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Lattice dynamics of cubic SrZrO3
ARTICLE IN PRESS
Journal of Physics and Chemistry of Solids 69 (2008) 876–879 www.elsevier.com/locate/jpcs
Lattice dynamics of cubic SrZrO3 R. Vali School of Physics, Damghan University of Basic Sciences, P.O. Box 36715/364, Damghan, Iran Received 8 August 2007; received in revised form 14 September 2007; accepted 26 September 2007
Abstract Using density functional perturbation theory, the optical dielectric constant, Born effective charges and phonon dispersion curves of cubic SrZrO3 have been calculated. The obtained dispersion curves show a soft phonon branch spreading from R to M points of the cubic Brillouin zone. An analysis based on the symmetry relationships indicates that the experimentally observed low-symmetry phases of SrZrO3 can be considered as results of the soft mode condensation at R and M points. r 2007 Elsevier Ltd. All rights reserved. Keywords: A. Oxides; C. Ab initio calculations; D. Lattice dynamics; D. Phonons
1. Introduction Strontium zirconate belongs to ABO3 perovskite oxides family and has many characteristics which are suitable for high-voltage and high-reliability capacitor applications. Recently, it has been investigated as a possible candidate material for high-k gate dielectrics [1]. In addition when SrZrO3 is doped with acceptor ions exhibits protonic conduction at high temperatures. This feature makes it suitable for use in high-temperature applications such as fuel cells, steam electrolysis, and hydrogen gas sensors [2–4]. SrZrO3 has an orthorhombic structure at room temperature and undergoes a sequence of structural phase transition. Using X-ray diffraction in conjunction with differential thermal analysis, Carlsson [5] deduced the following sequence: First, orthorhombic to pseudo-tetragonal at 700 1C, then pseudo-tetragonal to tetragonal at 830 1C, and finally tetragonal to cubic at 1170 1C. However, Carlsson did not suggest space groups for these different structures. Ahtee et al. [6], from neutron diffraction study, in an attempt to investigate the Carlsson’s pseudo-tetragonal and tetragonal phases concluded that the space groups were Cmcm (orthorhombic) and I4/mcm, respectively. Recently, Howard et al. [7] have studied the structures of SrZrO3 by very high-resolution neutron Tel.: +98 232 5233054; fax: +98 232 5244787.
E-mail address: [email protected] 0022-3697/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2007.09.022
powder diffraction in fine temperature steps from room temperature to 1230 1C. They concluded the following sequence. First, orthorhombic (Pnma) to pseudo-tetragonal (Imma) at about 750 1C, then to tetragonal (I4/mcm) at ¯ about 840 1C, and finally to cubic (Pm3m) at 1070 1C. In this work, we study lattice dynamics of cubic SrZrO3 within density functional theory. The electronic band structure and density of states of cubic SrZrO3 have been theoretically investigated by Mete et al. [8]. However, to our best knowledge this is the first ab initio study of cubic SrZrO3 using lattice dynamics. While, SrZrO3 has a rather high melting temperature of about 2647 1C and consequently it is cubic in a wide range of temperature where most of its applications take place. In this work, the Born effective charges, optical dielectric constant and phonon dispersion curves have been computed, using density functional perturbation theory. Our results indicate that the high-temperature cubic phase of SrZrO3 has soft modes at R and M reciprocal lattice points. Based on symmetry relationships, we find that the experimentally observed I4/mcm and Imma low-symmetry phases can be regarded as results of soft mode condensation at R and M points, respectively. Also, the Pmnb phase (Pmnb is equivalent to Pnma) can be considered as sequential condensations of two soft modes. However, the obtained dispersion curves do not show an instability at the X point and thus the Cmcm phase reported in Ref. [6] is unlikely to exist, as was concluded on the basis of experimental data in Ref. [7].
ARTICLE IN PRESS R. Vali / Journal of Physics and Chemistry of Solids 69 (2008) 876–879
This paper is organized as follows. Section 2 describes the method of calculations. Section 3 presents the results: the Born effective charges, the optical dielectric constant, and phonon dispersion curves. Section 4 concludes the paper. 2. Calculation method The ab initio calculations of cubic SrZrO3 were carried out using the density functional theory [9], thanks to the ABINIT code [10]. The exchange-correlation energy functional was evaluated within the local density approximation [11], using Ceperley–Alder homogeneous electron gas data [12]. The electronic wave functions were expanded in plane waves up to a kinetic energy cutoff of 55 hartree. Integrals over the Brillouin zone were approximated by sums on a 4 4 4 mesh of special k points [13]. We used norm conserving pseudopotentials [14,15] with Sr(4p, 5s), Zr(4s, 4p, 4d), and O(2s, 2p) levels treated as valence states. The optical dielectric constant, the Born effective charges, and the force constant matrix at selected q points of the Brillouin zone were computed within a variational formulation of the density functional perturbation theory [16–19]. The interpolation of phonon dispersion curves was carried out following the scheme described in Refs. [20,21]. In this approach, the long-range character of the dipole– dipole contribution was subtracted from the force constant matrix in reciprocal space. Then the short-range contribution to the interatomic force constants in real space is obtained from the remainder of the force constant matrix in q space using a discrete Fourier transformation [22]. 3. Results First the crystal structure was optimized. The positions of the atoms are imposed by symmetry: Zr occupies the 1a sites at (0, 0, 0), Sr 1b sites at (12, 12, 12) and O the 3d sites at (12, 0, 0). The only degree of freedom that must be relaxed is therefore the lattice constant a0. After minimization, we obtained for lattice constant a value of 4.164 A˚. It is in good agreement with the experimental value of 4.109 A˚. The discrepancy is about 1.3%, which mainly is due to the local density approximation. Our lattice dynamics calculations have been performed for both the experimental and theoretically optimized volumes. As SrZrO3 is an insulator, a complete characterization of its lattice dynamics requires knowledge of the optical dielectric tensor and Born effective-charge tensors for each ion, which determine the non-analytic contribution to the dynamical matrix in the limit q-0. Also, their knowledge allows identifying the long-range part of the interatomic force constants and makes the interpolation of phonon frequencies tractable. Table 1 presents the calculated Born effective charges and optical dielectric constant. The Sr and Zr atoms are located at centers of cubic symmetry, So that their effective charge tensors are isotropic. The O atoms are located at the
877
Table 1 The Born effective charges and optical dielectric constant of cubic SrZrO3 at the theoretically optimized (a0 ¼ 4.164) and experimental (a0 ¼ 4.109) lattice constant Atom
a0 ¼ 4.164
a0 ¼ 4.109
ZnSr ZnZr ZnOjjj
2.58 6.02 4.79
2.59 6.02 4.78
ZnO? 1 SCI 1
1.90 4.70 4.12
1.91 4.69 4.14
face centers and thus have two inequivalent directions either parallel or perpendicular to the Zr–O bond. The effective charge tensor of O atoms are thus diagonal with two independent components ZOjj and Z O? , corresponding to the displacement of the atom parallel and perpendicular to the Zr–O bond, respectively. The Born effective charges of Zr and Ojj are quite a bit larger than their nominal values. These indicate that a strong dynamic charge transfer takes place along the Zr–O bond as the bond length is varied. As can be seen from Table 1, the Born effective charges are less sensitive to the discrepancy between experimental and optimized lattice constant. In the cubic structure, the optical dielectric tensor reduces to scalar. As can be seen from Table 1, the values obtained at the experimental and theoretical lattice constant are respectively equal to 4.69 and 4.70. We are not aware of any reports on the corresponding experimental value. However, local density approximation usually overestimates the optical dielectric constant. The overestimation can be corrected in first approximation by using a scissor correction [23], which consists of a rigid shift of the conduction band with respect to the valence bands. When including in the calculation a scissor shift of 2.23 eV, that adjusts the calculated band gap 3.37 eV [8] to its experimental value of 5.6 eV [24], we find optical dielectric constant with values of 4.14 and 4.12 at the experimental and theoretical lattice constant, respectively. The calculated frequencies of G-point phonons are summarized in Table 2. At G-point there are three acoustic and 12 optical phonon modes: three modes of F1u symmetry and one mode of F2u symmetry, each of them triply degenerated. F1u and F2u modes are IR-active and silent, respectively. For IR-active modes there are LO–TO splittings due to the coupling of the atomic displacement with the long-range electric field by means of the Born effective charge tensors. As can be seen from Table 2, the phonon frequencies change by noticeable amount when going from experimental to the theoretically optimized volume. This behavior is similar to that in BaTiO3 [25]. The calculated phonon dispersion curves are plotted along high symmetry directions in Fig. 1. The GX, GM, and GR lines are along the /1 0 0S, /1 1 0S, and /1 1 1S directions, respectively. Instabilities occur at the R and M
ARTICLE IN PRESS R. Vali / Journal of Physics and Chemistry of Solids 69 (2008) 876–879
878
Table 2 The calculated G-point phonon frequencies (cm1) of cubic SrZrO3 at the theoretically optimized (a0 ¼ 4.164) and experimental (a0 ¼ 4.109) lattice constant Mode
a0 ¼ 4.164
a0 ¼ 4.109
F 1u ðTO1Þ F 1u ðTO2Þ F 1u ðTO3Þ F 1u ðLO1Þ F 1u ðLO2Þ F 1u ðLO3Þ F 2u
114 215 544 149 373 732 179
135 212 602 163 373 784 170
As can be seen from Fig. 1, the negative frequencies grow in magnitude when going from theoretically optimized to the experimental volume. Here, we consider the symmetry relationships. The soft mode irreducible representation induces a list of space ¯ cubic space group can be lowered by subgroups. The Pm3m the irreducible representations of the soft modes according to the following subduction diagrams [26]. ¯ Pm3mðz ¼ 1Þ ! ðG 4 ; e1 ¼ e2 ¼ 0; e3 a0Þ ! P4mmðz ¼ 1Þ (1) ¯ Pm3mðz ¼ 1Þ ! ðRþ 4 ; e1 ¼ e2 ¼ 0; e3 a0Þ ! I4=mcmðz ¼ 2Þ
(2) ðXÞ ðXÞ ðXÞ ¯ Pm3mðz ¼ 1Þ ! ðXþ 5 ; K x ¼ K y ¼ 0; K z a0;
900
e1 ¼ 0; e2 a0Þ ! Cmcmðz ¼ 2Þ
Frequency (cm-1)
750
¯ Pm3mðz ¼ 1Þ ! ðRþ 4 ; e1 ¼ e2 a0; e3 ¼ 0Þ ! Immaðz ¼ 2Þ (4)
600
ðMÞ ðMÞ ðMÞ ¯ Pm3mðz ¼ 1Þ ! ðMþ 3 ; K xz ¼ K yz ¼ 0; K xy a0Þ
450
! P4=mbmðz ¼ 2Þ
300 150 0 -150
Γ
X
M
Γ
R
M
900
Frequency (cm-1)
750 600 450 300 150 0 -150
ð3Þ
In Eqs. (1)–(5), the symbol e indicates the formal amplitude of the component of the corresponding irreducible representation. Irreducible representations of the space groups are labeled by wave-vector, called irreducible star. The symbol K indicates the arm of the irreducible star. These are notations used in group theory of space groups. For example, the irreducible star X, (Eq. (3)), has three arms Kx, Ky, and Kz and each of them has two components e1 and e2. There is only one arm of the irreducible star of ¯ the representation Rþ 4 of Pm3m. The ray representation of Rþ is three-dimensional. Thus, the order parameter has 4 three components. In Eq. (2) one component of the Rþ 4 is involved, while in Eq. (4) two components of Rþ 4 are involved. The irreducible star of M point consists of three arms; however, the ray representation of Mþ 3 is onedimensional. In Eq. (5), the symmetry lowering is due to condensation of a single arm K ðMÞ xy of the M irreducible star. In addition, the common symmetry elements of the space group from R and M points create another space group, since the intersection reads P4=mbmðz ¼ 2Þ \ Immaðz ¼ 2Þ ¼ Pmnbðz ¼ 4Þ
Γ
X
M
Γ
R
M
Fig. 1. Calculated phonon dispersion curves of cubic SrZrO3 at the (a) theoretically optimized and (b) experimental lattice constant. Exact phonon frequencies are at G, X, M, and R points.
points and along the zone edge from R to M. At the R point the triply degenerate mode Rþ 4 has the lowest frequency. At the M point the non-degenerate mode Mþ 3 has the lowest frequency. All modes at X point are stable.
ð5Þ
(6)
As we have seen, the high-temperature cubic phase of SrZrO3 has soft modes at R and M points which according to Eqs. (2) and (4) could lead to the experimentally observed I4/mcm and Imma phases. According to the Eq. (6), the Pmnb phase is a consequence of the intersection of Imma and P4/mbm space groups, being results of soft mode condensation at R, Eq. (2), and M, Eq. (5), points, respectively. Hence, the Pmnb phase can be considered as a result of sequential condensation of two soft modes. Our conclusion that the tetragonal (I4/mcm) phase is a ¯ result of condensation of Rþ 4 soft mode in Pm3m structure
ARTICLE IN PRESS R. Vali / Journal of Physics and Chemistry of Solids 69 (2008) 876–879
is consistent with the results of Ref. [27]. When the ¯ is lowered to the tetragonal space symmetry of Pm3m group I4/mcm, the triply degenerate Raman-inactive soft mode, Rþ 4 , splits into Raman-active Eg and A1g irreducible representations of I4/mcm. The temperature dependence of the Eg and A1g modes has been investigated by Fujimori et al. [27], using a continuous-wave ultraviolet Raman spectroscopic system designed to measure the Raman scattering from materials at high temperature. The optical Eg and A1g modes showed remarkable softening approached together and their peak intensities continuously decreased with increasing temperature. They disappeared between 1200 and 1220 1C. In this way, Fujimori et al. have ¯ confirmed that the cubic (Pm3m)-tetragonal (I4/mcm) phase transition was triggered by the collapse of R25 phonon branch at the R point of the high-temperature cubic Brillouin zone. According to the Eq. (3), the orthorhombic Cmcm phase might arise from a soft mode at the X point. However, as we have seen all modes at X point are stable, hence the Cmcm phase reported in Ref. [6] is unlikely to exist, as was concluded on the basis of experimental data in Ref. [7]. 4. Conclusions Lattice dynamics of cubic SrZrO3 have been investigated, using density functional perturbation theory. The optical dielectric constant, Born effective charges, and phonon dispersion curves at theoretically optimized and experimental volume have been calculated. The obtained results show that the Born effective charges and optical dielectric constant are essentially insensitive to the discrepancy between optimized and experimental lattice constant, while phonon frequencies change noticeably when going from theoretically optimized to experimental volume. The obtained phonon dispersion curves show a soft phonon branch spreading from R to M points. An analysis based on the symmetry relationships indicates that the experimentally observed low-symmetry phases of SrZrO3 can be considered as results of the soft mode condensation at R and M points. We, however, find that the Cmcm phase reported in Ref. [6] is unlikely to exist, as was concluded on the basis of experimental data in Ref. [7]. References [1] C. Chen, W. Zhu, T. Yu, X. Chen, X. Yao, Preparation of metalorganic decomposition-derived strontium zirconate dielectric thin films, Appl. Surf. Sci. 211 (2003) 244–249. [2] T. Higuchi, T. Tsukamoto, N. Sata, K. Hiramoto, M. Ishigame, S. Shin, Protonic conduction in the single crystals of SrZr0.95M0.05O3 (M ¼ Y, Sc, Yb, Er), Jpn. J. Appl. Phys. 40 (2001) 4162–4163. [3] T. Higuchi, T. Tsukamoto, S. Yamaguchi, N. Sata, K. Hiramoto, M. Ishigame, S. Shin, Protonic conduction in the single crystal of Scdoped SrZrO3, Jpn. J. Appl. Phys. 41 (2002) 6440–6442. [4] T. Yajima, K. Koide, H. Takai, N. Fukatsu, H. Iwahara, Application of hydrogen sensor using proton conductive ceramics as a solid electrolyte to aluminum casting industries, Solid State Ionic 79 (1995) 333–337.
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[5] L. Carlsson, High-temperature phase transitions in SrZrO3, Acta Crystallogr. 23 (1967) 901–905. [6] M. Ahtee, A.M. Glazer, A.W. Hewat, High temperature phases of SrZrO3, Acta Crystallogr. B 34 (1976) 752–758. [7] C.J. Howard, K.S. Knight, B.J. Kennedy, E.H. Kisi, The structural phase transitions in strontium zirconate revisited, J. Phys.: Condens. Matter 12 (2000) L677–L683. [8] E. Mete, R. Shaltaf, S. Ellialtioglu, Electronic and structural properties of a 4d perovskite: cubic phase of SrZrO3, Phys. Rev. B 68 (2003) 035119–035122. [9] W. Kohn, L.J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev. 140 (1965) A1133–A1138. [10] X. Gonze, J.-M. Beuken, R. Caracas, F. Detraux, M. Fuchs, G.-M. Rignanese, L. Sindic, M. Verstraete, G. Zerah, F. Jollet, M. Torrent, A. Roy, M. Mikami, Ph. Ghosez, J.-Y. Raty, D.C. Allan, Firstprinciples computation of material properties: the ABINIT software project, Comput. Mater. Sci. 25 (2002) 478–492. [11] P. Hohenberg, W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136 (1964) B864–B871. [12] D.M. Ceperley, B.J. Alder, Ground state of the electron gas by a stochastic method, Phys. Rev. Lett. 45 (1980) 566–569. [13] H.J. Monkhorst, J.D. Pack, Special points for Brillouin-zone integrations, Phys. Rev. B 13 (1976) 5188–5192. [14] N. Troullier, J.L. Martins, Efficient Pseudopotentials for plane-wave calculations, Phys. Rev. B 43 (1991) 1993–2006. [15] C. Hartwigsen, S. Goedecker, J. Hutter, Relativistic seprable dualspace Gaussian pseudopotentials from H to Rn, Phys. Rev. B 58 (1998) 3641–3662. [16] X. Gonze, D.C. Allan, M.P. Teter, Dielectric tensor, effective charges, and phonons in a-quartz by variational density functional pereturbation theory, Phys. Rev. Lett. 68 (1992) 3603–3606. [17] X. Gonze, First-principles responses of solids to atomic displacements and homogeneous electric fields: implementation of a conjugate-gradient algorithm, Phys. Rev. B 55 (1997) 10337–10354. [18] X. Gonze, C. Lee, Dynamical matrices, Born effective charges, dielectric permittivity tensors, and interatomic force constants from density functional perturbation theory, Phys. Rev. B 55 (1997) 10355–10368. [19] S. Baroni, S. de Gironcoli, A. Dal Corso, P. Giannozzi, Phonons and related properties from density-functional perturbation theory, Rev. Mod. Phys. 73 (2001) 515–562. [20] P. Giannozzi, S. de Gironcoli, P. Pavone, S. Baroni, Ab initio calculation of phonon dispersions in semiconductors, Phys. Rev. B 43 (1991) 7231–7242. [21] X. Gonze, J.-C. Charlier, D.C. Allan, M.P. Teter, Interatomic force constants from first-principles: the case of a-quartz, Phys. Rev. B 50 (1994) 13035–13038. [22] Ph. Ghosez, X. Gonze, J.-P. Michenaud, Ab initio phonon dispersion curves and interatomic force constants of barium titanate, Ferroelectrics 206 (1998) 205–217. [23] Z.H. Levine, D.C. Allan, Linear optical responses in silicon and germanium including self-energy effects, Phys. Rev. Lett. 63 (1998) 1719–1722. [24] Y.S. Lee, J.S. Lee, T.W. Noh, D.Y. Byun, K.S. Yoo, K. Yamaura, E. Takayama-Muromachi, systematic trends in the electronic structures of the 4d transition-metaloxides SrMO3 (M ¼ Zr, Mo, Ru and Rh), Phys. Rev. B 67 (2003) 113101–113104. [25] Ph. Ghosez, X. Gonze, J.-P. Michenaud, Coulomb interaction and ferroellectric instability of BaTiQ3, Europhys. Lett. 33 (1996) 713–718. [26] H.T. Stokes, D.M. Hatch, Isotropic Subgroups of the 230 Crystallographyic Space Groups, Gordon and Breach, New York, 1965. [27] H. Fujimori, M. Yashima, M. Kakihana, M. Yoshimura, Phase transition and soft phonon modes in SrZrO3 around 12001C by ultraviolet laser Raman spectroscopy, Phys. Rev. B 61 (2000) 3971–3974.
Journal of Physics and Chemistry of Solids 69 (2008) 876–879 www.elsevier.com/locate/jpcs
Lattice dynamics of cubic SrZrO3 R. Vali School of Physics, Damghan University of Basic Sciences, P.O. Box 36715/364, Damghan, Iran Received 8 August 2007; received in revised form 14 September 2007; accepted 26 September 2007
Abstract Using density functional perturbation theory, the optical dielectric constant, Born effective charges and phonon dispersion curves of cubic SrZrO3 have been calculated. The obtained dispersion curves show a soft phonon branch spreading from R to M points of the cubic Brillouin zone. An analysis based on the symmetry relationships indicates that the experimentally observed low-symmetry phases of SrZrO3 can be considered as results of the soft mode condensation at R and M points. r 2007 Elsevier Ltd. All rights reserved. Keywords: A. Oxides; C. Ab initio calculations; D. Lattice dynamics; D. Phonons
1. Introduction Strontium zirconate belongs to ABO3 perovskite oxides family and has many characteristics which are suitable for high-voltage and high-reliability capacitor applications. Recently, it has been investigated as a possible candidate material for high-k gate dielectrics [1]. In addition when SrZrO3 is doped with acceptor ions exhibits protonic conduction at high temperatures. This feature makes it suitable for use in high-temperature applications such as fuel cells, steam electrolysis, and hydrogen gas sensors [2–4]. SrZrO3 has an orthorhombic structure at room temperature and undergoes a sequence of structural phase transition. Using X-ray diffraction in conjunction with differential thermal analysis, Carlsson [5] deduced the following sequence: First, orthorhombic to pseudo-tetragonal at 700 1C, then pseudo-tetragonal to tetragonal at 830 1C, and finally tetragonal to cubic at 1170 1C. However, Carlsson did not suggest space groups for these different structures. Ahtee et al. [6], from neutron diffraction study, in an attempt to investigate the Carlsson’s pseudo-tetragonal and tetragonal phases concluded that the space groups were Cmcm (orthorhombic) and I4/mcm, respectively. Recently, Howard et al. [7] have studied the structures of SrZrO3 by very high-resolution neutron Tel.: +98 232 5233054; fax: +98 232 5244787.
E-mail address: [email protected] 0022-3697/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2007.09.022
powder diffraction in fine temperature steps from room temperature to 1230 1C. They concluded the following sequence. First, orthorhombic (Pnma) to pseudo-tetragonal (Imma) at about 750 1C, then to tetragonal (I4/mcm) at ¯ about 840 1C, and finally to cubic (Pm3m) at 1070 1C. In this work, we study lattice dynamics of cubic SrZrO3 within density functional theory. The electronic band structure and density of states of cubic SrZrO3 have been theoretically investigated by Mete et al. [8]. However, to our best knowledge this is the first ab initio study of cubic SrZrO3 using lattice dynamics. While, SrZrO3 has a rather high melting temperature of about 2647 1C and consequently it is cubic in a wide range of temperature where most of its applications take place. In this work, the Born effective charges, optical dielectric constant and phonon dispersion curves have been computed, using density functional perturbation theory. Our results indicate that the high-temperature cubic phase of SrZrO3 has soft modes at R and M reciprocal lattice points. Based on symmetry relationships, we find that the experimentally observed I4/mcm and Imma low-symmetry phases can be regarded as results of soft mode condensation at R and M points, respectively. Also, the Pmnb phase (Pmnb is equivalent to Pnma) can be considered as sequential condensations of two soft modes. However, the obtained dispersion curves do not show an instability at the X point and thus the Cmcm phase reported in Ref. [6] is unlikely to exist, as was concluded on the basis of experimental data in Ref. [7].
ARTICLE IN PRESS R. Vali / Journal of Physics and Chemistry of Solids 69 (2008) 876–879
This paper is organized as follows. Section 2 describes the method of calculations. Section 3 presents the results: the Born effective charges, the optical dielectric constant, and phonon dispersion curves. Section 4 concludes the paper. 2. Calculation method The ab initio calculations of cubic SrZrO3 were carried out using the density functional theory [9], thanks to the ABINIT code [10]. The exchange-correlation energy functional was evaluated within the local density approximation [11], using Ceperley–Alder homogeneous electron gas data [12]. The electronic wave functions were expanded in plane waves up to a kinetic energy cutoff of 55 hartree. Integrals over the Brillouin zone were approximated by sums on a 4 4 4 mesh of special k points [13]. We used norm conserving pseudopotentials [14,15] with Sr(4p, 5s), Zr(4s, 4p, 4d), and O(2s, 2p) levels treated as valence states. The optical dielectric constant, the Born effective charges, and the force constant matrix at selected q points of the Brillouin zone were computed within a variational formulation of the density functional perturbation theory [16–19]. The interpolation of phonon dispersion curves was carried out following the scheme described in Refs. [20,21]. In this approach, the long-range character of the dipole– dipole contribution was subtracted from the force constant matrix in reciprocal space. Then the short-range contribution to the interatomic force constants in real space is obtained from the remainder of the force constant matrix in q space using a discrete Fourier transformation [22]. 3. Results First the crystal structure was optimized. The positions of the atoms are imposed by symmetry: Zr occupies the 1a sites at (0, 0, 0), Sr 1b sites at (12, 12, 12) and O the 3d sites at (12, 0, 0). The only degree of freedom that must be relaxed is therefore the lattice constant a0. After minimization, we obtained for lattice constant a value of 4.164 A˚. It is in good agreement with the experimental value of 4.109 A˚. The discrepancy is about 1.3%, which mainly is due to the local density approximation. Our lattice dynamics calculations have been performed for both the experimental and theoretically optimized volumes. As SrZrO3 is an insulator, a complete characterization of its lattice dynamics requires knowledge of the optical dielectric tensor and Born effective-charge tensors for each ion, which determine the non-analytic contribution to the dynamical matrix in the limit q-0. Also, their knowledge allows identifying the long-range part of the interatomic force constants and makes the interpolation of phonon frequencies tractable. Table 1 presents the calculated Born effective charges and optical dielectric constant. The Sr and Zr atoms are located at centers of cubic symmetry, So that their effective charge tensors are isotropic. The O atoms are located at the
877
Table 1 The Born effective charges and optical dielectric constant of cubic SrZrO3 at the theoretically optimized (a0 ¼ 4.164) and experimental (a0 ¼ 4.109) lattice constant Atom
a0 ¼ 4.164
a0 ¼ 4.109
ZnSr ZnZr ZnOjjj
2.58 6.02 4.79
2.59 6.02 4.78
ZnO? 1 SCI 1
1.90 4.70 4.12
1.91 4.69 4.14
face centers and thus have two inequivalent directions either parallel or perpendicular to the Zr–O bond. The effective charge tensor of O atoms are thus diagonal with two independent components ZOjj and Z O? , corresponding to the displacement of the atom parallel and perpendicular to the Zr–O bond, respectively. The Born effective charges of Zr and Ojj are quite a bit larger than their nominal values. These indicate that a strong dynamic charge transfer takes place along the Zr–O bond as the bond length is varied. As can be seen from Table 1, the Born effective charges are less sensitive to the discrepancy between experimental and optimized lattice constant. In the cubic structure, the optical dielectric tensor reduces to scalar. As can be seen from Table 1, the values obtained at the experimental and theoretical lattice constant are respectively equal to 4.69 and 4.70. We are not aware of any reports on the corresponding experimental value. However, local density approximation usually overestimates the optical dielectric constant. The overestimation can be corrected in first approximation by using a scissor correction [23], which consists of a rigid shift of the conduction band with respect to the valence bands. When including in the calculation a scissor shift of 2.23 eV, that adjusts the calculated band gap 3.37 eV [8] to its experimental value of 5.6 eV [24], we find optical dielectric constant with values of 4.14 and 4.12 at the experimental and theoretical lattice constant, respectively. The calculated frequencies of G-point phonons are summarized in Table 2. At G-point there are three acoustic and 12 optical phonon modes: three modes of F1u symmetry and one mode of F2u symmetry, each of them triply degenerated. F1u and F2u modes are IR-active and silent, respectively. For IR-active modes there are LO–TO splittings due to the coupling of the atomic displacement with the long-range electric field by means of the Born effective charge tensors. As can be seen from Table 2, the phonon frequencies change by noticeable amount when going from experimental to the theoretically optimized volume. This behavior is similar to that in BaTiO3 [25]. The calculated phonon dispersion curves are plotted along high symmetry directions in Fig. 1. The GX, GM, and GR lines are along the /1 0 0S, /1 1 0S, and /1 1 1S directions, respectively. Instabilities occur at the R and M
ARTICLE IN PRESS R. Vali / Journal of Physics and Chemistry of Solids 69 (2008) 876–879
878
Table 2 The calculated G-point phonon frequencies (cm1) of cubic SrZrO3 at the theoretically optimized (a0 ¼ 4.164) and experimental (a0 ¼ 4.109) lattice constant Mode
a0 ¼ 4.164
a0 ¼ 4.109
F 1u ðTO1Þ F 1u ðTO2Þ F 1u ðTO3Þ F 1u ðLO1Þ F 1u ðLO2Þ F 1u ðLO3Þ F 2u
114 215 544 149 373 732 179
135 212 602 163 373 784 170
As can be seen from Fig. 1, the negative frequencies grow in magnitude when going from theoretically optimized to the experimental volume. Here, we consider the symmetry relationships. The soft mode irreducible representation induces a list of space ¯ cubic space group can be lowered by subgroups. The Pm3m the irreducible representations of the soft modes according to the following subduction diagrams [26]. ¯ Pm3mðz ¼ 1Þ ! ðG 4 ; e1 ¼ e2 ¼ 0; e3 a0Þ ! P4mmðz ¼ 1Þ (1) ¯ Pm3mðz ¼ 1Þ ! ðRþ 4 ; e1 ¼ e2 ¼ 0; e3 a0Þ ! I4=mcmðz ¼ 2Þ
(2) ðXÞ ðXÞ ðXÞ ¯ Pm3mðz ¼ 1Þ ! ðXþ 5 ; K x ¼ K y ¼ 0; K z a0;
900
e1 ¼ 0; e2 a0Þ ! Cmcmðz ¼ 2Þ
Frequency (cm-1)
750
¯ Pm3mðz ¼ 1Þ ! ðRþ 4 ; e1 ¼ e2 a0; e3 ¼ 0Þ ! Immaðz ¼ 2Þ (4)
600
ðMÞ ðMÞ ðMÞ ¯ Pm3mðz ¼ 1Þ ! ðMþ 3 ; K xz ¼ K yz ¼ 0; K xy a0Þ
450
! P4=mbmðz ¼ 2Þ
300 150 0 -150
Γ
X
M
Γ
R
M
900
Frequency (cm-1)
750 600 450 300 150 0 -150
ð3Þ
In Eqs. (1)–(5), the symbol e indicates the formal amplitude of the component of the corresponding irreducible representation. Irreducible representations of the space groups are labeled by wave-vector, called irreducible star. The symbol K indicates the arm of the irreducible star. These are notations used in group theory of space groups. For example, the irreducible star X, (Eq. (3)), has three arms Kx, Ky, and Kz and each of them has two components e1 and e2. There is only one arm of the irreducible star of ¯ the representation Rþ 4 of Pm3m. The ray representation of Rþ is three-dimensional. Thus, the order parameter has 4 three components. In Eq. (2) one component of the Rþ 4 is involved, while in Eq. (4) two components of Rþ 4 are involved. The irreducible star of M point consists of three arms; however, the ray representation of Mþ 3 is onedimensional. In Eq. (5), the symmetry lowering is due to condensation of a single arm K ðMÞ xy of the M irreducible star. In addition, the common symmetry elements of the space group from R and M points create another space group, since the intersection reads P4=mbmðz ¼ 2Þ \ Immaðz ¼ 2Þ ¼ Pmnbðz ¼ 4Þ
Γ
X
M
Γ
R
M
Fig. 1. Calculated phonon dispersion curves of cubic SrZrO3 at the (a) theoretically optimized and (b) experimental lattice constant. Exact phonon frequencies are at G, X, M, and R points.
points and along the zone edge from R to M. At the R point the triply degenerate mode Rþ 4 has the lowest frequency. At the M point the non-degenerate mode Mþ 3 has the lowest frequency. All modes at X point are stable.
ð5Þ
(6)
As we have seen, the high-temperature cubic phase of SrZrO3 has soft modes at R and M points which according to Eqs. (2) and (4) could lead to the experimentally observed I4/mcm and Imma phases. According to the Eq. (6), the Pmnb phase is a consequence of the intersection of Imma and P4/mbm space groups, being results of soft mode condensation at R, Eq. (2), and M, Eq. (5), points, respectively. Hence, the Pmnb phase can be considered as a result of sequential condensation of two soft modes. Our conclusion that the tetragonal (I4/mcm) phase is a ¯ result of condensation of Rþ 4 soft mode in Pm3m structure
ARTICLE IN PRESS R. Vali / Journal of Physics and Chemistry of Solids 69 (2008) 876–879
is consistent with the results of Ref. [27]. When the ¯ is lowered to the tetragonal space symmetry of Pm3m group I4/mcm, the triply degenerate Raman-inactive soft mode, Rþ 4 , splits into Raman-active Eg and A1g irreducible representations of I4/mcm. The temperature dependence of the Eg and A1g modes has been investigated by Fujimori et al. [27], using a continuous-wave ultraviolet Raman spectroscopic system designed to measure the Raman scattering from materials at high temperature. The optical Eg and A1g modes showed remarkable softening approached together and their peak intensities continuously decreased with increasing temperature. They disappeared between 1200 and 1220 1C. In this way, Fujimori et al. have ¯ confirmed that the cubic (Pm3m)-tetragonal (I4/mcm) phase transition was triggered by the collapse of R25 phonon branch at the R point of the high-temperature cubic Brillouin zone. According to the Eq. (3), the orthorhombic Cmcm phase might arise from a soft mode at the X point. However, as we have seen all modes at X point are stable, hence the Cmcm phase reported in Ref. [6] is unlikely to exist, as was concluded on the basis of experimental data in Ref. [7]. 4. Conclusions Lattice dynamics of cubic SrZrO3 have been investigated, using density functional perturbation theory. The optical dielectric constant, Born effective charges, and phonon dispersion curves at theoretically optimized and experimental volume have been calculated. The obtained results show that the Born effective charges and optical dielectric constant are essentially insensitive to the discrepancy between optimized and experimental lattice constant, while phonon frequencies change noticeably when going from theoretically optimized to experimental volume. The obtained phonon dispersion curves show a soft phonon branch spreading from R to M points. An analysis based on the symmetry relationships indicates that the experimentally observed low-symmetry phases of SrZrO3 can be considered as results of the soft mode condensation at R and M points. We, however, find that the Cmcm phase reported in Ref. [6] is unlikely to exist, as was concluded on the basis of experimental data in Ref. [7]. References [1] C. Chen, W. Zhu, T. Yu, X. Chen, X. Yao, Preparation of metalorganic decomposition-derived strontium zirconate dielectric thin films, Appl. Surf. Sci. 211 (2003) 244–249. [2] T. Higuchi, T. Tsukamoto, N. Sata, K. Hiramoto, M. Ishigame, S. Shin, Protonic conduction in the single crystals of SrZr0.95M0.05O3 (M ¼ Y, Sc, Yb, Er), Jpn. J. Appl. Phys. 40 (2001) 4162–4163. [3] T. Higuchi, T. Tsukamoto, S. Yamaguchi, N. Sata, K. Hiramoto, M. Ishigame, S. Shin, Protonic conduction in the single crystal of Scdoped SrZrO3, Jpn. J. Appl. Phys. 41 (2002) 6440–6442. [4] T. Yajima, K. Koide, H. Takai, N. Fukatsu, H. Iwahara, Application of hydrogen sensor using proton conductive ceramics as a solid electrolyte to aluminum casting industries, Solid State Ionic 79 (1995) 333–337.
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