# First-Principles Study of Electronic Structure and Optical Properties of Tetragonal PbMoO4

Authors: Qi-Jun Liu; Zheng-Tang Liu; Li-Ping Feng; Hao Tian
Journal: ISRN Condensed Matter Physics
 (2011)

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## Abstract

Using the plane-wave ultrasoft pseudopotential technique based on the first-principles density functional theory (DFT), we have studied the structural, electronic, chemical bonding, and optical properties of tetragonal PbMoO4. The obtained structural parameters are in good agreement with experiments. Band structure, density of states, and chemical bonding are calculated and shown. It is found that tetragonal PbMoO4 is an indirect band gap. The dielectric function, refractive index, extinction coefficient, reflectivity, absorption coefficient, loss function, and conductivity function are calculated for radiation up to 20 eV.


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## Body

## 1. Introduction

PbMoO4 has been the subject of great research interest both experimentally [1–8] and theoretically [9–13] due to its wide applications such as acousto-optic light deflectors, modulators, adjustable filters, surface acoustic wave devices, ionic conductors, and low-temperature scintillators, and  its superior properties such as high acousto-optic light figure of merit, low optical loss in the region 420–3900 nm, and good mechanical impedance for acoustic matching ([14–17] and the references therein).The crystal structure of tetragonal PbMoO4 belongs to the space group I41/a and the local symmetry C4h6. The dielectric constants [18], polarized reflectivity spectra [19], high-pressure Raman spectrum, and electrical properties [1, 7, 14] have been reported, which show that there is an anisotropy of optical properties and a transition from crystal to amorphous phase with increasing pressure. The framework of a fully relativistic self-consistent Dirac-Slater theory with a numerically discrete variational (DV-Xα) method [9, 13, 19], the ultrasoft pseudopotentials of generalized gradient approximations (GGAs) based on the density function theory (DFT) with CASTEP code [10, 12], the linearized-augmented-plane-wave method with WIEN97 code [11], and so forth have been used to study F type color centers, optical properties, electronic band structure, and so forth. Although these studied cases of PbMoO4 are well presented, it is not clear how electron transitions to influence optical properties is. Additionally, the chemical bonding of PbMoO4 should be explained.Hence, we study the structural parameters, electronic structure, chemical bonding, and optical properties of tetragonal PbMoO4 using the plane-wave ultrasoft pseudopotential technique based on the first-principles density functional theory. The rest of the work is organized as follows. In Section 2, we give a short description of the methods used in this paper. The results and discussion are shown in Section 3. We present our findings and give a brief summary in Section 4.


## 2. Computational Methodology

Density functional theory calculations are performed with plane-wave ultrasoft pseudopotential using the generalized gradient approximation (GGA) with the Perdew-Wang 1991 (PW91) functional [20] as implemented in the CASTEP code [21]. The ionic cores are represented by ultrasoft pseudopotentials for Mo, Pb, and O atoms. The Mo 4s24p64d55s1, Pb 5d106s26p2, and O 2s22p4 electrons are explicitly treated as valence electrons. The plane-wave cutoff energy is 380 eV, and the Brillouin zone integration is performed over the 5 × 5 × 6 grid sizes using the Monkhorst-Pack method for tetragonal structure optimization. This set of parameters assures the maximum force of 0.01 eV/Å, the maximum stress of 0.02 GPa, and the maximum displacement of 5.0×10-4 Å.


## 3. Results and Discussion

### 3.1. Geometry and Structure Optimization

The crystal structure of tetragonal PbMoO4 is shown in Figure 1. The optimized values of a and c for tetragonal PbMoO4 are listed in Table 1. The obtained structural parameters are in good agreement with the previous experimental data [2, 11, 16]. Of course, it can be seen that the GGA leads to overestimating our calculated data compared with experiments.Table 1
Calculated equilibrium lattice parametersa, c (in Å), and c/a compared with available experimental data [2, 11, 16] for tetragonal PbMoO4.
This workCastep GGA  (PW91)Expt. [11]Expt. [16]Expt. [2]a5.51735.424–5.43605.4335.418–5.464c12.330512.076–12.110712.11012.065–12.088c/a2.23492.226–2.229——Figure 1
The crystal structure of tetragonal PbMoO4.


### 3.2. Electronic Properties

The calculations of the electronic band structure along the symmetry lines of the Brillouin zone, the total and the partial density of states (DOSs and PDOSs) are shown in Figures2 and 3. The top of the valence band is taken as the zero of energy. In this compound, the valence band maximum (VBM) is located at 1 point (the valence band maximum of tetragonal PbMoO4 is not at high-symmetry point, but at the defined 1 point between X and Γ), whereas the conduction band minimum (CBM) is located at N point, resulting in indirect band gap of 2.838 eV. This value is in good agreement with the previous calculated data 2.59 eV [11] and 2.8 eV [12]. However, these results are all smaller than the experimental values of 2.94–4.7 eV [2] due to the well-known underestimation of conduction band energy in DFT calculations [22].Figure 2
Band structure of tetragonal PbMoO4 along with the high-symmetry points of the Brillouin zone.Total and partial density of states of tetragonal PbMoO4.
(a)(b)(c)(d)In order to further elucidate the nature of the electronic band structure, we have calculated and explained the DOSs and PDOSs. From the PDOSs, we can identify the angular momentum character of the different structures.  Structure (1) is mainly due to Mo-4 s electrons, structure (2) due to Mo-4p electrons, structure (3) due to O-2 s electrons, structure (4) due to Pb-5d electrons, structure (5) due to Pb-6 s electrons, structures (6) and (7) due to O-2p electrons with hybridization of Mo-4d electrons, and structure (8) due to O-2p electrons. The conduction bands are composed of Mo-4d and show the hybridization with O-2p, as well as the hybridization between Pb-6p and O-2p.To understand the chemical bonding of this material, we have poltted the charge density of (112) plane corresponding to the (1)–(8) located at Figure3 in Figure 4. The polt labeled (1) shows the isolated Mo-4 s, (2) the weak hybridization σ bonding between Mo-4p and O-2 s, (3) the weak hybridization σ bonding between O-2 s and Mo-4d, (4) the weak hybridization σ bonding between Pb-5d and O-2 s, (5) the weak hybridization σ bonding between Pb-6 s and O-2p, (6) and (7) the hybridization σ and π bonding between O-2p and Mo-4d, and (8) the nonbonding O-2Pπ. Hence, we can conclude that the bonding between Mo and O is mainly covalent and the bonding between Pb and O is mainly ionic. Additionally, the charge density of (220) plane and the results of population analysis have been shown in Figure 5 and Table 2, which are in good agreement with our analysis of chemical bonding.Table 2
Calculated results of population analysis.
AtomspdTotalCharge (e)O1.894.7806.68−0.68Mo2.246.514.0112.761.24Pb1.710.811012.521.48Total charge densities of (112) plane and charge densities located (1)–(8) in the (112) plane of tetragonal PbMoO4.Figure 5
Charge densities in the (220) plane of tetragonal PbMoO4.


### 3.3. Optical Properties

We need to calculate two dielectric tensor components to completely characterize the linear optical properties due to the tetragonal symmetry of PbMoO4. The imaginary and the real parts of the dielectric function are calculated using (1) [23–25]:(1)ε2(ω)=(4π2e2m2ω2)∑i,j∫〈i|M|j〉2fi(1-fi)δ(Ef-Ei-ω)d3k,ε1(ω)=1+2πP∫0∞ω′ε2(ω′)dω′(ω′2-ω2)
as well as the scissors operator approximation [25, 26] due to underestimating the energies of excitation with the density functional calculations. The good agreement with experiments are obtained for the optical properties like TiO2 [26], SrHfO3 [27], SrZrO3 [28], and HfO2 [29] using the scissors operator.Figures6 and 7 display the imaginary and the real parts of the dielectric function from (100) and (001) along with the calculated results from experimental data [19] for a radiation up to 20 eV. We can see that our results are consistent with the previous work [19]. The discrepancy between our results and the experiment [19] may be due to the different-temperature condition (0, K in our paper and 6, K in [19]). The imaginary parts exhibit four structures A–D of (100) and E–H of (001). Structures A and E originate mainly from transitions of O-2pπ into the conduction bands, and structures B and F from transitions of hybridization π bonding between O-2p and Mo-4d into the conduction bands, and structures C and G from transitions of hybridization σ bonding between O-2p and Mo-4d into the conduction bands, and structures D and H from transitions of Pb-6 s into the conduction bands. The calculated static dielectric constants are 5.337 and 4.910 from (100) and (001).Calculated imaginary parts of complex dielectric function of tetragonal PbMoO4 as well as calculated results from experimental data [19].
(a)(b)Calculated real parts of complex dielectric function of tetragonal PbMoO4 as well as calculated results from experimental data [19].
(a)(b)The refractive index and the extinction coefficient are displayed in Figure8. The static refractive index is found to have the values 2.310 and 2.216 from polarization vectors (100) and (001), which are in agreement with experimental data 2.28 and 2.40 [18] from (100) and (001). Figure 9 shows the calculated results on the reflectivity, absorption coefficient, loss function, and complex conductivity function from polarization vectors (100) and (001). We hope the calculated values can help to offer a theoretical basis for the experiment and application of tetragonal PbMoO4.Calculated refractive index and extinction coefficient of tetragonal PbMoO4 from (100) and (001).
(a)(b)Calculated reflectivity, absorption coefficient, loss function and complex conductivity function of tetragonal PbMoO4 from (100) and (001).
(a)(b)(c)(d)(e)


## 3.1. Geometry and Structure Optimization

The crystal structure of tetragonal PbMoO4 is shown in Figure 1. The optimized values of a and c for tetragonal PbMoO4 are listed in Table 1. The obtained structural parameters are in good agreement with the previous experimental data [2, 11, 16]. Of course, it can be seen that the GGA leads to overestimating our calculated data compared with experiments.Table 1
Calculated equilibrium lattice parametersa, c (in Å), and c/a compared with available experimental data [2, 11, 16] for tetragonal PbMoO4.
This workCastep GGA  (PW91)Expt. [11]Expt. [16]Expt. [2]a5.51735.424–5.43605.4335.418–5.464c12.330512.076–12.110712.11012.065–12.088c/a2.23492.226–2.229——Figure 1
The crystal structure of tetragonal PbMoO4.


## 3.2. Electronic Properties

The calculations of the electronic band structure along the symmetry lines of the Brillouin zone, the total and the partial density of states (DOSs and PDOSs) are shown in Figures2 and 3. The top of the valence band is taken as the zero of energy. In this compound, the valence band maximum (VBM) is located at 1 point (the valence band maximum of tetragonal PbMoO4 is not at high-symmetry point, but at the defined 1 point between X and Γ), whereas the conduction band minimum (CBM) is located at N point, resulting in indirect band gap of 2.838 eV. This value is in good agreement with the previous calculated data 2.59 eV [11] and 2.8 eV [12]. However, these results are all smaller than the experimental values of 2.94–4.7 eV [2] due to the well-known underestimation of conduction band energy in DFT calculations [22].Figure 2
Band structure of tetragonal PbMoO4 along with the high-symmetry points of the Brillouin zone.Total and partial density of states of tetragonal PbMoO4.
(a)(b)(c)(d)In order to further elucidate the nature of the electronic band structure, we have calculated and explained the DOSs and PDOSs. From the PDOSs, we can identify the angular momentum character of the different structures.  Structure (1) is mainly due to Mo-4 s electrons, structure (2) due to Mo-4p electrons, structure (3) due to O-2 s electrons, structure (4) due to Pb-5d electrons, structure (5) due to Pb-6 s electrons, structures (6) and (7) due to O-2p electrons with hybridization of Mo-4d electrons, and structure (8) due to O-2p electrons. The conduction bands are composed of Mo-4d and show the hybridization with O-2p, as well as the hybridization between Pb-6p and O-2p.To understand the chemical bonding of this material, we have poltted the charge density of (112) plane corresponding to the (1)–(8) located at Figure3 in Figure 4. The polt labeled (1) shows the isolated Mo-4 s, (2) the weak hybridization σ bonding between Mo-4p and O-2 s, (3) the weak hybridization σ bonding between O-2 s and Mo-4d, (4) the weak hybridization σ bonding between Pb-5d and O-2 s, (5) the weak hybridization σ bonding between Pb-6 s and O-2p, (6) and (7) the hybridization σ and π bonding between O-2p and Mo-4d, and (8) the nonbonding O-2Pπ. Hence, we can conclude that the bonding between Mo and O is mainly covalent and the bonding between Pb and O is mainly ionic. Additionally, the charge density of (220) plane and the results of population analysis have been shown in Figure 5 and Table 2, which are in good agreement with our analysis of chemical bonding.Table 2
Calculated results of population analysis.
AtomspdTotalCharge (e)O1.894.7806.68−0.68Mo2.246.514.0112.761.24Pb1.710.811012.521.48Total charge densities of (112) plane and charge densities located (1)–(8) in the (112) plane of tetragonal PbMoO4.Figure 5
Charge densities in the (220) plane of tetragonal PbMoO4.


## 3.3. Optical Properties

We need to calculate two dielectric tensor components to completely characterize the linear optical properties due to the tetragonal symmetry of PbMoO4. The imaginary and the real parts of the dielectric function are calculated using (1) [23–25]:(1)ε2(ω)=(4π2e2m2ω2)∑i,j∫〈i|M|j〉2fi(1-fi)δ(Ef-Ei-ω)d3k,ε1(ω)=1+2πP∫0∞ω′ε2(ω′)dω′(ω′2-ω2)
as well as the scissors operator approximation [25, 26] due to underestimating the energies of excitation with the density functional calculations. The good agreement with experiments are obtained for the optical properties like TiO2 [26], SrHfO3 [27], SrZrO3 [28], and HfO2 [29] using the scissors operator.Figures6 and 7 display the imaginary and the real parts of the dielectric function from (100) and (001) along with the calculated results from experimental data [19] for a radiation up to 20 eV. We can see that our results are consistent with the previous work [19]. The discrepancy between our results and the experiment [19] may be due to the different-temperature condition (0, K in our paper and 6, K in [19]). The imaginary parts exhibit four structures A–D of (100) and E–H of (001). Structures A and E originate mainly from transitions of O-2pπ into the conduction bands, and structures B and F from transitions of hybridization π bonding between O-2p and Mo-4d into the conduction bands, and structures C and G from transitions of hybridization σ bonding between O-2p and Mo-4d into the conduction bands, and structures D and H from transitions of Pb-6 s into the conduction bands. The calculated static dielectric constants are 5.337 and 4.910 from (100) and (001).Calculated imaginary parts of complex dielectric function of tetragonal PbMoO4 as well as calculated results from experimental data [19].
(a)(b)Calculated real parts of complex dielectric function of tetragonal PbMoO4 as well as calculated results from experimental data [19].
(a)(b)The refractive index and the extinction coefficient are displayed in Figure8. The static refractive index is found to have the values 2.310 and 2.216 from polarization vectors (100) and (001), which are in agreement with experimental data 2.28 and 2.40 [18] from (100) and (001). Figure 9 shows the calculated results on the reflectivity, absorption coefficient, loss function, and complex conductivity function from polarization vectors (100) and (001). We hope the calculated values can help to offer a theoretical basis for the experiment and application of tetragonal PbMoO4.Calculated refractive index and extinction coefficient of tetragonal PbMoO4 from (100) and (001).
(a)(b)Calculated reflectivity, absorption coefficient, loss function and complex conductivity function of tetragonal PbMoO4 from (100) and (001).
(a)(b)(c)(d)(e)


## 4. Conclusions

The paper reports detailed investigations on the structural, electronic, chemical bonding, and optical properties of tetragonal PbMoO4 using the plane-wave ultrasoft pseudopotential technique based on the first-principles density-functional theory (DFT). The calculated equilibrium lattice parameters are in agreement with experiments. Our calculated results of the band structure and DOSs show that this compound is an indirect band gap of 2.838 eV. The charge densities and population analysis are obtained and analyzed, which show that Mo and O are mainly covalent, whereas Pb and O are mainly ionic. The complex dielectric function has been shown, and the peaks position distributions of imaginary parts of complex dielectric function have been explained, which show electron transitions in the electronic bands.



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