Band gap and work function tailoring of SnO 2 for improved transparent conducting ability in photovoltaics

Transparent conducting oxides (TCOs) are an essential component in modern optoelectronic devices, such as solar panels and touch screens. Their ability to combine transparency and conductivity, two properties that are normally mutually exclusive, have made them the subject of intense research over the last 50 years. SnO 2 , doped with F or Sb, is a widely used and relatively inexpensive transparent conducting material, however, its electronic structure leaves scope for improving its properties for use in many TCO applications, especially in solar cell devices. Here we show using density functional theory that incorporation of Pb into SnO 2 reduces the band gap through lowering of the conduction band minimum, thereby increasing the electron aﬃnity. The electron eﬀective mass at the conduction band minimum decreases alongside the band gap, indicating improved charge carrier mobilities. Furthermore, the calculated optical absorption properties show the alloys retain their transparency in the visible spectrum. Our results suggest that alloying of PbO 2 with SnO 2 will enable improved electronic properties, including a highly tuneable workfunction, which will open up the material for other applications, such as hole injection layers in organic photovoltaics.


Introduction
Transparent conducting oxides (TCOs) are a class of materials that simultaneously possess the conflicting properties of optical transparency and conductivity. First documented over half a century ago, 1 TCOs are now an essential component in modern optoelectronic devices, including flat panel displays, touch-screen sensors and solar cells. [2][3][4][5] The industry standard n-type TCO is Sn-doped In 2 O 3 (In 2 O 3 :Sn or ITO), which possesses excellent optical and electronic properties, with carrier densities exceeding 10 21 cm −3 , resistivities below 10 −5 Ω cm and high optical transparency in the visible spectrum. 6 However, the low abundance of indium in the earth's crust together with massive demand for ITO has lead to increasing concerns over indium supply. 7,8 As such, the price of indium has fluctuated wildly in recent years and there are concerted efforts to eliminate its use in TCOs. 9,10 Alternative materials such as ZnO:Al (AZO), SnO 2 :Sb (ATO), and SnO 2 :F (FTO) have been employed in a range of devices but have so far been unable to replicate the high performance seen in ITO. 11 There are several properties necessary for an n-type TCO to achieve optimal performance. The optical band gap, E opt g , must be greater than 3.1 eV to provide transparency in the visible spectrum. Additionally, conductivity is dependent on the ability of the material to form a degenerate semiconductor upon donor doping, termed the dopability. In excellent n-type TCOs, donors will donate electrons directly into the conduction band (CB), leading to filled states at the conduction band minimum (CBM) and Moss-Burstein widening of the optical band gap. [12][13][14] The dopability is largely controlled by the position of the CBM relative to the vacuum level, i.e. the electron affinity (EA). [15][16][17][18] A large EA indicates that it is easier to get charge carriers into the system, effectively increasing carrier concentrations. To guarantee transparency after doping, a large separation from the first to the second conduction band (CBM to CBM+1) is necessary to prevent interband excitation of electrons. Finally, a highly disperse CB ensures a small carrier effective mass at the CBM. This generally arises from having a CB composed of metal-s-like orbitals allowing for high carrier mobilities. 19 The TCO deposited in the largest quantity, with regard to area, is F or Sb doped SnO 2 , for use in a variety of applications such as low-emissivity windows in buildings, electrochromic mirrors and defrosting windows in supermarkets. 20 Undoped SnO 2 is itself a prototypical TCO, with a large fundamental band gap of ∼3.6 eV, 21 up to 97% transparency in the visible spectrum and carrier densities approaching 10 21 cm −3 . 22 Importantly, as Sn is earthabundant (∼30-40 times more abundant than In in the Earth's crust 23 ), the raw materials needed for SnO 2 are less expensive than for ITO. 24 Manufacturing is also simplified due to the availability of chemical deposition methods, such as spray pyrolysis and atmospheric pressure chemical vapour deposition. [25][26][27][28] Similar to other TCO materials, a debate exists as to whether intrinsic oxygen vacancies (V O ) and tin interstitials (Sn i ) play a role in conduction. 29,30 The most recent studies, however, have indicated that they are deep donors or have restrictively high formation energies. 31,32 Instead, hydrogen acting as an unintentional donor (H i or H O ) has been identified both theoretically, 31,33 and experimentally 34,35 as a suitable defect to explain the conductivity seen in SnO 2 . Regardless, due to the propensity for native defects, hydrogen interstitials, and surface states to all be donor-like, King and Veal have argued that the charge neutrality level in n-type TCOs is likely to be above the CBM. 36 Thus, donor defect states remain energetically favourable even when the Fermi level appears inside the conduction band. In SnO 2 , due to the large gap between the Fermi level and the energy level of the first unoccupied states, any such defects do not cause vertical optical transitions in the visible range, enabling high carrier concentrations with little effect on transparency. 29 Comparison of the properties of In 2 O 3 and SnO 2 reveals many similarities: both are direct wide band gap semiconductors, have a CBM+1 greater than 3.1 eV above the CBM, and have highly dispersed conduction bands. 39 The EA of In 2 O 3 , however, is significantly larger than that of SnO 2 and, in fact, of all other TCOs (Fig. 1). 40 With the fundamental materials Of the group 14 oxides, PbO 2 is isostructural and isoelectronic to SnO 2 and therefore likely to allow for efficient alloying. Furthermore, it was recently identified as a narrow band gap semiconductor, with a conduction band dominated by low-lying Pb s states due to relativistic effects. 47 The electronic structure of the conduction band of PbO 2 is nearly ideal for a TCO, with a low electron effective mass of 0.18 m e and a very large separation between the CBM and the CBM+1. The fundamental band gap, however, is too small for transparency when undoped. Oxygen substoichiometry 48,49 and possibly adventitious hydrogen 50-52 cause the Fermi level to sit far above the CBM of PbO 2 , leading to high levels of conductivity. Furthermore, it has even been suggested that through tuning of the Fermi level position in the conduction band, PbO 2 could be transformed into a TCO itself. 53 Pb is also significantly more abundant and less expensive than Sn, 54 and as such PbO 2 was considered the ideal compound to incorporate into SnO 2 .
In this Article we propose incorporation of Pb as an efficient method of modulating the band gap of SnO 2 . Using hybrid density functional theory (DFT) we demonstrate that the 3.17 eV. The enthalpy of mixing is shown to be favourable at moderately high temperatures, suggesting the system can be achieved experimentally. Crucially, band gap modulation occurs primarily through lowering of the CBM relative to the vacuum level, thereby increasing the electron affinity. The ability to modulate the band gap -and consequently work function -has significant implications in the field of organic photovoltaics, in which the work function alignment of the cathode and hole injection layer is essential to form an Ohmic contact and increase the built-in potential of the interface.

Methodology
where m * is the effective mass, is the reduced Planck constant, and d 2 E dk 2 is the curvature of the band at the CBM.
In this work we investigate the thermodynamics of alloying and select the lowest energy alloy structures at particular compositions. Investigations into configuration effects in solid solutions are complicated by the large number of possible structures that can exist for a particular supercell. To avoid this problem we have followed the procedure implemented in the Site Occupancy Disorder (SOD) program developed by De Leeuw and co-workers. 70 Here, the complete configurational space for each supercell composition is generated, from which the subspace of symmetrically inequivalent configurations can be extracted. This method is able to reduce the computational complexity by several orders of magnitude, making previously prohibitive problems tractable. The process of calculating the configurational averages and entropies has been explained in more detail elsewhere in the literature, 71,72 but is based on the assumption that a Boltzmann-like probability can predict the extent of occurrence of a particular configuration. This takes into account both the energy, E m , of the configuration and its degeneracy, Ω m , i.e. how many times the configuration appears in the complete configurational space: where m = 1, . . . , M (M is the number of inequivalent configurations) and k B is Boltzmann's constant. From this, it can be shown that the average of any observable quantity at each composition, Q, can be estimated from the values of the quantity at each configuration, Q m , as: Finally, the configurational free energy, G, can be obtained directly from the partition function as:

Results and Discussion
SnO 2 and β-PbO 2 (mineral names, cassiterite and plattnerite) both crystallise in the rutile crystal structure, containing 6 atoms in a unit cell. 73 The cation is coordinated to six oxygen in a distorted octahedron (D 4h symmetry), with each oxygen coordinated to three cations by one short and two long bonds. The PBEsol calculated a and c lattice parameters for SnO 2 were 4.772 Å and 3.216 Å respectively. These are in close agreement (within 0.9 %) with neutron diffraction experiments. 74,75 In order to calculate the electronic properties of SnO 2 , the structure was relaxed using PBE0, after which the electronic structure was calculated, again using PBE0. The fundamental band gap of SnO 2 was found to be 3.67 eV. This is very similar to the experimentally observed fundamental band gap of 3.59 eV. [76][77][78] We note that this experimental measurement has not been performed in the traditional way (i.e. the ionisation potential -electron affinity, measured, for instance using inverse photoemission spectroscopy) but instead has been measured using two-photon spectroscopy and factors in the known exciton binding energy in SnO2 of 30 meV. 79 To investigate the effects of alloying, we have considered the substitution of Sn by Pb in a 2 × 2 × 2 (48 atom) supercell of SnO 2 containing 16 cation sites. Table 1  An analysis of the results reveals that, for each composition, the difference in energy between the most and least stable configurations is very small, at most 16 meV per atom. As such, a disordered alloy is more likely to form rather than an ordered solid solution.
To study the stability of the alloys against that of the individual components, the enthalpies and free energies of mixing were calculated as a function of composition, across a range of temperatures, as: respectively, where E(Sn 1−x Pb x O 2 ) is the average energy calculated according to eqn (3) and G(Sn 1−x Pb x O 2 ) is the configuration free energy of the composition calculated via eqn (4). miscibility gap disappears due to increasing contribution from the entropic term.
The dependence of the calculated lattice parameters on composition is shown in Figure   2b. Each point represents the average lattice constant across the entire configurational space for that composition, with the assumption of full disorder. This discounts any preference for ordering of the cations, however, as the difference in energy between configurations is small, the change in the average lattice parameters will be negligible. The lattice constants of the alloy display a linear increase with increasing Pb concentration, as expected due to the increase in atomic radius from Sn to Pb, and following Vegard's law. 88 In order to find the optimum doping levels, the band gap trend of the alloys was investigated. To calculate the electronic properties of almost 1000 structures accurately would have been prohibitively resource intensive. Instead, the lowest energy configurations at each composition were geometrically relaxed using the PBE0 functional, after which the electronic structures were calculated, again using PBE0. Figure 2c shows the band gaps of the lowest energy alloy structures at each composition. The alloy band gaps decrease monotonically with increasing Pb concentration, from 3.37 eV at x = 0.062 to 0.74 eV at x = 0.938. The decrease is not linear, indeed the band gap bowing parameter, defined from: shows a slight bowing of b = 0.81 eV, comparable with other ternary compounds. 89 This is in agreement with the band gap bowing parameter of 0.79 eV seen in experiment, 90 however, we note that whilst we provide the fundamental band gap bowing parameter, experimentally the optical band gap bowing parameter has been measured. The results predict a target region of ∼6.25-12.5 % Pb concentration, where the band gap is reduced but remains larger than the 3.1 eV needed to maintain transparency. As such, the lowest energy structures at these compositions (x = 0.062 and 0.125) were chosen for further analysis. To see how alloying affects the band structure, we have calculated the "effective" primitive cell band structure using the PBE0 functional and BandUp code, as described above. Figure   3 shows Having established that Pb incorporation decreases the fundamental band gap of SnO 2 , it is instructive to investigate the effects of the alloying on optical properties. As the inversion symmetry of the lattice results in disallowed transitions at the Γ point, the optical band gaps (E opt g ) of SnO 2 and PbO 2 are considerably widened relative to the fundamental band gaps. 47,79,81,100 The optical absorption spectra for SnO 2 , Sn 0.938 Pb 0.062 O 2 and Sn 0.875 Pb 0.125 O 2 , calculated using PBE0 from the frequency dependent dielectric matrix, are presented in Figure 5. We can clearly see that incorporation of Pb does not affect the disallowed nature of the optical band gaps, as in all cases the optical band gaps are significantly larger than the fundamental band gaps, indicating that the Sn 1−x Pb x O 2 alloys will retain high levels of optical transparency.

Conclusions
In this study we set out to tailor the band gap of SnO 2 in order to improve its performance as a transparent conducting oxide. Our approach was centred around reducing the band gap by decreasing the position of the conduction band minimum, thereby increasing the electron affinity and thus increasing the dopability. 18,46 To be effective, the fundamental band gap of the improved SnO 2 material must be greater than 3.1 eV whilst no detrimental effects on the effective mass at the conduction band minimum should be seen, in the interest of retaining high electron mobilities.
Through alloying with isoelectronic and isostructural PbO 2 , we have demonstrated that the band gaps of Sn 1x Pb x O 2 alloys can be tuned from 3.67 eV to 0.64 eV with increasing Pb content, arising from stabilization of the conduction band minimum relative to the vacuum level. We have found that Sn 0.875 Pb 0.125 O 2 displays a fundamental band gap that is just above 3.1 eV, possesses effective masses that are lower than for pure SnO 2 , and has an electron affinity 0.59 eV larger than SnO 2 . Furthermore, the optical transparency of this alloy remains extremely high. These properties should therefore, in principle, make SnO 2 :Pb a more efficient n-type transparent material and an ideal candidate for use in TCO applications.
Additionally, as lowering of the conduction band minimum results in an increase in the work function (provided the Fermi level remains near to the band edges), these results demonstrate the possibility of a single generic system, in which the work function can be finely tuned over a wide range, based only on single parameter. This poses significant advantages for organic solar cells, which require efficient alignment between the work functions of the cathode and hole injection layer in order to produce an Ohmic contact and maximise device efficiency. As such, we stress the pressing need for experimental verification of these results.

Acknowledgments
Discussions with J. Buckeridge are gratefully acknowledged. This work made use of the ARCHER UK National Supercomputing Service (http://www.archer.ac.uk), via our membership of the UK's HEC Materials Chemistry Consortium, which is funded by EPSRC (EP/L000202), the Iridis cluster, provided by the EPSRC funded Centre for Innovation (Grant codes EP/K000144/1 and EP/K000136/1) and the UCL Legion HPC Facility (Le-gion@UCL). AMG acknowledges Diamond Light Source for the co-sponsorship of a studentship.