Theoretical investigation of structural, electronic, dielectric and optical characteristics of cubic perovskite BaCeO 3

Structural, electronic and optical parameters of a cubic perovskite BaCeO 3 were calculated using FP-LAPW with WIEN2K code. The calculated band structure conﬁrmed the semiconducting behaviour of BaCeO 3 with an indirect band gap of 2.33 eV. The charge density distribution plot in the present study clearly indicates that a strong ionic bonding between Ba and O exists with a mixture of ionic and covalent bonding between Ce and O in the cubic phase BaCeO 3 . Further, the real and imaginary sections of the relative permittivity, index of refraction, coe ﬃ cients of absorption and reﬂection, energy loss function, coe ﬃ cient of extinction, and optical conductivity of BaCeO 3 were calculated using FP-LAPW in the photon energy range between 0 and 30 eV, and interesting results are reported. The maximum absorption of photons occurred at 17.33 eV, which is more favourable for optoelectronic devices application.


I. Introduction
A perovskite with general formula ABO 3 has a cubic lattice structure with a Pm3m space group. The utility of these materials in various sensing devices, solid oxide fuel cells (SOFC) and hydrogen separation membranes in renewable sources of the energy sector dragged the attention of many researchers [1][2][3][4][5]. BaCeO 3 perovskite is especially important since it could have relatively high proton and hole conductivity at high temperatures [6,7]. The crystal structure of BaCeO 3 changes with temperature in three steps [8]: orthorhombic Pmcn -orthorhombic Incn -rhombohedral F32/n -cubic Pm3m at transitional temperatures 563, 673 and 1173 K, respectively [9,10]. Studies of the use of differently doped barium cerates in a fuel cell are well conducted and relate the properties with their stability and transport mechanisms. However, optical property studies are neither explored nor reported to the best of our knowledge [11][12][13][14][15][16][17]. For instance, Zhang et al. [18] have reported the structural, lattice dynamical and thermal properties of orthorhombic, rhombohedral and cubic BaCeO 3 phases based on density functional theory. However, this research group did not report the dielectric and optical properties of BaCeO 3 .
In this paper, we present the theoretical investigation of structural, electronic, dielectric and optical properties of the cubic BaCeO 3 phase. Efforts have been made to identify the physical properties of BaCeO 3 using the full-potential linearised augmented plane wave method (FP-LAPW) in the density functional theory (DFT) by using WIEN2k code. The optical features include the relative permittivity with its real and imaginary values, index of refraction and coefficients of extinction and reflection. The remaining properties such as energy loss function, optical conductivity and absorption coefficient for the BaCeO 3 material are estimated with the firstprinciples approximation. To the best of our knowledge, no information is available in the literature about the dielectric and optical behaviours of cubic perovskite BaCeO 3 analysed by FP-LAPW using the WIEN2K code.

II. Computational method
In this work, we used DFT theory with WIEN2K code and all the calculations were conducted using FP-LAPW method and generalized gradient approximation (GGA) [19]. Both methods are powerful tools to estimate the chemical and physical properties of perovskite compounds, such as BaCeO 3 [20,21]. The structural, electronic and optical properties of cubic perovskite BaCeO 3 (Fig. 1) using full-potential linear augmented plane wave (FP-LAPW) method are investigated. The exchange-correlation energy within the local density approximation (LDA) and the generalized gradient approximation (GGA) are well-defined in the literature [22]. FP-LAPW includes the muffin-tin and the interstitial region between them and does not give any approximations about the estimations of the density of charge or potential. The core-to-valance levels partition was set at −9.0 Ry as cut-off energy. The radii of the spheres used for Ba, Ce and O were 2.2, 2.0 and 1.8 Å, respectively. The tetrahedron method helps to integrate the Brillouin zone [23] up to 35 k-points. The k-point sampling was converged to 0.0001 Ry for R MT K max = 7 in self-consistent calculations. In the plane wave expansion, the lowest value amongst all the muffin-tin radii was the R MT and the highest k-vector (radii) magnitude was K max . A relatively dense collection of 3500 k-points in the Irreducible Brillouin Zone (IBZ) was utilized to explore the sample's optoelectronic properties. The dielectric function, ε(ω) = ε 1 (ω) + i ε 2 (ω) was utilized to identify how BaCeO 3 responds to all the incident photons. The details of absorption obtained through the band structure of the material are present in the imaginary portion of the dielectric function and are mathematically given by [24,25]: for the dipole matrix M with initial and final states i and j, respectively. f i is the Fermi distribution function and E i is the i th level energy of electron with crystal wave propagation vector k. Using the Kramers-Kronig relation, the real portion ε 1 (ω) is extracted from the imaginary part [26,27]: where P is principal value integral.

Structural properties
Using the experimental value of lattice constant a 0 = 4.397 Å for cubic perovskite BaCeO 3 , the energy was optimized by using the Birch-Murnaghan equation taken from the literature [28], and found to be 34461.7243 Ry, as shown in Fig. 2. The equilibrium lattice constant a 0 is 1.72% higher than the experimental value, which is common in GGA. A similar effect has also been reported by Zhang et al. [29]. However, the obtained lattice constant (4.4729 Å) for cubic perovskite BaCeO 3 is slightly lower than the previously determined values of 4.49 Å [29] and 4.4788 Å [30].
The calculated values of the bulk modulus (B 0 ), pressure derivative (B ′ ) and ground-state energy (E 0 ) of BaCeO 3 are also summarized in Table 1.   Figure 3 shows the high symmetry lines within the Brillouin zone and Fig. 4 shows the density of states. Energy band structures in the above pictures give the electronic properties of BaCeO 3 . The ground state energy or zero energy corresponds to the valence band maximum (VBM) at the Γ point in the image. The conduction band minimum (CBM) is at the R point. Since CBM and VBM are at different symmetry points, the gap of 2.33 eV between them is indirect. As shown in   Moreover, it is identified that the hybridization between Ce and O is stronger than that between Ba and O and it is responsible for the covalent bonding (Fig.  5b). Thus, a strong ionic bonding between Ba and O exist with a mixture of ionic and covalent bonding between Ce and O in the cubic phase BaCeO 3 . On the other hand, the contour maps confirm that the Ce and O sites, and hence Ce-O bond direction have more charge along (100) and (110) planes, shown in 2D in Fig. 5a and 5b, respectively, and along (100) plane presented in 3D in Fig. 6.

Dielectric and optical characteristics
The internal structure of BaCeO 3 can be revealed by its optical properties (Figs. 7 and 8), which are studied   conductivity σ(ω) and absorption coefficient α(ω). All these parameters concern with the energy of photons in the 0-30 eV range. The imaginary part shows the threshold energy or the ground state energy E 0 = 2.34 eV and its absorption properties. In Fig. 7 Table 3. Values of refractive index n(0), reflectivity R(0), energy range for n(ω) < 1 for ω = 0, and n(ω), reflectivity R(ω) and optical conductivity σ(ω) of BaCeO 3  The lower part of Fig. 7 corresponds to the real portion of the dielectric function ε 1 (ω), associated with the compound's electronic polarizability. The dielectric constant starts from ω = 0 (with value 5.30), becomes maximum (8.53) at 2.82 eV and gradually falls to a negative value in the 13.32-24.03 eV and 25.57-30.00 eV ranges.
The minimum optical reflectivity R(0) is 15.45% and remains up to 3.07 eV, and the maximum is about 37.4% at 29.05 eV, as shown in Fig. 8c and Table 3. The loss of electron energy in the medium is L(ω) spectra, whose peak represents the plasma resonance at 24.27 eV, as shown in Fig. 8d. The optical conductivity σ(ω) starts from 3.06 eV and reaches a maximum at 17.23 eV with a value of 11178.2 Ω −1 cm −1 , as shown in Fig. 8e. The coefficient of absorption α(ω) behaves similarly up to 30 eV reaching a maximum at 17.33 eV, as shown in Fig. 8f.

IV. Conclusions
In this paper, the lattice constant, bulk modulus with its pressure derivative, the band energies, the density of energy levels and optical characteristics of a cubic perovskite BaCeO 3 were calculated using FP-LAPW with WIEN2K code. They agree well with the previously published literature. The indirect bandgap of BaCeO 3 was found to be 2.33 eV. Further, the real and imaginary portions of the relative permittivity, refractive index, reflection coefficient, energy loss function, absorption and extinction coefficients were thoroughly studied in the 0-30 eV energy range and agreed well with the previous ones.