Bulk modulus of cubic perovskites

Journal of Alloys and Compounds 541 (2012) 210–214

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Bulk modulus of cubic perovskites A.S. Verma a,⇑, A. Kumar b a b

Department of Physics, Panjab University, Chandigarh 160014, India Department of Physics, K. R. (P.G.) College, Mathura 281001, India

a r t i c l e

i n f o

Article history: Received 24 April 2012 Received in revised form 4 July 2012 Accepted 5 July 2012 Available online 16 July 2012 Keywords: Ferroelectrics Bulk modulus

a b s t r a c t In this paper, semiempirical formula for the bulk modulus (B in GPa) of perovskite structured solids are elaborated in terms of lattice constant (a in Å) and product of ionic charges (ZaZbZc) of the bonding. Values of bulk modulus, of the group A+1B+2X3, (X = F, Cl, Br), A+2B+4O3 and A+3B+3O3 cubic perovskites exhibit a linear relationship when plotted against the lattice constant (a) normalization, but fall on different straight lines according to the product of ionic charges of the compounds. The resulting expressions can be applied to a broad selection of perovskite (ABX3 = A: large cation with different valence, B: transition metal and X: oxides and halides) materials and their modulus predictions are in good agreement with the experimental data and those from ab initio calculations. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction Perovskite structured solids are of great interest in materials science because the relatively simple crystal structure displays many diverse electric, magnetic, piezoelectric, optical, catalytic and magnetoresistive properties. These solids are currently gaining considerable importance in the field of electrical ceramics, refractories, geophysics, astrophysics, particle accelerators, fission, fusion reactors, heterogeneous catalysis, etc. [1–4]. Additionally, they have received great attention as high-temperature proton conductors with the possibility of applications in fuel cells or hydrogen sensors. During the last few years, many experimental and theoretical investigations were devoted to the study of perovskite solids: typically ABX3 (A: large cation with different valence, B: transition metal, and X: oxides and halides). In perovskite structure, this is shown in Fig. 1, B cations are coordinates by six X anions, while A cations present a coordination number of 12 (also coordinated by X anions). The X anions have coordination number 2, being coordinated by two B cations, since the distance A–X is about 40% larger than the B–X bond distance. The cubic perovskite is called the ideal perovskite, which is the subject of this study. Their precise geometry depends on chemical composition, temperature, pressure and, in some cases, electric field [5–10]. The structural, elastic, dielectric and optical properties of the cubic perovskites are very important. Currently, most elastic modulus evaluations are carried out using the state-of-the-art ab initio calculation techniques [1]; however, one has to keep in mind that the rationalization of these first⇑ Corresponding author. Mobile: +91 9412884655. E-mail address: [email protected] (A.S. Verma). 0925-8388/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jallcom.2012.07.027

principles calculations often requires profound understanding of the nature of the chemical bonding and its attributes in various solid systems. But due to the long process, as well as complicated computational methods involving a series of approximations, such a method has always been complicated [11–14]. Empirical concepts such as valence, empirical radii, ionicity and plasmon energy are then useful [15–17]. These concepts are directly associated with the character of the chemical bond and thus provide means for explaining and classifying many basic properties of molecules and solids. As for covalent materials, it was found by Cohen [17] that their bulk modulus B (GPa) can be estimated by the following semiempirical expression:

B¼

N C ð1972  220kÞ 3:5 4 d

ð1Þ

where NC is the bulk coordination number, d is the bond length, and k is an empirical ionicity parameter that takes the values of 0, 1, and 2 for IV, III–V and II–VI group semiconductors, respectively. There have been a number of reports in the past of empirical relations describing the mechanical properties of solids [18,19]. Anderson and Nafe [20] first proposed an empirical relationship between bulk modulus B at atmospheric pressure and specific volume V0 of the form B–V0x. Recently, Verma and co-authors [21–24] has been evaluated the structural, electronic, mechanical and ground state properties of binary and ternary crystals with the help of ionic charge theory. In this paper, we improved this formula by replacing the ad hoc empirical ionicity parameter with more suitable product of ionic charge of the compounds. Both experimental data and theoretical calculations based on density functional theory follow the correlation.

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band gap energy is the dominant energy parameter in covalent solids. Recently, relationships connecting inherent traits such as thermal activation energies [27] and hardness [28] to the homopolar band gap energy were elaborated in the case of covalent crystals. These works further confirm that the intrinsic properties of covalent materials are predominantly dictated by Eh. Using a scaling argument, Eh can be expressed in terms of d as follows [16]:

Eh ¼

39:74

ð5Þ

2:5

d

where the units of Eh are in eV and d (nearest neighbor distance) is in Å. Since for small amounts of deformation, the strain parameter is a linear function of the nearest neighbor distance, from Eq. (5), it follows that

@2E @d2

1

a

ð6Þ

4:5

d

The cylindrical-shaped charge volume of covalent crystals is a linear function of the nearest neighbor distance, X  pð2aB Þ2 d (aB is Bohr radius) and it can be used in Eqs. (2) and (3), since it encloses the largest electron concentration [17]. Thus Eqs. (2)–(6) yield Fig. 1. Ideal cubic perovskite structure.

Ba 2. Theory Both bulk B and shear G moduli can be derived from the second derivative of the total energy E with respect to the appropriate deformation parameter at the equilibrium state as follows [17,25]:

@ 2 E B ¼ X 2 @X

X¼X0

1 @ 2 E G¼ X @d2

c¼c0

ð2Þ

ð3Þ

where X and d stand for volume and dimensionless deformation parameter, respectively. From Eqs. (2) and (3), it is evident that the first step in establishing the formulae for bulk and shear moduli is to approximate the energy derivatives in terms of chemical bonding parameters. Due to their spherical symmetry and tight-binding character, the core electrons are nearly unresponsive to low-energy perturbations [26] like those occurring under elastic deformation; while the valence electrons are completely affected by such phenomena. Consequently, the core electrons do not have a significant contribution to the elastic response of a material deforming within the limits of the elastic regime; whereas, the valence electrons are fully involved in the distortion process. Since the involvement of core electrons in the elastic deformation is insignificant, the variation of their energy is also negligible. Therefore, within the limits of the elastic regime, the second derivative of the total energy can be approximated by the variation of the valence electrons’ force. In the case of covalently bonded materials, as discussed by Philips [16] the band gap energy Eg provides an estimation of the valence bond strength and it results from homopolar and heteropolar or ionic contributions of the atoms to the bonds as follows:

E2g ¼ E2h þ E2c

ð4Þ

Here Eh refers to the homopolar or covalent contribution to the bonding, while Ec corresponds to the ionic contribution or the charge transfer to the bonds. In the case of purely covalent group IV crystals, such as diamond, silicon, or germanium, Eg is equal to Eh. Consequently, Eh characterizes the strength of the covalent bond. Cohen was the first to maintain [17] that Philips’ homopolar

1

ð7Þ

3:5

d

The author in previous research [13,24] found that substantially reduced ionic charges must be used to get better agreement with experimental values. Goldshmidt [29] has pointed out that a term A = ZaZc, where Za and Zc are the valence number of anion and cation, respectively, may be considered for a direct comparison of the hardness. Further it is well known that the hardness is closely related to the elastic properties of crystals [30] Ionic charge depends on the outermost-shell electrons of an atom. Thus, there must be a correlation between ionic charge and the properties of solids. Linear regression lines have been plotted for the bulk modulus (in GPa), which result in equation of the form

B ðGPaÞ ¼ S þ V

ðZ a Z b Z c Þ0:35 a3:5

ð8Þ

where Za,, Zb and Zc are the ionic charges on the A, B and X, respectively and a is lattice parameter in Å. The S and V are constants which depend on the crystal structure and the values are given in Table 1 along with the correlation coefficient (R) obtained from the regression analysis. The significance of the regression is given as the probability P of the null hypothesis (that there is no correlation). 3. Results and discussion The knowledge of elastic moduli is necessary in order to come to a better theoretical understanding of material properties that are essentially determined by the phonon density of states and lattice anharmonicity effects or by electron–phonon interaction processes mediated via deformation potentials. In view of the still unsatisfactory and contradictory data on the mechanical properties of the compounds on the one hand and of the importance of their knowledge for a comprehensive analysis of a wide variety of material characteristics on the other hand it was the aim of the present

Table 1 Linear regression results from the data for A+1B+2X3, (X = F. Cl, Br), A+2B+4O3 and A+3B+3O3 perovskites. Properties

S

V

R

P

Bulk modulus B

1.79 ± 4.15

5505.785 ± 186.86

0.97092

<0.0001

A.S. Verma, A. Kumar / Journal of Alloys and Compounds 541 (2012) 210–214

CaHfO3

180

+1 +2

BaNbO3

+4

Bulk Modulus (GPa)

A B O3 150

+3

CaSnO3 BiGaO3

+2

A B X3 (X=F, Cl, Br, I) BaZrO3

+3

A B O3

BaRuO3

SrZrO3

200

BaTiO3

KZnF3

90

RbMnF3

60

30

NaZnF3

RbZnF3

CsHgF3

CsCdF3

CsCaF3

CsPbCl3 CsPbBr3 CsCaCl3

KMgF3

NaMgF3

KMnF3

BaRuO3

SrZrO3

140

SrTiO3

BiGaO3 EuTiO3

BaTiO3

BaZrO3

BaSnO3 BaThO3

120 100

NaZnF3 RbZnF3

80

RbMnF3

60

20

CsCaF3 CsPbCl3

NaMgF3

KMnF3

CsHgF3 RbCaF3

CsPbF3

CsPbBr3

0 0.000

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

KZnF3 KMgF 3

CsCdF3

40

RbCaF3

CsPbF3

1/a

BaNbO3

160

BaSnO3

BiAlO3

CaHfO3

180

EuTiO3

BaThO3

120

CaSnO3

Y =1.79012+5505.78536 X

BiAlO3

Bulk Modulus (GPa)

212

CsCaCl3

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

3.5

(ZaZbZc)

Fig. 2. Plot of bulk modulus B (GPa) and lattice constant (a in Å) for the group A+1B+2X3, (X = F. Cl, Br), A+2B+4O3 and A+3B+3O3 perovskites and found at three positions, which is depending upon the product of ionic charges. In this figure bulk modulus and lattice constant values are taken from Reference [1–10,31–38].

study to critically evaluate and review related experimental and theoretical data reported in the literature so far. It has been verified [17] that elastic moduli assume a decreasing linear trend with increasing lattice parameter. Therefore, based on Eq. (7), B is linear function of lattice parameter. The bulk modulus is expected to exhibit the explicit dependences on lattice parameter (a) and ionic charge (Z). As an example to verification of ionic charge theory, we have plotted the curves between B Vs a3.5 (B = bulk modulus in GPa, a = lattice parameter in Å) for group A+1B+2X3, (X = F. Cl, Br), A+2B+4O3 and A+3B+3O3 cubic perovskites and data are presented in Fig. 2. We observe that in the plot of B Vs a3.5 normalization, the group A+1B+2X3, (X = F. Cl, Br), A+2B+4O3 and A+3B+3O3 cubic perovskites exhibit three positions in this Figure This effects induced by the ionic charges of the compounds in the case of group A+1B+2X3, (X = F. Cl, Br), A+2B+4O3 and A+3B+3O3 perovskites. If all

0.35

/a

3.5

Fig. 3. Bulk modulus B (GPa) for perovskites (A+1B+2X3; X = F, Cl, Br, A+2B+4O3 and A+3B+3O3) as a function of (ZaZbZc)0.35/a3.5. This line shows linear relationship as determined by regression analysis. In this figure bulk modulus and lattice constant values are taken from Reference [1–10,31–38].

data of bulk modulus (in GPa) plots with product of ionic charges (ZaZbZc) of group A+1B+2X3, (X = F. Cl, Br), A+2B+4O3 and A+3B+3O3 perovskites and found a straight line for all groups, which are presented in Fig. 3. From the comparison of Figs. 2 and 3, we see that three lines found in Fig. 2 is due to different ionic charges of the compounds. In Fig. 3, it has been verified that bulk modulus assume a linear trend with lattice parameter. Bulk modulus is linear function of product of ionic charges and 1/a3.5. We note the procedure for determining B (in GPa) from these techniques considers a difference between experimental measurements, and this could magnify errors. In the Tables 2–4, we have presented experimental and theoretical bulk modulus values evaluated by different researchers [1–4,7–10,31–38] for the sake of comparison. The simple trend

Table 2 Values of bulk modulus (B in GPa) defined by Eq. (8) obtained for A+1B+2X3, (X = F. Cl, Br), perovskites. The value of product of ionic charge (ZaZbZc) is 6. Solids

a (Å) [5]

B (GPa) [1–4]

B (GPa) this work

Solids

a (Å) [5]

CsCdF3 CsCaF3 CsHgF3 CsSrF3 CsEuF3 CsPbF3 CsYbF3 CsCaCl3 CsCdCl3 CsPbCl3 CsHgCl3 CsEuCl3 CsTmCl3 CsYbCl3 CsPbBr3 CsPbI3 RbZnF3 RbCoF3 RbVF3 RbFeF3 RbMnF3 RbCdF3 RbCaF3 RbHgF3 RbPdF3 RbYbF3

4.47 4.523 4.57 4.75 4.78 4.8 4.61 5.396 5.21 5.605 5.41 5.627 5.476 5.437 5.874 6.29 4.11 4.141 4.17 4.174 4.24 4.398 4.452 4.47 4.298 4.53

58 49 55

54.6 52.4 50.5 44.1 43.2 42.6 49.0 28.2 31.9 24.7 28.0 24.4 26.8 27.5 21.0 16.5 73 71.3 69.6 69.4 65.7 57.8 55.4 54.6 62.6 52.1

RbPbF3 KCaF3 KCdF3 KMgF3 KNiF3 KZnF3 KCoF3 KVF3 KFeF3 KMnF3 NaVF3 NaMgF3 NaZnF3 NaCoF3 AgMgF3 AgNiF3 AgZnF3 AgCoF3 AgMnF3 LiBaF3 TlCoF3 TlFeF3 TlMnF3 TlCdF3 TlPdF3 TlMnCl3

4.79 4.38 4.293 3.989 4.013 4.056 4.071 4.1 4.121 4.189 3.94 3.84 3.88 3.9 3.918 3.936 3.972 3.983 4.03 3.992 4.138 4.188 4.26 4.4 4.301 5.02

36 23.4 25.8

23.5 19.8 86.9

67.5 47.3

B (GPa) [3,4]

82.7 85.5

64.9 87.9 89.8

B (GPa) this work 42.9 58.6 62.9 81.3 79.6 76.7 75.7 73.9 72.6 68.5 84.9 92.9 89.6 88.0 86.6 85.2 82.5 81.8 78.5 81.1 71.5 68.6 64.6 57.7 62.5 36.4

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Table 3 Values of bulk modulus (B in GPa) defined by Eq. (8) obtained for A+2B+4O3 perovskites. The value of product of ionic charge (ZaZbZc) is 48 and values of bulk modulus are tabulated as experimental (E) for comparison. Solids

a (Å) [5]

B (GPa) [7–10,31]

B (GPa) this work

Solids

a (Å) [5]

B (GPa) [32–35]

B (GPa) this work

BaMnO3 BaFeO3 BaMoO3 BaNbO3 BaSnO3 BaHfO3 BaZrO3 BaIrO3 BaPbO3 BaTbO3 BaPrO3 BaCeO3 BaAmO3 BaNpO3 BaUO3 BaPaO3 BaThO3 BaTiO3 BaRuO3 BaPuO3 SrMnO3

3.846 3.994 4.04 4.08 4.116 4.171 4.193 4.1 4.265 4.285 4.354 4.397 4.357 4.384 4.387 4.45 4.48 4.012 4.05 4.36 3.806

196

191.3 167.6 161.0 155.6 150.9 144.0 141 152.9 133.2 131.1 123.9 119.7 123.6 121.0 120.7 114.8 112.2 165.0 159.7 123.3 198.4

SrVO3 SrFeO3 SrTiO3 SrTcO3 SrMoO3 SrNbO3 SrSnO3 SrHfO3 SrTbO3 SrAmO3 SrPuO3 SrCoO3 SrZrO3 SrRuO3 SrThO3 CaVO3 CaHfO3 CaZrO3 CaRuO3 CaSnO3 CaTiO3

3.89 3.85 3.905 3.949 3.975 4.016 4.034 4.069 4.18 4.23 4.28 3.85 4.104 3.93 4.42 3.767 3.99 4.01 3.84 3.95 3.84

181.5

183.9 190.6 181 174.4 170.5 164.4 161.9 157.1 142.9 137.1 131.6 190.6 152.4 177.4 117.6 205.7 168.2 165.3 192.4 174.3 192.4

171 145.8E 127E

124 162E 167.9

E

179

173 164

151

191.6

191.8 175

Table 4 Values of bulk modulus (B in GPa) defined by Eq. (8) obtained for A+3B+3O3 perovskites. The value of product of ionic charge (ZaZbZc) is 54. Solids

a (Å) [5]

B (GPa) [36,37]

B (GPa) this work

Solids

a (Å) [5]

EuTiO3 EuAlO3 EuCrO3 EuFeO3 EuGaO3 EuInO3 CeAlO3 GdAlO3 GdCrO3 GdFeO3 LaAlO3 LaCrO3 LaFeO3 LaGaO3 LaRhO3 LaTiO3 LaVO3 LaInO3 NdAlO3 NdGaO3 NdCoO3

3.905 3.725 3.803 3.836 3.84 4.03 3.772 3.71 3.795 3.82 3.778 3.874 3.92 3.874 3.94 3.92 3.91 4.11 3.752 3.86 3.777

172.6

189.0 223.0 207.4 201.2 200.5 169.3 213.4 226.1 208.9 204.2 212.2 194.4 186.5 194.4 183.2 186.5 188.2 158.0 217.4 196.9 212.4

NdCrO3 NdFeO3 NdMnO3 PrAlO3 PrCrO3 PrFeO3 PrGaO3 PrMnO3 PrVO3 PrCoO3 SmAlO3 SmCoO3 SmVO3 SmFeO3 YAlO3 YCrO3 YFeO3 BiAlO3 BiGaO3 BiInO3 BiScO3

3.835 3.87 3.8 3.757 3.852 3.887 3.863 3.82 3.89 3.78 3.734 3.75 3.89 3.845 3.68 3.768 3.785 3.802 3.905 4.111 3.974

168.7

when a larger lattice constant leads to a smaller bulk modulus. It has been demonstrated also for different perovskites ABO3 [39]. In the present work it is shown that analogous relation exists for the perovskite materials, which can be successfully employed to estimate the bulk modulus from their ionic charges. We note that the evaluated values of bulk modulus by the proposed relation are in close agreement with the experimental data as compared to the values reported by previous researchers so far. For example, the results for bulk modulus differ from experimental by 1% (SrTiO3), 1.8% (BaTiO3), 3.5% (BaSnO3) and 11% (BaZrO3) in the current study. Consequently, the reliability and accuracy of proposed formula is well verified.

B (GPa) [37,38]

130

218.5 178.6 158.4 178.5

B (GPa) this work 201.4 195.1 207.9 216.4 198.3 192.1 196.3 204.2 191.6 211.8 221.1 217.8 191.6 199.6 232.7 214.2 210.8 207.6 189.0 157.9 177.8

tionship when plotted against the lattice parameter, but fall on different straight lines according to the product of ionic charge of the compounds, which are presented in Figs. 2 and 3. From the results and discussion obtained by using the proposed empirical relation, it is quite obvious that the bulk modulus reflecting the elastic properties can be expressed in terms of product of ionic charge and lattice parameter of these materials, which are basic parameters. A good agreement between the author’s calculated values of these material properties and the values reported by different researchers has been found. The method presented in this work will be helpful to material scientists for finding new materials with desired elastic properties among a series of structurally similar materials.

4. Summary and conclusions Acknowledgements There are several methods in determining elastic properties in semiconductors, but due to the small changes of the unit cell dimensions, the accuracy of determining these parameters always has been unpredictable. Furthermore, we found that in the compounds investigated here, the bulk modulus exhibit a linear rela-

First author (Dr. Ajay Singh Verma, PH/08/0049) is thankful to University Grant Commission, New Delhi, India, for supporting this research under the scheme of U.G.C. Dr. D.S. Kothari Post Doctoral Fellowship.

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