Stability and bandgaps of layered perovskites for one- and two-photon water splitting

Direct production of hydrogen from water and sunlight requires stable and abundantly available semiconductors with well positioned band edges relative to the water red-ox potentials. We have used density functional theory (DFT) calculations to investigate 300 oxides and oxynitrides in the Ruddlesden–Popper phase of the layered perovskite structure. Based on screening criteria for the stability, bandgaps and band edge positions, we suggest 20 new materials for the light harvesting photo-electrode of a one-photon water splitting device and 5 anode materials for a two-photon device with silicon as photo-cathode. In addition, we explore a simple rule relating the bandgap of the perovskite to the number of octahedra in the layered structure and the B-metal ion. Finally, the quality of the GLLB-SC potential used to obtain the bandgaps, including the derivative discontinuity, is validated against G0W0@LDA gaps for 20 previously identified oxides and oxynitrides in the cubic perovskite structure.


Introduction
The direct conversion of solar light into chemical fuels through electrochemical reactions represents a clean, sustainable and potentially cheap alternative to fossil fuels. The simplest reaction of this kind is the water splitting (WS) reaction in which water is split into hydrogen and oxygen. In the most basic photo-electrochemical device, each photon is harvested by a single semiconductor and the created electron-hole pair is used to evolve hydrogen and oxygen. The maximum efficiency of such a one-photon device is only around 7% when overpotential of the reactions and losses are accounted for [1]. Significantly higher energy conversion efficiencies of up to 27% [2] can be achieved using a combination of two or more semiconductors with appropriately aligned band edges.
First attempts of photocatalytic WS were made in the 1970s using TiO 2 as the light harvesting and hydrogen evolving material [3]. Due to the large bandgap of TiO 2 and its poor catalytic properties the devices had extremely low efficiencies. Since then several materials have been proposed as light harvesting photo-electrodes for WS both in the UV and in the visible range [4]. Still, the efficiencies of the WS devices are significantly lower than the alternative combination of electrolysis driven by standard photovoltaic cells.
The search for new materials can be guided by ab initio quantum mechanical calculations, avoiding expensive 'trial and error' experimental processes. Recently, high-throughput materials design of stable binary and ternary alloys [5], carbon capture and storage [6], batteries [7], photovoltaic [8,9] and WS materials [2,10] have been reported. A number of databases have been also implemented to store and to analyze the huge amount of computed data generated. Some examples are the Materials Project database 4 , the AFLOWLIB consortium [5] and the Computational Materials Repository 5 .
In previous studies [2,10], we have investigated the cubic perovskite structure and proposed 20 materials for the one-photon WS process and 12 others for the anode in a twophoton device with a Si cathode. Several of those were unknown in the WS community, but some of them have been already successfully tested. In particular the oxynitrides, like BaTaO 2 N, SrTaO 2 N, CaTaO 2 N, LaTiO 2 N and LaTaON 2 , gives good results in term of oxygen and/or hydrogen evolution in presence of sacrificial agents [11]. The perovskite is a very versatile structure with materials exhibiting a large variety of properties and applications [12]. In addition to the standard cubic and low-symmetry perovskites, two cubic perovskites with general formula ABO 3 can be combined together in the so-called double perovskite [13]. The double perovskite structure was recently explored with the focus of finding new materials for WS and engineering of the bandgap by combining two perovskites with different electronic properties [14,15].
In the present work, we investigate the layered perovskite structure. A layered perovskite is composed of two-dimensional (2D) slabs of ABO 3 cubic perovskite separated by a motif of metal atoms. There are several phases of layered perovskites which differ in the thickness and the relative displacement of the cubic perovskite slabs and in the motifs. The phase studied here is the Ruddlesden-Popper with general formula A n+1 B n O 3n+1 , where n = 1, . . . , ∞ is the number of BO 6 octahedra forming the 2D slabs and the upper limit for n correspond to the cubic phase. Figure 1 shows the structures under investigation: we consider the cases of n = 1 and n = 2 (A 2 BO 4 and A 3 B 2 O 7 , respectively) and one oxygen replacement in favor of nitrogen in the case of n = 1 (A 2 BO 3 N). 6 There are several distinct oxygen sites that can be replaced by nitrogen. To avoid large distortions in the octahedron, we replace the oxygen between the Aand B-atoms leaving the x y-plane of the octahedron unchanged. This unit cell does not allow to investigate possible symmetry lowering in the structure (i.e. octahedra tilting and Jahn-Teller distortions). In order to do that a larger unit cell is required, which would increase dramatically computational time, making the screening unaffordable. In any case we should remark that the use of a cubic symmetry is well justified since this symmetry lowering usually disappears at high temperatures. Indeed in many cases perovskites recover a cubic-like symmetry at room temperature [16,17].

Method
In the one-photon WS device, one material is responsible for the evolution of both oxygen and hydrogen. First a photon is harvested creating an electron-hole pair. Next, the electron and hole reach two different points on the surface where the evolution of O 2 and H 2 takes place. Any material that should function as photo-electrode for WS should meet a number of criteria: (i) chemical and structural stability; (ii) a bandgap in the visible range; (iii) well positioned band edges with respect to the red-ox levels of water; and (iv) high mobility for electrons and holes. In addition, low cost and non-toxicity are required. Here we focus on the criteria (i)-(iii).
The stability of a material is evaluated with respect to around 2000 reference systems. The reference systems constitute stable phases of the possible materials into which the considered layered perovskite can be decomposed, and are taken from the experimental ICSD database 7 andtheMaterialsProjectdatabase(seefootnote4).Amaterialisconsideredstableonlyif the energy difference between the layered perovskite phase and the most stable alternative combination of reference systems is below 0.2 eV atom −1 . We use this finite threshold energy to avoid the exclusion of potential candidates due to inaccuracies in the calculations [18] and to take into account metastability of the investigated structure. Each combination is fully relaxed using the revised Perdew-Burke-Ernzerhof (RPBE) functional [19] implemented in the density functional theory (DFT)-GPAW code [20,21].
As an example, the stability of the Ba 2 TaO 3 N compound is given by where c i are positive and sum up to assure the correct stoichiometry of the A 2 BO 3 N layered structure and the chemical formulae indicate the DFT total energies of the references. All the references are in their solid state phase except for N 2 and O which are in the gas phase. Note that the energy of oxygen is calculated from H 2 and H 2 O due to the well known problems associated with the DFT description of the O 2 triplet ground state. We note that the present stability analysis does not include corrosion of the materials. Extended stability analysis including this effect were recently found to be of some importance [22,23]. However, we leave this for a future study.
It is well known that the Kohn-Sham eigenvalues from DFT systematically underestimate bandgaps of extended semiconductors due to the approximate nature of the exchange-correlation functionals and the missing derivative discontinuity [24]. On the other hand, many-body methods like the GW approximation give better bandgaps, but are computationally too expensive to be used in a screening project of several hundred materials. Here, we used the Gritsenko, van Leeuwen, van Lenthe and Baerends potential (GLLB) improved for solids (-SC) The agreement between the GLLB-SC potential and G 0 W 0 is very good. potential [25,26], which includes an explicit estimation of the derivative discontinuity. In a previous work [10], the GLLB-SC bandgaps of around 40 metal oxides were compared to experiments. On average the GLLB-SC bandgaps were found to lie within 0.5 eV of the experimental gaps. Figure 2 shows the bandgaps of the 20 earlier identified cubic perovskites for onephoton WS [2] here calculated using different methods and xc-functionals. Standard DFT functionals, like local density approximation (LDA) and PBE (not shown in the figure), seriously underestimate the bandgap, while the hybrid PBE0 tends to overestimate the gaps. We have performed G 0 W 0 calculations using a recent implementation in the GPAW code [27]. In these calculations the G 0 W 0 self-energy is evaluated using LDA energies and wave functions and the frequency dependence of the dielectric function is fitted to a plasmon pole model. We use a 150 eV plane wave cut off for the representation of the dielectric matrix and include empty bands up to the same energy cut off. A 7 × 7 × 7 k-point mesh is used for both DFT and G 0 W 0 calculations 8 . The GLLB-SC and G 0 W 0 bandgaps are quite similar with a mean absolute difference of 0.3 eV. This is consistent with the results of a similar comparison made for a set of ten well characterized simple semiconductors and insulators [27]. Based on this we conclude that the bandgaps obtained with the GLLB-SC are sufficiently accurate for the purpose of materials screening.
In addition to the bandgap, the position of the conduction and valence band edges with respect to the water red-ox levels is crucial: the energy of electrons at the surface should be above the hydrogen evolution potential and the energy of holes must be below the oxygen evolution level. It is not trivial to calculate the band edge positions at a semiconductor-water interface from first principles. Although some methods have been proposed [28,29], they are computationally rather demanding and not suited for screening studies. Instead we obtain the position of the band edges simply using an empirical equation [30,31] that gives the center of the bandgap in terms of the geometrical average of the electronegativities in the Mulliken scale of the pure chemical elements, χ M , forming the compound. The band edge positions are then calculated by adding and subtracting half of the bandgap, E gap . In formula, for the A 2 BO 4 layered perovskite: where E 0 is the difference between the normal hydrogen electrode (NHE) and the vacuum (E 0 = −4.5 eV). More details about the method and its validation are available in [2,10]. It has been recently shown for the cubic perovskite structure [32], that the search for new materials can be guided using chemical-based rules. The three chemical rules used here are: • Valence balance rule: in a realistic material, the sum of the possible oxidation states of its elements must be zero.
• Even-odd electrons rule: the number of electrons per unit cell should be an even number.
A material with an odd number of electrons has partially occupied bands at the Fermi level and is thus a metal. There are some exceptions to this rule, but they correspond to magnetic and/or strongly correlated materials which require more advanced theoretical descriptions than the one used here.
• Ionic radii of the atoms: the metals that can occupy the A-and B-ion position in the perovskite depend on the ionic radii of the neutral atoms. Usually alkali and alkaline earth metals occupy the A-ion side, and post-transition metals the B-ion position [12]. We implement this rule by having separate allowed chemical elements for the A-and B-sites 9 .
The use of these rules drastically reduce the number of calculations from around 8000 to 300 different materials. The screening parameters for the one-photon WS device are summarized in table 1. In addition to stability and a bandgap in the visible range, the band edges should straddle the red-ox levels of water plus the required overpotentials (0.1 eV for hydrogen and 0.4 eV for oxygen [33]).
The ideal efficiency of the one-photon device can be up to 7% [1]. Higher efficiencies can be achieved using a two-photon, or tandem, device, consisting of two semiconductors forming a pn-junction. Electrons generated in the cathode move to the surface and evolve hydrogen Table 1. Screening parameters (in eV) used for the one-and two-photon WS devices. The red-ox levels of water with respect to the NHE are 0 and 1.23 eV for the hydrogen and oxygen evolution, respectively. The criterion for the position of the band edges include also the overpotentials for hydrogen (0.1 eV) and for oxygen (0.4 eV) [33]. Silicon is used as cathode in the two-photon WS device (VB Si edge = 0.86 eV). while the holes move toward the pn-junction and recombine with electrons from the anode. Similarly, the holes generated in the anode move toward the surface and evolve oxygen. Thus it takes two photons to generate one H 2 . The efficiency of the device depends strongly on the relative sizes of the bandgaps of the two materials. Assuming ideal band edge line up, the best performance (around 25%) is obtained by combining two materials with bandgaps of 1.1 eV and 1.7, respectively [2]. The sum of the two bandgaps should not be smaller than 2.8 eV: 1.23 eV is the energy required per electron to split water and approximately 0.5 eV is required to account for the overpotentials of the oxidation and reduction reactions. In addition, the relevant quasi-Fermi levels are located ≈ 0.25 eV below (above) the conduction (valence) bands 10 . Finally, the quasi-Fermi level corresponding to the conduction band of the cathode should be at least 0.1 eV above the quasi-Fermi level corresponding to the valence band of the anode to ensure efficient charge recombination at the interface. As it turns out, silicon is an ideal photo-cathode for WS since it has a bandgap of 1.1 eV well positioned with respect to the reduction potential of hydrogen and because of the mature fabrication technology. A more detailed explanation of the two-photon device is available in [2]. In the case of two-photon devices we therefore limit our search to photo-anode materials assuming a silicon cathode and use the screening criteria summarized in table 1.

Trends in stability and bandgaps
The simplest layered perovskite is the Ruddlesden-Popper phase. The two phases studied here have the general formula A 2 BO 4 and A 3 B 2 O 7 . The former is composed of slabs of cubic perovskite ABO 3 offset by a translation of (1/2, 1/2) with respect to each other, terminated by oxygens to complete the octahedra, and separated by a layer of A metal atoms. In the latter structure each slab of perovskite is formed by two octahedra BO 6 units. The unit cells 10 The quasi-Fermi level describes the population electrons and holes in a semiconductor when their populations are not in equilibrium, for example when the semiconductor is under illumination. When an electron-hole pair is created, the density of both electrons and holes are above their equilibrium values and the populations of the carriers cannot be described by a single Fermi level. One of the consequences of the quasi-Fermi levels is that the effective energy of the electrons and holes does not correspond anymore to the conduction and valence band but the quasi-Fermi levels have to be considered. Materials of the form A 2 BO 3 N, in which an oxygen atom has been replaced by nitrogen, have been considered. In general, oxides tend to have valence band edges that are somewhat deep compared to the oxidation potential of water. Nitrogen is slightly less electronegative compared to oxygen and the valence bands of the oxynitrides are consequently shifted up compared to the oxides [34]. The smaller electronegativity of nitrogen leads also to a reduction in the size of the bandgaps. In addition, a nitrogen replacement breaks the symmetry of the cubic perovskite and creates an excess of charge in one of the corners of the octahedron. This leads to a reduction in the stability of the obtained materials, as shown in figure 3. Figure 3 shows the calculated heat of formation for the A 2 BO 4 , A 2 BO 3 N and A 3 B 2 O 7 perovskites plotted against the GLLB-SC bandgap. The bandgaps of both types of oxides range from 0 to above 6 eV. The oxides are in general highly stable especially the materials with larger bandgaps. As expected, the oxynitrides are less stable and show smaller gaps. Despite of this, around 8 oxynitrides are found to fulfill the criteria on bandgap and stability for the onephoton WS device. A further nitrogen replacement is likely to lower the bandgaps and reduce the stability even further [2], and therefore has not been investigated here.
All the bandgaps of the stable A 2 BO 4 layered perovskites are plotted in figure 4. There is some degree of correlation between the chemical elements and the size of the bandgaps. First of all it can be seen that the gap is mainly determined by the B-ion while the A-ion has less influence. Hf and Zr in the B-ion position generate large bandgap insulators with gaps above 6 eV. Ti, Ge and Sn lead to structures with gaps above 3 eV while W and Zn produce structures with gaps around 2 eV. Bandgaps of the stable compounds in the A 2 BO 4 structure. The gaps range from 0 to more than 6 eV. The chemical elements are sorted for similarity so that atoms that generate layered perovskites with similar bandgaps are close together.
Most of the oxides investigated have valence bands formed by the 2p levels of the O 2− ions. Since the nature of the bonds in these materials is highly ionic, this implies that the top of the valence band is roughly the same for all the oxides 11 . The conduction bands are mainly composed of the lowest unoccupied molecular orbitals (LUMO) of the A and B metal ions. This means that the size of the bandgap should correlate with the lowest lying LUMO level of the A and B cations. The relevant LUMO level is, however, not that of the cations in vacuum, but rather a cation sitting in the electrostatic field from the rest of the lattice, V R . To analyze this, we follow an Ewald-Evjen scheme, assuming that all the ions keep their nominal charges. This scheme is described elsewhere in the literature [35]. Table 2 shows how the LUMO energies change drastically passing from the free cations to the cations under the effect of V R . V R destabilizes much more the energies of the B-site ions than Table 2. LUMO energies (in eV) of several B 4+ ions in vacuum and in the presence of the crystal electrostatic field V R , respectively. V R is calculated for ASnO 4   and Hf 4+ . This difference is strongly reduced when V R is considered due to the smaller ionic radius of Ti 4+ (10 pm smaller than Zr 4+ or Hf 4+ ionic radii [36]), which has O 2− ions closer. In the case of the carbon group (Ge 4+ and Sn 4+ ), their compounds have similar bandgaps due to a compensation of effects: Ge 4+ s level in vacuum is lower in energy than Sn 4+ s level (by ≈ 7 eV), but Ge 4+ ionic radii is 16 pm smaller than that of Sn 4+ . When Cd 2+ occupies the A site, its LUMO (the s level) is more stable than the LUMO of B 4+ ions, contrary to the case of an alkaline-earth on A position. This leads to compounds with a conduction band dominated by the Cd 2+ s level and consequently to a small bandgap. A similar investigation has been performed also for the A 3 B 2 O 7 layered perovskites with similar results. Since nitrogen has an electronegativity very close to the one of oxygen, we expect that these results are still valid for the oxynitrides. The main difference will be that the valence band is formed by an hybridization of the oxygen and nitrogen 2p orbitals.
A couple of A-and B-ions can be used for both the A 2 BO 4 and the A 3 B 2 O 7 perovskites. For these cases, we have investigated the changes in the size of the bandgaps depending on the number of octahedra forming the 2D slab. There is a weak trend that correlates the bandgaps to the B-ions: when the B-ion position is occupied by a p-metal (e.g. Ge and Sn) the gaps are generally reduced when the slab thickness is increased, while for d-metals (e.g. Ti, Zr and Hf), the gaps seem to increase with slab thickness. In average, for the p-metals the bandgaps are reduced by 0.6 eV from the n = 1 to n = 2 structures and by 0.4 eV from the n = 1 to n = ∞ structures. For the d-metals the bandgaps are reduced by 0.03 and 0.2 eV, respectively. There are some exceptions to this trend, like Sr n+1 Ti n O 3n+1 . Lee et al [37] have recently shown that bandgaps of the Sr n+1 Ti n O 3n+1 Ruddlesden-Popper phase is reduced with an increase of the thickness and we have seen the same effect. In general, these trends could be used to tune the bandgap by increasing or reducing the number of octahedra within the layers.

Candidates for water splitting
The screening criteria of table 1 have been applied to the calculated structures. Out of the 300 investigated materials 20 fulfill the criteria for one-photon WS, see figure 5. Ba 2 TaO 3 N and Sr 3 Sn 2 O 7 are known to exist experimentally in the layered perovskite structure. Ba 2 TaO 3 N has also recently been proposed as a WS material by Wu et al [38] who used a computational screening procedure similar to the one proposed here. Y 2 BiO 3 N and the other A 3 B 2 O 7 compounds are already known in other stoichiometries and most of the A 2 BO 4 materials and Sr 2 TaO 3 N have been synthesized in other crystal structures with the same stoichiometry as the layered perovskite. It might not be an issue that some compounds are known in another crystal structure. In fact, layered perovskites can often be grown epitaxially even if they are not globally stable. Eight of the identified perovskites have not been investigated experimentally to our knowledge. In particular, the oxynitrides are interesting candidates for WS. In fact, the cubic perovskites ABO 2 N with A = Ba, Sr and Ca, and B = Ta and Nb are known to evolve hydrogen and/or oxygen in the presence of a sacrificial agent [11,39]. We expect that the layered perovskites containing these elements conserve the good properties in terms of activity already present in the cubic phase.
For the two-photon device with a silicon cathode, our screening identifies five layered perovskites as candidates for the anode material. All five are experimentally known in other structures/stoichiometry, but none of them has been used so far as photocatalyst.

Conclusions
With the aim of identifying stable and abundantly available semiconductors for light harvesting photo-electrodes for WS, we have screened 300 oxides and oxynitrides in the layered perovskite structure with stoichiometry A 2 BO 4 , A 3 B 2 O 7 and A 2 BO 3 N. The stability and bandgaps were calculated using DFT and the band edge alignment relative to the water red-ox potentials was estimated using an empirical formula. The accuracy of the calculated bandgaps, which were obtained with the GLLB-SC potential, was validated by comparing with state of the art G 0 W 0 calculations for 20 oxides in the cubic perovskite structure.
We have identified 20 candidate materials for one-photon WS, and additional 5 materials for photo-anode in a two-photon device with a silicon cathode. A few of these materials are already experimentally known, but none of them have been used so far for photoelectrocatalysis.
The trends in stability and bandgap have been studied in some detail. We have found that the bandgap of the layered perovskites is mainly determined by the position of the most stable LUMO of the A-and B-ion. Furthermore, our results indicate that the bandgap can be tuned to some extent by varying the number of octahedra forming the layered structure. Specifically, for a d-metal (p-metal) at the B-ion position the effect is to increase (decrease) the gap with the number of octahedra within a layer.