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First-principles study of structural, vibrational and dielectric properties of double perovskites Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y)
Journal of Alloys and Compounds 547 (2013) 81–85
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First-principles study of structural, vibrational and dielectric properties of double perovskites Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) Karandeep, H.C. Gupta ⇑, S. Kumar Department of Physics, Indian Institute of Technology, Hauz Khas, New Delhi 110 016, India
a r t i c l e
i n f o
Article history: Received 30 July 2012 Received in revised form 23 August 2012 Accepted 23 August 2012 Available online 31 August 2012 Keywords: Rare earth alloys and compounds Crystal structure Phonons Dielectric response Computer simulations
a b s t r a c t The structural, vibrational and dielectric properties of Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) type double per m have been investigated for the first time using the pseudopotential ovskites in the space group Fm 3 plane wave method based on density functional theory (DFT) under local density approximation (LDA) and linear response formalism. The lattice constants, along with the Raman and the infrared wavenumbers at zone center are calculated and assigned. The calculated values of the lattice constants, Raman and the infrared wavenumbers are in very good agreement with respective experimental values. The bulk modulus, pressure derivative, Born effective charges and dielectric constants have also been calculated and discussed. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Double perovskites of type A2BB0 O6 have been studied for many years because of their scientific and technological importance. In 1995 Kurian et al. [1] have found that double perovskites of type Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) can be used as substrate for the high temperature superconducting materials. Since then a number of research groups have studied different properties of these compounds [2–11]. X-ray diffraction studies have established that Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) belongs to cubic m space group [2–5]. Recently, Konopka et al. [6] have studFm 3 ied the variation of dielectric constant of Ba2YSbO6 with temperature. Further Vijayakumar et al. [7,8] have measured dielectric constants of nanocrystalline Ba2GdSbO6 and Ba2DySbO6 oxides. Properties of these materials are affected quite strongly by the distribution of Ln and Sb cations over the octahedral sites. Raman and infrared spectroscopy is an attractive technique to investigate these effects. Thus, Vijayakumar et al. [10] studied the Raman and Infrared phonons of nanocrystalline Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y). Earlier, we had calculated the force constants in Ba2YSbO6 and Ba2SmSbO6 compounds using Wilson GF-matrix method [11]. To our best knowledge no theoretical studies of lattice constants, phonons and dielectric constants based on first principle m formalism has been done in these materials belonging to Fm 3 space group. First-principles calculations offer one of the most ⇑ Corresponding author. Tel.: +91 11 26591345; fax: +91 11 26862037. E-mail address: [email protected] (H.C. Gupta). 0925-8388/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jallcom.2012.08.106
powerful tools for theoretical studies of these properties. Hence in this paper an attempt has been made to investigate the structural, vibrational and dielectric properties of Ba2LnSbO6 (Ln = Sm, m space group by pseudopotential Gd, Dy and Y) belonging to Fm 3 plane wave method within the framework of density functional theory (DFT) under the local density approximation (LDA) and linear response formalism using quantum ESPRESSO code [12]. 2. Computational details All calculations are performed by using quantum ESPRESSO distribution, which is based on density functional theory with planewave pseudopotential method [12]. Ultrasoft pseudopotentials with local density approximation (LDA) as parameterized by Perdew and Zunger [13] are used in the calculations. Ultrasoft pseudopotentials for yttrium(Y) and oxygen (O) are taken from the Vanderbilt group site [14], whereas ultrasoft pseudopotentials for barium (Ba), antimony (Sb) and lanthanides (Ln = Sm, Gd, Dy) are generated by considering 6s 6p, 5s 5p 5d and 5s 6s 5p 6p 5d, as valance states respectively. These pseudopotentials include nonlinear core correction to account for large overlap between core and valance states. Kinetic energy cutoffs for wavefunctions and charge density are 80 Ry, 800 Ry respectively and a 4 4 4 Monkhorst–Pack k-point mesh [15] is used for Brillouin zone samplings of electronic states. The structures are relaxed by taking convergence threshold for forces and pressure to be 0.0001 eV/Å and 0.5 kbar respectively. Born effective charge tensors, phonon frequencies and dielectric permittivity tensors are obtained as second-order linear-response
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the increase in B cation ionic radii for both the experimental and calculated values is observed, this variation is shown in the Fig. 2. All calculated values of lattice constants are smaller than the observed values, which is consistent with the fact that LDA overbinds the crystal and underestimates the lattice constant. Calculated and observed bond distances are also in excellent agreement with each other. The similar values of lattice constants and bonding distances in Ba2YSbO6 and Ba2DySbO6 is due to similar ionic radii of Y+3 (0.900) and Dy+3 (0.912) ions. Bulk modulus and its pressure derivative are obtained by fitting a third-order BirchMurnaghan equation of state defined as [18]: 8" 9 #3 " 2 #2 " 2 23 #= 9V 0 B0 < V 0 3 V0 3 V0 0 EðVÞ ¼ E0 þ 1 B0 þ 1 64 ; 16 : V V V
derivatives of the total energy with respect to external electric or atomic displacements within the framework of density functional perturbation theory (DFPT) [16,17].
3. Results and discussion 3.1. Crystal structure The Ideal double perovskites A2BB0 O6 have cubic structure with m space group with one formula unit in primitive unit cell. In Fm 3 this structure B, B0 cations occupy octahedral sites 4a(0,0,0) and 4b(0.5,0,0) respectively and A cations, oxygens occupy the 8c(0.25,0.25,0.25) and 24e(x,0,0) Wyckoff sites respectively. Struc m space group is ture of Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) in Fm 3 shown in Fig. 1. The calculated and observed lattice constants, oxygen 24e(x) parameter and bond distances along with calculated bulk modulus and its first derivative are given in Table 1. It is clear from this table that lattice parameters of Ba2YSbO6, Ba2DySbO6, Ba2GdSbO6 and Ba2SmSbO6 differ from observed value by 0.32%, 0.24%, 0.33% and 0.19% respectively, whereas difference in internal parameter x for oxygen atoms in Ba2YSbO6, Ba2DySbO6 and Ba2SmSbO6 is 0.0019, 0.0027, and 0.0067 respectively. Clearly the calculated values of lattice constants and oxygen internal parameters are in excellent agreement with the observed values. An increase in lattice constant of double perovskites with
ð1Þ where V 0 is the equilibrium volume and B0 is bulk modulus evaluated at V 0 , B00 is the pressure derivative of bulk modulus also evaluated at V 0 . The fitted energy versus volume curves for Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) are shown in Fig. 3 and the calculated values of bulk modulus and pressure derivative are given in Table 2. No experimental or theoretical results are available to compare the calculated bulk modulus and its pressure derivatives. However, Lufaso et al. [19] have found bulk modulus of Ba2YTaO6 to be m structure and the present calculated values 157(16) GPa in Fm 3 of bulk modulus lie in this range. The values of bulk modulus of Ba2YSbO6 and Ba2DySbO6 are almost same as these compounds have almost same volume and the decrease in values of bulk modulus in going from Ln = Dy to Gd to Sm compounds is due to the increase in volumes of respective compounds. 3.2. Vibrational modes The primitive unit cell of Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) contains one formula unit, thus there are thirty modes of vibration m structure. In this structure, Ba atoms occupy 8c sites of in Fm 3 Td symmetry, Ln and Sb atoms occupy 4a and 4b sites of Oh symmetry and oxygen atoms are in 24e sites of C4m symmetry. The total m space group is number of zone center modes for Fm 3
Ccryst vib ¼ CBa þ CLn þ Csb þ CO ¼ ðF1u þ F2g Þ þ F1u þ F1u þ ðA1g þ Eg þ F1g þ F2g þ F2u þ 2F1u Þ ¼ A1g þ Eg þ F1g þ 2F2g þ F2u þ 5F1u
ð2Þ
Further, out of these normal modes A1g, Eg and 2F2g are Raman active and 4F1u modes are infrared active. Also, one F1u is acoustical and F1g, F2u modes are optically inactive. Vijayakumar et al. [10] observed four Raman active modes (A1g, Eg and 2F2g) as strong or medium intense bands at 760, 572, 375
Fig. 1. Conventional unit cell of Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) in Fmbar3m (O5h ) space group (Z = 4, ZB = 1). Blue, gray, green and red balls represent Ba, Ln, Sb and O atoms respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Table 1 Calculated and observed [2–5] structural parameters along with calculated bulk modulus, B and pressure derivative of bulk modulus, B0 of Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) in Fmbar3m (O5h ) space group (Z = 4, ZB = 1). Ba2YSbO6
a(Å) O(x) Sb–O(Å) Ln–O(Å) Ba–O(Å) B (in GPa)a B0 a
Ba2DySbO6
Ba2GdSbO6
Ba2SmSbO6
Obs.
Cal.
Obs.
Cal.
Obs.
Cal.
Obs.
Cal.
8.4240 0.2636 1.9914 2.2206 2.9805 – –
8.3966 0.2617 2.0012 2.1971 2.9703 161.0 4.80
8.4247 0.2646 1.9832 2.2292 2.9811 – –
8.4049 0.2619 2.0012 2.2012 2.9733 161.8 4.82
8.474 – – – – – –
8.4461 0.2632 1.9998 2.2232 2.9882 159.2 4.86
8.50908 0.258 2.0591 2.1953 3.0091 – –
8.49312 0.2647 1.9985 2.2481 3.0054 156.1 4.83
structure). Lufaso et al. [19] have observed bulk modulus of Ba2YTaO6 to be 157(16) GPa (a similar compound with Fm3m
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Table 2 Calculated and observed [10] wavenumbers (in cm1) of zone center phonons in Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) in Fmbar3m (O5h ) space group (Z = 4, ZB = 1). Species
A1g Eg F2g(1) F2g(2) F1u(1) F1u(2)a F1u(3) F1u(4) F2u F1g
Ba2YSbO6
Ba2DySbO6
Ba2GdSbO6
Ba2SmSbO6
Obs.
Cal.
Obs.
Cal.
Obs.
Cal.
Obs.
Cal.
760 572 380 108 634 – – – – –
759.9 585.5 388.8 111.8 627.6 334.2 246.5 133.0 247.2 98.4
761 574 375 108 629 – – – – –
762.3 595.9 388.0 111.1 627.6 331.0 207.0 125.2 246.8 98.0
762 575 375 105 627 – – – – –
763.4 600.4 383.7 107.6 630.2 329.5 206.2 121.7 239.9 79.8
759 566 372 102 623 – – – – –
764.0 604.6 378.7 103.6 632.7 327.9 205.5 117.2 231.8 52.5
a Lavat and Baran [20] observed a peak at 357, 359 cm1 for Ba2InSbO6 and m structure). Ba2ScSbO6 respectively (similar compounds with Fm 3
Fig. 2. Variation of Lattice constant with B cation ionic radii of Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) in Fmbar3m (O5h ) space group (Z = 4, ZB = 1).
and 108 cm1 respectively and one strong infrared absorption mode around 630 cm1 along with several weak bands. The presence of weak bands can be ascribed to the impurities, local symmetry breakdown, incomplete ordering and lowering of site symmetry of octahedral site cations [20]. The calculated and observed Raman and infrared wavenumbers given in Table 2 are in excellent agreement with each other. It is important to mention here that, in our earlier study of Ba2YSbO6 and Ba2SmSbO6 compounds a normal coordinate analysis was performed to calculate the inter atomic force constants by taking the Raman and infrared frequencies as input parameters [11]. The input infrared frequencies taken in the fitting process were the weak infrared modes which appear in the Raman scattering due to lowering of site symmetry of octahedral site cations as given by Vijayakumar et al. [10] rather than the single absorption mode that appears in the FT-IR spectrum. The present calculations clearly do not justify our earlier approach. The displacement pattern of all atoms in Raman active modes is shown in Fig. 4. It is clear from Eq. (2) and Fig. 4 that only oxygen atoms move in the A1g and Eg modes while all cations are at rest. Thus wavenumbers corresponding to these modes are determined by the Sb–O and Ln–O bonding forces and distances. An increase in the wavenumbers of A1g and Eg modes in going from Ln = Dy to Gd to Sm is due to the decrease in Sb–O bonding distances. Also from
Eq. (2), we conclude that in F2g modes both barium and oxygen atoms move, but from Fig. 4 it is clear that displacement of oxygen atoms dominate in F2g(1) mode and displacement of barium atoms dominate in F2g(2) mode. The wavenumbers corresponding to the F2g(1), F2g(2) modes are influenced by the O–Sb–O and O–Ln–O bending forces and Ba–O stretching forces respectively. The decrease in F2g(1), F2g(2) wavenumbers from Ba2YSbO6 to Ba2SmSbO6 is due to the increase in Ln–O and Ba–O bonding distance from Ba2YSbO6 to Ba2SmSbO6, respectively. The situation is more complex in the infrared modes, since all atoms move in the F1u modes. Vijayakumar et al. [10] observed only one infrared mode (F1u(1)) at 630 cm1 whereas Lavat and Baran [21] studied infrared spectra of Ba2BSbO6 (B = In, Sc) and found two bands near at 659–615and 357–359 cm1. It is obvious from Table 2 that the calculated F1u(1), F1u(2) modes lie in this range. A small decrease in wavenumbers of F1u(2), F1u(3) and F1u(4) modes as the Ln cation changes from Y to Sm indicates that these modes can be attributed to the increase in Ln–O and Ba–O bonding distances. A noticeable decrease in wavenumbers F1u(3) from Ba2YSbO6 and Ba2LnSbO6 (Ln = Sm, Gd, Dy) indicates the influence of higher masses of lanthanides than yttrium. Thus these modes are determined mainly by the Ln–O bonds. It is also important to mention here that the calculated trends of the changes in going from Ln = Y to Dy to Gd to Sm for the A1g, Eg and F1u(1) modes in case of Ba2SmSbO6 are different from that of the experimental ones, this discrepancy in the trend is possibly due to the existence space of Ba2SmSbO6 at phase transition from rhombohedral R3 m space group as mentioned by Vijayakugroup to the cubic Fm 3 mar et al. [10].
Fig. 3. Fitted energy versus volume curves along with data points for Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) in Fmbar3m (O5h ) space group (Z = 4, ZB = 1).
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Fig. 4. Displacement pattern of all atoms in Raman active mode of Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) in Fmbar3m (O5h ) space group (Z = 4, ZB = 1). Here only primitive cell is shown. Blue, gray, green and red balls represent Ba, Ln, Sb and O atoms respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
3.3. Dielectric properties The dispersion theory [22,23] shows that static dielectric permittivity tensor of a crystal within harmonic approximation is given by $ s
¼
$
$
1 þ ionic
ð3Þ
$ 1
where is the electronic dielectric tensor, tric tensor given by
ionic ab ¼
X
l ¼
l
X Z la Zla 2 l v 0 m0 xl
$ ionic
is the ionic dielec-
ð4Þ
where l labels the zone center (q = 0) infrared-active modes of vibrations with frequencies xl l the oscillator strength of mode l, Zla their effective charges in Cartesian direction a, and V the volume per unit cell. Mode effective charges in Cartesian direction a for a given mode l is defined as
ðZ l Þa ¼
X Z iac ðm0 =mi Þ1=2 ðal Þic
ð5Þ
iac
where Z i is Born effective charge tensor for ion i, mi its mass, ðal Þic } the component of normalized dynamical matrix eigenvector for mode l involving ion i in the c direction, and m0 an arbitrary mass, which is cancelled by the denominator of Eq. (2). We choose m0 = 1 amu in this work. Finally the Born effective charge tensor for a given ion i is defined as:
@Pa Z iac ¼ V @uic
tive charge tensor of oxygen (O) is anisotropic due to different environment around oxygen. The Born effective charge of oxygen (O), along Ln–O–Sb bond is higher in magnitude than the nominal ionic charge of 2, whereas Born effective charge transverse to Ln– O–Sb bond is smaller than the nominal charge 2. The anomalies are larger in Sm, Dy, Gd, Y and O ions than the Sb and Ba ions. Finally the Born effective charges of Ba, Sb and O ions are almost independent of the B-site cations (Y, Dy, Gd, Sm) with variations 0.32%, 0.24%, and 0.33% for Ba, Sb and O ions respectively. The contribution of the individual modes to the static dielectric are given in Table 4. It is constant l and mode effective charges Z l clear from Table 4 that out the four F1u modes, F1u(3) has highest mode effective charge and maximum contribution towards the static dielectric constant. High frequency modes, F1u(1) and F1u(2) despite having higher mode effective charges contribute less than the low frequency mode F1u(4). The calculated electronic dielectric tensors along with the static dielectric tensor are also shown in Table 4. While the dielectric constants of Ba2YSbO6 and Ba2DySbO6 are almost same, an increase in the dielectric constant from Dy to Gd to Sm is mainly due to the F1u(4) mode. Konopka et al. [6] have measured the dielectric constants in Ba2YSbO6 and Vijayakumar et al. [7,8] have measured dielectric constants in nanocrystalline Ba2DySbO6 and Ba2GdSbO6. The difference between the calculated and observed dielectric constants of Ba2YSbO6, Ba2DySbO6 and Ba2GdSbO6 are 4.38%, 30.42% and 47.44% respectively. Interestingly the experimental values of dielectric constants are found to be higher than corresponding DFT calculated values and difference between experimental and theoretical values is very
ð6Þ
where P is the polarization of the system and uic is displacement of ion i in direction c. The calculated Born effective charges for Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) are given in Table 3. The Born effective charge of barium (Ba), lanthanides (Sm, Gd, Dy and Y) are larger than the nominal ionic charges +2 and +3. The antimony (Sb) Born effective charge is smaller than the nominal ionic charge +5. The Born effec-
Table 3 m (O5 ) Calculated Born effective charges of Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) in Fm 3 h space group (Z = 4, ZB = 1). Materials
Z Ba
Z Ln=c
Z Sb
Z 0
Ba2YSbO6 Ba2DySbO6 Ba2GdSbO6 Ba2SmSbO6
2.36923 2.37191 2.37628 2.38165
4.09210 4.10519 4.12995 4.15228
4.78243 4.77787 4.81405 4.85110
[3.39170, [3.39844, [3.42230, [3.44634,
1.70602, 1.70613, 1.71156, 1.71690,
1.70602] 1.70613] 1.71156] 1.71690]
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Table 4 Calculated mode effective charges Z l and contribution of different infrared phonon modes to static dielectric constant for Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y). Calculated values of l and s are also given. Experimental values ofl at room temperature are given in parentheses [6–8]. Modes
Ba2YSbO6 Z l
F1u(1) F1u(2) F1u(3) F1u(4)
1 s
4.4848 4.2878 5.0739 3.0172 3.6663 12.9090(13.5)
Ba2DySbO6
u 1.0012 1.8269 3.7246 2.6900
Z l 4.6822 4.5754 4.6822 2.8641 3.6709 13.0349(17)
Ba2GdSbO6
u
Z l
0.9709 2.3170 4.1201 1.9560
large in nanocrystalline Ba2DySbO6 and Ba2GdSbO6. The probable reasons are: (1) We have calculated the dielectric constants at absolute zero temperature whereas the experimental values are reported at room temperature. Although Konopka et al. [6] have studied the variation of static dielectric constant with temperature in Ba2YSbO6 oxide only. A steep increase in static dielectric constant towards zero Kelvin indicates a new phase in Ba2YSbO6 oxide. (2) As we have discussed earlier that in these materials due to the lowering of site symmetries Raman modes become infrared active and appear as weak bands in infrared absorption spectra [10,19]. Higher values of the experimental static dielectric constants may be due to the contributions of these weak bands towards static dielectric tensors. Indeed these modes are also found by Vijayakumar et al. [7,8] during their study of the dielectric properties of nanocrystalline Ba2DySbO6 and Ba2GdSbO6. Small difference in calculated and experimental values of dielectric constants in Ba2YSbO6 compared to Ba2DySbO6 and Ba2GdSbO6 indicates the more site symmetry breaking in later compounds. 4. Conclusions The structural and vibrational properties of double perovskites Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) have been studied for the first time by using first-principles plane wave pseudopotential method within the linear density approximation (LDA). The following conclusions are obtained: (1) The calculated lattice parameters and the internal parameter x for oxygen atoms in these materials are in excellent agreement with the experimental values. (2) The calculated values of bulk modulus and pressure derivative of bulk modulus for Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) lie in the range of 156–162 GPa and 4.78–4.87 respectively. (3) The calculated vibrational modes are in excellent agreement with the experimental values. (4) The Born effective charge of barium (Ba), lanthanides (Sm, Gd, Dy and Y) are larger than the nominal ionic charges +2 and +3. The antimony (Sb) Born effective charge is smaller than the nominal ionic charge +5 and Born effective charge of oxygen (O) is anisotrophic. Born effective charges of Ba, Sb and O ions are almost independent of the B-site cations (Y, Dy, Gd, Sm).
4.4889 4.5304 4.7087 3.0636 3.6896 13.5650(20)
Ba2SmSbO6
u 0.9788 2.1978 4.1686 2.5302
Z l 4.5632 4.4751 4.7341 3.2638 3.7100 14.2795
u 0.9859 2.0632 4.1923 3.3281
(5) Out the four F1u modes, F1u(3) has highest mode effective charge and maximum contribution towards the static dielectric constant. (6) Due to lowering of site symmetries Raman modes become infrared active and appear as weak bands in infrared absorption spectra. Higher values of the experimental static dielectric constants may be due to the contributions of these weak bands towards static dielectric tensors in Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y).
Acknowledgements Two of the authors, Karandeep and S. Kumar gratefully acknowledge the financial assistance from Council of Scientific and Industrial Research (CSIR)-India. References [1] J. Kurian, J. Koshy, P. Wariar, Y. Yadava, A. Damodaran, J. Solid State Chem. 116 (1995) 193–198. [2] P.G. Casado, A. Mendiola, I. Rasines, Z. Anorg. Allg. Chem. 510 (1984) 104–198. [3] H. Karunadasa, Q. Huang, B. Ueland, P. Schier, R. Cava, PNAS 100 (2003) 8097– 8102. [4] W. Fu, D. IJdo, J. Solid State Chem. 178 (2005) 2363–2367. [5] P.J. Saines, B.J. Kennedy, M.M. Elcombe, J. Solid State Chem. 180 (2007) 401– 409. [6] J. Konopka, R. Jose, M. Wolcyrz, Physica 435 (2006) 53–58. [7] C. Vijayakumar, H.P. Kumar, S. Solomon, J. Thomas, P. Wariar, A. John, J. Kohsy, Bull. Mater. Sci. 31 (2008) 719–722. [8] C. Vijayakumar, H.P. Kumar, S. Solomon, J. Thomas, P. Wariar, A. John, J. Kohsy, J. Alloys Comp. 475 (2009) 778–781. [9] F. Fernandez-Martinez, J.L. Montero, I. Carrillo, C. Colon, J. Alloys Comp. 538 (2012) 34–39. [10] C. Vijayakumar, H.P. Kumar, S. Solomon, J. Thomas, P. Wariar, A. John, J. Alloys Comp. 480 (2009) 167–170. [11] H.C. Gupta, J. Singh, S. Kumar, N. Rani, Physica 405 (2010) 410–412. [12] P. Giannozzi, J. Phys. 21 (2009) 395502. [13] J.P. Perdew, A. Zunger, Phys. Rev. 23 (1981) 5048–5079. [14] D. Vanderbilt, Phys. Rev. 41 (1990) 7892–7895. URL: http:// www.physics.rutgers.edu/~dhv/uspp. [15] H.J. Monkhorst, J.D. Pack, Phys. Rev. 13 (1976) 5188–5192. [16] S. Baroni, S. de Gironcoli, A.D. Corso, P. Giannozzi, Rev. Mod. Phys. 73 (2001) 515–562. [17] P. Giannozzi, S. de Gironcoli, P. Pavone, S. Baroni, Phys. Rev. 43 (1991) 7231– 7242. [18] F. Birch, Phys. Rev. 71 (1947) 809–824. [19] M.W. Lufaso, R.B. Macquart, Y. Lee, T. Vogt, H.-C. zur Loye, J. Phys. 18 (2006) 8761–8780. [20] N. Ramadass, J. Gopalakrishnan, M. Sastri, J. Inorg. Nucl. Chem. 40 (1978) 1453–1454. [21] A.E. Lavat, E.J. Baran, Vib. Spectrosc. 32 (2003) 167–174. [22] M. Born, K. Huang, Dynamical Theory of Crystal Lattices, First ed., Oxford, 1954. [23] E. Cockayne, B.P. Burton, Phys. Rev. 62 (2000) 3735–3743.
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Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jalcom
First-principles study of structural, vibrational and dielectric properties of double perovskites Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) Karandeep, H.C. Gupta ⇑, S. Kumar Department of Physics, Indian Institute of Technology, Hauz Khas, New Delhi 110 016, India
a r t i c l e
i n f o
Article history: Received 30 July 2012 Received in revised form 23 August 2012 Accepted 23 August 2012 Available online 31 August 2012 Keywords: Rare earth alloys and compounds Crystal structure Phonons Dielectric response Computer simulations
a b s t r a c t The structural, vibrational and dielectric properties of Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) type double per m have been investigated for the first time using the pseudopotential ovskites in the space group Fm 3 plane wave method based on density functional theory (DFT) under local density approximation (LDA) and linear response formalism. The lattice constants, along with the Raman and the infrared wavenumbers at zone center are calculated and assigned. The calculated values of the lattice constants, Raman and the infrared wavenumbers are in very good agreement with respective experimental values. The bulk modulus, pressure derivative, Born effective charges and dielectric constants have also been calculated and discussed. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Double perovskites of type A2BB0 O6 have been studied for many years because of their scientific and technological importance. In 1995 Kurian et al. [1] have found that double perovskites of type Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) can be used as substrate for the high temperature superconducting materials. Since then a number of research groups have studied different properties of these compounds [2–11]. X-ray diffraction studies have established that Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) belongs to cubic m space group [2–5]. Recently, Konopka et al. [6] have studFm 3 ied the variation of dielectric constant of Ba2YSbO6 with temperature. Further Vijayakumar et al. [7,8] have measured dielectric constants of nanocrystalline Ba2GdSbO6 and Ba2DySbO6 oxides. Properties of these materials are affected quite strongly by the distribution of Ln and Sb cations over the octahedral sites. Raman and infrared spectroscopy is an attractive technique to investigate these effects. Thus, Vijayakumar et al. [10] studied the Raman and Infrared phonons of nanocrystalline Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y). Earlier, we had calculated the force constants in Ba2YSbO6 and Ba2SmSbO6 compounds using Wilson GF-matrix method [11]. To our best knowledge no theoretical studies of lattice constants, phonons and dielectric constants based on first principle m formalism has been done in these materials belonging to Fm 3 space group. First-principles calculations offer one of the most ⇑ Corresponding author. Tel.: +91 11 26591345; fax: +91 11 26862037. E-mail address: [email protected] (H.C. Gupta). 0925-8388/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jallcom.2012.08.106
powerful tools for theoretical studies of these properties. Hence in this paper an attempt has been made to investigate the structural, vibrational and dielectric properties of Ba2LnSbO6 (Ln = Sm, m space group by pseudopotential Gd, Dy and Y) belonging to Fm 3 plane wave method within the framework of density functional theory (DFT) under the local density approximation (LDA) and linear response formalism using quantum ESPRESSO code [12]. 2. Computational details All calculations are performed by using quantum ESPRESSO distribution, which is based on density functional theory with planewave pseudopotential method [12]. Ultrasoft pseudopotentials with local density approximation (LDA) as parameterized by Perdew and Zunger [13] are used in the calculations. Ultrasoft pseudopotentials for yttrium(Y) and oxygen (O) are taken from the Vanderbilt group site [14], whereas ultrasoft pseudopotentials for barium (Ba), antimony (Sb) and lanthanides (Ln = Sm, Gd, Dy) are generated by considering 6s 6p, 5s 5p 5d and 5s 6s 5p 6p 5d, as valance states respectively. These pseudopotentials include nonlinear core correction to account for large overlap between core and valance states. Kinetic energy cutoffs for wavefunctions and charge density are 80 Ry, 800 Ry respectively and a 4 4 4 Monkhorst–Pack k-point mesh [15] is used for Brillouin zone samplings of electronic states. The structures are relaxed by taking convergence threshold for forces and pressure to be 0.0001 eV/Å and 0.5 kbar respectively. Born effective charge tensors, phonon frequencies and dielectric permittivity tensors are obtained as second-order linear-response
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the increase in B cation ionic radii for both the experimental and calculated values is observed, this variation is shown in the Fig. 2. All calculated values of lattice constants are smaller than the observed values, which is consistent with the fact that LDA overbinds the crystal and underestimates the lattice constant. Calculated and observed bond distances are also in excellent agreement with each other. The similar values of lattice constants and bonding distances in Ba2YSbO6 and Ba2DySbO6 is due to similar ionic radii of Y+3 (0.900) and Dy+3 (0.912) ions. Bulk modulus and its pressure derivative are obtained by fitting a third-order BirchMurnaghan equation of state defined as [18]: 8" 9 #3 " 2 #2 " 2 23 #= 9V 0 B0 < V 0 3 V0 3 V0 0 EðVÞ ¼ E0 þ 1 B0 þ 1 64 ; 16 : V V V
derivatives of the total energy with respect to external electric or atomic displacements within the framework of density functional perturbation theory (DFPT) [16,17].
3. Results and discussion 3.1. Crystal structure The Ideal double perovskites A2BB0 O6 have cubic structure with m space group with one formula unit in primitive unit cell. In Fm 3 this structure B, B0 cations occupy octahedral sites 4a(0,0,0) and 4b(0.5,0,0) respectively and A cations, oxygens occupy the 8c(0.25,0.25,0.25) and 24e(x,0,0) Wyckoff sites respectively. Struc m space group is ture of Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) in Fm 3 shown in Fig. 1. The calculated and observed lattice constants, oxygen 24e(x) parameter and bond distances along with calculated bulk modulus and its first derivative are given in Table 1. It is clear from this table that lattice parameters of Ba2YSbO6, Ba2DySbO6, Ba2GdSbO6 and Ba2SmSbO6 differ from observed value by 0.32%, 0.24%, 0.33% and 0.19% respectively, whereas difference in internal parameter x for oxygen atoms in Ba2YSbO6, Ba2DySbO6 and Ba2SmSbO6 is 0.0019, 0.0027, and 0.0067 respectively. Clearly the calculated values of lattice constants and oxygen internal parameters are in excellent agreement with the observed values. An increase in lattice constant of double perovskites with
ð1Þ where V 0 is the equilibrium volume and B0 is bulk modulus evaluated at V 0 , B00 is the pressure derivative of bulk modulus also evaluated at V 0 . The fitted energy versus volume curves for Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) are shown in Fig. 3 and the calculated values of bulk modulus and pressure derivative are given in Table 2. No experimental or theoretical results are available to compare the calculated bulk modulus and its pressure derivatives. However, Lufaso et al. [19] have found bulk modulus of Ba2YTaO6 to be m structure and the present calculated values 157(16) GPa in Fm 3 of bulk modulus lie in this range. The values of bulk modulus of Ba2YSbO6 and Ba2DySbO6 are almost same as these compounds have almost same volume and the decrease in values of bulk modulus in going from Ln = Dy to Gd to Sm compounds is due to the increase in volumes of respective compounds. 3.2. Vibrational modes The primitive unit cell of Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) contains one formula unit, thus there are thirty modes of vibration m structure. In this structure, Ba atoms occupy 8c sites of in Fm 3 Td symmetry, Ln and Sb atoms occupy 4a and 4b sites of Oh symmetry and oxygen atoms are in 24e sites of C4m symmetry. The total m space group is number of zone center modes for Fm 3
Ccryst vib ¼ CBa þ CLn þ Csb þ CO ¼ ðF1u þ F2g Þ þ F1u þ F1u þ ðA1g þ Eg þ F1g þ F2g þ F2u þ 2F1u Þ ¼ A1g þ Eg þ F1g þ 2F2g þ F2u þ 5F1u
ð2Þ
Further, out of these normal modes A1g, Eg and 2F2g are Raman active and 4F1u modes are infrared active. Also, one F1u is acoustical and F1g, F2u modes are optically inactive. Vijayakumar et al. [10] observed four Raman active modes (A1g, Eg and 2F2g) as strong or medium intense bands at 760, 572, 375
Fig. 1. Conventional unit cell of Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) in Fmbar3m (O5h ) space group (Z = 4, ZB = 1). Blue, gray, green and red balls represent Ba, Ln, Sb and O atoms respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Table 1 Calculated and observed [2–5] structural parameters along with calculated bulk modulus, B and pressure derivative of bulk modulus, B0 of Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) in Fmbar3m (O5h ) space group (Z = 4, ZB = 1). Ba2YSbO6
a(Å) O(x) Sb–O(Å) Ln–O(Å) Ba–O(Å) B (in GPa)a B0 a
Ba2DySbO6
Ba2GdSbO6
Ba2SmSbO6
Obs.
Cal.
Obs.
Cal.
Obs.
Cal.
Obs.
Cal.
8.4240 0.2636 1.9914 2.2206 2.9805 – –
8.3966 0.2617 2.0012 2.1971 2.9703 161.0 4.80
8.4247 0.2646 1.9832 2.2292 2.9811 – –
8.4049 0.2619 2.0012 2.2012 2.9733 161.8 4.82
8.474 – – – – – –
8.4461 0.2632 1.9998 2.2232 2.9882 159.2 4.86
8.50908 0.258 2.0591 2.1953 3.0091 – –
8.49312 0.2647 1.9985 2.2481 3.0054 156.1 4.83
structure). Lufaso et al. [19] have observed bulk modulus of Ba2YTaO6 to be 157(16) GPa (a similar compound with Fm3m
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Table 2 Calculated and observed [10] wavenumbers (in cm1) of zone center phonons in Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) in Fmbar3m (O5h ) space group (Z = 4, ZB = 1). Species
A1g Eg F2g(1) F2g(2) F1u(1) F1u(2)a F1u(3) F1u(4) F2u F1g
Ba2YSbO6
Ba2DySbO6
Ba2GdSbO6
Ba2SmSbO6
Obs.
Cal.
Obs.
Cal.
Obs.
Cal.
Obs.
Cal.
760 572 380 108 634 – – – – –
759.9 585.5 388.8 111.8 627.6 334.2 246.5 133.0 247.2 98.4
761 574 375 108 629 – – – – –
762.3 595.9 388.0 111.1 627.6 331.0 207.0 125.2 246.8 98.0
762 575 375 105 627 – – – – –
763.4 600.4 383.7 107.6 630.2 329.5 206.2 121.7 239.9 79.8
759 566 372 102 623 – – – – –
764.0 604.6 378.7 103.6 632.7 327.9 205.5 117.2 231.8 52.5
a Lavat and Baran [20] observed a peak at 357, 359 cm1 for Ba2InSbO6 and m structure). Ba2ScSbO6 respectively (similar compounds with Fm 3
Fig. 2. Variation of Lattice constant with B cation ionic radii of Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) in Fmbar3m (O5h ) space group (Z = 4, ZB = 1).
and 108 cm1 respectively and one strong infrared absorption mode around 630 cm1 along with several weak bands. The presence of weak bands can be ascribed to the impurities, local symmetry breakdown, incomplete ordering and lowering of site symmetry of octahedral site cations [20]. The calculated and observed Raman and infrared wavenumbers given in Table 2 are in excellent agreement with each other. It is important to mention here that, in our earlier study of Ba2YSbO6 and Ba2SmSbO6 compounds a normal coordinate analysis was performed to calculate the inter atomic force constants by taking the Raman and infrared frequencies as input parameters [11]. The input infrared frequencies taken in the fitting process were the weak infrared modes which appear in the Raman scattering due to lowering of site symmetry of octahedral site cations as given by Vijayakumar et al. [10] rather than the single absorption mode that appears in the FT-IR spectrum. The present calculations clearly do not justify our earlier approach. The displacement pattern of all atoms in Raman active modes is shown in Fig. 4. It is clear from Eq. (2) and Fig. 4 that only oxygen atoms move in the A1g and Eg modes while all cations are at rest. Thus wavenumbers corresponding to these modes are determined by the Sb–O and Ln–O bonding forces and distances. An increase in the wavenumbers of A1g and Eg modes in going from Ln = Dy to Gd to Sm is due to the decrease in Sb–O bonding distances. Also from
Eq. (2), we conclude that in F2g modes both barium and oxygen atoms move, but from Fig. 4 it is clear that displacement of oxygen atoms dominate in F2g(1) mode and displacement of barium atoms dominate in F2g(2) mode. The wavenumbers corresponding to the F2g(1), F2g(2) modes are influenced by the O–Sb–O and O–Ln–O bending forces and Ba–O stretching forces respectively. The decrease in F2g(1), F2g(2) wavenumbers from Ba2YSbO6 to Ba2SmSbO6 is due to the increase in Ln–O and Ba–O bonding distance from Ba2YSbO6 to Ba2SmSbO6, respectively. The situation is more complex in the infrared modes, since all atoms move in the F1u modes. Vijayakumar et al. [10] observed only one infrared mode (F1u(1)) at 630 cm1 whereas Lavat and Baran [21] studied infrared spectra of Ba2BSbO6 (B = In, Sc) and found two bands near at 659–615and 357–359 cm1. It is obvious from Table 2 that the calculated F1u(1), F1u(2) modes lie in this range. A small decrease in wavenumbers of F1u(2), F1u(3) and F1u(4) modes as the Ln cation changes from Y to Sm indicates that these modes can be attributed to the increase in Ln–O and Ba–O bonding distances. A noticeable decrease in wavenumbers F1u(3) from Ba2YSbO6 and Ba2LnSbO6 (Ln = Sm, Gd, Dy) indicates the influence of higher masses of lanthanides than yttrium. Thus these modes are determined mainly by the Ln–O bonds. It is also important to mention here that the calculated trends of the changes in going from Ln = Y to Dy to Gd to Sm for the A1g, Eg and F1u(1) modes in case of Ba2SmSbO6 are different from that of the experimental ones, this discrepancy in the trend is possibly due to the existence space of Ba2SmSbO6 at phase transition from rhombohedral R3 m space group as mentioned by Vijayakugroup to the cubic Fm 3 mar et al. [10].
Fig. 3. Fitted energy versus volume curves along with data points for Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) in Fmbar3m (O5h ) space group (Z = 4, ZB = 1).
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Fig. 4. Displacement pattern of all atoms in Raman active mode of Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) in Fmbar3m (O5h ) space group (Z = 4, ZB = 1). Here only primitive cell is shown. Blue, gray, green and red balls represent Ba, Ln, Sb and O atoms respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
3.3. Dielectric properties The dispersion theory [22,23] shows that static dielectric permittivity tensor of a crystal within harmonic approximation is given by $ s
¼
$
$
1 þ ionic
ð3Þ
$ 1
where is the electronic dielectric tensor, tric tensor given by
ionic ab ¼
X
l ¼
l
X Z la Zla 2 l v 0 m0 xl
$ ionic
is the ionic dielec-
ð4Þ
where l labels the zone center (q = 0) infrared-active modes of vibrations with frequencies xl l the oscillator strength of mode l, Zla their effective charges in Cartesian direction a, and V the volume per unit cell. Mode effective charges in Cartesian direction a for a given mode l is defined as
ðZ l Þa ¼
X Z iac ðm0 =mi Þ1=2 ðal Þic
ð5Þ
iac
where Z i is Born effective charge tensor for ion i, mi its mass, ðal Þic } the component of normalized dynamical matrix eigenvector for mode l involving ion i in the c direction, and m0 an arbitrary mass, which is cancelled by the denominator of Eq. (2). We choose m0 = 1 amu in this work. Finally the Born effective charge tensor for a given ion i is defined as:
@Pa Z iac ¼ V @uic
tive charge tensor of oxygen (O) is anisotropic due to different environment around oxygen. The Born effective charge of oxygen (O), along Ln–O–Sb bond is higher in magnitude than the nominal ionic charge of 2, whereas Born effective charge transverse to Ln– O–Sb bond is smaller than the nominal charge 2. The anomalies are larger in Sm, Dy, Gd, Y and O ions than the Sb and Ba ions. Finally the Born effective charges of Ba, Sb and O ions are almost independent of the B-site cations (Y, Dy, Gd, Sm) with variations 0.32%, 0.24%, and 0.33% for Ba, Sb and O ions respectively. The contribution of the individual modes to the static dielectric are given in Table 4. It is constant l and mode effective charges Z l clear from Table 4 that out the four F1u modes, F1u(3) has highest mode effective charge and maximum contribution towards the static dielectric constant. High frequency modes, F1u(1) and F1u(2) despite having higher mode effective charges contribute less than the low frequency mode F1u(4). The calculated electronic dielectric tensors along with the static dielectric tensor are also shown in Table 4. While the dielectric constants of Ba2YSbO6 and Ba2DySbO6 are almost same, an increase in the dielectric constant from Dy to Gd to Sm is mainly due to the F1u(4) mode. Konopka et al. [6] have measured the dielectric constants in Ba2YSbO6 and Vijayakumar et al. [7,8] have measured dielectric constants in nanocrystalline Ba2DySbO6 and Ba2GdSbO6. The difference between the calculated and observed dielectric constants of Ba2YSbO6, Ba2DySbO6 and Ba2GdSbO6 are 4.38%, 30.42% and 47.44% respectively. Interestingly the experimental values of dielectric constants are found to be higher than corresponding DFT calculated values and difference between experimental and theoretical values is very
ð6Þ
where P is the polarization of the system and uic is displacement of ion i in direction c. The calculated Born effective charges for Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) are given in Table 3. The Born effective charge of barium (Ba), lanthanides (Sm, Gd, Dy and Y) are larger than the nominal ionic charges +2 and +3. The antimony (Sb) Born effective charge is smaller than the nominal ionic charge +5. The Born effec-
Table 3 m (O5 ) Calculated Born effective charges of Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) in Fm 3 h space group (Z = 4, ZB = 1). Materials
Z Ba
Z Ln=c
Z Sb
Z 0
Ba2YSbO6 Ba2DySbO6 Ba2GdSbO6 Ba2SmSbO6
2.36923 2.37191 2.37628 2.38165
4.09210 4.10519 4.12995 4.15228
4.78243 4.77787 4.81405 4.85110
[3.39170, [3.39844, [3.42230, [3.44634,
1.70602, 1.70613, 1.71156, 1.71690,
1.70602] 1.70613] 1.71156] 1.71690]
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Table 4 Calculated mode effective charges Z l and contribution of different infrared phonon modes to static dielectric constant for Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y). Calculated values of l and s are also given. Experimental values ofl at room temperature are given in parentheses [6–8]. Modes
Ba2YSbO6 Z l
F1u(1) F1u(2) F1u(3) F1u(4)
1 s
4.4848 4.2878 5.0739 3.0172 3.6663 12.9090(13.5)
Ba2DySbO6
u 1.0012 1.8269 3.7246 2.6900
Z l 4.6822 4.5754 4.6822 2.8641 3.6709 13.0349(17)
Ba2GdSbO6
u
Z l
0.9709 2.3170 4.1201 1.9560
large in nanocrystalline Ba2DySbO6 and Ba2GdSbO6. The probable reasons are: (1) We have calculated the dielectric constants at absolute zero temperature whereas the experimental values are reported at room temperature. Although Konopka et al. [6] have studied the variation of static dielectric constant with temperature in Ba2YSbO6 oxide only. A steep increase in static dielectric constant towards zero Kelvin indicates a new phase in Ba2YSbO6 oxide. (2) As we have discussed earlier that in these materials due to the lowering of site symmetries Raman modes become infrared active and appear as weak bands in infrared absorption spectra [10,19]. Higher values of the experimental static dielectric constants may be due to the contributions of these weak bands towards static dielectric tensors. Indeed these modes are also found by Vijayakumar et al. [7,8] during their study of the dielectric properties of nanocrystalline Ba2DySbO6 and Ba2GdSbO6. Small difference in calculated and experimental values of dielectric constants in Ba2YSbO6 compared to Ba2DySbO6 and Ba2GdSbO6 indicates the more site symmetry breaking in later compounds. 4. Conclusions The structural and vibrational properties of double perovskites Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) have been studied for the first time by using first-principles plane wave pseudopotential method within the linear density approximation (LDA). The following conclusions are obtained: (1) The calculated lattice parameters and the internal parameter x for oxygen atoms in these materials are in excellent agreement with the experimental values. (2) The calculated values of bulk modulus and pressure derivative of bulk modulus for Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y) lie in the range of 156–162 GPa and 4.78–4.87 respectively. (3) The calculated vibrational modes are in excellent agreement with the experimental values. (4) The Born effective charge of barium (Ba), lanthanides (Sm, Gd, Dy and Y) are larger than the nominal ionic charges +2 and +3. The antimony (Sb) Born effective charge is smaller than the nominal ionic charge +5 and Born effective charge of oxygen (O) is anisotrophic. Born effective charges of Ba, Sb and O ions are almost independent of the B-site cations (Y, Dy, Gd, Sm).
4.4889 4.5304 4.7087 3.0636 3.6896 13.5650(20)
Ba2SmSbO6
u 0.9788 2.1978 4.1686 2.5302
Z l 4.5632 4.4751 4.7341 3.2638 3.7100 14.2795
u 0.9859 2.0632 4.1923 3.3281
(5) Out the four F1u modes, F1u(3) has highest mode effective charge and maximum contribution towards the static dielectric constant. (6) Due to lowering of site symmetries Raman modes become infrared active and appear as weak bands in infrared absorption spectra. Higher values of the experimental static dielectric constants may be due to the contributions of these weak bands towards static dielectric tensors in Ba2LnSbO6 (Ln = Sm, Gd, Dy and Y).
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