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ISSN 1063 7834, Physics of the Solid State, 2013, Vol. 55, No. 9, pp. 1941­1945. © Pleiades Publishing, Ltd., 2013. Original Russian Text © A.I. Lebedev, I.A. Sluchinskaya, 2013, published in Fizika Tverdogo Tela, 2013, Vol. 55, No. 9, pp. 1825­1829.

PHASE TRANSITIONS

Structural Instability in BaZrO3 Crystals: Calculations and Experiment
A. I. Lebedev* and I. A. Sluchinskaya
Moscow State University, Moscow, 119991 Russia * e mail: swan@scon155.phys.msu.ru
Received March 4, 2013

Abstract--The phonon spectrum of cubic barium zirconate is calculated from first principles using the den sity functional theory. An unstable R25 phonon mode observed in the phonon spectrum indicates an instability of the BaZrO3 structure with respect to the oxygen octahedra rotations. It is shown that the symmetry of the ground state structure of the crystal is I4/mcm. The local structure of BaZrO3 is studied by the EXAFS spec troscopy at the BaLIII absorption edge at 300 K to search for the instability predicted by calculations. Anom alously high values of the Debye­Waller factor for the Ba­O atomic pairs ( 1 ~ 0.015 å2) are attributed to the appearance of this structural instability. The average amplitude of the octahedra rotations caused by ther mal vibrations is estimated from the measured 1 value to be ~4° at 300 K. The closeness of the calculated energies of various distorted phases resulting from the condensation of the R25 mode suggests a possible for mation of the structural glass state in BaZrO3 as the temperature is lowered. It explains the origin of the dis agreement between the results of calculations and diffraction experiments. DOI: 10.1134/S1063783413090229
2 2

1. INTRODUCTION Crystals of the perovskite family are widely used in modern technology and applications. This is due to the lability of their crystal structure which allows them to undergo various (ferroelectric, ferroelastic, mag netic, and superconducting) phase transitions. Barium zirconate BaZrO3 having the perovskite structure is widely used in microwave electronics. Its solid solutions with BaTiO3 are relaxors with a very high electric field tuning of the dielectric constant and are promising for applications in tunable filters, gener ators, phase shifters, and phased array antennas [1, 2]. When doped with yttrium, BaZrO3 becomes an ionic conductor with high proton conductivity promising for applications in fuel cells [3]. According to X ray and neutron diffraction studies, barium zirconate retains its cubic structure up to the lowest temperatures (2 K) [4] and exhibits no phase transitions. However, these structural data disagree with the results of the first principles calculations of the BaZrO3 phonon spectrum (see [4­7] and also [8]), which reveal an instability of the R25 mode in the phonon spectrum and predict an instability of the BaZrO3 cubic structure with respect to the octahedra rotations. An indirect evidence of the possible phase transformation in BaZrO3 can be its very low thermal expansion coefficient at temperatures below 300 K [4]. It was supposed that the disagreement between theory and experiment is caused by the fact that the long

range order in the octahedra rotations is not estab lished for some reasons in these crystals even at low temperatures. In particular, in [6] it was suggested that zero point lattice vibrations can be such a reason. The aim of this work was to calculate the phonon spectrum and the ground state structure of BaZrO3 from first principles and to find experimental evi dences for the structural instability in these crystals. Our idea was that even if the long range order in this material is absent for some reason, the structural insta bility (if it exists) should manifest itself in a change of the local structure. To search for such changes, we used the extended X ray absorption fine structure tech nique (EXAFS spectroscopy). This is one of the pow erful modern experimental techniques which provides a detailed information primarily on the local crystal structure [9]. We expected to obtain evidences of the BaZrO3 structural instability from an anomalous behavior of Debye­Waller factors which characterize the local displacements and amplitude of thermal vibrations. 2. CALCULATION TECHNIQUE AND RESULTS The calculations were performed from first princi ples within the density functional theory using the ABINIT program. Pseudopotentials for Ba, Zr, and O atoms were borrowed from [10, 11]. The maximum energy of plane waves in the calculations was 30 Ha

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LEBEDEV, SLUCHINSKAYA

additional instability at the M point found in [6] was not observed in our calculations. To determine the ground state structure, we calcu lated the energies of various distorted phases which can arise from the cubic perovskite structure as a result of condensation of the unstable triply degenerate R25 mode, taking into account its degeneracy. If we write the expansion of the total energy of a crystal in a power series of the distortion amplitudes in the spirit of the Landau expansion, E
X M R M
tot 2

= E tot ( 0 ) + a ( x + y + z )
2 22 4 4 4

2

2

2

+ b ( x + y + z ) + c ( x + y + z ) + d ( x + y + z ) + e x y z + ... ,
2 2 23 2 2 2

(1)

Fig. 1. Phonon spectrum of BaZrO3 in the cubic Pm3m phase.

(Hartree) (816 eV). Integration over the Brillouin zone was performed on the 8 â 8 â 8 Monkhorst­Pack mesh. The lattice parameters and equilibrium posi tions of atoms were calculated from the condition of a decrease of the Hellmann­Feynman forces below 5 â 10­6 Ha/Bohr (0.25 meV/å) while the total energy calculation accuracy was better than 10­10 Ha. The phonon spectra were calculated using the formulas derived in the perturbation theory and the interpola tion technique described in [10]. The calculated lattice parameter for cubic BaZrO3 (4.1659 å) is in good agreement with the experimental value obtained at 10 K (4.191 å [4]). The minor disagreement of these values is caused by the systematic underestimation of the lattice parameter, typical of the local density approximation (LDA) used in this work. The phonon spectrum of cubic BaZrO3 with the perovskite structure (space group Pm3m) is shown in Fig. 1. An unstable phonon of the R25 symmetry pre senting in this spectrum (the imaginary values of the phonon energy are given by negative numbers in the figure) suggests that the cubic structure is unstable with respect to the octahedra rotations. The results of these calculations agree with the results of previous studies [4­8]; however, it should be noted that an
Table 1. Energy of various distorted BaZrO3 structures (the energy of the cubic Pm3m phase is taken as the energy reference point) Unstable mode R R R
25 25 25

where Etot(0) is the total energy of the cubic phase, x, y, and z are the rotation angles about three fourfold axes, it becomes evident that, to search for the ground state, it is sufficient to calculate the energies of the phases described by the order parameters (, 0, 0), (, , 0), and (, , ) at which the Etot minimum is reached for various combinations of signs of the expansion coefficients c and e. These order parameters correspond to the Glazer rotation systems (a0a0c­), (a0b­b­), and (a­a­a­) and result in space groups I4/mcm, Imma, and R 3 c , respectively. As follows from Table 1, the I4/mcm phase has the lowest energy among these phases. This phase appears from the Pm3m cubic structure by out of phase octahedra rota tions about one of the fourfold axes. The energy of this phase is lower than the energy of the cubic phase by ~10 meV, which corresponds to an approximate phase transition temperature of ~120 K. It should be noted that among the obtained solu tions there is no P 1 phase which was considered as the ground state in [5]. This phase corresponds to the Glazer rotation system (a­b­c­) in which the rotation angles about three fourfold axes of the cubic structure are different. A check of this solution showed that the P 1 structure slowly relaxes to the above I4/mcm solu tion; however, the convergence required more than 400 iterations. This shows that the use of information about the crystal symmetry in the calculation of the total energy makes it possible to significantly (more than tenfold) reduce the calculation time needed to search for the ground state structure and guarantees accurate results. The lattice parameters and atomic coordinates in the ground state structure of barium zirconate are given in Table 2.

Glazer rotations a­a­a­ a0b­b­ a0a0c­

Space group R3c Imma I4/mcm

Energy, meV ­9.17 ­9.47 ­10.01

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STRUCTURAL INSTABILITY IN BaZrO3 CRYSTALS Table 2. Calculated lattice parameters and atomic coordinates in the BaZrO3 ground state structure Phase I4/mcm Lattice parameters, å a = 5.8722 c = 8.3577 Atom Ba Zr O O Position 4b 4c 4a 8h x 0 0 0 0.22315 y 0.50000 0 0 0.72315 z

1943

0.25000 0 0.25000 0

Table 3. Structural parameters for the first three shells in cubic BaZrO3 at 300 K obtained from EXAFS data analysis Measurement technique Fluorescence Transmission R1, å 2.916 2.914 1, å 0.0155 0.0145
2

R2, å 3.640 3.635

2 , å 0.0079 0.0068

2

R3, å 4.244 4.244

3 , å 0.0101 0.0087

2

3. SAMPLES AND EXPERIMENTAL TECHNIQUE The samples of barium zirconate were prepared by the solid state reaction method. The starting compo nents were BaCO3 and microcrystalline ZrO2 obtained by the decomposition of ZrOCl2 · 8H2O at 300°C. Components were dried at 600°C, weighted in appro priate proportions, ground in acetone, and annealed in air at 1100°C for 6 h. The prepared powders were ground again and repeatedly annealed at a tempera ture of 1500°C for 3 h. The single phase nature of the samples was confirmed by X ray diffraction. The EXAFS spectra were recorded by simulta neous measurement of the transmission and X ray flu orescence at the KMC 2 station of the BESSY syn chrotron radiation source (the beam energy is 1.7 GeV; the beam current is up to 300 mA) at the BaLIII edge (5.247 keV) at 300 K. The intensity of the radiation incident on the sample was measured using an ionization chamber; the intensity of the radiation passed through the sample was measured using a sili con photodiode, and the intensity of the fluorescence excited in the sample was measured using a RæNTEC silicon energy dispersive detector. A total of four spec tra were recorded. The BaLIII edge was chosen because the predicted structural instability accompanied by the oxygen octa hedra rotations should most strongly manifest itself in the EXAFS measurements of the Ba­O interatomic distances, rather than of the Zr­O interatomic dis tances, since the latter remain almost unchanged when the octahedra are rotated. In principle, the octa hedra rotations could also be observed in the changes of the multiple scattering contributions to the EXAFS spectra at the Zr K edge; this technique is described in [12]. The EXAFS spectra were independently processed in the conventional way [13]; the results obtained were then averaged. According to our theoretical evalua
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tions, the temperature of the expected phase transition is below 300 K; therefore, in the data analysis we con sidered the BaZrO3 structure as cubic and assumed that the structural instability in it will manifest itself as overestimated Debye­Waller factors. 4. EXPERIMENTAL RESULTS AND DISCUSSION The typical EXAFS spectrum of the BaZrO3 sam ple recorded at the BaLIII edge at 300 K is shown in Fig. 2. The data analysis shows that the spectra are in good agreement with a model in which the structure of the samples under study is cubic. The structural parameters (interatomic distances Ri and Debye­
k(k) 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 2 3 4 5 k, å 6
-1

7

8

9

Fig. 2. Typical EXAFS spectrum of the BaZrO3 sample obtained at the BaLIII edge at 300 K (squares) and the result of its fitting (solid curve). The limited wavenumber k range is caused by the closeness of the BaLIII and BaLII absorption edges.

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Waller factors i ) for the first three shells are given in Table 3. One can see that the structural parameters determined by two different measurement techniques (transmission and fluorescence) are almost identical. In Table 3, attention should be drawn to an overes timated Debye­Waller factors for the first shell, which are significantly larger than those for the second and third shells (usually, this factor monotonically increases with increasing interatomic distance). The Debye­Waller factor is usually controlled by two components: static distortion of the structure and thermal vibrations. Since our measurements are per formed above the temperature of the expected phase transition, no static distortions should be observed. Further, the optical modes responsible for the ZrO6 octahedron deformation are the highest frequency ones in the BaZrO3 phonon spectrum and are almost not excited at 300 K. Therefore, we can consider this octahedron as a rigid one. The thermal motion of the center of this octahedron with respect to the Ba atom 2 is described by the Debye­Waller factor 2 . Since, by virtue of the octahedron rigidity, the oxygen atoms are strongly bound to Zr atoms, the only way to explain 2 2 why 1 2 is to take into account the octahedra rotations. Assuming that the Debye­Waller factors for Ba­O and Ba­Zr bonds in the absence of the octahe dra rotations are close (0.008 å2), we can attribute the 2 excess part 1 of the Debye­Waller factor in the first shell to the thermal rotational vibrations. Then, using the formula a = 0 12
2 1 22

which describes the dependence of 1 on the octa hedra rotation angle about the fourfold axis of the cubic structure, we can estimate the amplitude of rota tional vibrations. For the lattice parameter a0 4.2 å, it is 4°. For comparison, in SrTiO3, in which the structural phase transition occurs at 105 K, the exper imental oxygen octahedra rotation angle at 50 K is 2.01° [14] which is comparable to the amplitude of rotational vibrations in BaZrO3 determined in our study. We now discuss the causes of the absence of the long range order in the octahedra rotations in BaZrO3 at low temperatures. As noted in the Introduction, the suppression of the long range order by zero point lat tice vibrations was considered as one of the causes of such behavior. In [15], a criterion was proposed which enables us to estimate the stability of a structure with respect to zero point vibrations and is based on a set of parameters calculated completely from first princi ples. According to this criterion, zero point vibrations
2

2

make the system localization in one of the local min ima of the potential energy impossible at h/E0 > 2.419, where is the frequency of the unstable phonon and E0 is the depth of the local minimum of the poten tial energy. For the unstable R25 mode in BaZrO3, h = 81.2 cm­1 10 meV and E0 10 meV (Table 1); hence, zero point vibrations should not suppress the struc tural distortions in barium zirconate. The fact that the influence of zero point vibrations on the phase transi tion in BaZrO3 is weaker than in SrTiO3 is not surpris ing since the masses of both metal atoms in barium zir conate are larger. Thus, the explanation of the sup pression of the long range order in BaZrO3 by zero point vibrations proposed in [6] seems unlikely. The theoretical calculations carried out in this work allow us to propose another explanation of the absence of the long range order in BaZrO3 exhibiting the structural instability. The closeness of the energies of the R 3 c , Imma, and I4/mcm phases (Table 1) which can emerge from the condensation of the R25 mode, on the one hand, and the fact that the transitions between these phases are of the first order, on the other hand, suggest that the structural glass state appears in the crystal upon cooling. Internal strains and defects in the samples result in the condensation of phases with different octahedral rotation patterns at different points of the sample, which obviously leads to the absence of the long range order in their rotations. Furthermore, the oxygen vacancies, which are rather easily formed in BaZrO3, break the three dimensional connectivity of the perovskite structure and, hence, promote the formation of the structural glass state. The uniqueness of barium zirconate among other crystals with the perovskite structure is that the energy difference between the phases with different rotation patterns is only 0.84 meV (Table 1), which is much smaller than that in other crystals we have studied (3.4 meV in SrTiO3, 136 meV in CaTiO3, 371 meV in CdTiO3 [10, 15], and 69 meV in SrZrO3). This explains the tendency of BaZrO3 to the formation of the structural glass state. 5. CONCLUSIONS The first principles calculations of the BaZrO3 phonon spectrum and the experimental observation of anomalously high Debye­Waller factors for the Ba­O bonds in EXAFS measurements suggest that the struc tural instability with respect to the oxygen octahedra rotations indeed exists in barium zirconate. It was shown that zero point lattice vibrations are not the factor which prevents the formation of the long range order in the octahedra rotations. The extreme close ness of the energies of phases with different octahedral rotation patterns which can result from the condensa tion of the R25 phonon in BaZrO3 enables us to suggest the transition to the structural glass state upon cooling.
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This explains the apparent contradiction between pre dictions of the calculations and the experimentally observed absence of the long range order in the octa hedra rotations. Further experimental studies of BaZrO3, such as the study of the temperature depen dence of diffraction line width, diffuse scattering, elas tic properties, and other physical phenomena sensitive to the octahedra rotations could yield additional infor mation about the causes of disagreement between the ory and experiment and prove or disprove the hypoth esis proposed in this work. ACKNOWLEDGMENTS The calculations presented in this work were per formed on the laboratory computer cluster (16 cores). The authors are grateful to the BESSY staff for help in preparation of experiment and to the Russian­Ger man laboratory for financial support during their stay at BESSY. This work was partially supported by the Russian Foundation for Basic Research (project no. 13 02 00724). REFERENCES
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Translated by A. Kazantsev

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