Full potential calculation of structural, elastic and electronic properties of BaZrO3 and SrZrO3

phys. stat. sol. (b) 242, No. 5, 1054–1062 (2005) / DOI 10.1002/pssb.200402142

© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Full potential calculation of structural, elastic and electronic properties of BaZrO3 and SrZrO3

R. Terki*, H. Feraoun, G. Bertrand, and H. Aourag

Laboratoire d’Etude et de Recherche sur les Matériaux, les Procédés et les Surfaces, Université de Technologie de Belfort-Montbeliard, Site de Sevenans, 90010 Belfort, France

Received 29 March 2004, revised 30 September 2004, accepted 20 December 2004 Published online 2 February 2005

PACS 61.50.Ah, 71.15.Ap, 71.20.Ps, 77.84.Dy

Ab initio calculations have been performed on the structural and electronic properties of perovskite-type compounds BaZrO3 and SrZrO3 ceramics. The Kohn–Sham equations were solved by applying the full-potential linearized augmented-plane-wave (FP-LAPW) method. In this approach, the generalized gradi-ent approximation was used for the exchange-correlation potential. The ground state properties such as lattice parameter, elastic constants, bulk modulus and its pressure derivative were calculated and the re-sults are compared with previous calculations and experimental data when available. The SrZrO3 perovskite should exhibit higher hardness and stiffness than BaZrO3. Furthermore, the electronic structure calculations show that these materials are weakly ionic and exhibit indirect and wide band gaps, which are typical of insulators.


1 Introduction

In the last decade, many experimental and theoretical investigations were devoted to the study of perovskite-type oxides: typically ABO3 (A: large cation with different valence and B: transition metal). This class of materials has great potential for a variety of device applications due to their simple crystal structures and unique ferroelectric and dielectric properties. Among the perovskite oxides studied most intensively are zirconates of alkaline earth metals, particularly BaZrO3 and SrZrO3. Indeed, they are cur-rently gaining considerable importance in the field of electrical ceramics, refractories and heterogeneous catalysis. Additionally, they have received great attention as high temperature proton conductors with the possibility of applications in fuel cells or hydrogen sensors [1, 2]. Recently, theses two zirconate com-pounds have been investigated for their potential use as thermal barrier coatings (TBCs) materials due to their high melting points, chemical and mechanical stability in a wide range of temperature. However, there is a lack of fundamental understanding at the microscopic level on most of the structural ceramics including BaZrO3 and SrZrO3. This could be partially explained by difficulties in using computer model-ling to reproduce the unusual physical properties of the ceramic materials. In order to understand the fundamental spectroscopic and thermodynamical properties of BaZrO3 and SrZrO3 compounds, it is im-perative that the structural and electronic properties of these materials can be thoroughly understood. Until now, very few calculations have been reported on the electronic structure of both BaZrO3 and SrZrO3 ceramics [3, 4]. In this study, we present the structural, elastic and electronic properties of barium zirconate and stron-tium zirconate calculated using the full potential linearized augmented plane wave method (FP-LAPW) based on density functional theory implemented in the WIEN2K code [5]. The exchange–correlation

* Corresponding author: e-mail: [email protected]

phys. stat. sol. (b) 242, No. 5 (2005) / www.pss-b.com 1055


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Paper

term was determined within the GGA parametrized by Perdew–Burke–Ernzerhof [6] (PBE). The paper is organized in the following way. Section 2 gives a description of the method, as well as some details of the calculations. The calculated ground-state and electronic properties are presented, and discussed in Section 3. Finally, the conclusion is given in Section 4.

2 Computational method

In our calculation, we use the WIEN2k code [5] based on the all-electron full-potential linearized aug-mented plane wave (FP-LAPW) method within the framework of density functional theory (DFT) [7]. In the FP-LAPW method, one employs Slater’s old idea of muffin-tin spheres to divide space into two regions. Near the atoms all quantities of interest are expanded in spherical harmonics and in the intersti-tial region they are expanded in plane waves. The first type of expansion is defined within a so-called muffin-tin sphere of radius RMT around each nucleus. The rate of convergence of these expansions de-pends on the choice of sphere radii, but this only affects the speed of the calculation. The exchange-correlation potential was calculated by the generalized gradient approximation (GGA-92) using the scheme of Perdew et al. [6] (PBE). The semirelativistic approximation without spin-orbit effects was employed in the calculation of the valence states, whereas the core levels were treated fully relativistically [8] and were updated at each iteration. The muffin tin radii were chosen to be 1.9, 1.8, 1.8 and 1.6 a.u. for the Ba, Sr, Zr, and O atoms respectively. In the calculation, 3878 and 3623 planes waves have been used for BaZrO3 and SrZrO3 respectively. The maximum l value for the wave function expansion inside the atomic spheres was confined to lmax = 10. Outside the muffin-tin spheres, the charge density is expanded in a Fourier series. The dependence of the total energy on the number of k-points in the irreducible wedge of the first Brillouin zone (BZ) has been explored within the linearized tetrahedron scheme [9] by performing the calculation for 64 k-points (grid of 7 × 7 × 7 meshes, equivalent to 500 k-points in the entire Brillouin zone) and extrapolating to an infinite number of k-points. A satisfactory degree of convergence was achieved by considering a number of FP-LAPW basis functions up to RMTKmax = 7 (where RMT is the minimum radius of the muffin-tin spheres and Kmax gives the magnitude of the largest K vector in the plane wave expansion. In order to keep the same degree of convergence for all the lattice constants studied, we kept the values of the sphere radii and Kmax constant over all the range of lattice spacings considered. Concerning the structural aspect, the BaZrO3 compound shows a cubic perovskite-type crystal struc-ture. The space group (Pm3m:221) contains 48 symmetry operations including inversion. The Wyckoff positions of the atoms are Ba 1a (0.0, 0.0, 0.0), Zr 1b (0.5, 0.5, 0.5) and O 3c (0.0, 0.5, 0.5). On the other hand, there are different opinions about the crystal structure of SrZrO3; it is generally believed that SrZrO3 is orthorhombic at room temperature (space group 16

2hD -Pnma) [10, 11], but in some papers, a cubic cell is reported [12]. Other authors pointed out that “true” SrZrO3 exhibits a cubic structure. How-ever, the crystal structure is very sensitive to lattice defects caused by impurities, small deviations from the nominal stochiometric composition or changes in synthesis temperatures which could result in differ-ent crystal structures and phase transitions [13]. In the range 700–1200 °C, the SrZrO3 perovskite lattice has three structural transformations [11, 14]. At high temperatures, there is a cubic and at low tempera-ture an orthorhombic modification of the perovskite unit cell. In the intermediate region, however, two tetragonal phases with c/a > 1 and c/a < 1 were reported. No data of the SrZrO3 structure at temperatures above 1200 °C are given. We restrict ourself to study only the cubic structure due to a lack of crystallo-graphic parameters (as atomic coordinates) of the other structures of SrZrO3.

3 Results and discussion

3.1 Structural properties

The ground state properties of BaZrO3 and SrZrO3 are studied in this work; the total energy is obtained as a function of volume and fitted to a Murnaghan equation of state [15] to obtain the equilibrium lattice constant and other ground state properties. The variation in the total energy as a function of volume is shown in Fig. 1.

1056 R. Terki et al.: Full potential calculation of properties of BaZrO3 and SrZrO3


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Fig. 1 Calculated total energy as a function of the volume for cubic perovskite (a) BaZrO3 and (b) SrZrO3.

In Table 1, the equilibrium lattice constant (a0), bulk modulus (B) and its first pressure derivative (B′) values are compared with experiment and previous theoretical results. It is observed that a larger lattice constant leads to a smaller bulk modulus. This result can be explained by considering the radius of A cations (Br and Sr). The lattice constant increases when replacing Sr with the larger Ba ion. Thus, the compressibility decreases as the size of the ion decreases. Vanderbilt’s paper [3] also showed this simple trend in different ABO3. Furthermore, our estimated bulk modulus of BaZrO3 is lower than that of SrZrO3. A strong correlation between hardness and a bulk modulus value has been confirmed in a num-ber of recent papers [16, 17]. For cubic compounds, the bulk modulus should be a quite good indicator of hardness, since it is related to an isotropic deformation [18]. Therefore, we expect that BaZrO3 should exhibit lower hardness than SrZrO3.

3.2 Elastic properties

Elastic constants are believed to be related to the strength of materials; indeed, these microscopic quanti-ties are related to the macroscopic parameters as the shear modulus G and the Young’s modulus E which are frequently measured for polycrystalline materials when investigating their hardness. The elastic moduli require knowledge of the derivative of the energy as a function of a lattice strain [19]. In the case

Table 1 Calculated lattice constant a (Å), equilibrium volume Veq (Å3), minimum total energy E0 (eV),

bulk modulus B (GPa) and its pressure derivative (B′) for both BaZrO3 and SrZrO3. Our results are com-pared with experimental data and with previous LDA calculations, when available.

material method Veq (Å3) a0 (Å) B (GPa) B′ –E0(eV)

BaZrO3

GGA-92 LDA(a) expt.(b)

74.47 71.68 73.66

4.207 4.154 4.192

157 175.3 –

4.236 – –

–325610.91227 – –

SrZrO3

GGA-92 expt.(c)

72.51 68.97

4.17 4.101

160 –

3.742 –

–190651.19967 –

(a) [3], (b) [25], (c) [10].

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Table 2 Calculated elastic properties of both BaZrO3 and SrZrO3, compared to experimental data and previous LDA calculations, when available.

material method C11 (GPa) C12 (GPa) C44 (GPa) G (GPa) E (GPa)

GGA-92 322.5 79 70 91 229 BaZrO3 LDA(a) 336 95 89.5 102 256 expt – – – 103(b) 243(b), 181 ± 11(c) SrZrO3 GGA-92 338.6 71 77 100 248

(a) [3], (b) [25], (c) [26]. of a cubic lattice, it is possible to choose this strain so that the volume of the unit cell is preserved. Thus, for the calculation of the modulus C11 –C12, we have used the volume-conserving orthorhombic strain tensor

( )

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Application of this strain changes the total energy from its unstrained value to E(δ) = E (0) + (C11–C12)Vδ

2. Where V is the volume of the unit cell and E(0) is the energy of the unstrained lattice at volume V. For the elastic modulus C44, we used the volume-conserving monoclinic strain tensor

( )

2

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d

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which changes the total energy to E (δ) = E (0) + 12 C44Vδ

2. For an isotropic cubic crystal, the bulk modulus is given exactly by: B = 1

3 (C11 + 2C12) [20]. On the other hand, the shear and the Young’s moduli are related to the microscopic elastic constants by means of the following equations: E = 9BG/(3B + G) and G = (C11 – C12 + 3C44)/5. Table 2 gives our elastic properties and also reports the experimental and theoretical data for comparison when available. For the BaZrO3 compound, the elastic constants obtained are smaller than those reported from Vanderbilt’s paper [3]; whereas the calculated shear and Young’s moduli tend to support the experimental data. It is important to note that the meas-urements of G and E are taken from polycrystalline samples including defects and porosity. In our case, however, all calculations are related to a perfect crystal. From Table 2, we also notice that SrZrO3 (hav-ing a largest shear and Young’s moduli) is stiffer than BaZrO3. On the one hand, the macroscopic pa-rameters (E and G) are related to the strength of the lattice; on the other hand, the shear modulus is in general proportional to the bulk modulus. Thus, there is a proportionality between hardness and stiffness of these ceramics.

3.3 Electronic structure calculations

3.3.1 The density of states (DOS)

The density of states is computed by the tetrahedron method [21] (which requires many k-points). From the eigenvalues and eigenvectors solved at a sufficient number of k-points in the Brillouin-zone, the total

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Energy (eV) DOS can be projected into its partial components or partial DOS (PDOS) with respect to the different atoms. The densities of states for the two perovskite compounds have been computed at their equilibrium lattice constants. There is an overall topological resemblance of the present DOS for the two zirconates. Figures 2–4 display the computed results for BaZrO3 and SrZrO3; since the partial densities of states of Zr and O atoms in BaZrO3 are similar to those in SrZrO3. We only present the results of the first com-pound. For both BaZrO3 and SrZrO3 compounds, it is found that the upper valence bands (between –5 and 0 eV), are essentially dominated by O–2p states, with a minor admixture from Zr–4d states. Moreover, the presence of this latter in the bonding region suggests a covalent bonding contribution in these materi-als. The s, p states of Sr, Zr atoms and Ba–s states are also contributing to the valence bands, but the values of the corresponding densities of states are quite small compared to Zr–4d and O–2p states as shown in Figs. 3 and 4. The total DOS of BaZrO3 shows also a narrow band peaked at –11 eV; this band is formed by the Ba–5p states with small contribution of O–2s states. The bottom of the conduction bands is dominated by Zr–4d states which hybridize with Sr–5s and Sr–4d states in the SrZrO3 and with Ba–6s and Ba–5d states in the BaZrO3. All of these states are dis-tributed in a wide energy range of 4.0–12.0 eV. On the other hand, there is an energy gap between the

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Fig. 3 Total and partial density of states (PDOS) of Ba (a) atom in BaZrO3 and Sr (b) atom in SrZrO3.

Fig. 2 Calculated total density of states (DOS) of BaZrO3 and SrZrO3, at the predicted equilibrium lattice constant.

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Fig. 4 Total and partial density of states (PDOS) of Zr (a) and O (b) atoms in BaZrO3. occupied O–2p states and the unoccupied Zr–4d states, as seen from the analysis of the partial densities of states. Our total DOS calculations of both BaZrO3 and SrZrO3 compare well with other theoretical works [3, 4].

3.3.2 The electronic band structure

The calculated electronic band structures of the cubic phases of both BaZrO3 and SrZrO3 are shown in Fig. 5. The two compounds are found to have indirect band gaps; the valence band maximum occurs at the L-point (k = π/a(111)) in the Brillouin-zone and has an energy of –0.0349 eV for BaZrO3 and –0.1477 eV for SrZrO3, whereas, the conduction band minimum is at the Γ (000) or X (k = 2π/a(100)) point in the Brillouin-zone and with an energy of 3.199 eV for BaZrO3 and 3.189 eV for SrZrO3. The band gaps are large with values of 3.23 eV (exp. 5.33 eV [22]) and 3.34 eV (exp. 6 eV [3]) for BaZrO3 and SrZrO3 respectively. The width of gap indicates the presence of an insulating feature which is in good agreement with experimental data [22]. This result can be explained by the presence of Zr atoms; indeed, both BaTiO3 or SrTiO3 have small band gap; but when Zr replaces Ti, the band gap increases by 2 eV, because it is controlled by Zr–d states [22]. On the other hand, our band gaps are underestimated in comparison with experiment. This is not surprising as it is well known that the GGA calculations systematically underestimate the fundamental gap of semiconductors.

3.3.3 Charge density

Contours maps of the charge density in barium zirconate and strontium zirconate are shown in Fig. 6. This latter shows that our choice for the muffin-radii is about right, since we capture most of the spheri-cal regions inside the muffin-tin spheres. Both BaZrO3 and SrZrO3 compounds are found to have very similar charge densities. The value of the charge density between an oxygen atom and zirconium atom is 0.07 electrons per bohr3 which is similar to the value of the charge density found between the atoms in a metal and about one-third of the value in diamond. Therefore, the bonding has a significant covalent character due to the hybridization effect between Zr–4d and O–2p orbitals. This result should explain the high bulk moduli of these ceramics. The value between an oxygen atom and barium atom is 0.026 electron per bohr3 while the value between an oxygen atom and strontium atom is 0.04 bohr3. These numbers can be compared with ionic reference NaCl, where the charge density between the sodium and chlorine atoms is about 0.01 electron per bohr3. The Ba–O and Sr–O bonds seem to have an ionic char-

a) b)



-15

-10

-5

0

5

10

L ΧΧΧΧΓΓΓΓ ∆∆∆∆ ΖΖΖΖ W

En

ergy

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0 5 10 15 20

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ΚΚΚΚΖΖΖΖΧΧΧΧ∆∆∆∆ΓΓΓΓΛΛΛΛL WWDOS (states/eV)

Ene

rgy

(eV

)

0 2 4 6 8 10 12

EF

Fig. 5 Calculated electronic band structure and total DOS of BaZrO3 (a) and SrZrO3 (b). acter as seen in perovskite BaTiO3 [23]. Moreover, the charge transfer occurs mainly from the other atomic species towards the oxygen atoms. Furthermore, the Ba–O bond is more ionic than the Sr–O bond. Indeed, as you move down a column for a group (e.g. alkaline earth), the sizes of atoms increase. The electrons in larger atoms are not held as strongly as those in smaller atoms. Therefore, the larger the atom, the less likely it will attract electrons from other atoms. Hence, electronegativity decreases. Addi-tionally, when a chemical bond is formed with another element and the difference of electronegativity is large, the bond tends to be more ionic [24]. Our results show that both BaZrO3 and SrZrO3 are weakly ionic as previously reported by King–Smith et al. [3].

4 Conclusion

In this paper, we have performed calculations on structural, elastic and electronic properties of BaZrO3 and SrZrO3. We use the full-potential linearized augmented Plane Wave (FP-LAPW) method, in the framework of density functional theory (DFT) with the generalized gradient approximation (GGA). The

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Fig. 6 Contours of equal charge density at the equilibrium lattice parameters, in the (110) plane for BaZrO3 (a) and SrZrO3 (b).

ground-state properties including lattice constants, bulk modulus and elastic constants were calculated. The results show that both BaZrO3 and SrZrO3 have relatively high bulk moduli, which is a result of strong covalent bonding between d-states of transition metal and oxygen p-states. Furthermore, SrZrO3

should exhibit higher hardness and stiffness than BaZrO3, which can be explained using the simple chemical idea based on the size of ions. On the other hand, the electronic structure calculations showed that both BaZrO3 and SrZrO3 exhibit wide and indirect band gaps, but their values are underestimated which is typical of GGA calculations. The analysis of the DOS also revealed that the conduction band is mainly composed of Zr–4d with some mixture of Sr–5s and Sr–4d states for SrZrO3 and with Ba–6s and Ba–5d states for BaZrO3, while the valence band is essentially dominated by O–2p states. Further-more, the results of the charge densities compare well with experiment and other calculations. The Zr–O bond has a significant covalent character while the Ba–O and Sr–O bonds are typically ionic.

References

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Documents

Full potential calculation of structural, elastic and electronic properties of BaZrO3 and SrZrO3