Conducting Behavior of Crystalline α-PbO₂ as Revealed by DFT Calculations

Abstract

PbO₂ is one material that has recently emerged as potential transparent conducting oxide for applications in the modern opto-electronic industry. In this work the electronic structure of the α-PbO₂ polymorph has been investigated, aiming to contribute to the understanding of its high levels of conductivity. DFT calculations using B3LYP hybrid density functional and considering long range interactions among the atoms have been performed. A direct band gap of 0.90 eV has been found, compatible with high conductivity values. Although the stoichiometric material is somewhat transparent, the band structure indicates that appropriated modifications in the material Fermi level can be performed in order to decrease the absorption of light. Charge distribution plus overlap population analysis show that the material is predominantly ionic. The charge distribution throughout the material is strongly dependent on the crystal direction. Results suggest that α-PbO₂ can be potentially more interesting for opto-electronic purposes than the β polymorph.

Keywords:
Lead dioxide; transparent conducting oxides; density functional theory; optical band gap

1. Introduction

The modern optoelectronic technology is highly dependent on transparent conducting oxides (TCOs), compounds which combine transparency and conductivity, properties that normally do not coexist. Initially that technology was mainly based on In₂O₃, a rather expensive material due to the scarcity of indium in the earth (In is obtained as a by-product of refining other metals like Zn and Pb, for example, and the amount of In available in the minerals is not known). Because of that it has been intensely sought alternative materials to replace it.¹ SnO₂ and ZnO are among the materials that have proved to be quite competitive as substituents to In₂O₃. Low coast and recent progress in improving performance make them attractive substitutes for the indium based TCOs.² However, the investigation has not over and new materials are insistently sought after. Among the new investigated materials, PbO₂ arises as a very promising one, which electrical properties have been explored very recently.^3-8 PbO₂ has a high chemical stability and because of the lower coast of Pb compared to In, achieving optical transparency with high electrical conductivity could have important reflexes in optoelectronic industry in the medium-term. Even though Pb is not environmentally friendly, it is highly likely that the economic benefits outweigh the expense of the logistics involved in recycling. Above all, it is highly significant, from a social point of view, finding new and cheaper materials that can guarantee to countries with lower purchasing power access to the most modern technologies.

An orthorhombic (columbite) α, a tetragonal (rutile) β, and a high-pressure γ modification polymorphs of PbO₂ have been identified. Both the α-phase (space group Pbcn), and β-phase (space group P42/mnm) are stable under normal conditions, the β-phase being the most stable in acid medium and the α-PbO₂ prevailing in neutral/alkaline solutions.⁹ The active material of positive plate of the lead-acid batteries is a mixture of α- and β-PbO₂.¹⁰

Most of the TCO's are n-type semiconductors having a band gap of the order of ~3 eV, optical transmittance in the visible region higher than 80%, and electrical resistance less than or equal to 10-3 Ω.cm.¹¹ From the carried out studies reported up to now is very well established that β-PbO₂ is intrinsically metallic or a narrow band gap (NB) semiconductor,^6-8 with a band gap of 0.61 eV, and a carrier concentration about 10²¹ cm^-1.¹² NB semiconductors are colored or black when closed to stoichiometry, which is the case of β-PbO₂, but may potentially become transparent when donor doped to high carrier concentrations, as it has been found for CdO,^13,14 thanks to the blue-shift of the optical band gap, following the Moss-Burstein effect.^15,16 The shift arises because the Fermi energy lies in the conduction band for heavy n-type doping and, consequently, the filled states block optical excitation. Thus, the band gap determined from the onset of interband absorption moves to higher energy. However, achieving optical transparency also depends on the conduction band structure. The optical transparency is guaranteed since the transitions from the filled conduction band states to the next highest conduction band lie above the threshold for visible light absorption.^17,18

β-PbO₂ typically displays degenerate n-type conductivity attributed to oxygen vacancies. The oxygen stoichiometry is critical to the conductivity, since each lack of oxygen contribute with two free electrons. The calculated Fermi energy lies deep in the conduction band, which is consistent with high levels of electrical conductivity.⁸

However, if on one hand, there are still many gaps to be filled concerning to the electronic properties of β-PbO₂, on the other hand no study was found in the literature related to those properties of α-PbO₂. In this paper are being reported the first results related to a systematic investigation on the α-PbO₂ electronic properties, as revealed by calculations based in density functional theory (DFT), aiming to evaluate its potential to be applied as TCO.

2. Model and Computational Method

As already stated in the introduction, α-PbO₂ has an orthorhombic symmetry with 4 formula units per unit cell. Neighbor unit cells share non-opposing edges, resulting in the formation of zigzag chains, according to Figure 1. The material consists of Pb or O interleaved planes in the direction [001] and a sequence of PbO₂ planes in the direction [111] with Pb atoms hexa-coordinated and O atoms tetra-coordinated. The computational simulations were performed in the framework of the periodic DFT implemented in the crystalline orbital computer code CRYSTAL14.¹⁹ DFT is a powerful methodology to simulate structural, energetic and electronic properties of solid state materials with high accuracy, thanks to the advent of functionals based on the generalized gradient approximation.^20,21 A deep analysis on the reproducibility of DFT calculations of solids has been published recently.^22,23 Crystalline orbitals were represented as a linear combination of Bloch functions defined in terms of local functions (atomic orbitals) using Gaussian-type basis set.²⁴

Figure 1
The unit cell and zigzag chain of α-PbO₂ (some atom labels are numbered for later discussion).

Taking into account that the choice of the proper functional is fundamental for the quality of the results, two of them have been tried for us: the B3LYP,^25,26 and the HSE06²⁷ ones. B3LYP is a very popular functional used with DFT for studying solids. It is a type of approximation to the exchange-correlation energy, based in the scheme introduced by Axel Becke in 1993.²⁵ A comprehensive analysis of the performance of B3LYP functional for a broad set of molecules has been reported by Jorgensen and Tirado-Rives.²⁸

On the other hand, HSE06 is a functional derived from the PBE one,^29-31 that apply a screened Coulomb potential to the exchange interaction in order to screen the long-range part of the Hartree-Fock exchange.²⁷ This functional was implemented into CRYSTAL in the 2014 version. When using the B3LYP functional the long range interactions were considered by adopting the Grimme approach^32,33 (in this case the functional is named B3LYP-D).³⁴ In this approaching the DFT total energy is augmented with a semi-empirical dispersion term based on an additive atom-atom damped empirical potential, according to:³⁴

(1) ${\mathit{E}}_{\mathit{disp}}=-{\mathit{s}}_6\sum _{\mathit{i}=1}^{{\mathit{N}}_{\mathit{at}}-1}\sum _{\mathit{j}=\mathit{i}+1}^{{\mathit{N}}_{\mathit{at}}}\sum _{\mathit{g}}\frac{{\mathit{c}}_6^{\mathit{ij}}}{{\mathit{R}}_{\mathit{ij},\mathit{g}}^6}{\mathit{f}}_{\mathit{dmp}}\left({\mathit{R}}_{\mathit{ij},\mathit{g}}\right)$

where the summation is over all atom pairs and g lattice vectors excluding the i = j contribution for g = 0, s₆ is a scaling factor related to the functional used, c₆^ij is the dispersion coefficient for the pair ij computed by using a geometric mean and R_ij,g is the interatomic distance between atoms i in the reference cell and j in the neighboring cells at distance ||g||. The dumping function f_dmp is used to avoid singularities for small R_ij,g distances. Such function is implemented in the CRYSTAL14 package as:³⁴

(2) ${\mathit{f}}_{\mathit{dmp}}\left({\mathit{R}}_{\mathit{ij},\mathit{g}}\right)=\frac{1}{1+{\mathit{e}}^{-\mathit{d}\left({\mathit{R}}_{\mathit{ij},\mathit{g}}/{\mathit{R}}_{\mathit{vdW}}-1\right)}}$

were d is the constant that determines the steepness of the dumping function, and R_vdW is the sum of van der Waals atomic radii for atom pair ij. For the calculations the dispersion term used was s₆ = 1.05 (concerning to the B3LYP hybrid functional) and the steepness of the dumping function d = 20. Quite recently a new routine has been implemented for Grimme (named D3), where the atomic radius of each atom is adjusted accordingly to the chemical surroundings and the atomic coordination number.^20,21 This methodology has not been used in the present simulations, since it is not implemented in the commercial version of CRYSTAL14, but may be applied in the future, which may lead to better results. A cutoff radio of 25.0 Å was used to truncate the summation over the lattice vectors.³⁵ The R_vdW and c₆^ij parameters used for Pb were those optimized for PbO by Canepa et al.³⁶ Although these parameters have not been specifically optimized for α-PbO₂ and so do not account exactly its chemical environment, they improved the description of the α-PbO₂ properties compared to not consider dispersion factor. The C₆ and R_vdW parameters used for Pb were, respectively, 63.16 and 1.767,³⁶ and for O were 0.70 and 1.342.³³

A full optimization procedure was carried out to determine the equilibrium geometry. All structures were optimized by the use of analytical energy gradients with respect to atomic coordinates and unit cell parameters. Convergence was checked on gradient components and nuclear displacements with tolerances on their root mean square set to 0.0001 and 0.0009 a.u., respectively. The level of accuracy of the calculation of Coulomb and the exchange series is controlled by five parameters. The 8, 8, 8, 8, and 14 parameters were chosen for the Coulomb overlap, Coulomb penetration, exchange overlap, first exchange pseudo-overlap, and second exchange pseudo-overlap, respectively. The shrinking (Monkhorst-Pack and Gilat)³⁷ was set up to 6, which corresponds to 40 independent k-points in the irreducible part of the Brillouin zone integration.

Mulliken population analysis was used. This choice was based on the simplicity, frequent application in theoretical calculations of similar systems, and convenient interpretation of the results on the light of chemical intuition. The XCrysDen program was used to draw the draft of the band structure, density of states (DOS) graphics, and maps of charge density distribution.³⁸

Since the choice of the optimal basis set is difficult for such crystalline materials, several of them have been tested to simulate the material. Two all-electron basis sets have been tried for oxygen: a 6-31G* basis set optimized by Gatti et al.³⁹ and the Triple Zetta Plus Polarization (TZVP) reported by Peintinger et al.⁴⁰ In turn Pb was described for an effective core pseudopotential DB31G* (where DB stands for Durand-Barthelat's large effective core potential),⁴¹ the HAYWLC-211(1d)G,⁴² and the ECP60MDF-6111(51d)G.⁴³ Pb was treated in the framework of core pseudopotential approximation, because its number of electrons makes the simulations very expensive. The lattice parameters reported by Scanlon et al.⁴ were fitted to decide on the better computational conditions for the simulations. To the best of our knowledge these are the only experimental properties that have been published for the material. Even the value of the band gap, which would be extremely useful to judge about the quality of results, is unknown. The analysis of the results permits to conclude that the B3LYP-D functional with oxygen 6-31G* and the DB31G* pseudopotential for lead were the computational conditions which provided the best agreement between the theoretical and experimental crystalline structure and were used for calculation of the reported results. Just to give an idea of the divergence, results obtained for α-PbO₂ with B3LYP and HSE06 functionals using those basis set are shown in the Table 1.

Table 1
Experimental⁴ and calculated (oxygen 6-31G* and lead DB31G*) equilibrium lattice parameters and density for α-PbO₂. Percentage deviation (%) in parenthesis.

	a (Å)	b (Å)	c (Å)	d(g.cm-3)
Experimental	5.001	5.931	5.443	9.820
B3LYP	5.100(2.00)	5.931(0.00)	5.393(0.92)	9.939(1.21)
HSE06	5.224(4.48)	5.517(6.98)	5.259(3.38)	10.624(8.19)

3. Results and Discussion

The experimental lattice parameters for α-PbO₂, have been recently obtained using neutron diffraction at ISIS Spallation Neutron Source - Rutherford-Appleton Laboratory (UK),⁴ facility which has also been used for us to investigate the structure of other systems.^44-47 Table 1 shows the set of cell parameters of experimental and calculated results obtained for the oxide.

As it can be noticed from the Table, the parameters calculated with B3LYP-D functional show a better agreement with the experimental data available than the results obtained with HSE06 one, which determined the choose of the first functional to perform the simulations. This functional has been very successful in describing electronic and structural properties of several oxides and other materials.^34,48-50 It deserves to point out that calculations without considering long range weak interactions give cell parameters in better agreement with the experimental ones (calculations were performed using the same input script excepting the GRIMME calculation routine). However, it was opted for keeping this routine in the simulations since describing better the interactions between the atoms must reflect in a better quality of the calculated electronic properties (also the comparison with HSE06 functional is more appropriated, since this functional considers long range weak interactions in its formalism).

The band structure of α-PbO₂ obtained from the calculations is shown in Figure 2. Both the top of valence band and the bottom of the conduction band are situated on the same Г point, resulting in a direct band gap of 0.90 eV. It is worth remembering that pure density functionals like GGA and LDA usually underestimate band gaps, while hybrid functionals, as the B3LYP used in this work, usually provide a good agreement with experimental values.^34,48-50 However, calculations performed in parallel for β-PbO₂ using both B3LYP and HSE06 functionals gave a band gap value of 0.21 eV, while the experimental value is 0.61 eV.⁴ Thus, both the functionals tried for us underestimated the β-PbO₂ band gap for about 66% respecting to the experimental value. Taking into account this result it can be considered that the α-PbO₂ band gap is at least 0.90 eV. In other words, can be inferred that the α-PbO₂ is less metallic than β-PbO₂, which is a sufficient reason to suppose that α-PbO₂ would be a better candidate for TCO than β-PbO₂. Interestingly, it has been reported that β-PbO₂ (plattnerite) is black and opaque, while α-PbO₂ (scrutinyite) is brown and somewhat transparent.⁵¹

Figure 2
Band structure of α-PbO₂, with the energy scale set so that the Fermi level corresponds to 0.0 eV. (left and right graphics differ in the plotted energy range).

The topography of the α-PbO₂ band structure on the Г point is very similar to that of β-PbO₂, however, while in the β-PbO₂ there is a range of about 6 eV between the bottom of the conduction band to the next highest portion of the band (E_intra), in the α-PbO₂ that range is only about 4.7 eV. Thus, according to what has been discussed for Walsh et al.³ it should be easier to manipulate the transparency in α-PbO₂ (controlling the oxygen vacancies or doping, for instance), thanks to move the electron chemical potential inside the conduction band, than to β-PbO₂. As it can be seen in the graphic, the theoretical valence band width is roughly 8.75 eV. Theoretical values reported for β-PbO₂ are 8.02 eV and 8.45 eV, depending on the functional used.⁴ The conduction band shows a partition into two distinct regions separated by about 3 eV, which will be commented below when discussing the features of the density of states.

Total DOS profile and the most relevant projected DOS (PDOS) for Pb and O atoms are depicted in Figure 3. Following the notation adopted by Payne et al for β-PbO₂,⁷ it has been identified three main features (labeled I - III) in the valence states of the total DOS profile. Region III is basically composed by a mix of 50% Pb 6s and 50% O 2p orbitals, showing that the states of lower energy are predominantly occupied for electrons equally distributed for those orbitals (it is usual to speak in terms of orbital hybridization). Region II is predominantly composed by oxygen 2p orbitals, with a residual contribution of the lead 6p orbitals, while in the highest energy band closest to the Fermi surface (region I), the contribution of lead 6p orbitals is, proportionally, even smaller, indicating a strong asymmetry in the electron distribution among the atoms. Thus, the energy states near the top of the valence band are predominantly occupied for electrons located in oxygen 2p orbitals. It deserves to pay attention that in the case of PbO₂, both lead 6s and 6p orbitals are, stoichiometric speaking, empty. Thus, it is understandable there are significant differences between the DOS distribution of this material and PbO, as it has been found, for example, by Watson and Parker.⁵²

Figure 3
Total and partial DOS diagram for α-PbO₂.

On the other hand, the conduction band shows two well defined regions, being that the empty states in the bottom (region i) are composed by equal contributions of O 2p and Pb 6s orbitals, while the states at higher energy (region ii) are predominantly composed of Pb 6p orbitals. It is remarkable the partition of the conduction band states into two distinct regions separated by a range of about 3 eV. Clearly this separation is related to that seen in the band structure profile, which has been highlighted above.

For a deeper comprehension of the electronic distribution throughout the material it has been plotted maps of electronic density distribution for some planes of α-PbO₂ (Figure 4). Some other planes have also been investigated, presenting the same profile. As a general behavior, it is noticed that the density isolines profiles show a well-defined charge distribution around the Pb and O atoms, an indication of the ionic character of the atomic interactions, accordingly to analysis done above. Thus, there is an asymmetry on the charge distribution through the material depending on the crystal direction. Aiming to detail further these founds it has been calculated the Pb - O overlap populations, which results are listed in the Table 2 (see unit cell shown in Figure 1 for atom numbering). Firstly, the values of the overlap population are about of that obtained for other studied oxide, ZnO (143 m|e|),⁵³ while the C - C overlap population of carbon nanotubes, a typical covalent bond is bigger than 400 m|e|.⁵³ This result corroborates that Pb - O bond in α-PbO₂ has a ionic character. Secondly, there is a distinct difference among three of the oxygens (O1, O3 and O4) and O2 concerning to the Pb1 - O bond. The same behavior is observed also in the Pb - O bonds evolving Pb2, namely, the O2 - Pb1 and O5 - Pb2 bonds present a character yet slightly more ionic than the other bonds studied. This result is directly related to the bond lengths, shorter than the other Pb - O bonds, leading to an different charge distribution in each case.

Figure 4
Charge density maps for 3 different α-PbO₂ planes (red denotes an excess of negative charge and blue a deficiency of it). a: oxygen only; b: oxygen and lead; c: lead only.

Table 2
Average Pb–O bond length (Å) and overlap population (m|e|) of α-PbO₂.

	Pb1	Pb2
O1	2.123 (126)	-
O2	2.503 (58)	-
O3	2.137 (122)	-
O4	2.123 (126)	2.137 (122)
O5	-	2.441 (63)
O6	-	2.137 (122)

Finally, the partial charges on the atoms calculated using Mulliken population analysis results -0.95 for oxygen and 1.90 for lead, yet indicating a predominate ionic character of the bonds. Obviously, these results should be viewed with caution, since the choice of Mulliken partition is arbitrary and the results depend on the basis set used. Besides, there is no unique method of performing the partition of the charge density. Even so, the choice of a particular scheme is still useful to support an argument.

4. Concluding Remarks

Electronic structure of α-PbO₂ has been studied by periodic DFT calculations. α-PbO₂ presented a theoretical band gap of 0.90 eV. The topography of the band structure suggests that transparency and conductivity can be achieved. Significant population of states at the top of the valence band arises from oxygen 2p states. The empty states are virtually composed of lead 6p and 6s, with a predominance of 6p orbitals. Maps of charge density for several planes shown that the material is predominantly ionic, which is confirmed by the overlap populations analysis. The charge distribution throughout the material depends on the crystal direction. Taking into account all these aspects, it may be expected a better suitability of α-PbO₂ to be applied as TCO than β-PbO₂. Several experimental techniques should be able of confirm these theoretical founds. The methodology used in the present case has also been used for studies of surfaces, doping and adsorption process, for example, and that will be the next goal in the investigation of the properties of both PbO₂ phases. The results obtained could suggest new properties and functionalities of these materials.

5. Acknowledgments

This work is supported by Brazilian Funding Agencies: CNPq (46126-4), CAPES (787027/2013, 8881.068492/2014-01), FAPESP (2013/07296-2, 2016/07476-9). The computational facilities were supported by resources supplied by Molecular Simulations Laboratory, São Paulo State University, Bauru, Brazil, and Laboratory of Computational Chemistry, São Paulo State University, Ilha Solteira, Brazil. JMMC is deeply grateful to Dr. Armando Beltrán, from University Jaume I, Castellon de la Plana, Spain, for his kind and fruitful discussions.

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Publication Dates

• Publication in this collection
  18 Dec 2017
• Date of issue
  2018

History

• Received
  11 July 2017
• Reviewed
  30 Oct 2017
• Accepted
  10 Nov 2017