Indirect control of band gaps by manipulating local atomic environments using solid solutions and co-doping

The ability to tune band gaps of semiconductors is important for many optoelectronics applications including photocatalysis. A common approach to this is doping, but this often has the disadvantage of introducing defect states in the electronic structure that can result in poor charge mobility and increased recombination losses. In this work, density functional theory calculations are used to understand how co-doping and solid solution formation can allow tuning of semiconductor band gaps through indirect effects. The addition of ZnS to GaP alters the local environments of the Ga and P atoms, resulting in shifts in the energies of the P and Ga states that form the valence and conduction band edges, and hence changes the band gap without altering which atoms form the band edges, providing an explanation for previous experimental observations. Similarly, N doping of ZnO is known from previous experimental work to reduce the band gap and increase visible-light absorption; here we show that, when co-doped with Al, the Al changes the local environment of the N atoms, providing further control of the band gap without introducing new states within the band gap or at the band edges, while also providing an energetically more favourable state than N-doped ZnO. Replacing Al with elements of different electronegativity is an additional tool for band gap tuning, since the different electronegativities correspond to different effects on the N local environment. The consistency in the parameters identified here that control the band gaps across the various systems studied indicates some general concepts that can be applied in tuning the band gaps of semiconductors, without or only minimally affecting charge mobility.


Introduction
Semiconductors with direct band gaps that correspond to the energy of visible light (1.7-3.0 eV) have many applications in optoelectronics, such as diodes, solar cells and photocatalysts, e.g. for clean fuel generation and pollutant degradation using solar energy.The latter of these has proven hard to achieve at a commercially-viable level due to the difficulty of finding bulk materials that are cost-effective and possess the required combination of properties for high solar energy conversion efficiency, including low band gap and high charge separation and transport efficiency [1].For example, binary metal oxides such as TiO 2 and ZnO that have been extensively investigated for photocatalysis due to their beneficial characteristics including stability and earth-abundance generally have wide band gaps (>3 eV) that correspond to UV wavelengths [2,3].This is problematic because UV light accounts for only 4% of the energy in sunlight, while visible light comprises 53% [1].
Doping has been extensively investigated as a means of enhancing visible-light absorption, but this is often achieved by the dopants introducing localised, mid-gap states that do not always overlap well with the valence or conduction band states, so the enhanced light absorption often comes at the cost of decreased charge mobility and increased recombination losses [4,5].Furthermore, doping often leads to defect formation, e.g.due to charge compensation, which can also increase recombination losses and limit the dopant concentration and hence the extent of visible-light absorption that can be achieved while maintaining good crystal quality and charge mobility [6].Thus, lowering band gaps to achieve strong absorption of visible light while maintaining good charge mobility is challenging.Exploring approaches to control band gaps by shifting the existing band states rather than introducing new states may lead to better outcomes.
As alternatives to doping with single elements, co-doping and solid solution formation have emerged as potential approaches to achieving high-efficiency visible-light photocatalysts, where the charge compensation provided by the introduction of multiple elements can potentially overcome the deficiencies of single-element doping.For example, co-doping of TiO 2 with nickel and either tantalum or niobium, and co-doping SrTiO 3 with nickel and tantalum, have been found to give enhanced visible-light photocatalytic activity compared with doping with only nickel, due to the charge compensation between the two dopants giving a reduction in charge trapping [7].Co-doping TiO 2 with chromium and antimony [8] or tantalum and nitrogen [9] gave enhanced visible-light photocatalytic activity attributed to reduced recombination due to charge compensation in the former case and more dispersed, delocalised valence band states, in contrast to isolated states formed above the valence band for N-doped TiO 2 , in the latter case.Similarly, co-doping ZnO with aluminum and nitrogen has been found experimentally to decrease the band gap and increase the photoactivity compared with N-doped ZnO [10], with the charge-compensating aluminum increasing the amount of nitrogen that can be incorporated and improving the crystallinity, thus reducing recombination losses.
Solid solutions also provide this charge compensation effect, and offer the capacity to tune the width and position of band gaps by varying the composition (e.g. the ratio of the end-members in the solid solution); non-linear variations in band gap are sometimes seen, and band gaps lower than those of both end-members can be achieved.For example, both GaN and ZnO have band gaps higher than 3.0 eV, but their solid solutions have experimentally-measured band gaps of 2.6-2.8eV depending on the composition [11].Similar to co-doping ZnO with Al and N, these GaN-ZnO mixtures can have enhanced crystallinity, lower recombination and increased photoactivity compared with N-doped ZnO [12].We have shown computationally that ZnS-GaP solid solutions have tunable direct band gaps in the energy range of visible light and in the ideal range for solar water splitting, with good band dispersion suggesting high charge mobility [13], overcoming the limitations associated with both the wide band gap of ZnS and the indirect band gap of GaP.The visible-light activity of ZnS-GaP mixtures has been confirmed experimentally; ZnS-GaP nanowires have reported band gaps of 2.4-3.7 eV depending on the composition [14], while ZnS-GaP multilayered thin films have shown visible-light activity that was both enhanced compared with pure ZnS and extended to a considerably longer wavelength (650 nm) than that at which either pure ZnS or GaP showed photoactivity [15], with bonding between ZnS and GaP at the interfaces found to play a major role in achieving these effects.Therefore, charge-compensated approaches of solid solution formation and co-doping are promising for achieving the combination of strong visible-light activity, good charge mobility, and low recombination.
Dopants and compounds for combining in solid solutions are often chosen because they are expected to change the band gap by directly modifying the valence or conduction band states.For example, when nitrogen is added as a dopant to an oxide, the valence band maximum (VBM) will usually be formed by nitrogen states [16].However, it is possible that dopants may be able to play a role in indirectly controlling band gaps even if the atoms contributing states at the VBM and conduction band minimum (CBM) are not changed, e.g. by changing local atomic environments and thus their electrostatic potentials, and hence altering electronic energies.This indirect approach of controlling the band gap through altering local atomic environments, rather than directly changing the valence or conduction band states, potentially has the advantage of eliminating localised dopant states within the band gap and at the band edges, and hence reducing recombination losses.Co-doping may also improve the solubility of dopants, reducing disordering and allowing higher dopant concentrations and hence stronger light absorption to be achieved.
In our previous density functional theory (DFT) work on ZnS-GaP [13], we showed that the solid solution has a band gap that varies non-linearly with composition and also depends on atomic ordering; direct band gaps lower than those of both pure ZnS and GaP are seen, an effect confirmed in our experimental work [15,17].These results are somewhat surprising, since, based on the type-I band alignment of pure ZnS and GaP [18], it could be assumed that the valence band of their mixtures would be formed by P states and the conduction band by Ga states, as in pure GaP, so no significant change in band gap compared with pure GaP would be expected.However, this is not what has been observed both computationally and experimentally.Similarly, other co-doped and solid solution systems including the AlN-ZnO and GaN-ZnO systems discussed above have shown non-linear variations in band gap and band gaps lower than either of the constituents.
In order to understand the origins of these non-linear effects and the indirect effects of element additions on the electronic properties of semiconductors, here we investigate several representative and varied systems for which experimental results have been reported-ZnS-GaP mixtures, both in the form of solid solutions and multilayered thin films, and co-doped ZnO.
For ZnO-based solid solutions, simulation cells contained 32 atoms, constructed as a 2 × 2 × 2 expansion of the conventional wurtzite ZnO unit cell.For ZnS-GaP multilayered structures, the simulation cells varied in size depending on the layer thickness, with the smallest containing 12 atoms and the largest 64 atoms.In the plane of the interface, the supercells were constructed by 1 × 1 and √2 × √2 expansions of the conventional zinc blende unit cell and a 1 × 1 expansion of the primitive zinc blende unit cell for (100), (110) and (111) interfaces, respectively.For (ZnS) x (GaP) 1−x solid solutions, the simulation cell used for most calculations contained 16 atoms, constructed as a 2 × 2 × 2 expansion of the primitive zinc blende unit cell; however, in order to access a greater range of compositions and orderings, some calculations for x ⩽ 0.125 were done with supercells containing 54 atoms (3 × 3 × 3 expansion of the primitive zinc blende unit cell) or 64 atoms (2 × 2 × 2 expansion of the conventional zinc blende unit cell).
The Monkhorst-Pack grid for k-point sampling was set to 12 × 12 × 12 for the Brillouin zone; convergence with respect to the number of k-points was confirmed 5 .Densities of states and electrostatic potentials were calculated after a full geometry optimisation of all lattice parameters and atomic positions.This method was found to accurately reproduce the experimentally-measured band gaps and lattice parameters of the binary compounds considered in this work (table 1).
Formation enthalpies, ∆E f , for the ZnS-GaP multilayered structures and enthalpies of mixing, ∆E mix , for solid solutions and co-doped systems were calculated by: where is the enthalpy of a ZnS-GaP mixture or ZnO co-doped with a metallic element M and nitrogen, and E (AB) and E (CD) are the enthalpies of the pure constituents, i.e.ZnS, GaP, ZnO and the metal nitride MN.

ZnS-GaP multilayered structures
Simulation cells with equal amounts of ZnS and GaP but with different atomic thicknesses of alternating ZnS and GaP layers were created.Atomic thicknesses between one and eight for each layer were investigated for three low-index interfacial planes: (111), (100) and (110).Figure 1 shows, as examples, the structures with layer thicknesses of four atomic layers for each of the three interfacial planes.A key difference between the interfacial planes is that (111) and (100) are polar, meaning that any layer of atoms parallel to the interface contains either cations or anions, but not both, and the bonds at a given interface are either all Ga-S or all Zn-P; in contrast, (110) is a non-polar interface-each layer of atoms parallel to the interface contains an equal number of cations and anions and interfacial bonds alternate between Ga-S and Zn-P at both interfaces.The band gaps of the heterostructures are plotted as a function of the thickness of each ZnS/GaP layer in figure 2. The band gaps generally decrease as the thickness of the ZnS/GaP layers increases; the structures with (100) interfaces show the steepest decrease in band gap with increasing layer thickness.The structures with (100) and (111) interfaces become conducting if the number of alternating layers is higher than four and five, respectively.When the interfacial plane is (110), the decrease in band gap with layer thickness is considerably less steep, likely related to the non-polar nature of this interface.Significantly, the band gaps of all structures are smaller than for bulk ZnS and smaller in many cases than for bulk GaP.Most band gaps are direct, with just a few cases showing an indirect band gap slightly smaller than the direct band gap.These results are consistent with experimental findings that the photoactivity of ZnS-GaP multilayered structures under visible-light wavelengths is higher than that of pure ZnS [15].Note that the structure with a (110) interface in which each ZnS and GaP layer is one atomic layer thick shows anomalous behaviour-it has the same band gap as the (100) structure with layers of the same thickness.This is because both structures have the same bonding environments at the interfaces as shown in supporting information, figure S1.
For all three interfacial planes and all layer thicknesses, the largest contribution to the density of states at the VBM is always from P atoms, while the largest contribution at the CBM is from Ga atoms (figure 3 and supporting information, figure S2), consistent with our previous work on ZnS-GaP solid solutions [13].Taking the structure with (100) interfaces and four alternating layers of ZnS and GaP as an example, it can be seen that the VBM is specifically formed predominantly by states on P atoms that are bonded to Zn (i.e.P atoms at the interface with ZnS, figure 3), while the largest contribution to the density of states at the CBM is from Ga atoms that are bonded to S (i.e.Ga atoms at the interface with ZnS, figure 3).This also applies to structures with (111) interfaces (supporting information, figure S2(a)).The structures with (110) interfaces do not have a single layer that dominates the contribution to either the VBM or CBM.Instead, all GaP layers have indistinguishable densities of states and contribute equally at the VBM and CBM, with a larger contribution than the ZnS layers (supporting information, figure S2(c)).
To understand why atoms in particular layers predominantly form the VBM and CBM and consequently to explain why the band gaps decrease with increasing layer thickness, the electrostatic potentials at each atom for all the structures were calculated.It was found that the most positively charged Ga environment in every structure is always for the Ga bonded to S at the interface; as discussed above, it is this Ga bonded to S at the interface that contributes most to the density of states at the CBM, and hence the Ga in the most positively charged environment (i.e. the Ga at the interface) forms and determines the energy of the CBM.When the thickness of the alternating layers of ZnS/GaP increases, the calculated electrostatic potential at the Ga bonded to S increases (i.e. the negative potential value becomes smaller in magnitude, figure 4(a)), lowering the energy of electronic states on this Ga atom, so the CBM energy and the band gap decrease with increasing layer thickness.Correspondingly, the atom that contributes the most to the VBM density of states is the P bonded to Zn at the interface, and the electrostatic potential at this P atom is always the most negative compared with other P atoms in each structure.Hence the electronic energies of this P atom are higher than for other P atoms.As shown in figure 4(b), as the thickness of the layers increases, the electrostatic potential at these P atoms becomes more negative, so the VBM increases and the band gap decreases.
Thus, the changes in band gap are closely related to the interfacial atomic environments.These local environments are so important that differences in the behaviour for the three different interfacial planes can correspondingly be explained by the differences in the interfacial bonding arrangements.At the (100) interfaces, there are two Zn-P bonds per Zn and P atom, and correspondingly two Ga-S bonds per Ga and S atom, while at the (111) interface, there is just one such interfacial bond per atom (figure 1).Due to this larger number of Zn-P/Ga-S bonds at the (100) than ( 111) interfaces, the electrostatic potential at the interfacial Ga atoms is more positive (i.e.smaller in magnitude) and the conduction band energy is lower for (100) structures (for the same number of atomic layers in each ZnS/GaP layer, figure 4(a)).Similarly, the electrostatic potential at the interfacial P atoms is more negative, and the valence band energy is higher, for structures with a (100) interface than with a (111) interface (figure 4(b)).Thus, for a given thickness of ZnS/GaP layers, the band gap is lower for structures with a (100) interface than (111), corresponding to the steeper decrease in band gap with increasing layer thickness seen for structures with (100) interfaces (figure 2).
At the (110) interfaces, there is one Zn-P/Ga-S bond per atom when there are two or more layers of ZnS alternating with GaP, as for the (111) interface.However, the non-polar nature of the (110) interface, i.e. there are Zn-P bonds neighbouring Ga-S bonds at each interface (figure 1), limits the extent to which the electrostatic potentials at the interfaces change with increasing thickness of the ZnS/GaP layers (figure 4).This results in much smaller changes in the CBM and VBM with increasing layer thickness and hence smaller changes in the band gap than for the structures with (111) and (100) interfaces (figure 2).No clear trends between layer thickness, VBM energy and electrostatic potential at P atoms are seen for structures with (110) interfaces (figure 4(b)).Again, the (110) structure with only one layer of ZnS alternating with GaP is anomalous; this structure has two Zn-P/Ga-S bonds at the interface per atom as for the structure with a (100) interface and one layer of ZnS alternating with GaP (figure S1, supporting information), and correspondingly these two structures have the same electrostatic potentials for atoms at the interfaces (figure 4), the same VBM and CBM energies and the same band gaps (figure 2).
Overall, it is seen from these results that band energies and hence band gaps are determined by electrostatic potentials, which are in turn related to local bonding environments.Zn and S states do not form either the VBM or the CBM in any structure; the effect of the presence of ZnS is only to alter the local environments of Ga and P atoms and hence shift the energies of Ga and P states.

Mixing at interfaces
The ZnS-GaP multilayered structures with (100) and (111) interfaces show a steep decrease in band gap with increasing ZnS/GaP layer thickness (figure 2) and become conducting for layer thicknesses higher than four and five, respectively.These changes are driven by the strong polarity of the (100) and (111) interfaces, and may not be realistic outcomes experimentally.Furthermore, it has been shown by aberration-corrected scanning transmission electron microscopy and energy-dispersive x-ray spectroscopy that the interfaces of ZnS-GaP multilayered structures contain structural defects and an interdiffused region (∼5 nm thick) of ZnS and GaP, essentially making the interface region a solid solution of ZnS and GaP [36].The presence of this solid solution at the interfaces is likely to be the origin of the visible-light activity of the multilayered structures [15].
To investigate the potential for mixing to occur across interfaces, every interface in the structures with either three or four alternating ZnS/GaP layers for the (111) and (100) interfaces was changed into a mixed layer with equal amounts of ZnS and GaP, as shown in figure 5(b) for the (100) structure with four alternating layers.The band gaps of these structures with interfacial mixing are found to be direct and higher than their counterparts with pure ZnS/GaP interfaces (table 2).However, they are still significantly smaller than for pure ZnS, and also smaller than for pure GaP in most cases, so this mixing does not change the conclusion that ZnS/GaP mixed structures should have good visible-light activity.The formation enthalpies  of the mixed structures are slightly higher than for the structures with pure interfaces (supporting information, figure S3), but the differences are small and mixing is often controlled by kinetic and entropic effects.
Figure 5 shows the densities of states per layer for the (100) structure with four alternating ZnS/GaP layers and mixed interfaces.Unlike its counterpart with pure interfaces, this structure shows a more uniform distribution of states at the VBM and CBM across several layers-all layers containing Ga bonded to S contribute significantly to the CBM and all layers containing P bonded to Zn contribute significantly to the VBM.In terms of the electrostatic potentials, the most positive value (i.e.smallest in magnitude) at a Ga atom in the structures with mixed interfaces is lower than in their counterparts with pure interfaces and the same number of alternating layers and the most negative value at a P atom is higher, corresponding to the wider band gaps.

ZnS-GaP solid solutions
It is now interesting to reconsider our previous work on ZnS-GaP solid solutions, to see if the same correlations between band gaps and electrostatic potentials of local atomic environments also apply and to see if such correlations can explain the strong non-linear effects on the band gaps of these solid solutions with variations in composition that have been previously reported [13].
Here, we focus initially on solid solutions with a 50:50 ratio of ZnS to GaP, in line with the multilayered structures discussed above.A 16-atom supercell is used and, for this composition, several orderings are possible, which can be considered as intermediates between two extremes, one in which the numbers of Zn-P and Ga-S bonds are minimised (each Zn atom is bonded to one P and three S atoms, figure 6(a)) and another where the numbers of Zn-P and Ga-S bonds are maximised (each Zn atom is bonded to one S and three P atoms, figure 6(b)).Thus, the atomic orderings can be quantitatively described by the average number of Zn-P bonds per Zn atom.
The band gap decreases as the number of Zn-P bonds increases (figure 7(a)) and the changes in the band gap correspond to differences in the electrostatic potentials.For the ordering in which the number of Zn-P bonds is maximised, giving the minimum band gap, the electrostatic potential at the Ga atoms is higher (smaller in magnitude) and the electrostatic potential at the P atoms is more negative than in the other orderings, corresponding to a lower CBM and a higher VBM energy, respectively (figures 7(b) and (c)).The smallest magnitude (i.e.most positive value) for the electrostatic potential at any Ga atom progressively decreases as the number of Zn-P bonds decreases, reaching a minimum for the ordering in which the number of Zn-P bonds is minimised, corresponding to the highest CBM energy and largest band gap; similarly, the most negative electrostatic potential at any P atom reaches a maximum for the ordering in which the number of Zn-P bonds is minimised.
In orderings where there are symmetrically non-equivalent atomic environments, the top of the valence band is formed predominantly by states on the P atoms with the lowest electrostatic potential, i.e. those with the most bonds to Zn (figure 7(d)), consistent with the results presented earlier for ZnS-GaP multilayered structures.Similarly, the bottom of the conduction band is formed by states on Ga atoms with the highest (smallest magnitude) electrostatic potential, which are those with the most S neighbours.
Turning to compositions other than (ZnS) 0.5 (GaP) 0.5 , figure 8 shows the band gap as a function of composition for (GaP) 1−x (ZnS) x .Since the band gap is strongly influenced by atomic ordering, the trends with composition are shown separately for orderings for which the average number of Zn-P bonds per Zn atom (or the number of Ga-S bonds per Ga atom for x ⩾ 0.5) is one, two or three.The band gap varies almost linearly with composition when there is an average of only one Zn-P bond per Zn atom (i.e. three Zn-S bonds), but there is large negative deviation from linearity when the number Zn-P/Ga-S bonds is increased.This indicates that the non-linear behaviour of solid solution band gaps is related to ordering, and that stronger non-linear behaviour and better band gap tunability of semiconductor solid solutions is associated with good mixing of the semiconductors (i.e. a large number of Zn-P and Ga-S bonds).
In general, the variation of band gap, E g (x), as a function of composition, x, for a solid solution (AB) x (CD) (1−x) can be described by [37]: where b is the band gap bowing parameter, which quantifies the deviation from linearity, and E g (AB) and E g (CD) are the band gaps of the binary constituents AB and CD, respectively.Following Bernard and Zunger [37], the bowing parameter b can be decomposed into three parts: The volume deformation component (b VD ) is the sum of the changes in the band gaps of the binary constituents due to compression/dilation when the volume is changed to that of the mixed compound (assuming linear variation of the lattice parameters with composition).The charge exchange component (b CE ) is obtained by substituting atoms in the deformed binary constituents to form the mixed compound with no relaxation of the bond lengths.The final component (b SR ) is the change in the band gap that occurs upon structural relaxation of the mixed compound.These three components of the bowing parameter have been calculated for three atomic orderings for composition (GaP) 0.5 (ZnS) 0.5 (table 3).The volume deformation component of the bowing parameter, b VD , is very small, because the band gap increase caused by volume compression of GaP is similar to the band gap decrease caused by volume expansion of ZnS.Furthermore, the band gap changes due to these volume deformations are small because the lattice parameters of pure GaP and ZnS are very similar (∼1% mismatch [32]).The value of b CE is large and positive for all atomic orderings, and it increases as the number of Zn-P bonds increases.The value of b SR is smaller in magnitude than b CE and negative for all atomic orderings.This gives overall positive values of b that increase as the number of Zn-P bonds increases, corresponding to the negative deviation from linearity seen in figure 8.
Table 3.Values of the band gap bowing parameter, b, and its components for (GaP)0.5(ZnS)0.5 for atomic orderings for which the number of Zn-P and Ga-S bonds per atom is one, two or three.Note that, for the structure with 3 Zn-P bonds per Zn/P atom, the band gap becomes zero upon substituting atoms in the deformed binary constituents with no relaxation of the bond lengths, so the values of bCE and bSR are given as limiting minimum and maximum values, respectively.These results indicate that the charge exchange component is the most important in determining the bowing parameter and hence the band gaps of these solid solutions.The better the mixing of ZnS and GaP, the more Zn-P and Ga-S bonds there are, and the larger the value of b CE and the decrease in the band gap.Obtaining low band gaps in synthesised materials will therefore require good mixing, such that there are sufficient Zn-P and Ga-S bonds.The dominance of the b CE component is consistent with the importance discussed above of electrostatic potentials in determining the valence and conduction band energies and, hence, the variation in the band gap for different atomic orderings.As for the multilayered structures, it is seen that the addition of ZnS changes the band gap indirectly by altering the local environments and hence electrostatic potentials of Ga and P atoms; the relative positions of Zn and S to Ga and P in the structures, i.e. the atomic ordering, correspond to different effects on local environments, hence different values of b CE and different extents of non-linearity in the band gap variation.

ZnO-based solid solutions
We now investigate further the relationship between electronic properties and local atomic environments, particularly electrostatic potentials, in solid solutions based on ZnO.We start by examining the effect of atomic ordering on the band gap, using (ZnO) 0.9375 (AlN) 0.0625 as an example.Seven different orderings of the atoms in the 32-atom simulation cell are considered, each with a different value for the minimum separation between the Al and N atoms; four structures are shown as representative examples in figure 9, and all orderings are shown in figure S4, supporting information.Even at this low concentration of AlN, the band gap is significantly reduced compared to pure ZnO for all orderings (table 4).The densities of states show that this is due to an increase in the energy of the VBM, caused by the introduction of a N 2p state at the top of the valence band (figure 10) that is at higher energies than the O states that form the VBM in undoped ZnO, consistent with previous work on the effects of N doping in ZnO [38,39].As in undoped ZnO, the conduction band is formed predominantly by Zn states.Aluminum states do not contribute significantly to either the VBM or CBM, and are predominantly found at higher energies than the Zn states in the conduction band.Thus Al does not significantly contribute to and hence directly affect either the VBM or CBM.
The band gaps differ significantly for different atomic orderings (table 4).The densities of states show that the differences in the band gaps are mainly due to changes in the energy of the N states (e.g. the separation between the N states at the VBM and the O states which are predominantly at lower energies, figure 10) and hence changes in the energy of the VBM (figure 10).Since it is a N state that forms the top of the valence band and it is the energy of this state that changes most significantly between the different orderings, we anticipate a relationship between the VBM, the band gap and the electrostatic potential at the N atom.In general, looking at all orderings, as the electrostatic potential at the N atom decreases, the energy of the VBM increases (figure 11(a)).This decrease in the electrostatic potential at the N atom is correlated with an increase in the minimum Al-N separation (figure 11(b)).Although the changes in the energy of the CBM are much smaller than in the energy of the VBM, the CBM energy does show a correlation with the electrostatic potential at the Zn atom that makes the largest contribution to the states at the bottom of the conduction band (i.e. the Zn with the most positive electrostatic potential, figure 11(c)) excluding one structure for which these Zn atoms are not equivalent to those in the other structures.
It is clear once again that the electrostatic potentials at the atoms contributing the most states to the valence and conduction bands are crucial in determining the band energies and hence the band gap.The Al atom does not contribute states to directly affect the energy of either the VBM or CBM, but it evidently indirectly influences the band gap by affecting the electrostatic potential at the N atom.Therefore, in co-doped systems and solid solutions, the introduced elements can have indirect effects on the band gap and their electronegativities are expected to be important in determining the magnitudes of the band gaps.To explore this, we compare replacing the Al with two other metals with different electronegativities-Ga and Sc.To provide an additional example of the effect of substituting atoms of different electronegativity, the effects of simultaneously replacing one O with N and one O with F were also examined.To consider only the effect of the different electrostatic potentials at the N atom, the atomic ordering used is Ordering 7; in this ordering, the nitrogen and co-dopant atom are not directly bonded to each other (figure 9), thus eliminating effects on the valence and conduction band energies arising from different bonding interactions.For these calculations, the simulation cell was doubled to 64 atoms, with one O replaced with N and either one Zn replaced with Al, Ga or Sc, or another O replaced with F. This gives a total concentration of dopant elements of 3.125 at%; the lower dopant concentration compared with the previous calculations was used so that oxygen atoms sufficiently far from the dopant atoms could be used as a reference, allowing band energies and electrostatic potentials for the different compositions to be aligned.The structures used in these calculations are shown in supporting information, figure S5.
In all cases, the densities of states show that the top of the valence band is formed predominantly by N 2p states (supporting information, figure S6), consistent with the results of Di Valentin for (ZnO) 1−x (GaN) x [40] and the results presented above for (ZnO) 1−x (AlN) x .The conduction band is always formed predominantly by Zn states, except when the co-dopant is Ga, in which case Ga states also make a significant contribution to the conduction band (supporting information, figure S7).
The addition of elements with different electronegativities as co-dopants with N results in different charges on the atoms; these different charges change the local atomic environments and hence the valence and conduction band energies.It can be seen that there is, for example, a correlation between the band gap and the charge on the dopant metal atom (figure 12(a)).This trend can be attributed to the differences in the electrostatic potentials, which are in turn due to the charge distribution over all atoms.Thus, there are corresponding trends between the electrostatic potential at the N atom and the energy of the VBM (figure 12(b)), and between the highest (smallest magnitude) electrostatic potential at a Zn atom and the energy of the CBM (figure 12(c)).When co-doped with Ga, the Ga states do contribute significantly to the conduction band, whereas in the other cases the conduction band is formed predominantly by Zn states; this Ga contribution to the conduction band changes the band energy, and hence the CBM energy for (ZnO) 0.96875 (GaN) 0.03125 lies below the general trend in figure 12(c).Similarly, for co-doping with N and F, the data point lies above the trend in the energy of the VBM shown in figure 12(b); this can be related to other factors influencing the VBM energy, such as the value of the most negative electrostatic potential at an O atom (figure 12(d)).
These results further demonstrate that dopants and co-dopants can be used to control the band gaps of semiconductors, not only by directly altering the valence and conduction band states, but by changing the local atomic environments and hence the electrostatic potentials of the atomic environments.In this way, it may be possible to tune band gaps without introducing states at the band edges or in the mid-gap region that are localised on dopant atoms, with the corresponding potential for poor charge mobility and increased recombination losses.

Thermodynamic stability
The formation enthalpies of the ZnS-GaP multilayered structures and mixing enthalpies of ZnO-AlN solid solutions reported here are low, at <12 kJ mol −1 (figure S3, supporting information) and <2 kJ mol −1 (table 3 and figure S8, supporting information), respectively, as expected based on the small lattice mismatch of both systems.Mixing enthalpies are similarly low for the ZnO-GaN and ZnO-ScN solid solutions (1.4 kJ mol −1 and 0.2 kJ mol −1 , respectively).As discussed in detail in our previous work [13,41], mixing enthalpies of ZnS-GaP solid solutions are also low.For ZnO-AlN, the mixing enthalpies increase as the separation between the Al and N atoms increases (figure S8, supporting information), corresponding to the increase in enthalpy of mixing as the average number of Zn-P bonds per atom increases for ZnS-GaP solid solutions seen in our previous work [13], i.e. enthalpies of mixing tend to increase as the interaction between elements from the same parent compound decreases.Nevertheless, using appropriate synthesis conditions, For the ZnS-GaP multilayered structures, the (111) interfacial plane gives the lowest formation enthalpies up to a layer thickness of three atomic layers, then (110) becomes lower in formation enthalpy for larger thicknesses, although the formation enthalpy of (111) remains low at ∼3 kJ mol −1 with increasing thickness (figure S3, supporting information).This is promising in terms of achieving visible-light absorption in practice with such structures, since band gaps of ∼2 eV are seen in the thickness range in which (111) interfaces are most favourable.These observations are consistent with experimental work showing the dominance of (111) planes and good visible-light activity for ZnS-GaP multilayered structures [15].
For ZnO-AlN, an analysis of the energetics allows a benefit of the co-dopant to be seen additional to the indirect control of the band gap and band energies discussed above.Two plausible reactions for forming the solid solution are considered, i.e. formation by the oxides of Zn and Al reacting with N 2 , or formation by mixing ZnO and AlN: Each of these reactions can be broken down into two steps, the first for addition of N and the second for addition of Al: In these reactions, the single dopant cases are charge compensated with either a Zn or O vacancy as appropriate from electroneutrality considerations.Equivalent reactions can be considered where charge compensation is provided by either an O or a Zn interstitial.The enthalpy changes for each of these reaction steps was calculated, using a 32-atom supercell with 2 Al and/or 2 N atoms.For the single-doped structures, several orderings were tested and the ordering with the lowest enthalpy is used in the calculations.For the AlN co-doped structure, results for two orderings are used-the one found to give the lowest enthalpy but having almost the same band gap as undoped ZnO, and another with moderately higher enthalpy (by 2.9 kJ mol −1 ) that has a band gap of 2.1 eV, in the ideal range for solar absorption.
The results are shown in table 5.It can be seen that the reaction enthalpies are mostly endothermic for addition of N and then exothermic for addition of Al as a co-dopant.This indicates that addition of the two compensating dopants is a more promising synthetic route than addition of N as a single dopant, and so reducing the band gap of ZnO by co-doping is more feasible than adding N on its own.

Band structures
We have previously reported the band structures for ZnS-GaP solid solutions [13]; these solid solutions show significant band dispersion and have no isolated states within the band gap, suggesting that good charge mobility and hence good performance in photocatalytic and other optoelectronic applications is expected.This is true regardless of composition.In general, when doping is used to reduce band gaps, isolated states can be particularly problematic at low dopant concentrations, which are most feasible to achieve experimentally.However, addition of even small amounts of GaP to ZnS should give significant reductions in band gap, depending on the synthesis conditions (figure 8 and [41]), without producing isolated states (figure S9(a), supporting information).The same is also seen for ZnS-GaP multilayered structures (e.g.no isolated states are seen in the densities of states shown in figures 3, 5 and S2, supporting information).The lack of isolated states arises from achieving band gap tuning not by introducing new states at the band edges but by shifting the energy of the existing states, i.e. the valence and conduction bands are formed by the same atomic orbitals as in pure GaP, and the addition of ZnS only shifts those states by altering the local environments of the Ga and P atoms.
The same is not true for the ZnO-MN mixtures.In this case, the band gap is altered directly by N states that are introduced at the top of the valence band and, depending on the ordering, can be isolated from the other VB states (figure 10).While this may give sub-optimal charge transport behaviour, it is important to note that the N state is contiguous with the O states in the valence band for the orderings with the lowest enthalpies (figure 10), but when N is added on its own as a dopant in ZnO, there is no ordering effect and the N states are always somewhat isolated from the O states in the valence band (figure S9(b), supporting information).

Conclusions
For both ZnS-GaP and ZnO-AlN semiconductor mixtures, the local atomic environments, particularly their electrostatic potentials which in turn are related to their bonding environments, are key to determining the valence and conduction band energies and hence the band gaps.For ZnS-GaP, this means that the valence and conduction band states are always formed by P and Ga, respectively, and the ZnS plays only an indirect role by shifting the energies of these states to change the band gap.Thus, the band gap can be tuned across a wide range of values without introducing the localised states that often lead to increased recombination losses in other approaches to band gap tuning.For ZnO-AlN, the N does introduce new states at the top of the valence band resulting in a band gap reduction, but the role of Al is to only modify the N environment and hence the band gap.Further band gap tunability is obtained by swapping Al with other elements of different electronegativity.Thus, the work presented here provides some explanation of the band gap tunability and non-linear behaviour that has been observed in experimental work on both ZnS-GaP and ZnO-AlN systems.Specific orderings are studied here to allow trends and hence factors that influence the band gaps to be identified, but in practice, synthesised materials would almost certainly contain a mixture of orderings and the experimentally-measured band gap would be a multi-configurational average, as discussed in our previous work [41].
The consistency of the effects and key parameters found here that determine the band gap across chemically (i.e.ZnS-GaP, ZnO-AlN) and physically (solid solutions, heterostructures) varied systems indicates at least some generality, although the extent of this cannot yet be concluded.However, it can be seen that co-doping and combining semiconductors, whether in multilayered structures or solid solutions, based on the principle of selecting multiple charge-compensated elements on their ability to alter the electrostatic potentials of local environments, rather than selecting elements that will introduce new states at the band edges, is a viable route to band gap tunability and hence potentially achieving enhanced light absorption with reduced defect formation, improved crystal quality, and thus lower recombination losses than doping with single elements or other approaches that can result in localised electronic states.In particular, systems that rely purely on indirect effects, i.e. shifting band energies without introducing new states at the band edges such as the case of adding ZnS to GaP, can avoid the introduction of localised mid-gap states.

Figure 2 .
Figure 2. Band gap as a function of ZnS/GaP layer thickness (in atomic layers) for multilayered structures with interfacial planes of (100), (110) and (111).The value given for zero thickness is that of pure bulk ZnS.

Figure 3 .
Figure 3. (a) Projected densities of states (DOS) per layer of the ZnS-GaP multilayered structure with (100) interfaces and four atomic layers of ZnS alternating with four atomic layers of GaP (32-atom simulation cell).The two layers of ZnS and GaP that are not at an interface have very similar DOS, and hence only their average DOS is shown.The dotted vertical lines indicate the VBM and CBM for each layer.(b) Corresponding structure with layer numbering scheme indicated.

Figure 4 .
Figure 4. (a) CBM energy as a function of the most positive (smallest in magnitude) electrostatic potential at a Ga atom (i.e. a Ga atom at the interface with ZnS), and (b) VBM energy as a function of the most negative electrostatic potential at a P atom (i.e. a P atom at the interface with ZnS), for the ZnS-GaP multilayered structures with different interfacial planes and ZnS/GaP layer thicknesses.The numbers inside the data point symbols indicate the thickness (in number of atomic layers) of each ZnS/GaP layer.

Figure 5 .
Figure 5. (a) Projected densities of states (DOS) per layer of the ZnS-GaP multilayered structure with a (100) interface and four atomic layers of ZnS alternating with four layers of GaP, with mixing of ZnS and GaP in the interfacial layers (32-atom simulation cell).The dotted vertical lines indicate the VBM and CBM for each layer.(b) Corresponding structure with layer numbering scheme indicated.

Figure 6 .
Figure 6.Two orderings of a 16-atom supercell (shown here for 64 atoms for clarity) for a (ZnS)0.5(GaP)0.5solid solution, which can be considered to be the two limiting cases in which the number of Zn-P and Ga-S bonds is (a) minimised and (b) maximised.

Figure 7 .
Figure 7. (a) Band gap as a function of the average number of Zn-P bonds per P (and Zn) atom (corresponding to the average number of Ga-S bonds per S and Ga atom), (b) CBM energy as a function of the most positive (smallest magnitude) electrostatic potential at a Ga atom, and (c) VBM energy as a function of the most negative electrostatic potential at a P atom, for different orderings of a (ZnS)0.5(GaP)0.5solid solution.In (b) and (c), each data point is labelled with the corresponding average number of Zn-P bonds per Zn atom in the structure.Dashed lines are linear fits of the data and serve only to guide the eye.(d) Projected densities of states of one ordering of (ZnS)0.5(GaP)0.5(16-atom supercell) with varying local atomic environments (average of 1.5 Zn-P bonds per P/Zn atom).The vertical dashed line indicates the VBM.

Figure 8 .
Figure 8. Band gap as a function of composition for a (ZnS)x(GaP) 1−x solid solution, with trends shown separately for different values of the average number of Zn-P bonds per Zn atom or Ga-S bonds per Ga atom for x ⩾ 0.5.The dotted lines are fits of the data to equation (2), with b values of 12.6 for 4 Zn-P/Ga-S bonds, 10.4 for 3 Zn-P/Ga-S bonds, 3.3 for 2 Zn-P/Ga-S bonds and 0.9 for 1 Zn-P/Ga-S bond.The band gaps shown here are the direct band gaps at the Γ point.

Figure 9 .
Figure 9. Four orderings of the 32-atom supercell used for the (ZnO)0.9375(AlN)0.0625solid solution: (a) Ordering 1 (Al and N are nearest neighbours); (b) Ordering 2 (Al and N are nearest neighbours, but with a different orientation of the Al-N bond from Ordering 1); (c) Ordering 3; (d) Ordering 7 (Al and N are as far apart as possible).

Figure 10 .
Figure 10.Total densities of states and projected densities of states for the N and Al atoms, as well as the sum of all O atoms, in the energy region of the band gap for the 32-atom (ZnO)0.9375(AlN)0.0625supercell: (a) Ordering 1; (b) Ordering 2; (c) Ordering 7. The vertical dashed lines indicate the VBM.

Figure 11 .
Figure 11.For (ZnO)0.9375(AlN)0.0625:(a) VBM energy as a function of the electrostatic potential at the N atom; (b) the electrostatic potential at the N atom as a function of the minimum Al-N separation; (c) CBM energy as a function of the electrostatic potential at the Zn atom contributing the most states to the bottom of the conduction band (excluding Ordering 4, for which there are a larger number of Zn atoms with the same electrostatic potential and, in the structure, these Zn atoms are offset from the Al atom in the direction parallel to the c lattice direction, in contrast to all the other structures for which the Al atom is in the same ab-plane as the Zn atoms with the most positive electrostatic potentials).Dashed lines are linear fits of the data and serve only to guide the eye.In (b), two data points lying below the trend are excluded from the fit; in the structures corresponding to these data points, the Al-N separation is parallel to the c lattice direction (figures 9(a) and (c)) in contrast to the other structures, and the second shortest Al-N distance is longer than in the other structures, indicating some influence from the second-neighbour N atom of Al.

Figure 12 .
Figure 12.(a) Band gap as a function of Mulliken charge on the metal dopant atom (Al, Ga, or Sc) for ZnO co-doped with N and a metal; for ZnO co-doped with N and F, the charge on the Zn atom at the equivalent site is used.Each data point is labelled with the co-dopant element added with N. (b) Energy of the VBM as a function of the electrostatic potential at the N atom, (c) energy of the CBM as a function of the highest electrostatic potential at a Zn atom, and (d) energy of the VBM as a function of the lowest electrostatic potential at an O atom, for ZnO co-doped with N and either a metal on a Zn site or F on a O site.Dashed lines are third (a) and (d) or second degree polynomial (b) and (c) fits of the data and serve only to guide the eye.

Table 1 .
Calculated and experimental values of band gaps and lattice parameters for ZnS, GaP, ZnO, AlN, GaN and ScN.

Table 2 .
Band gaps of ZnS-GaP multilayered structures with (100) and (111) interfaces, and either three or four atomic layers of ZnS alternating with GaP, for both pure and mixed interface layers.

Table 5 .
Calculated enthalpies for addition of Al and N together to ZnO (i.e.overall formation of a ZnO-AlN solid solution), with enthalpies for step-wise addition of N and Al as single dopants, based on reactions 6 and 7. Enthalpies are given per mol of Al/N dopant added.For step 2 and the overall reaction enthalpy, two values are given, for two different orderings of the ZnO-AlN solid solution.