Structural, electronic, optical and elastic properties of the cubic perovskite PbHfO3 through modified Becke–Johnson potential

Accepted Manuscript

Structural, electronic, optical and elastic properties of the cubic perovskite PbHfO3 through modified Becke-Johnson potential D. Chenine, Z. Aziz, A. Abbad, B. Bouadjemi, O.K. Youb, T. Lantri, O. Lakel, S. Bentata PII: DOI: Reference:

S0577-9073(17)30500-2 10.1016/j.cjph.2017.09.001 CJPH 338

To appear in:

Chinese Journal of Physics

Received date: Revised date: Accepted date:

27 April 2017 25 July 2017 1 September 2017

Please cite this article as: D. Chenine, Z. Aziz, A. Abbad, B. Bouadjemi, O.K. Youb, T. Lantri, O. Lakel, S. Bentata, Structural, electronic, optical and elastic properties of the cubic perovskite PbHfO3 through modified Becke-Johnson potential, Chinese Journal of Physics (2017), doi: 10.1016/j.cjph.2017.09.001

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Highlights • The structural, optoelectronic and elastic properties of PbHfO3 were calculated. • TB-mBJ scheme found to be an efficient approximation for improving the values of the band gap over GGA value. • The semiconductor character was found by using GGA as well as TB-mBJ approaches.

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• PbHfO3 is an attractive and promising material for UV optoelectronic applications.

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• Mechanical stability criteria are satisfied for the studied material and classified as ductile in nature.

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Structural, electronic, optical and elastic properties of the cubic perovskite PbHfO3 through modified Becke-Johnson potential D. Cheninea,b,, Z. Aziza , A. Abbada , B. Bouadjemia , O.K. Youba , T. Lantria , O. Lakelc , S. Bentataa a Laboratoire

de Technologie et Propri´et´es du solide (LTPS), Universit´e Abdelhamid Ibn Badis, BP 227, Mostaganem 27000, Algeria de Structure, Elaboration et Applications des Mat´eriaux Mol´eculaires (SEA2M), D´epartement de Chimie, Universit´e Abdelhamid Ibn Badis, B.P. 981, R.P., Mostaganem 27000, Algeria c Modelling and Simulation in Materials Science Laboratory, Djillali Liab´ es University of Sidi Bel-Abb´es, 22000 Sidi Bel-Abbes, Algeria

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b Laboratoire

Abstract

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The structural, optoelectronic, and elastic properties of cubic perovskite lead hafnate, PbHfO3 , have been investigated by the full potential linearized augmented plane wave (FP-LAPW) method with the generalized gradient approximation (GGA), and the Tran-Blaha modified Becke-Johnson potential approximation (TB-mBJ) as coupled with GGA. All calculations are based on the Density Functional Theory (DFT). The results found in the course of this study are in good agreement with experimental and analytical data. The band gap value estimated by TB-mBJ approach was found to be 3.75 eV, which is very close to the reported experimental value (3.4 eV) in contrast to the value of 2.59 eV given by GGA showing thus the semiconductor character of the material .The compound has significant absorption in large range of photon energies and could have potential use in UV Optoelectronic applications. Also, we investigated the elastic constants (Ci j ), Young’s modulus (Y), Poisson’s ratio γ and anisotropic factor (A) and we found that our compound is elastically stable, ductile and rigid in nature.

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Keywords: First-principles calculations, Structural properties, Optoelectronic properties, Elastic constants, GGA, TB-mBJ.

1. Introduction

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Many perovskites-oxides with the nominal composition ABO3 are well known recently, and they reveal a large class of materials that display a remarkable variety of interesting technological applications, since they show many attractive properties such as ferroelectricity [1, 2], semiconductivity [3, 4], superconductivity [5], ionic conduction characteristics [6], piezoelectric [7] and antiferroelectric [8]. Up to now, these materials have been broadly used in many fields, such as spintronic devices, optical wave guides, laser-host crystals, high temperature oxygen sensors, surface acoustic wave devices and dynamic random access memories. As a consequence of the various motivating properties mentioned above, the perovskites have been intensively explored theoretically and experimentally. Among these materials, one of the most interesting is the antiferroelectric perovskite oxides such as PbHfO3 single crystal. Since its discovery in 1953 [9], only a few studies have been done about the synthesis of PbHfO3 , despite the fact that it is structurally similar to lead zirconate PbZrO3 at room temperature [10, 11, 12]. For PbHfO3 , structural studies are very limited, including only simple symmetry investigations (Dernier and Remeika [13]), analysis by polycrystalline X-ray diffraction (Zaitsev et al. [14]) and high-temperature induced phase transitions (Shirane and Pepinsky [9]; Leont’ev et al. [15]). To the best of our knowledge there are no systematic studies on the optoelectronic, and elastic properties of PbHfO3 . Furthermore, there are a few theoretical reports about this compound by density functional theory (DFT), although the first-principles calculations offer one of the most powerful tools for carrying out theoretical studies of these properties. The aim of the present work is to report the results of a systematic theoretical study of PbHfO3 compound. So, we have carried out first-principle calculations to study the different ground states properties using the full-potential Email address: [email protected] (D. Chenine) Preprint submitted to Chinese Journal of Physics

September 8, 2017

Figure 1: Crystal structure of PbHfO3 .

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linearized augmented plane wave method (FP-LAPW), within GGA and TB-mBJ approaches in order to compare between them. The calculations are performed at the experimental lattice constant a0 =4.13 A [13]. Firstly, we studied the ideal cubic perovskite structure in ferromagnetic (FM), anti-ferromagnetic (AFM) and non magnetic (NM) configurations. In Sect. 2 of this paper, we describe the calculation method as well as we discuss our results in Sect. 3. Finally, we will give conclusions in Sect. 4. 2. Computational methodology

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The calculations reported here were done by means of the Full Potential Linearized Augmented Plane Wave (FP-LAPW) method [16] within the frame work of the density functional theory (DFT) [17] as implemented in the wien2k software [18]. The exchange-correlation functional was performed by the generalized gradient approximation (GGA) in the scheme of Perdew-Burke-Ernzerhof (PBE) [19]. Since the GGA usually underestimates the band gap of materials, we have employed the Tran-Blaha modified Becke-Johnson potential (TB-mBJ) [20, 21, 22, 23, 24] which has been shown to improve efficiently the computations of most properties calculated within FP-LAPW method, and more particularly, the electronic and optical properties [25]. In FP-LAPW method, the space is partitioned into nonoverlapping muffin-tin (MT) spheres separated by an interstitial region (IR). The region of nonoverlapping atomic spheres is treated as a linear combination of radial functions times spherical harmonics, whereas in the interstitial region a plane wave expansion is used. In all configurations studied the non-spin polarized (non-magnetic), anti-ferromagnetic and ferromagnetic, the parameter Rmt × Kmt is equal to 8 (where Rmt is the smallest of all atomic sphere radii and Kmax is the maximum value of the wave vector K) and the expansion of the wave functions was set up to lmax = 10 inside of muffin-tin. We have chosen the muffin-tin radii to be 2.50, 1.98 and 1.71 a.u. for Pb, Hf and O, respectively. The integration of the full Brillouin zone is performed with 11×11×11 k-points mesh, which corresponds to 56 k-points in the irreducible wedge. The self-consistent cycles calculations are considered to be converged when the total energy is stable within 0.1mRy (10 −4 Ry). And the value of Gmax =14, where Gmax is defined as the magnitude of largest vector in charge density Fourier expansion. The cut off energy, which defines the separation of valence and core states, was chosen as -10 Ry. For the cubic crystal phases with 5 atoms per unit cell, PbHfO3 belong to the idealized cubic structure with space group Pm3m, the atomic distribution is as follows: Pb atom is located at cube corner position (0, 0, 0), Hf atom at body centre position (1/2, 1/2, 1/2), and three O atoms at face centered positions (1/2, 1/2, 0); (1/2 ,0, 1/2); (0, 1/2, 1/2) forming a regular octahedron as shown in Fig. 1. At low temperatures, we specify that PbHfO3 assumes an orthorhombic structure with space group Pnma at 0 GPa pressure (see Fig. 2). But from 23 GPa, the material is cubic even for low temperatures as shown in Fig. 2.

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Pnma Pm3m

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PbHfO 3

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Figure 2: The variations of the total energy as function of volume for Pm3m and Pnma phase of PbHfO3 .

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3. Results and discussions 3.1. Structural properties

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The PbHfO3 have been studied in cubic phase and the obtained results for the equilibrium structural parameters are computed by fitting the data for energy versus volume into the birch-Murnaghan’s equation of state (EOS) [26]. Table.1 summarizes the calculated minimum energy E0 , the equilibrium lattice constant a0 , the bulk modulus B0 and its pressure derivative B00 , for ferromagnetic (FM), anti-ferromagnetic (AFM) and non-magnetic (NM) configurations. The optimization plots of the calculated total energy versus reduced volume of each state using GGA-PBE approximation are given in Fig. 3. We can see from this figure that the performed calculations predict the non-magnetic order (NM) to be the more stable since it has the lower energy. The value of the lattice parameter for the non-magnetic phase of PbHfO3 is a0 = 4.16 A (see Table.1). As compared to the experimental data (a0 = 4.13A, see Dernier and Remeika [13]), our value corresponds to the lattice constant deviating by 0.72% from experiment, so it is in good agreement with the experimental. The lattice constant of our material is also calculated by ionic radii method using

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Table 1: Calculated structural equilibrium lattice constant a0 (in A), ground state energies (E0 ), bulk modulus B0 and its pressure derivatives B00 of the cubic PbHfO3 using the GGA calculation in comparison with analytical and experimental work. (a Dernier et al. [13], b Ubic [27] and c Shannon [28])

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a0 (A)

B0 (GPa)

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PbHfO3

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FM NM AFM — —

4.17 4.16 4.17 4.13a 4.21b

163.08 163.71 162.89 — —

4.56 4.56 4.53 — —

-72504.208718 -72504.208721 -72504.208720 — —

Pb 1.49 b Hf 0.71 b O 1.40 b — —

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Figure 3: The calculated total energy (Ry) as a function of volume (A3 ) in the cubic perovskite PbHfO3 with ferromagnetic (FM), anti-ferromagnetic (AFM) and non-magnetic (NM) states by using GGA approximation.

the following empirical formula [27]:

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a0 = α + β (rPb + rO ) + γ(rH f + rO )

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where α (0.06741), β (0.4905) and γ(1.2921)[27] are constants however; rPb , rH f and rO are the respective ionic radii of Pb cation, Hf cation and O anion in the appropriate coordination of number 12 and 06 for Pb/Hf. By using these parameters instead of tabulated given by Shannon [28], our calculated lattice constant by GGA as well as ionic radii method (a0 = 4.21 A) is 1.90% larger than the experimental value. This overestimation may be due to the fact that the theoretical lattice constant has been calculated in the cubic phase at 0 K. From Table.1 we can see that our results are in good agreement with theoretical results found by Dernier and Remeika [13] and Guang-Xin et al.[29]. For the bulk modulus B0 there is no experimental data; hence our work will be a remarkable indication for further investigations.

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3.2. Electronic band structure and density of states (DOS) The electronic band structure of PbHfO3 compound are calculated at the theoretical lattice constant along highsymmetry directions in the first Brillouin zone, as shown in Fig. 4. To solve the problem of underestimation of the band-gap, we have employed the TB-mBJ approache without spin orbit configuration. The calculated electronic band structures using both methods are found to be similar except for the the value of their band gap as presented in Fig. 4. Using GGA approximation, the valence band maximum (VBM) is 0 eV at the M point but the conduction band minimum (CBM) is 2.59 eV at the X point; which confirms that PbHfO3 has an indirect band gap equal to 2.59 eV. However, by using the TB-mBJ, the conduction band minimum (CBM) as well as the valence band maximum (VBM) are located at the same symmetry point (X-X). Therefore, PbHfO3 has a direct band gap of 3.75 eV which is in good agreement with a previous theoretical study of Guang-Xin et al. [29] and very closer to the experimental one (3.4eV) as obtained in the work of Prokopalo et al. [30]. Using TB-mBJ approximation, our results agree well with the experimental data compared to GGA results. To further elucidate the nature of the electronic band structure, we have also calculated the total (TDOS) and partial densities of states (PDOS) of PbHfO3 . These are displayed in Fig. 5. The Fermi level EF is set at 0 eV. The states Pb (5p6 , 4f14 , 5d10 , 6s2 , 6p2 ), Hf (5s2 , 5p6 , 4f14 , 5d2 , 6s2 ), and O (2s2 , 2p4 ) are treated as valence electrons. In Fig. 5, it can be seen that the peak situated around -6 eV and -5.6 eV by GGA and TB-mBJ, respectively, consists of Pb s orbitals. The states located between -4.5 eV (GGA) and -4.4 eV (TB-mBJ) up to 0 eV are mainly due to O 2p 5

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Figure 4: The calculated band structures for PbHfO3 by using GGA and TB-mBJ for non-spin polarized state.

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Figure 5: The calculated GGA and TB-mBJ total and partial density of states of cubic PbHfO3 .

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electrons, with small contribution of Hf 5d electrons, while the upper valence bands with a width of 2.5 eV (GGA) and 3.6 eV (TB-mBJ), are essentially formed by Pb 6p states and Hf 5d states. The positions of these peaks are in good agreement with the results of Luo et al. [31]. The conduction bands ranging from 2.5 to 7.9 eV (GGA) and 3.8 to 8.8 eV (TB-mBJ) are composed predominantly of Pb 6p states. Furthermore in Fig. 5, the region beyond 7.9 eV (GGA) and 8.8 eV (TB-mBJ) is due to the Hf d-eg states. O 2p states and Hf d-t2g states also exhibit a strong hybridization character as is discussed by [31]. In addition, the Hf d-t2g state is also discovered with a major contribution to the conduction bands, as illustrated in Fig. 5. So in this case the PbHfO3 have a semi-conductor behaviour and its density of states is dominated by the Pb-s, Hf-d (d-eg , d-t2g ) and O-p.

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3.3. Charge densities

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In order to give more information on the bonding system, the electronic charge densities are calculated. The distribution of charge density in the (101) plane is shown in Fig. 6. From this figure, we find that there is covalent bonding between hafnium and oxygen Hf-O with a weak degree of ionicity, this bonding is due to the interaction of charges between Hf -d and O-p hybridization effect. The near spherical charge distribution around the Pb site is negligible and as a result the Pb atom is fairly isolated which could indicate that the bonding between Pb and HfO3 is mainly ionic. Therefore, we find that in cubic PbHfO3 , there is a bonding anisotropy as reported by Kuroiwa et al. [32]. The ionic bonding is also present due to the oxygen elements. A difference between the GGA and TB-mBJ is always present; we see clearly that the TB-mBJ charge density is denser than GGA. GGA

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Figure 6: Contour plot of the total valence charge density in the (101) plane for PbHfO3 within GGA and TB-mBJ approaches.

3.4. Optical properties

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The knowledge of optical properties of cubic PbHfO3 compound is important for optoelectronic devices. We have investigated the optical properties using TB-mBJ approach, because of the accurate band gap [25]. The optical properties can be obtained from the complex dielectric function ε(ω) = ε1 (ω) + iε2 (ω), where ω is the angular frequency. The real transition between the occupied and unoccupied electronic states commonly depends on the imaginary part ε2 and we can derive the real part ε1 (ω) of the dielectric function using the Kramers-Kroning relations. In general, there are two ways of contributions to ε(ω) namely intraband and interband transitions. The intraband transition is important only for metals [33]. The calculation of the optical reflectivity, refractive index and absorption coefficient are represented in Figs. 7 and 8 in the energy range up to 40 eV. For more details about these properties we refer to [34]. The dispersion of real part ε1 (ω) of the dielectric function is reported in Fig. 7a. At zero energy, the static dielectric constant at zero frequency ε1 (0) which is inversely related to the band gap [35] is equal to 4.80. The curve starts increasing from this frequency, attains its maximum value at 5.05 eV, and then decreases to attain negative values which implies that the incident electromagnetic radiation are reflected from the medium and the compound has a metallic behaviour in these energy ranges [36, 37]. Fig. 7b shows the variation of the ε2 (ω) spectra, there are mainly four peaks 5.40, 7.14, 11.00 and 35.79 eV. The first peaks (A) is due to the transition of electrons from 2p-O states of the valence band (VB) to the 6p-Pb state in the conduction band (CB). The second peak (B) appear due to 7

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3.5. Elastic properties

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the electronic transition of 2p-O states of the VB to the Hf-5d-t2g in the CB. The third peak (C) is due to the transition of 6s-Pb states of the VB to the Hf-d-t2g in the CB and the last peak (D) comes from the transition from 2p-O states in the valence band to the Hf-deg in the conduction band. The calculated reflectivity R(ω) is displayed in Fig. 8a. We remark that the minimum reflectivity occurs in the energy range from 0 - 9.75 eV. The spectra exhibits high reflection peaks at energies 5.31; 7.30 and 8.96 eV similar to those in the dielectric function. In addition, the static refractive index n(ω) √ can be estimated as square root of the dielectric function at zero energy level, it can be inferred that nT B−mBJ = 4.80 = 2.19n as displayed in Fig. 8b. The refractive index increases in the transparent region. In Fig. 8c, we present the variation of absorption coefficient as a function of energy. The first critical point occurs at 3.75 eV (TB-mBJ), which is associated to the fundamental band gap, as was ever notice in Fig. 4, this energy gap XV -XC originates from a transition from 2p-O states of the VB to the Hf-5d-t2g state in the CB. It is clear that a wide range of absorption region is observed with high absorption peaks at high energies. Therefore our material could be potential candidate for ultraviolet optoelectronic applications.

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In order to confirm the stability of the studied compound, the elastic constants have been calculated using the stress-strain method. These constants are important in providing important informations for many technological applications. They also describe the resistance of material against external force. The elastic constants contain significant informations which can be obtained from ground state total energy calculations [38]. For a Cubic system, there are only three independent elastic constants Ci j , namely C11 , C12 and C44 . Table. 2, lists the calculated elastic constants and elastic moduli at the level of GGA-PBE. To the best of our knowledge; no experimental or theoretical data are available in the literature for comparison. Therefore, we hope that our results could provide reference data for future investigations. It is noticed that for a cubic crystal, the necessary mechanical stability conditions should be (C11 − C12 > 0); (C11 + 2C12 > 0; C44 > 0 and C12 < B0 < C11 [38]. The obtained Ci j values are summarized in Table.2, the above criteria are satisfied. So PbHfO3 is mechanically stable. We remark that the calculated bulk modulus from the elastic constants such as B0 = (1/3)(C11 +2C12 ) has nearly the same value as obtained from fits of Birch Murnaghan equation of state EOS (see Table.1). In addition, we have also investigate other macroscopic parameters such as the anisotropy factor A, shear modulus G, the Poisson’s ratio γ and the Young’s modulus Y . They are calculated from the elastic constants according to the following expressions [39]: A=

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Figure 8: The calculated: (a) reflectivity R(ω), (b) refractive index n(ω) and (c) absorption coefficient of PbHfO3 compound.

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9B0 G 3B0 + G 3B0 −Y γ= 6B0

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The Reuss and Voigt limits [40, 41, 42] for the Shear modulus G are given by 1 GV = (C11 −C12 + 3C44 ) 5 5C44 (C11 −C12 ) 4C44 + 3(C11 −C12 ) G=

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Table 2: Calculated elastic constants Ci j and the bulk modulus B0 (in GPa), anisotropy factor A, shear modulus G (in GPa), Young’s modulus Y (in GPa) and the Poisson’s ratio γ of the cubic PbHfO3

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First, if we search the ductile or brittle nature of our material, we must calculate two parameters which provide us the difference: the ratio B0 /G and Cauchy pressure (C12 −C44 ) given by [43, 44]. So If (C12 −C44 ) < 0, the material is considered to be brittle; whereas if (C12 −C44 > 0), the material is ductile which is the case of our compound. We know that there is a limit value of B0 /G ratio which separates the ductility and brittleness. As a rule, the material will be brittle (ductile) if B0 /G ratio is low (high) than 1.75. The value of the B0 /G for PbHfO3 in cubic phase is found to be 1.76 this indicates that our material is ductil in nature. The value of Poisson’s ratio γ provides information about characteristic of the bonding nature of solid materials, it has been reported by Haines et al. [45] that for central force solids the lower and upper limit values are 0.25 and 0.5 respectively. The Poisson’s ratio γ for covalent materials has a small value; 0.1 and is larger than or equal to 0.25 for ionic compounds. The calculated γ value for PbHfO3 compound is 0.12, which confirms the ionic bonding for this compound found before in the study of charge densities. We have also calculated the elastic anisotropy factor A . If it has a value equal to unity the material is isotropic, while any value smaller or greater than unity indicates anisotropy. From Table.2, we can see that the value of the anisotropy factor A is 0.58 with GGA approach. This indicates that our material is anisotropic in nature. Furthermore, we have the Young’s modulus (Y ) (indicates 386.98 GPa), which is used to provide a measure of the stiffness of the solid, higher the value of Young’s modulus (Y ) stiffer will be the compound. 4. Conclusion

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In summary, we have performed first-principles calculations, within FP-LAPW method to evaluate the structural, optoelectronic and elastic properties of PbHfO3 perovskite compound. This compound has a cubic crystal structure with space group Pm3m for pressures higher than 23 GPa. The calculated results indicate that the non-magnetic ordering is energetically more favorable than ferromagnetic and anti-ferromagnetic phases. For the electronic band structure and the density of states the TB-mBJ approximation improves the value of the band gap comparing to the GGA, and we obtained a value which is close to the experimental one. Moreover, PbHfO3 has a semiconducting behavior. The calculated optical properties indicate that this compound has a considerable absorption in wide range of photon energies; therefore it can be effectively used in UV based optoelectronic devices. Finally, the calculated elastic constants obey to the cubic stability conditions and according to B0 /G ratio our material is ductile in nature. So, we are not aware of any published data for the optical and elastic properties of PbHfO3 consequently, our results may be considered as predictions. 10

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References References

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