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Phonon properties of protonic conductor SrZrO3 in cubic phase
ARTICLE IN PRESS Physica B 404 (2009) 1187–1189
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Phonon properties of protonic conductor SrZrO3 in cubic phase M.M. Sinha a,, Anupamdeep Sharma b a b
Department of Physics, Sant Longowal Institute of Engineering and Technology, Longowal, Sangrur, Punjab 148106, India Sant Baba Bhag Singh Institute of Engineering and Technology, Padhiana, Jalandhar, Punjab, India
a r t i c l e in fo
abstract
Article history: Received 1 November 2008 Accepted 20 November 2008
Strontium zirconate has an orthorhombic structure at room temperature and undergoes a sequence of structural phase transition. 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, we are reporting the results of our theoretical investigation on the phonon properties of SrZrO3 in its cubic phase by using lattice dynamical simulation method based on de Launey angular force (DAF) constant model to understand the role of phonon in this system. The calculated zone center frequencies agree well with available results. The phonon dispersion curves of SrZrO3 in cubic phase are also drawn. The present calculation gives rise to softening of acoustical mode at zone boundary along [qqq] direction. The calculated results are compared and analyzed with other results. & 2008 Elsevier B.V. All rights reserved.
Keywords: Lattice dynamics Phonon dispersion Proton conductor Soft modes
1. Introduction There is an increasing interest in material research with potential technological applications, such as hydrogen sensors, fuel sensors and non-volatile random access memory devices. Among these materials, AZrO3 (A ¼ Ba, Sr, Pb) perovskites crystals have attracted considerable attention due to their ferroelectric and piezoelectric properties, yielding attractive models for experimental and theoretical academic research. In recent years, zirconate perovskites have been the focus of experimental studies. Notwithstanding, substantial theoretical research has not been devoted to these materials. In nuclear safety studies, SrZrO3 plays an important role as it is formed in the UO2 fuel by reaction between the fission products in the fuel matrix and during coreconcrete interactions between the fission products and the oxidized zircaloy cladding. The thermal properties and phase transitions are thus of special interest as they influence the behavior of hazardous fission products. The thermal properties and phase transitions are thus of special interest as they influence the behavior of hazardous fission products. High-temperature phase transitions of SrZrO3 were studied by using powder neutron diffraction [1] and X-ray diffraction [2]. This compound has a high melting temperature of about 2923 K [3], so it is cubic in a wide range of temperature where most of its useful applications take place. Shende et al. [4] suggested that this material could be used in high-voltage capacitor applications due to their high breakdown strengths as well as high dielectric constant. In order to fully take advantage of the properties of SrZrO3, a thorough Corresponding author. Tel.: +911672 280169; fax: +911672 280057.
E-mail address: [email protected] (M.M. Sinha). 0921-4526/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2008.11.095
investigation of its lattice dynamical properties is necessary. The calculation of vibrational properties of solids, surfaces and nanocrystals plays an important role in the structural characterization of matter. Besides the possibility to compare the vibrational frequencies with experimental data, phonon modes can be used also to determine the structural stability of a system [5–8]. This requires a reliable description of the vibrational properties of the system. In view of this, we shall employ short range force constant model [9] to calculate the zone center phonons and phonon dispersion curves of SrZrO3 in three symmetric directions.
2. Crystal structure At room temperature SrZrO3 has an orthorhombic phase as revealed by structural studies that date back to the 1950s and 1960s [10,11]. Later the existence of two additional phases at high temperature was proposed by Carlsson to be both tetragonal [12], however, more recent studies [1,2,13] on high temperatures have shown that SrZrO3 undergoes three structural phase transitions summarized as follows: First, orthorhombic (Pnma) to orthorhombic (Cmcm) at 970 K, then to tetragonal (I4/mcm) at 1100 K and finally to cubic (Pm3m) at 1400 K. A number of workers have used such changes to infer the presence of thermally induced phase transitions [14]. SrZrO3 has a rather high melting temperature of about 2920 K [3] and consequently it is cubic in a wide range of temperature where most of its applications take place. Therefore, in this work, we are studying vibrational properties of SrZrO3 in its cubic phase. In cubic perovskite-SrZrO3, the strontium atoms sit at the corners of a simple cubic unit cell, the oxygen atoms sit on the faces and the zirconium atom sits at
ARTICLE IN PRESS 1188
M.M. Sinha, A. Sharma / Physica B 404 (2009) 1187–1189
Table 1 Values of force constants (103 dyn cm1). Force constant
SrZrO3
a1 (Zr–O) a01 (Zr–O) a2 (Sr–O) a02 (Sr–O) a3 (Sr–Zr) a03 (Sr–Zr) a4 (O–O) a04 (O–O)
123.0 6.0 11.3 9.8 18.0 4.5 4.2 1.0
Fig. 1. Cubic phase of SrZrO3. Table 2 Calculated zone center phonon frequencies in cm1.
the center of the body as shown in Fig. 1. From the point of view of lattice dynamics, the unit cell contains five atoms that give rise to 15 phonons (three acoustics and 12 optic). The symmetry of these phonons at the G point (in terms of the Oh representation) is
ZC-phonons
Present work
Vali [16]
T1u(TO1) T1u(TO2) T1u(TO3) T2u
136 212 602 170
135 212 602 170
GðOh Þ ¼ 4T1u þ T2u where T1u and T2u represent the normal modes with triple degeneracy. One T1u mode is acoustic and the rest are optical modes. The T2u mode is inactive, while T1u modes are only IR active. The compound in cubic phase has no Raman mode.
3. Theoretical methodology In order to calculate the phonon properties of SrZrO3, a de Launey angular force (DAF) constant model [9] has been applied in the present investigation. The phonons in the SrZrO3 have been calculated to check whether softening of acoustic phonons are there or not in some crystal symmetry directions as it is supposed to be connected to phase changes in these compounds. In DAF model, the relative displacement of the reference atom and one of the neighbors is considered. The restoring force on the reference atom is taken to be proportional to the component of the relative displacement perpendicular to the line joining the two atoms at their equilibrium positions. The forces due to all neighbors are calculated separately and summed up together. Different force constants are used for the various categories of neighbors and the net force on the reference atom is obtained by summing over the contribution from all the neighbors. The present calculation involves four central force constants a1, a2, a3, a4 and four angular force constants a01 , a02 , a03 and a04 between Zr–O, Sr–O, Sr–Zr and O–O atoms, respectively, up to the third nearest neighbor. The calculated dynamical matrix of (15 15) is reduced to three matrices of the order (5 5) at zone center (ZC). The matrix elements are given in our earlier paper [15]. Recently Vali [16] has studied lattice dynamics of cubic phase of SrZrO3 within density functional theory. In the present calculation the interatomic force constants are obtained by fitting the calculated results of Vali [16] at the ZC for transverse infrared active phonon frequencies. The force constants thus calculated are listed in Table 1. Taking these force constants as input parameters, the dynamical matrix is solved at the ZC as well as along three symmetric directions [qoo], [qqo] and [qqq]. The ZC phonons thus obtained are listed in Table 2. The phonon dispersion curves thus obtained in three symmetric directions are shown in Fig. 2.
4. Results and discussion It is clear from Table 1 that the calculated force constant a1 between Zr–O is strongest among all other interatomic interactions and is followed by a3 (Sr–Zr) and is in agreement with finding of Wakamura [17]. This suggests that the covalent bonding between (Zr–O) is strongest than that between (Sr–Zr), (O–O) and (Sr–O). The larger force constant corresponds to stronger bond giving rise to larger value of frequency (TO3). One of the central force constants a2 between Sr–O is negative. The negative force constant corresponds to motion along modes that lead to energy lowering. Insofar as the frequencies are computed from the square roots of the force constants, this leads to an imaginary frequency. Frequency calculations are thus diagnostics as to nature of stationary points. All positive frequencies imply a (local) minimum, one imaginary frequency implies a transition state structure. The negative force constant also suggests electronic instability in SrZrO3. Due to large differences in mass, the phonon dispersion curves (PDC) of SrZrO3 (Fig. 2) decomposes into two well separated parts, a low-frequency region with predominantly Sr and Zr modes and the vibration of the light O atom around 602 cm1. It is obvious from Fig. 2 that phonon branches are distributed almost uniformly up to about 250 cm1 in all symmetric directions. The PDC shows that there are three regions in which phonon modes are distributed. The calculated eigenfrequencies and eigenvectors suggest that the top most region about 602 cm1 consisting of three phonon branches are due to oxygen atom vibrations, the middle three branches at about 250 cm1 are due to Sr atom vibrations and the lower nine branches are due to Sr and Zr atom vibrations. The main issue of the present calculation is to verify the occurrence of softening of acoustical modes at the boundary of Brillouin Zone (BZ) along major symmetry directions in SrZrO3 compound as obtained by Vali [16]. The unstable modes, which determine the nature of the phase transitions and the dielectric and piezoelectric responses of the compounds, have imaginary frequencies. Their dispersion is shown below the zero frequency line. The character of these modes also has significant implications for the properties of the system. The present calculation is not able to produce the unstable modes at zone boundary points along [qoo], [qqo] but the present calculation shows softening of acoustical mode along
ARTICLE IN PRESS M.M. Sinha, A. Sharma / Physica B 404 (2009) 1187–1189
X 600
M
Γ [q00]
[qq0]
1189
R
Γ [qqq]
Frequency (cm-1)
500
400
300
200
100
0
-100 0.5 0.4 0.3 0.2 0.1 0.0 0.0 0.1
0.2 0.3 0.4 Wavevector
0.5 0.0 0.1 0.2 0.3 0.4 0.5
Fig. 2. Phonon dispersion curves of SrZrO3.
[qqq] directions of BZ. The instability of phonon modes appearing along [qqq] is in agreement with DFT calculation [16]. The softening of phonons and instability of modes are the features which are supposed to be connected to phase transitions in these compounds. References [1] B.J. Kennedy, C.J. Howard, B.C. Chakoumakos, Phys. Rev. B 59 (1999) 4023. [2] T. Matsuda, S. Yamanaka, K. Kurosaki, S. Kobayashi, J. Alloys Compd. 351 (2003) 43. [3] D. Souptel, G. Behr, A.M. Balbashov, J. Cryst. Growth 236 (2002) 583. [4] R.V. Shende, D.S. Krueger, G.A. Rosetti, S.J. Lombardo, J. Am. Ceram. Soc. 84 (2001) 1648.
[5] L.J. Zhang, Y.L. Niu, T. Cui, Y. Li, Y. Wang, Y.M. Ma, Z. He, G.T. Zou, J. Phys. Condens. Matter 18 (2006) 9917. [6] Y. Li, L. Zhang, T. Cui, Y. Ma, G. Zou, Phys. Rev. B 74 (2006) 54102. [7] H.R. Xia, S.Q. Sun, X.F. Cheng, S.M. Dong, H.Y. Xu, L. Gao, D.L. Cui, J. Appl. Phys. 98 (2005) 3351. [8] K. Ga0 al-Nagy, D. Strauch, Phys. Rev. B 73 (2006) 014117. [9] J. de Launey, Solid State Phys. Ser. 2 (1956) 219. [10] R.S. Roth, J. Res. Natl. Bur. Stand. 58 (1957) 75. [11] H.E. Swanson, M.I. Cook, T. Isaacs, E.H. Evans, Natl. Bur. Stand. Circ. (US) 539 (1960) 51. [12] L. Carlsson, Acta Crystallogr. 23 (1967) 901; L. Carlsson, J. Mater. Sci. 5 (1970) 325. [13] D. de Ligny, P. Richet, Phys. Rev. B 53 (1996) 3013. [14] X. Liu, R.C. Liebermann, Phys. Chem. Miner. 20 (1993) 171. [15] M.M. Sinha, K. Wakamura, Solid State Ionics 136–137 (2000) 1345. [16] R. Vali, J. Phys. Chem. Solids 69 (2008) 876. [17] K. Wakamura, Solid State Ionics 145 (2001) 315.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Phonon properties of protonic conductor SrZrO3 in cubic phase M.M. Sinha a,, Anupamdeep Sharma b a b
Department of Physics, Sant Longowal Institute of Engineering and Technology, Longowal, Sangrur, Punjab 148106, India Sant Baba Bhag Singh Institute of Engineering and Technology, Padhiana, Jalandhar, Punjab, India
a r t i c l e in fo
abstract
Article history: Received 1 November 2008 Accepted 20 November 2008
Strontium zirconate has an orthorhombic structure at room temperature and undergoes a sequence of structural phase transition. 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, we are reporting the results of our theoretical investigation on the phonon properties of SrZrO3 in its cubic phase by using lattice dynamical simulation method based on de Launey angular force (DAF) constant model to understand the role of phonon in this system. The calculated zone center frequencies agree well with available results. The phonon dispersion curves of SrZrO3 in cubic phase are also drawn. The present calculation gives rise to softening of acoustical mode at zone boundary along [qqq] direction. The calculated results are compared and analyzed with other results. & 2008 Elsevier B.V. All rights reserved.
Keywords: Lattice dynamics Phonon dispersion Proton conductor Soft modes
1. Introduction There is an increasing interest in material research with potential technological applications, such as hydrogen sensors, fuel sensors and non-volatile random access memory devices. Among these materials, AZrO3 (A ¼ Ba, Sr, Pb) perovskites crystals have attracted considerable attention due to their ferroelectric and piezoelectric properties, yielding attractive models for experimental and theoretical academic research. In recent years, zirconate perovskites have been the focus of experimental studies. Notwithstanding, substantial theoretical research has not been devoted to these materials. In nuclear safety studies, SrZrO3 plays an important role as it is formed in the UO2 fuel by reaction between the fission products in the fuel matrix and during coreconcrete interactions between the fission products and the oxidized zircaloy cladding. The thermal properties and phase transitions are thus of special interest as they influence the behavior of hazardous fission products. The thermal properties and phase transitions are thus of special interest as they influence the behavior of hazardous fission products. High-temperature phase transitions of SrZrO3 were studied by using powder neutron diffraction [1] and X-ray diffraction [2]. This compound has a high melting temperature of about 2923 K [3], so it is cubic in a wide range of temperature where most of its useful applications take place. Shende et al. [4] suggested that this material could be used in high-voltage capacitor applications due to their high breakdown strengths as well as high dielectric constant. In order to fully take advantage of the properties of SrZrO3, a thorough Corresponding author. Tel.: +911672 280169; fax: +911672 280057.
E-mail address: [email protected] (M.M. Sinha). 0921-4526/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2008.11.095
investigation of its lattice dynamical properties is necessary. The calculation of vibrational properties of solids, surfaces and nanocrystals plays an important role in the structural characterization of matter. Besides the possibility to compare the vibrational frequencies with experimental data, phonon modes can be used also to determine the structural stability of a system [5–8]. This requires a reliable description of the vibrational properties of the system. In view of this, we shall employ short range force constant model [9] to calculate the zone center phonons and phonon dispersion curves of SrZrO3 in three symmetric directions.
2. Crystal structure At room temperature SrZrO3 has an orthorhombic phase as revealed by structural studies that date back to the 1950s and 1960s [10,11]. Later the existence of two additional phases at high temperature was proposed by Carlsson to be both tetragonal [12], however, more recent studies [1,2,13] on high temperatures have shown that SrZrO3 undergoes three structural phase transitions summarized as follows: First, orthorhombic (Pnma) to orthorhombic (Cmcm) at 970 K, then to tetragonal (I4/mcm) at 1100 K and finally to cubic (Pm3m) at 1400 K. A number of workers have used such changes to infer the presence of thermally induced phase transitions [14]. SrZrO3 has a rather high melting temperature of about 2920 K [3] and consequently it is cubic in a wide range of temperature where most of its applications take place. Therefore, in this work, we are studying vibrational properties of SrZrO3 in its cubic phase. In cubic perovskite-SrZrO3, the strontium atoms sit at the corners of a simple cubic unit cell, the oxygen atoms sit on the faces and the zirconium atom sits at
ARTICLE IN PRESS 1188
M.M. Sinha, A. Sharma / Physica B 404 (2009) 1187–1189
Table 1 Values of force constants (103 dyn cm1). Force constant
SrZrO3
a1 (Zr–O) a01 (Zr–O) a2 (Sr–O) a02 (Sr–O) a3 (Sr–Zr) a03 (Sr–Zr) a4 (O–O) a04 (O–O)
123.0 6.0 11.3 9.8 18.0 4.5 4.2 1.0
Fig. 1. Cubic phase of SrZrO3. Table 2 Calculated zone center phonon frequencies in cm1.
the center of the body as shown in Fig. 1. From the point of view of lattice dynamics, the unit cell contains five atoms that give rise to 15 phonons (three acoustics and 12 optic). The symmetry of these phonons at the G point (in terms of the Oh representation) is
ZC-phonons
Present work
Vali [16]
T1u(TO1) T1u(TO2) T1u(TO3) T2u
136 212 602 170
135 212 602 170
GðOh Þ ¼ 4T1u þ T2u where T1u and T2u represent the normal modes with triple degeneracy. One T1u mode is acoustic and the rest are optical modes. The T2u mode is inactive, while T1u modes are only IR active. The compound in cubic phase has no Raman mode.
3. Theoretical methodology In order to calculate the phonon properties of SrZrO3, a de Launey angular force (DAF) constant model [9] has been applied in the present investigation. The phonons in the SrZrO3 have been calculated to check whether softening of acoustic phonons are there or not in some crystal symmetry directions as it is supposed to be connected to phase changes in these compounds. In DAF model, the relative displacement of the reference atom and one of the neighbors is considered. The restoring force on the reference atom is taken to be proportional to the component of the relative displacement perpendicular to the line joining the two atoms at their equilibrium positions. The forces due to all neighbors are calculated separately and summed up together. Different force constants are used for the various categories of neighbors and the net force on the reference atom is obtained by summing over the contribution from all the neighbors. The present calculation involves four central force constants a1, a2, a3, a4 and four angular force constants a01 , a02 , a03 and a04 between Zr–O, Sr–O, Sr–Zr and O–O atoms, respectively, up to the third nearest neighbor. The calculated dynamical matrix of (15 15) is reduced to three matrices of the order (5 5) at zone center (ZC). The matrix elements are given in our earlier paper [15]. Recently Vali [16] has studied lattice dynamics of cubic phase of SrZrO3 within density functional theory. In the present calculation the interatomic force constants are obtained by fitting the calculated results of Vali [16] at the ZC for transverse infrared active phonon frequencies. The force constants thus calculated are listed in Table 1. Taking these force constants as input parameters, the dynamical matrix is solved at the ZC as well as along three symmetric directions [qoo], [qqo] and [qqq]. The ZC phonons thus obtained are listed in Table 2. The phonon dispersion curves thus obtained in three symmetric directions are shown in Fig. 2.
4. Results and discussion It is clear from Table 1 that the calculated force constant a1 between Zr–O is strongest among all other interatomic interactions and is followed by a3 (Sr–Zr) and is in agreement with finding of Wakamura [17]. This suggests that the covalent bonding between (Zr–O) is strongest than that between (Sr–Zr), (O–O) and (Sr–O). The larger force constant corresponds to stronger bond giving rise to larger value of frequency (TO3). One of the central force constants a2 between Sr–O is negative. The negative force constant corresponds to motion along modes that lead to energy lowering. Insofar as the frequencies are computed from the square roots of the force constants, this leads to an imaginary frequency. Frequency calculations are thus diagnostics as to nature of stationary points. All positive frequencies imply a (local) minimum, one imaginary frequency implies a transition state structure. The negative force constant also suggests electronic instability in SrZrO3. Due to large differences in mass, the phonon dispersion curves (PDC) of SrZrO3 (Fig. 2) decomposes into two well separated parts, a low-frequency region with predominantly Sr and Zr modes and the vibration of the light O atom around 602 cm1. It is obvious from Fig. 2 that phonon branches are distributed almost uniformly up to about 250 cm1 in all symmetric directions. The PDC shows that there are three regions in which phonon modes are distributed. The calculated eigenfrequencies and eigenvectors suggest that the top most region about 602 cm1 consisting of three phonon branches are due to oxygen atom vibrations, the middle three branches at about 250 cm1 are due to Sr atom vibrations and the lower nine branches are due to Sr and Zr atom vibrations. The main issue of the present calculation is to verify the occurrence of softening of acoustical modes at the boundary of Brillouin Zone (BZ) along major symmetry directions in SrZrO3 compound as obtained by Vali [16]. The unstable modes, which determine the nature of the phase transitions and the dielectric and piezoelectric responses of the compounds, have imaginary frequencies. Their dispersion is shown below the zero frequency line. The character of these modes also has significant implications for the properties of the system. The present calculation is not able to produce the unstable modes at zone boundary points along [qoo], [qqo] but the present calculation shows softening of acoustical mode along
ARTICLE IN PRESS M.M. Sinha, A. Sharma / Physica B 404 (2009) 1187–1189
X 600
M
Γ [q00]
[qq0]
1189
R
Γ [qqq]
Frequency (cm-1)
500
400
300
200
100
0
-100 0.5 0.4 0.3 0.2 0.1 0.0 0.0 0.1
0.2 0.3 0.4 Wavevector
0.5 0.0 0.1 0.2 0.3 0.4 0.5
Fig. 2. Phonon dispersion curves of SrZrO3.
[qqq] directions of BZ. The instability of phonon modes appearing along [qqq] is in agreement with DFT calculation [16]. The softening of phonons and instability of modes are the features which are supposed to be connected to phase transitions in these compounds. References [1] B.J. Kennedy, C.J. Howard, B.C. Chakoumakos, Phys. Rev. B 59 (1999) 4023. [2] T. Matsuda, S. Yamanaka, K. Kurosaki, S. Kobayashi, J. Alloys Compd. 351 (2003) 43. [3] D. Souptel, G. Behr, A.M. Balbashov, J. Cryst. Growth 236 (2002) 583. [4] R.V. Shende, D.S. Krueger, G.A. Rosetti, S.J. Lombardo, J. Am. Ceram. Soc. 84 (2001) 1648.
[5] L.J. Zhang, Y.L. Niu, T. Cui, Y. Li, Y. Wang, Y.M. Ma, Z. He, G.T. Zou, J. Phys. Condens. Matter 18 (2006) 9917. [6] Y. Li, L. Zhang, T. Cui, Y. Ma, G. Zou, Phys. Rev. B 74 (2006) 54102. [7] H.R. Xia, S.Q. Sun, X.F. Cheng, S.M. Dong, H.Y. Xu, L. Gao, D.L. Cui, J. Appl. Phys. 98 (2005) 3351. [8] K. Ga0 al-Nagy, D. Strauch, Phys. Rev. B 73 (2006) 014117. [9] J. de Launey, Solid State Phys. Ser. 2 (1956) 219. [10] R.S. Roth, J. Res. Natl. Bur. Stand. 58 (1957) 75. [11] H.E. Swanson, M.I. Cook, T. Isaacs, E.H. Evans, Natl. Bur. Stand. Circ. (US) 539 (1960) 51. [12] L. Carlsson, Acta Crystallogr. 23 (1967) 901; L. Carlsson, J. Mater. Sci. 5 (1970) 325. [13] D. de Ligny, P. Richet, Phys. Rev. B 53 (1996) 3013. [14] X. Liu, R.C. Liebermann, Phys. Chem. Miner. 20 (1993) 171. [15] M.M. Sinha, K. Wakamura, Solid State Ionics 136–137 (2000) 1345. [16] R. Vali, J. Phys. Chem. Solids 69 (2008) 876. [17] K. Wakamura, Solid State Ionics 145 (2001) 315.