen from table 4, the charge redistributions can be still observed in the outer layers of each slab. If we take the atomic charges of 8

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Figure 5.  Computed band structure of the O-rich termination of the SrZrO3 (1 1 0) surface (a) and a schematic of the oxygen dimmer configuration (b). Only the valence-band top and conduction-band bottom are drawn in the band structure. In the case of an isolated O22 −group, the two π* orbits are completely filled.

Figure 4.  Projected bulk band structure and electronic surface states of the SrZrO3 (1 1 0) polar surface. (a) The Brillouin zone of the bulk and SrZrO3 (1 1 0) surface. The surface Brillouin zone is the projection of the bulk Brillouin zone in the [1 1 0] direction; (b) projection of the bulk band structure into the O-deficient (1 1 0) termination; (c) projection of the bulk band structure into the O-rich (1 1 0) termination. The bulk states are shown by the shaded areas.

the five outermost layers into consideration, the total values are 0.86, 0.84 and −0.85 e, respectively, which are close to half of the bulk plane charge. Therefore, equation (1) is fulfilled and the polarity compensation condition is achieved. Under the circumstances, the SrO-rich, SrO-deficient, and stoichiometric (1 1 0) terminations can be regarded as selfcompensated slabs. 3.3.  Thermodynamic stabilities of polar terminations

The cleavage energy corresponds to the energy required to split a crystal into 2 complementary terminations. If the cleavage energy is very high, the termination should be unstable. The relaxed cleavage energy (E clrel) can be computed according to equation (15). When the SrZrO3 crystal is split into two parts, two groups of complementary terminations, with O-rich and O-deficient terminations on one hand and SrO-rich, SrO-deficient terminations on the other, are formed. The remaining stoichiometric (1 1 0) termination is considered to be self-complementary. From the computed cleavage

Figure 6.  Computed band structure of bulk SrZrO3.

energies summarized in table  5, we know that the stoichiometric termination possesses the lowest cleavage energy (0.74 J m−2) among all terminations, whereas O-rich and O-deficient terminations have the highest (2.81 J m−2) ones. It can also be noted that the cleavage energies of the SrOrich and SrO-deficient terminations are moderate (1.43 J m−2). According to these values, it can be concluded that changes in 9

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Figure 7.  SDOS of the first and second layers in SrO-deficient (a), stoichiometric (b) and SrO-rich (c) terminations. Table 4.  Atomic charges (in e)a and layer excess charges (Δe, in e)b of the SrO-rich, SrO-deficient,and stoichiometric terminations.

SrO-rich termination Layer

Sr

1 2 3 4 5 6 Sum c

1.15

Zr

1.32

1.08

1.40

1.16

0.86

SrO-deficient termination

O

Δe

−0.91 −0.86 −0.86 −0.85 −0.85

−0.54 −0.12 −0.15 −0.02 0.02 0.00

Sr

Zr

O

Δe

1.48

−0.89 −0.81 −0.85 −0.85 −0.85 −0.85

−1.10 0.08 0.20 0.00 −0.01 −0.00

1.37

1.37

1.38

1.15

0.84

stoichiometric termination Sr

Zr

1.25

1.35

1.32

1.21

1.34 −0.85

1.16

O −0.85 −0.88 −0.83 −0.85 −0.85 −0.84

−0.86 −0.86

Δe −0.85 0.03 0.01 −0.01 −0.01 −0.03

a

  Calculated charges of the Sr, Zr and O atoms in bulk crystal are 1.39, 1.15 and −0.85 e, respectively.   Δe, layer excess charge (elayer ‒ ebulk‒plane), which are calculated by layer and bulk plane charges. c   The sum is the total charge of five outer layers. b

surface stoichiometry are more effective for the polar surfaces to reduce their energy in static conditions. To take the effect of chemical environments into account, the SGP method was introduced and used to evaluate the relative stabilities of the investigated terminations. According to the computational methods described in the section  above, once the intervals of ΔμSr and ΔμO are determined, the SGP values of different SrZrO3 (1 1 0) terminations can be obtained and relevant discussion concerning the relative stabilities of these terminations in various chemical environments becomes available. In this calculation, the obtained formation energy of bulk SrZrO3 is  −18.60 eV, which is somewhat smaller than PbTiO3 (−13.83  eV [59]), SrTiO3 (−16.38  eV [57]) and BaTiO3 (−17.70 eV [58]). In this context, the deduced ΔμSr and ΔμO are restricted within the intervals of [−18.60 eV, 0.00 eV] and [−6.20 eV, 0.00 eV], respectively. The accessible SGPs of different terminations in different Sr and O external environments are displayed in figure 8. It should be emphasized that if the SGP values are not positive, the surfaces will form spontaneously and the crystal will be destroyed [85].

The results in figure 8(b) indicate that the SrO-rich, SrOdeficient and O-rich terminations of SrZrO (1 1 0) surfaces are stable in oxygen-rich environments (Δμo = 0 eV). With decreasing Δμo, the slopes of the computed SGP lines remain unchanged, whereas their positions move significantly, as indicated by the arrows in figure 8(b). If Δμo falls to −0.64 eV, Ω(stoichiometric) becomes equivalent to Ω(O-rich), and the relevant SGP line of the stoichiometric termination coincides with that of the O-rich termination. If Δμo is smaller than the critical value above, the stability interval of the O-rich termination will disappear, accompanied by the appearance of the stability interval of the stoichiometric termination, as illustrated in figure 8(c). Therefore, the most stable configurations change from the SrO-rich, SrO-deficient and O-rich terminations to the SrO-rich, SrO-deficient, and stoichiometric terminations. The results predicted that four out of five possible terminations were thermodynamically stable and only the O-deficient termination could not be stabilized. Furthermore, it should be pointed out that there may be some other possible configurations within the (1  ×  1) supercell, even 10

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J. Phys.: Condens. Matter 26 (2014) 395002

Table 5.  Cleavage energies (E

E rel cl ϕa

rel cl )

and ϕ (J m−2).

O-deficient

O-rich

SrO-deficient

SrO-rich

stoichiometric

2.81 5.30

2.81 0.32

1.43 5.48

1.43 −2.62

0.74 0.74

 The ϕ values are calculated by equation (9).

a

for the same composition. For example, maybe the peroxo group is perpendicular to the surface rather than parallel to it. Fortunately, we located its geometry and the calculated results show that the latter is 3.82 eV lower in total energy than the former. Therefore, the O-rich termination with a peroxo group perpendicular to the surface is less stable. It can also be found that the energy difference (368.399 kJ mol−1) between the two surface configurations can be comparable with the bonding energy of a single Zr-O bond (410.244 kJ mol−1) in ZrO2 crystal [86]. Therefore, when determining which configuration is preferable among all possible surfaces with the same composition, the condition of achieving maximum bonding on the surface should be considered. The discussion above mainly concerns the surface stability with respect to the precipitation of metals and oxygen on surfaces. However, it was reported that the precipitation of small mono-metal oxide crystals on the surface is also possible [85] because it can affect the stability areas of surfaces. To prevent the precipitation of materials on the surface as small mono-metal oxides but not as perovskite surface layers, the following boundary conditions should be considered:

3.4.  Comparison with other ABO3 (1 1 0) polar terminations

As discussed in the previous section, for SrZrO3 (1 1 0) polar surfaces, the SrO-rich, SrO-deficient, O-rich and stoichiometric terminations can be stabilized, while the O-deficient termination is highly unstable. This situation is markedly different from ATiO3 (A = Sr, Ba) and BaZrO3 compounds, in which the O-deficient (1 1 0) termination is found to be stable in some specific regions, whereas the O-rich termination is confirmed to be unstable over all areas [57, 58]. A better understanding of these different behaviors requires deeper analysis. According to equations (8) and (9) and figure 8, the SGP summation of the O-deficient and O-rich terminations is constant and independent of ΔμO and ΔμSr , as they are complementary slabs. The SrO-deficient and SrO-rich terminations are another two complementary surfaces. The relative position of the SGP line is expected to change significantly and lead to different stability diagrams if the total energy of a slab can be reduced effectively. Table 5 gives the calculated ϕ values of different terminations. For SrZrO3, the ϕ value of the O-rich termination is much smaller than that in other systems [57, 58, 85]. If the B site atom changes from Ti to Zr, the average ϕ value of the O-rich and stoichiometric terminations exhibits a decreasing trend [57]. Furthermore, it is found that the ϕ value of the O-rich termination is even smaller than the stoichiometric termination, which is contrary to the behavior of SrTiO3, BaTiO3 and BaZrO3 perovskite materials [57, 58, 85]. These special characteristics result in an observable stability area for the O-rich (1 1 0) surface of SrZrO3 crystal. However, the behavior is absent in other ABO3 compounds [57, 58, 85]. As emphasized above, the position of the SGP line of a system depends on its total energy; thus, understanding the factors that significantly lower the total energy is of particular importance. We believe rotational relaxation plays an important role in reducing the energy of the O-rich SrZrO3 (1 1 0) polar surface. The results from figures  1 and 2 highlight that the oxygen atoms on the O-rich and stoichiometric terminations possess very large rotation displacements along the y- and z-axes. For the stoichiometric terminations of ATiO3 (A = Ba, Pb) and BaZrO3 (1 1 0) polar surfaces, the displacements of O atoms were relatively small and restricted to several outermost layers along these directions, whereas the rotational displacements of O atoms in the O-rich terminations were almost negligible [57, 58, 85]. Accordingly, we hypothesize that the large rotational displacements of oxygen provide a key reason for the decrease in total energy of the surface. To further elucidate the origin of surface stability, we also calculated the phonon dispersion relation of cubic SrZrO3. The results shown in figure 10 indicate a possible low-temperature

bulk , μSr + μO ≤ ESrO (17)

ΔμSr + ΔμO ≤ Ef (SrO), (18) ΔμSr + ΔμO ≥ Ef (SrZrO3) − Ef (Zr O2). (19)

Figure 9 indicates that the region where pure SrZrO3 surfaces can be obtained is restricted to the narrow strip between the SrO precipitation line and the ZrO2. Under this restriction, the O-rich and stoichiometric (1 1 0) terminations remain stable. The results suggested that the O-rich and stoichiometric terminations could be prepared within a narrow strip, and the growth of the ZrO2 and SrO clusters on the perovskite surface was also possible outside the particular region. Moreover, it should be noted that the (0 0 1) surfaces of II-IV perovskite compounds are generally believed to be the most stable, so a comparison concerning the stabilities of different oriented surfaces becomes important. To further clarify this issue, we performed some relevant calculations and the results are also presented in figure  9. When considering the surface stability with respect to Sr and O environments, ZrO2 (0 0 1) and SrO (0 0 1) surfaces are found to be stable too; however, when considering the oxides’ precipitation condition, their stability regions lie outside the narrow strip. In conclusion, the competition among different SrZrO3 surface configurations depends sensitively on the chemical environment. Therefore, to obtain well-controlled SrZrO3 (1 1 0) polar terminations, the effect of the environment should be carefully considered in further experiments. 11

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Figure 8.  Stability diagram of the (1  ×  1) SrZrO3 (1 1 0) polar surface. For (b), ΔμO = 0.0 eV; for (c), ΔμO = −2.0 eV. In sections I and II,

Ω(SrO-rich) and Ω(SrO-deficient) are smaller than zero and O-rich and stoichimotric terminations may become stable depending on their SGP values.

Figure 9.  Stability diagrams of the SrZrO3 (1 1 0) surface. Oxide precipitation on the surface and comparison with the (0 0 1) orientation are

included.

AFD transition associated with the I4/mcm symmetry in cubic SrZrO3, which is in good agreement with the available experiments [5, 70–74]. Furthermore, it is also an interesting question of whether the AFD transition of bulk SrZrO3 can affect the relaxation patterns of the corresponding (1 1 0) polar surfaces and therefore the surface stabilities. To rationalize this issue, one should distinguish the surface-related and instabilityrelated relaxations that have occurred on the surface and then a detailed investigation of (1 1 0) polar surfaces of AFD compounds is needed. It is widely known that the surface effect is

limited to several top layers and after those the surface relaxation is damped effectively. According to this criterion, the surface- and instability-related relaxations that have occurred at the surface can be clarified. Eglitis and coworkers performed some theoretical calculations of the (1 1 0) surfaces of SrTiO3, CaTiO3, SrZrO3 and PbZrO3 compounds [34, 87–89]. They found that the AO-rich and AO-deficient (1 1 0) terminations (A = Sr, Ca, Pb) have no rotational relaxation, but the stoichiometric (1 1 0) terminations possess large y-axis relaxations [34, 87–89], which may be due to the AFD instability of the 12

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J. Phys.: Condens. Matter 26 (2014) 395002

and Sr-moderate conditions. The O-deficient termination was unstable in all possible regions. Therefore, it can be anticipated that the polar surfaces with vacancies predominate over the cleavage or growth of the SrZrO3 crystal along [1 1 0] orientation. Furthermore, when the [0 0 1] orientation is taken into account, the ZrO2 and SrO (0 0 1) terminations are also stable in some certain areas. If an oxide precipitation condition is further considered, only the O-rich and stoichiometric terminations are stable. In comparison to other ABO3 compounds, it can be clarified that the stability domain of the O-rich (1 1 0) termination of SrZrO3 originates from the AFD distortions. Although the present calculation has revealed the electronic structures and stabilities of the (1  ×  1) (1 1 0) polar terminations of SrZrO3, it should be pointed out that there are still many possibilities. For example, the superstructures of (2  ×  1) and (2  ×  2) et al may be possible. To obtain a fair insight into a larger set of the SrZrO3 polar terminations, further theoretical investigations are still needed if the computational resources and costs are within a controllable scope.

Figure 10.  Phonon dispersion relation of cubic SrZrO3.

bulk compounds. Nevertheless, as the relaxation data given by Eglitis et al was restricted to the three outermost layers and both O-rich and O-deficient (1 1 0) terminations were excluded in the literature [34, 87–89], further deduction is still impossible at present. But our simulated results of SrZrO3 clearly indicated that the O-rich and stoichiometric (1 1 0) terminations did have large rotational relaxations, which were never damped until the very center of the slabs. In consideration of the facts that no rotational relaxations were reported for the O-rich (1 1 0) terminations of the ABO3 compounds in which low-temperature AFD or AFE transitions were absent [58, 59, 85, 90, 91] and that the rotational displacements for the stoichiometric (1 1 0) terminations of ATiO3 (A = Ba and Pb) and BaZrO3 compounds are damped quickly [58, 59, 85, 89–91], it is reasonable to deduce that the abnormal rotational relaxations found on the O-rich and stoichiometric (1 1 0) terminations of SrZrO3 can be attributed to the AFD instability of the bulk layer of the slab, which leads to the observable difference in the stability diagram.

Acknowledgments The authors are grateful for the financial support from the National Natural Science Foundation of China (nos. 21173072 and 21301052), Program for Innovative Research Team in University (The Ministry of Education of China, grant no. IRT-1237), Specialized Research Fund for the Doctoral Program of Higher Education (no. 20132301120001), and Natural Science Foundation (grant no. B201003) and Educational Commission Foundation (grant no. 11551340) of Heilongjiang Province of China. References

4. Conclusion

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We have investigated the structural relaxations, polarity compensation mechanisms and surface stabilities of five SrZrO3 (1 1 0) polar terminations by the first-principles DFT method at GGA+U level. The calculated results indicate that the polarity compensation criterion for the considered terminations is fulfilled. The electronic structures of O-rich and O-deficient terminations were changed significantly in response to the charge redistributions. A considerable filling of CB states was observed for the O-deficient surface, whereas the formation of a peroxo group on the O-rich surface led to the depletion of VB states. For SrO-deficient, SrO-rich and stoichiometric terminations, the changes of surface composition were helpful for the polarity compensations and their electronic structures were qualitatively similar to that of the bulk crystal, which allowed them to retain their insulating characteristics. By applying the SGP technique, three stable (1 1 0) terminations were found. It was observed that the SrO-deficient termination was stable in Sr- and O-poor conditions, and the stoichiometric termination showed a stability domain in moderate Sr and O regions. The O-rich termination could be stabilized only within a small domain corresponding to O-rich 13

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