First-principles calculations of structural, electronic ...

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JOURNAL OF PHYSICS D: APPLIED PHYSICS

J. Phys. D: Appl. Phys. 41 (2008) 000000 (6pp)

doi:10.1088/0022-3727/41/1/000000

First-principles calculations of structural, electronic and optical properties of orthorhombic CaPbO3 J M Henriques1 , C A Barboza1 , E L Albuquerque1,4 , E W S Caetano2 , V N Freire3 and J A P da Costa3 Departamento de Física, Universidade Federal do Rio Grande do Norte 59072-970 Natal-RN, Brazil Centro Federal de Educaça˜ o Tecnológica do Ceará, Avenida 13 de Maio 2081, Benfica, 60040-531 Fortaleza, Ceará, Brazil 3 Departamento de F´ısica, Universidade Federal do Ceará, Centro de Ciências, Caixa Postal 6030, Campus do Pici, 60455-760 Fortaleza, Ceará, Brazil 1 2

E-mail: [email protected]

Received 24 September 2007, in final form 17 December 2007 Published Online at stacks.iop.org/JPhysD/41 Abstract Calculations within the density functional theory approach were performed to obtain structural parameters, electronic band structure, carrier effective masses and optical absorption spectra in orthorhombic CaPbO3 . Both local density and generalized gradient approximations, LDA and GGA, respectively, were considered. A comparison reveals good agreement of the calculated lattice parameters with experimental results. A direct → one-electron energy band gap of 0.84 eV (0.94 eV) was obtained within the GGA (LDA) level of calculation, in contrast to a previous interpretation of experimental data pointing to a gap of only 0.43 eV. Confronting our results with band gap energies previously obtained for CaXO3 crystals (X = C in calcite, X = Si in wollastonite and X = Ge,Sn,Pb in the orthorhombic phase), we note that the energy gap oscillates but with an overall trend to decrease, as the atomic number of the X atomic species increases.

1. Introduction

On the other hand, structural and electrical properties of perovskite calcium plumbate (CaPbO3 ) have been investigated to improve the understanding of the role of PbO6 octahedra in copper-free superconducting oxides [9]. Ab initio density functional theory (DFT) calculations of the structural, electronic and optical properties were already performed for all CaCO3 polymorphs (calcite [10], aragonite [11] and vaterite [12]), triclinic CaSiO3 [13], orthorhombic CaGeO3 [14] and CaSnO3 [15], considering both local density and generalized gradient approximations (LDA and GGA, respectively). However, no similar result is known, to the best of the authors’ knowledge, for orthorhombic CaPbO3 . The calculated band structures for the above-mentioned CaXO3 compounds within the LDA approximation reveal an oscillating behaviour of the main band gap as we switch the X element down through group 14 of the periodic table, but with a trend to decrease the band gap for heavier elements according to the sequence CaCO3 calcite (∼5.0 eV)→ CaSiO3 (∼5.4 eV)→ CaGeO3

The past years witnessed strong interest in the CaXO3 (X = C, Si, Ge, Sn, Pb) class of materials due to their role in biological (calcite and aragonite CaCO3 are the main constituents of mollusk shells) and Earth sciences (between 6% and 12% of the Earth lower mantle weight are constituted of perovskite CaSiO3 [1]) and promising industrial applications (plastic, rubbers, papers, paints, etc) [2]. The production of calcitebased implants was already reported [3–5] and CaSnO3 was proposed as a high capacity anode material for Li-ion batteries [6] and phosphorescent material for traffic signs, interior decoration and light sources [7]. CaGeO3 , quenched to ambient condition, has been studied mainly to understand the mechanisms behind the stabilization of the various perovskite structures [8]. 4

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(∼2.3 eV)→ CaSnO3 (∼2.9 eV). Whether this band gap shrinking will go on for CaPbO3 is a question to be answered. It is important also to note that only the energy gap of calcite was measured (by reflection electron energy-loss spectroscopy in surface), and its value is 6.0 ± 0.35 eV [16], about 20% higher than the value estimated by first-principles calculations [10, 17]. CaPbO3 presents two crystalline variants [9]: one resulting from thermal decomposition of CaPb(OH)6 at low temperatures 400–500 ◦ C is a trigonal form with an ilmenitetype structure; the other, which is the focus of this work, is an orthorhombic perovskite phase with space group Pbnm and Z = 4 molecules per unit cell. This phase is more interesting due to its similarity to BaPbO3 and SrPbO3 crystals (mother phases of superconducting oxides) and was synthesized from a mixture of Ca2 PbO4 and PbO2 at high temperature and pressure. In the case of orthorhombic CaPbO3 , Yamamoto et al [9] suggested that its band structure is close to those of BaPbO3 and SrPbO3 due to their structural similarity. They have estimated an energy gap of 0.43 eV for orthorhombic CaPbO3 by assuming thermal-activation-type conduction and have argued that its energy gap arises from the decrease in the overlap of Pb-6s and O-2p orbitals (in comparison with an undistorted structure) as a result of marked structural distortion and possibly from the increase in the covalency of the Pb–O bonds. The semiconductive character of CaPbO3 is revealed by its calculated energy band gap of 0.84 eV (GGA) and 0.94 eV (LDA); besides the band structure shows the absence of overlapping bands in the Fermi level region. This conclusion is reinforced when we consider the increasing tilting of the PbO6 octahedra with decreasing ionic radius at the A site [18]. It is the purpose of this work to carry out ab initio quantum mechanical calculations of the structural, electronic and optical properties of orthorhombic CaPbO3 . Our theoretical model is based on the realms of DFT [19, 20], aimed to put forward first-principles based estimates for its energy gap.

Figure 1. Crystalline unit cell of orthorhombic CaPbO3 . Top: ball and stick model showing different views of the unit cell. Bottom: tilted PbO6 octahedra revealed.

in the DFT Hamiltonian. The absence of such contribution, we believe, is only relevant for the electronic band structure, not for the structural calculations. Therefore, the electronic band structure presented in this work should be considered with some caution. On the other hand, we note that there is a report [27] on the DFT band structure of functional oxides using both the screened exchange and the weighted density approximation which calculated with good accuracy the band gap of PbTiO3 without taking into account the spin–orbit interaction. The valence states taken into account in our calculations were Ca-3s2 3p6 4s2 , Pb-5d10 6s2 6p2 and O-2s2 2p4 . The planewave basis set cutoff energy was 500 eV. A Monkhorst–Pack scheme for integration in the first Brillouin zone [28, 29] with a 4 × 4 × 3 grid was chosen, and the total and partial densities of states (DOS) were computed according to [29–31]. Aiming to find the atomic positions and cell unit parameters that minimize the total energy, a cycle of geometry optimizations was performed using the Broyden–Fletcher– Goldfarb–Shannon algorithm [32]. CASTEP does not include some additional contributions to the optical matrix elements when ultrasoft potentials are used, so we have adopted norm-conserved pseudopotentials to calculate the optical properties. A self-consistent iteration was performed with these norm-conserved pseudopotentials and a plane-wave basis set cutoff energy of 900 eV. The complex dielectric function, (ω) = 1 (ω) + i2 (ω), 1 (ω) and 2 (ω) being its real and imaginary parts, respectively, and the optical absorption α(ω) of orthorhombic CaPbO3 were calculated according to [13–15]. Effective masses at the extrema of the valence and conduction bands were estimated by quadratic interpolation of the corresponding band curves [14, 15].

2. Computational details The CaPbO3 perovskite lattice parameters and atomic positions obtained by the Rietveld method [9] were used to prepare the input files of geometry optimization. As seen in figure 1, orthorhombic CaPbO3 is built up from tilted and distorted PbO6 octahedra. The calculations were carried out using the CASTEP code [21]. Core electron states were not explicitly taken into account, being replaced by ultrasoft Vanderbilt pseudopotentials [22], while valence states were expanded in a plane-wave basis set. Both LDA and GGA exchangecorrelation potentials were considered in our simulations. In particular, the GGA approach tries to improve over the LDA, which tends to perform well only for systems where the electron density varies slowly [23]. In this work, the LDA exchange-correlation energy follows the CA-PZ parametrization [24, 25] while the GGA approach is performed using the Perdew–Burke–Ernzerhof (PBE) [26] exchangecorrelation functional. As Pb is a heavy element, we can expect a significant degree of spin–orbit coupling. The CASTEP code, however, does not include any spin–orbit energy term

3. Numerical results We present the lattice parameters and unit cell volume for orthorhombic CaPbO3 in table 1, including the experimental Rietveld refinement from neutron diffraction data for the 2

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sake of comparison. LDA results show an underestimation of the lattice parameters a and c by −0.87% and −0.44%, respectively, which is consistent with the well-known LDA overbinding trend. The b lattice parameter, however, is practically the same obtained from the experimental measurements. The LDA unit cell volume is smaller by −1.28% in comparison with the experimental data. The GGA results, on the other hand, predict a larger unit cell (GGA approaches tend to overestimate interatomic distances and unit cell size), with lattice parameters a, b and c larger than the experimental ones by +1.32%, +1.67% and +2.09%. The unit cell volume is larger by +5.16%. The electronic partial density of states (PDOS) for orthorhombic CaPbO3 , considering the GGA approximation is shown in figure 2. The Fermi level is taken to be 0 eV. There are four energy bands at −37.6 eV originating from Ca-3s levels (left side of figure 2), 12 bands near −19.0 eV with the main contribution from Ca-3p levels and 32 bands between −17.0 and −13.0 eV mainly due to Pb-5d levels. There are 36 valence bands from −7.6 to 0 eV, originating from O-2p states, with smaller contributions from Pb-6s (between −7.5 and −5 eV) and Ca-3d levels (between −2.5 and 0 eV, see the right side of figure 2). Close to the Fermi level, O-2p and Pb-6s contributions can be noted, with the former much larger than the latter, indicating a small overlap between these electronic states at the top of the valence band, as suggested by Yamamoto et al [9]. Looking at the conduction band, 12 energy bands were calculated. The first bands, approximately 1 eV above the Fermi level, originate mainly from Pb-6s and O-2p states. For energy bands above 5 eV, the main contribution is from

Ca-3d and Pb-6p orbitals, being the former contribution more pronounced. As we observed for CaSnO3 [15] and CaSiO3 [13], there are Ca-3d contributions to the conduction bands of CaPbO3 , a few electron volts above the Fermi level. This should be compared with the absence of such contributions in orthorhombic CaGeO3 [14]. As we will show, the existence of these Ca-3d bands leads to increased optical absorption in the 6–8 eV range (absent in CaGeO3 ). The band structure was determined according to the orthorhombic Brillouin zone [33]. A close-up of the band structure near the main band gap is shown in figure 3, where GGA (solid) and LDA (dotted) curves are shown. Our results indicate that orthorhombic CaPbO3 has a direct band gap of 0.84 eV (GGA) and 0.94 (LDA). An indirect band gap of 0.85 eV (0.98 eV) is observed between the S point at the valence band and the point at the conduction band, for both GGA and LDA data, being very close to the direct transition energy threshold. Due to the approximations assumed in the construction of the DFT functionals, the calculated band gaps are very rough and in general much smaller than the experimental data. This should be compared with those estimated by Yamamoto et al who proposed a band gap of only 0.43 eV for orthorhombic CaPbO3 . As the DFT band gap is always smaller than the real band gap, we presume that the assumption of Yamamoto and its coworkers (on the existence of thermal-activation-type conduction in CaPbO3 ) is not correct or incomplete at least. Further experimental studies, therefore, are necessary to clarify this matter. Nevertheless, we can infer that there is a trend in the energy band gap behaviour of CaXO3 (X = C, Si, Ge, Sn, Pb) compounds to decrease (but oscillating) as the atomic number of the X species increases, changing their character from insulator to semiconductor [10, 13–15]. Indeed, the semiconductor character of CaPbO3 is reinforced by the effective mass calculations presented in table 2. Holes are heavier than electrons and their masses are very anisotropic. Electron masses, on the other hand, are much less anisotropic according to the LDA calculations. However, except for those masses along the directions → T,

Table 1. Lattice parameters and unit cell volume for orthorhombic CaPbO3 using the LDA and GGA approaches. Experimental data are shown for comparison.

LDA GGA Exp.

a (Å)

b (Å)

c (Å)

V (Å3 )

5.62 5.74 5.67

5.88 5.98 5.88

8.11 8.31 8.15

268.62 286.15 272.09

Figure 2. CaPbO3 partial density of states (PDOS) for the full electronic energy spectrum (left) and near the main band gap (right), showing the contributions from each atomic species. Solid, dashed and dotted lines correspond to s, p and d contributions, respectively.

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Table 3. Carrier effective masses of orthorhombic CaPbO3 along a few chosen high symmetry directions. All masses are given in terms of the free electron mass m0 . LDA

–Z –T –Y –U –X –R –S

Figure 3. Band structure near the main band gap for orthorhombic CaPbO3 for both GGA (solid) and LDA (dashed) calculations.

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Exp.

LDA

GGA

O1 –Pb–O2 O1 –Pb–O (i) O1 (iii)–Pb–O2 O2 –Pb–O2 (i) O2 (i)–Pb–O2 (ii)

86.35 92.38 93.64 88.94 91.05

85.67 93.06 94.32 88.61 91.38

86.06 93.30 93.93 88.32 91.67

Electron

Hole

Electron

Hole

0.32 0.58 0.47 0.42 0.48 0.52 0.51

2.60 2.30 1.50 1.34 0.72 1.07 1.05

0.41 0.62 0.72 0.73 0.48 1.18 0.53

2.66 2.35 1.87 1.60 1.05 2.25 1.65

band (in this region, electronic quantum states originate mainly from Pb-6s levels). For energies above 6 eV, transitions involving mainly O-2p valence states to mainly Ca-3d conduction states abruptly increase the absorption intensity (see figure 2). The same behaviour was observed in all previously studied CaXO3 compounds, except for CaGeO3 , which lacks Ca-3d bands [11, 13–15]. Distinct absorption peaks appear around 3.5, 4.6, 4.8 and 7.7 eV. In general, optical absorption, as a function of electric field alignment with respect to crystalline axes, for the electronic energies considered in this work, is more intense in the [1 1 1] crystal direction, followed by the [1 1 0] and [1 0 0] ones. Finally, in figure 5 we see the real and imaginary parts of the calculated dielectric function for a polycrystalline sample of CaPbO3 according to the GGA approach. The imaginary part of the dielectric function is related to the optical absorption previously discussed. The real part, on the other hand, is related to the response of electronic dipoles in phase with incident electromagnetic radiation on CaPbO3 . The most intense resonance of Re(ε) occurs for 0.8 eV, at the onset of absorption. A secondary peak can be seen at 1.2 eV. Smaller and isolated peaks appear at 4.2, 5.0 and 6.2 eV. Re(ε) becomes negative in the 7.0–11.0 eV range, with two distinct peaks at 7.7 and 8.9 eV. For values grater than 8.9 eV, we see smaller peaks at 14.1, 21.2, 23.2 and 25.5 eV. In the limit of zero energy, Re(ε) tends to 6.8. In the visible spectrum energy range (between 2 and 3 eV), Re(ε) varies smoothly from 6 to 2.2 approximately.

Table 2. Selected bond angles (degree) in PbO6 octahedra, considering the following symmetry codes: (i) x, y, −z + 1/2, (ii) x − 1/2, −y + 1/2, z + 1/2, (iii) x + 1, y, z. Bond

GGA

→ X and → S, GGA electron effective masses are clearly larger and anisotropic than the ones calculated using the LDA approach. In particular, along → R, the electron GGA mass is more than 100% larger than the LDA one. LDA electron masses starting from the point vary from 0.32 ( → Z) to 0.52 ( → R), while GGA electron masses vary from 0.41 ( → Z) to 1.18 ( → R). Indeed, a larger discrepancy between LDA and GGA effective mass data is always observed along the → R direction for both electrons and holes. The → R hole has a mass of 1.07 in the LDA calculation and 2.25 in the GGA calculation. We must remember that some caution is necessary with these estimates because DFT band structure calculations sometimes do not predict accurate values of carrier effective masses [34–37]. Yet, due to the lack of experimental data on the carrier effective masses of orthorhombic CaPbO3 , we hope our results will stimulate experimental efforts in this direction (table 3). The theoretically calculated optical absorptions for CaPbO3 taking into account different incident light polarizations ([1 0 0], [1 1 0] and [1 1 1], with respect to the crystalline axes), using the CASTEP software, are shown in figure 4. Light incident on a polycrystalline sample is also considered. The CASTEP mimics an experiment on a polycrystalline sample through an average over all polarization directions. The onset of absorption is almost linear for all directions considered, being more sharper for the [1 1 1] direction and smoother for the [1 0 0] and 110 directions, in increasing order, respectively. The polycrystalline sample follows closely the absorption behaviour along 100. We identify two distinct absorption regimes in CaPbO3 , for energies below and above 6 eV. For energies below 6 eV, absorption is weaker due to the small density of states at the bottom of the conduction

4. Conclusion We have calculated the structural, electronic and optical properties of orthorhombic CaPbO3 using first-principles calculations within the DFT framework. Geometry optimization was performed in both DFT-LDA and DFT-GGA levels. The GGA calculation overestimates the experimental lattice parameters of orthorhombic CaPbO3 by 2% at most. A much better agreement with experiment is obtained for the LDA calculations, with differences always smaller than 1% (in particular, for the b parameter the calculated and experimental values are practically the same). We compare this result with the difference between the LDA and experimental lattice parameters obtained in previous works for orthorhombic CaCO3 (maximum difference of 3.8% between theory and experiment for the b parameter) [11], triclinic CaSiO3 4

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is indirect, while for X = Sn, Pb it is direct. The CaSiO3 compound has the largest band gap in the family (in order to explain this we remember that the CaSiO3 gap was calculated for a triclinic phase, while the other materials considered are orthorhombic). Effective masses calculated in this work reinforce the belief that orthorhombic CaPbO3 is a semiconductor. Electron effective masses are smaller than the free electron mass and practically isotropic in the LDA framework, but show a very pronounced anisotropy according to the GGA data along the → R direction. Hole effective masses, on the other hand, are heavier and anisotropic in both approaches. The onset of optical absorption is practically linear for incident light polarized along the [1 0 0], [1 1 0] and [1 1 1] directions (relative to the crystal axes), rising more sharply along [1 1 1] and more smoothly for the [1 0 0] and [1 1 0] directions. The polycrystalline sample optical absorption mimics closely the absorption observed for 100-polarized incident light. For energies below 6.0 eV, the optical absorption intensity is smaller due to the scarcity of available states (mainly from Pb-6s levels) at the bottom of the conduction band. For energies above 6.0 eV, the absorption intensity increases due to the existence of conduction bands with a strong contribution from Ca-3d states in the 6–8 eV energy range. Such enhancement in optical absorption related to Ca-3d bands is also observed in CaCO3 aragonite [11], CaSiO3 [13], CaSnO3 [15], but not in CaGeO3 , where the Ca-3d bands are absent. Absorption intensities are stronger for light polarized along the [1 1 1] crystal direction, followed by the [1 1 0] and [1 0 0] ones. However, optical absorption anisotropy is not very pronounced in comparison, for example, with that calculated for CaCO3 aragonite [11] between the [1 0 1] and [0 1 0] directions. The real part of the dielectric function has its most intense peak at 0.8 eV, with a secondary peak at 1.2 eV, becoming negative in the 7.0–11.0 eV range. In the limit of zero energy, Re(ε) tends to 6.8 and in the visible spectrum energy range, Re(ε) decreases from 6 to 2.2.

Figure 4. Optical absorption of orthorhombic CaPbO3 calculated for different incident light polarizations and for a polycrystalline sample.

Figure 5. Real and imaginary parts of the dielectric function for a polycrystalline sample of CaPbO3 .

(-4.6%, b) [13], orthorhombic CaGeO3 (-1.9% a and c) [14] and orthorhombic CaSnO3 (-1.8%, a) [15]. So, the unit cell parameters of CaXO3 compounds calculated using the DFT-LDA approach are improved (with the exception of CaGeO3 ) as we move down through the corresponding column in the periodic table. The smallest energy gap between valence and band conduction is direct, corresponding for LDA and GGA calculations to 0.94 eV and 0.84 eV, respectively. An indirect band gap of 0.85 eV (GGA) and 0.98 eV (LDA) between the S point at the valence band and the point at the conduction band is predicted. We compared this result with the value proposed by Yamamoto et al of only 0.43 eV. As the DFT band gap is in general smaller than the real band gap, we assume that the interpretation of experimental data presented by Yamamoto and its coworkers is incomplete or incorrect, probably due to the existence of defects or impurities in their samples. We can also conclude that the DFT-LDA energy band gap of CaXO3 compounds, as we move down through this group of atoms in the periodic table, tends to an overall decrease with some oscillation: 3.96 eV (X = C), 5.43 eV (X = Si), 2.31 eV (X = Ge), 2.92 eV (X = Sn), 0.94 eV (X = Pb) [11, 13–15]. We note also that for X = C, Si and Ge the main band gap

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Acknowledgments VNF and ELA are senior researchers from the Brazilian National Research Council CNPq and would like to acknowledge the financial support obtained during the development of this work from the grants CNPq-CTENERG 504801/2004-0 and CNPq-Rede NanoBioestruturas 555183/2005-0. JMH was sponsored by a Postdoctoral Fellowship from CAPES at the Physics Department of the Universidade Federal do Rio Grande do Norte.

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