Arsenic doping and diffusion in CdTe: a DFT study of bulk and grain boundaries

The doping of CdTe with As is a method which is thought to increase cell efficiency by increasing electron hole concentrations. This doping relies on the diffusion of As through CdTe resulting in AsTe substitution. The potential effectiveness of this is considered through kinetic and electronic properties calculations in both bulk and Σ3 and Σ9 grain boundaries using Density Functional Theory. In bulk zinc-blende CdTe, isolated As diffuses with barriers <0.5 eV and with similar barriers through wurtzite structured CdTe, generated by stacking faults, suggesting that As will not be trapped at the stacking faults and hence the transport of isolated As will be unhindered in bulk CdTe. Substitutional arsenic in bulk CdTe has little effect on the band gap except when it is positively charged in the AX-centre position or occurring as a di-interstitial. However in contrast to the case of chlorine, arsenic present in the grain boundaries introduces defect states into the band gap. This suggests that a doping strategy whereby the grain boundaries are first saturated with chlorine, before single arsenic atoms are introduced, might be more beneficial.


Introduction
Polycrystalline CdTe research cells have recently been demonstrated with efficiencies above 22% [1], however, this is still well below the theoretical maximum. This discrepancy is mainly attributed to low carrier concentration and short carrier lifetime, resulting in a lower V OC (open-circuit voltage) * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. [2]. Improvement to the V OC has been slow, with the highest V OC demonstrated in the high efficiency cells being 900 mV which is short of what is possible in a material with a band gap of 1.4-1.5 eV [3,4]. However, recent work in single crystal CdTe indicates that efficiency towards 25% in polycrystalline CdTe would be possible with a V OC > 1 V. This could be achieved if the hole density were to be improved to ≃10 16 cm −3 [5,6]. In recent decades typical hole concentrations have remained around 10 14 -10 15 cm −3 [2].
To increase hole concentration, one must dope the material p-type, which can be done in two different ways. The first is by substituting Cd for a group 11 element under Te-rich conditions. Cu is typically used for this process, however, in Te-rich conditions, the antisite defect Te Cd occurs readily which acts as a major recombination centre, lowering carrier lifetime and therefore mitigating the increased carrier lifetime caused by the group 11 doping [7][8][9][10]. The alternative is doping with a group-V element in Cd-rich conditions in order to form the substitution of the group-V element on the Te site. This proves favourable for increasing V OC and can result in hole concentrations exceeding 10 16 cm −3 [11][12][13].
Arsenic (As) and phosphorous (P) have emerged as particularly good candidates for p-type doping under Cd-rich conditions [14]. Initially, phosphorous was thought to be the best choice due to its lower ionisation energy and its predicted lower tendency to exhibit AX centre behaviour [13][14][15]. However, it has been shown that phosphorous diffuses faster than arsenic and therefore the required substitution, P Te , may be problematic. It is now believed that arsenic is a safer alternative as it also exhibits a low defect formation energy in bulk CdTe and the required As Te may be more stable [10,16,17]. The ability for As to form this substitution in bulk is dependant on the diffusion characteristics in CdTe which can be hindered by the structure of CdTe layer growth [18].
CdTe's grain boundary (GB) structures can also disrupt As migration by acting as 'sinks' for interstitial atoms demonstrated previously with chlorine [19]. Interstitial and substitutional defects in these GB regions play a significant role in both the electronic properties of the resultant film and the structure of the grain interior [19,20]. Therefore, understanding the preferred arrangements for arsenic in the bulk and its propensity to segregate to GBs will be key to furthering the understanding of arsenic's role in CdTe.
We consider As diffusion in CdTe by finding diffusion barriers not only in zinc-blende but also in the wurtzite structure which arises locally from the high concentration of stacking fault layers in as-deposited CdTe [21]. The stacking faults have previously been shown to inhibit diffusion of interstitial inert gas atoms [18]. We use two different GB structures (Σ9 and Σ3) separated by a region of bulk CdTe to discuss the segregation energies of arsenic and analyse electronic properties of arsenic, both at a Σ3 GB and in bulk regions.

Methodology
The VASP package was used for all density functional (DFT) calculations [22]. Calculations were undertaken with projector augmented wave pseudopotentials and a plane wave electronic basis set with the PBEsol exchange correlation functional. This formulation has been shown to perform well when predicting CdTe structures in previous work [19,20,23]. Bulk CdTe calculations were carried out in a 64-atom CdTe cell which was kept at constant pressure with a 4 × 4 × 4 Γ-centred k-point grid. All relaxations and nudged elastic band (NEB) calculations were performed to a force convergence of 0.01 eV Å −1 with six NEB images and a NEB string parameter of 4 eV Å −2 . GB structure relaxations use a 4 × 1 × 4 Γ-centred k-point grid due to the elongated axis which spans the grain interior. Calculations were performed using Σ3 and Σ9 GBs [23]. These GB structures were chosen as representative lowangle GBs which have the advantage of allowing for periodicity in a relatively small simulation box allowing for computationally expensive HSE06 hybrid DFT treatment. A grain interior length between adjacent boundaries of 31 Å was used chosen such that the defect formation energy of arsenic plateaued in the centre to give a bulk-like environment at the midpoint.
A study of arsenic's effects on GBs used hybrid DFT (HSE06) which is required for good understanding of electronic effects, however, the large size of the cell used for the calculations (276 atoms, 4×1×4 k-point grid) makes HSE06 calculations prohibitively expensive. An equivalent cell with reduced grain width and depth, measuring 9.14 Å × 46.30 Å × 11.25 Å and containing 136 atoms, was therefore used for electronic properties only. Static calculations with 4 × 1 × 2 kpoint grids were used in this smaller cell to minimise computational expense. This strategy has been used successfully to understand these complex systems when examining chlorine in similar GB structures [19,20]. Bader charge analysis was also performed, to complement density of states (DOS) calculations, by using the method described in [24]. These results can be found in the supplementary information.

Arsenic stable sites and diffusion in bulk CdTe
Arsenic is predicted to be stable in CdTe in three charge states, namely, neutral, 1+ and 1− with the stability of these depending on the Fermi energy [6]. Therefore, diffusion in both zincblende and wurtzite structures as well as at stacking fault interfaces are discussed in these states. Due to the advantageous stability of As Te substitutions, these defects are not expected to contribute to diffusion and therefore we focus only on As interstitials [6].
In zinc-blende, neutral and 1− arsenic interstitials each form a split-interstitial with Te, which can be seen in figure 1(a). The 1+ charge state, however, is unstable in the split interstitial with Te and instead prefers to find the most open structure in the lattice causing distortion to surrounding lattice atoms, which can be seen in figure 1 The split interstitials perform a 'split to split' transition shown in figure 1(a). This mechanism has energy barriers of ≃0.27 and ≃0.45 eV for neutral and 1− charge states, respectively. The split interstitials can also perform a rotation with barriers of ≃0.1 and ≃0.25 eV for neutral and 1− charge states (images 8-10 in figure 1(c)), respectively. This does not lead to As migration but will occur, in both charge cases ≃6 times more often than a split to split transition at the As doping temperatures of ≃1100 K [25-27] calculated using Arrhenius' equation and assuming both pathways have a comparable exponential prefactor. This rotation allows the As to diffuse in three dimensions due to these rotations occurring more often than a transition.
The 1+ charge state diffuses through the most open structure in the lattice and has a lower diffusion barrier of ≃0.05 eV. The transition pathway and barrier can be seen in figures 1(b) and (c), respectively. In general, the barriers found here are lower than seen in previous work by Colegrove et al [6] who found barriers of 0.2-0.3 eV and ≃0.6 eV for neutral and 1− charge states, respectively. This difference is attributed to the fact that Colegrove et al used hybrid functionals for their work and a looser cut-off for their minimisation convergence of 0.05 eV Å −1 .
Migrating interstitials in CdTe have previously been shown to cluster, inhibiting their diffusivity even with small cluster of two atoms [28]. An arsenic di-interstitial in an otherwise clean CdTe structure forms a tri-interstitial with Te, seen in figure 2(a). The formation of this tri-interstitial is associated with a decrease in system energy of ≃1.8 eV compared with isolated As split interstitials. Therefore, since this formation cannot migrate as an entity, the diffusion of As will be restricted by the formation of these tri-interstitials. However, the kinetics of this mechanism were not investigated and it is assumed that at equilibrium, the tri-interstitials configurations will be seen due to the overall energy decrease associated with their formation.
An alternative formation of an arsenic dimer can occur over a Te vacancy with an As-As split interstitial which can be seen in figure 2(b). The structure is unable to diffuse as an entity but can also perform a 'break-away' mechanism, releasing the As dimer, creating an As Te substitution and an As-Te split interstitial. The energy increase associated with this break away is ≃0.3 eV.
In wurtzite structured CdTe, all charge states sit as split interstitials with Te which can be seen in figures 3(a) and (b). This split interstitial structure in wurtzite is in contrast to previous work with inert gas where it was found that the  inert gas would sit in the largest open space in a wurtzite structure and result in large diffusion barriers >1.2 eV [18]. Again, the split interstitial based 'passing' mechanism remains in wurtzite but two planes of diffusion are possible. One transitions the As atom across the (111) structure and is referred to as an an 'across' transition, shown in figure 3(a) with energy barriers in figure 3(c). The 'down' transition into the (111) structure results in the structure in figure 3(b) with barrier given in figure 3(d).
Note, in the 'down' diffusion, the neutral and 1+ charge states overcome one barrier to reach the end state, however, the 1− charge state overcomes two. The first is the actual split-tosplit transition where the arsenic moves to the next split interstitial position. The second barrier is a rotational one within that split interstitial to reach the final state. This motion is not present in other charge states.
The 'across' transition has only one low barrier for each charge state. However, this transition is very short moving <2 Å from its start position to the end. This means for the arsenic to move from one hexagon to another, three of these 0.15-0.45 eV barriers would be required. However, due to the low barriers, arsenic can be expected to conduct this motion ≃300 times before a 'down' transition in the neutral charge state, again calculated at 1100 K with Arrhenius' equation and assuming comparable prefactors between the two mechanisms. Therefore, wurtzite structured layers in CdTe could allow for fast migration of As atoms 'across' the (111) plane, in stark contrast to previous work for inert gas diffusion [18,28].
The defect formation energy for neutral arsenic shows there is a ≃0.1 eV decrease in energy for As in wurtzite compared to zinc-blende. This is confirmed by the NEB results in figure 4 which shows the migration pathway and barrier of an arsenic atom from a zinc-blende structure (Image 0) to a wurtzite structure (Image 15), induced by a stacking fault indicated by the black rectangle. Figure 4(a) shows the pathway for 0/−1 charge state and figure 4(b) shows the pathway for +1 charge state.
First, for the 0/−1 charge states, in figure 4(a) the first barrier is a split-to-split transition in zinc-blende. However, comparing the barrier to the bulk values shows that the close proximity of the stacking fault layer reduces the migration barriers by ≃25% and results in a ≃0.1 eV energy decrease which is due to the split interstitial now being adjacent to the stacking fault layer. The second barrier is a rotational one around the same Te atom and results in a further 0.1 eV energy decrease as the arsenic becomes adjacent to the stacking fault layer. The final barrier represents a pathway which is approximately the same as the wurtzite 'down' transition ( figure 3(b)) and has comparable barriers associated with it. The lowest energy configuration is when arsenic is adjacent to the stacking fault layer. This energy decrease and the lower barriers in the vicinity of the stacking fault compared to bulk values indicates that arsenic may be promoted towards stacking faults. In a charge state of 1+, As initially occupies the open volume of the structure as described in figure 1(b), however, once the As reaches the wurtzite structure, it transitions to a split interstitial, consistent with figure 3. Overall, this implies that, in an opposite behaviour to inert gas diffusion, stacking faults and the resulting structure changes do not significantly hinder the migration of As in these regions.

GB segregation
Grain boundaries can provide a 'sink' for interstitial defects allowing them to accumulate in these regions. We discuss two GB structures and investigate the potential for neighbouring substitutional and interstitial As atoms to move into these regions.
The Σ9 and Σ3 GB structures investigated are shown in figures 5(a) and (b). Both have a Te-core and a Cd-core separated by a region of bulk-like CdTe. The positions of the As atoms are changed, starting from the Cd-Core, through the bulk-like structure and into the Te-Core for both interstitial and substitutional sites. The structures are relaxed in each position and the energies are shown relative to the most stable position for each charge state.

Σ9 GB.
Results for the Σ9 GB structure can be seen in figure 6. The energies for the interstitial atoms show the same trend and indeed occupy the same sites between the charge states. The Cd-core has the most stable interstitial site, by between 3 and 3.7 eV compared to bulk depending on the charge state, and is ≃2 eV more stable than the analogous site in the Te core.
The Cd-core is a more open structure and therefore, the presence of an interstitial As in the boundary causes less distortion than in the tighter Te-core, hence the lower energy. This open structure also extends into the bulk-like region close to the core and results in more stable interstitial sites in this  region due to small reconstructions aided by the presence of an openly structured GB. If As is within ≃12 Å of the Cd-Core then diffusion away from the core would be unlikely due to a ≃1.2 eV energy increase. This would also be the case if the As atom reached the Cd-core. In this case the energy decrease would be ≃1.4 eV and therefore diffusion out of the core would be unlikely.
The most stable substitutional sites in both cores are the centre sites, which in the Cd-core is triangular coordinated, whereas in the Te-core this site is tetragonal pyramidal. The outer site, the site closest to bulk CdTe, is higher in energy by ≃1.9 and ≃0.5 eV for the Cd-core and Te-core, respectively, which both have a distorted tetrahedral structure. As atoms in substitutional sites in the bulk will not segregate to the GB due to the large substitution diffusion barrier [29] coupled with the relatively flat energy profile of the substitutional sites in the bulk region.

Σ3 GB.
Results of the energies in each charge state for the Σ3 GB structure can be seen in figure 7.
Once again, the energies for the interstitial sites show that there is only a small difference in the trend between the charge states. However, in the Σ3 boundary, the Te-core is shown to be the most stable for the As interstitial due to this being the more open structure. This is in agreement with the analysis of Cl interstitials at Σ3 GB's which found the Cd-core to be more densely packed than the Te-core, requiring fewer Cl atoms to saturate the structure [20]. The Cd-core has a wider area of influence on the structure, causing energy decreases around 10 Å from the centre of the GB culminating in a ≃1.3 eV energy decrease compared to bulk.
The substitutional energies remain flat through the bulk region but there is a noticeable shift in trend between the charge states at both of the boundary cores. In the Cd-core, substitutional sites at ≃2 and ≃3 Å from the centre site (0 Å) follow the same trend between the charge states, however, the trend of energies at the centre site do not agree between the charge states. This is due to the interaction of charged atoms with dangling Cd bonds located either side of the grain boundaries central position. This is also seen in the Te-core at a distance of ≃29 Å since the dangling bond interaction occurs at this position in the Te core rather than at the centre site.

Electronic properties
The Σ3 structure shown in figure 5(a) is used to investigate the electronic effect of As in a GB. Figure 8(a) shows the DOS of the clean GB structure. Three main defects are observed in the band gap. At 0.0 and −0.2 eV there are two prominent defect states which are smeared together to show a large DOS peak. This peak is towards the mid-gap, indicating that it could be a centre for Shockley-Read-Hall electronic recombination [30] which would impact cell efficiency significantly. The third defect can be seen as the flat line extending from the valence band. While this defect is further from the mid-gap it still may provide a site for electronic recombination.
The DOS peak between 0.0 and −0.2 eV is caused by a cross-GB Te-Te interaction in the Te-core and the flat defect extending from the valence band minimum is caused by an under-coordinated Cd atom in the Cd-core. These findings agree well with previous work where it was found that Cl could be used to passivate these defects by substituting Te atoms in the Te-core involved in the cross-GB interaction thereby removing the −0.2 to 0.0 eV defect and also by adding an interstitial Cl to the Cd-core to remove the flat defect [19]. Figure 8(b) shows the DOS plot of the GB structure where As atoms have been substituted into the Te site responsible for the cross-GB Te-Te interaction in the Te-core. In a similar way to Cl, As has removed this mid-gap defect by blocking this cross-boundary interaction. However, unlike Cl, As has introduced DOS peaks into the conduction band causing significant band-gap shortening. This is likely to mitigate the reduction in electronic recombination due to a reduced bandgap. Figure 8(c) shows the DOS of the structure in which the As Te substitutions in the Te-core remain and an interstitial As has been placed in the Cd-core to passivate the undercoordinated Cd atom in this region. In this case large defects appear throughout the DOS plot. This is in sharp contrast to the same analysis with Cl which showed a clean band-gap on addition of this substitution in the Te and the interstitial in the Cd-core. Figure 9 shows the DOS plots of As in various sites. Figure 9(a) shows the DOS of a clean 64-atom zinc-blende CdTe cell and is shown for comparison. Its band-gap remains defect free and the band gap is estimated to be ≃1.6 eV. Arsenic defects are added to this structure and the DOS plots generated. Figure 9(b) shows the DOS of the structure with a single As Te substitution which is the desired defect for increased cell efficiency in CdTe cells. As can be seen, the band-gap again remains clean with no defects introduced into this region. There is some evidence of band-gap shortening which is common in heavily doped systems such as this (1.5% As concentration due to cell size restrictions) [31]. Finally, there is evidence that the system is doped p-type since there is an apparent drop in Fermi level compared to the clean reference.
It is reported that positively charged As can form an AX centre in the vicinity of Te vacancies which can act as a site of electronic recombination [13][14][15]. Figure 9(c) gives the DOS plot of this defect showing that the As defect introduces electronic defects into the band gap. This is similar to the effect of an interstitial which can be seen in figure 9(d). The formation of the split interstitial once again introduces band gap defects which could allow for electron-hole recombination. The same is true for the di-interstitial defects discussed in figure 2 which can be seen in figures 9(e) and (f). Therefore, defects other than the standard As on Te substitution would be detrimental to cell performance due to these mid gap defects.
Finally, to provide some insight as to the charge transfer between the dopant and the neighbouring sites in the bulk and at the GB which give rise to the DOS plots in figures 8 and 9, bader charge analysis is presented in the supplementary information. This allows one to see both in visual and tabular form that there is real charge transfer between the dopant and the previously unsaturated bond. In the GB scenarios, the analysis suggests a net gain in neighbouring Te charge for As Te whereas there is a drop in neighbouring Cd charge in the case of As interstitials for figure 8. The Te core does not seem to be affected as much. See figure and table S1. In the bulk, the charged As defects show a gain in net charge of neighbouring Te sites as shown in figure and table S2.

Conclusion
Single As interstitial diffusion barriers are low regardless of the charge state or the local CdTe structure so diffusion would easily occur even at room temperature. Since the main stacking faults observed in CdTe are twins, which can be regarded as a local change from zinc-blende to wurtzite structure these do not create an interstitial trap for single As atoms as the barriers to move between the two structures and those within the structures themselves are <0.8 eV. However, stacking fault layers do provide a low energy pathway 'across' the (111) structure with barriers <0.4 eV. Arsenic di-interstitial diffusion will only occur at high temperatures as a large ≃1.8 eV 'break-away' barrier is required for this to happen.
Analysis of GB sites shows that these are energetically preferred compared to single interstitial As in the bulk but if the As atom is substitutional, there is little difference in energy for the Σ3 GB. For the Cd-core in the Σ9 GB substitutional As was 1.5 eV more favourably situated than in the bulk but since substitutional As in the bulk requires a barrier of ≃2.7 eV to be overcome before it can escape, a substitutional atom in the bulk is unlikely to diffuse to this site. Stacking faults are often observed spanning the length of grain interiors and terminating at grain boundaries, therefore, the low barrier pathways that the stacking fault layers provide may allow mass migration of As from bulk CdTe to grain boundaries.
CdTe GB structures contain electronic defects which form centres for electronic recombination. It has been shown that when grain boundaries are saturated with interstitial and substitutional Cl these defects are passivated [19,20]. An examination of the DOS plots for As in the grain boundaries revealed that while substitutional As passivated some defects, overall substitutional and interstitial As in the grain boundaries produced heavily defected band gaps which would result in increased electronic recombination and therefore a decrease in CdTe solar cell efficiency.
Clean grain interiors have a defect free band gap as does CdTe with a single As Te substitution so this could provide the required p-type doping. While we cannot provide conclusive evidence of p-type doping using these DFT methods we have clearer shown that, unlike any other As defect, As Te substitution does not negatively impact the band-gap. However, the alternative form of the positively charged As substitution, referred to as an AX-centre, introduces defects into the band gap, as does any defect containing at least one interstitial As.
As di-interstitials in bulk CdTe and As interstitials at the grain boundaries should therefore be avoided. As incorporation strategies which rely on diffusion of As through the surface of CdTe will probably not produce the required p-type doping since As dimers and hence di-interstitials are likely to be present. In addition, diffusion into the material directly along the grain boundaries could occur, as is seen with Cl treatment [32]. Furthermore, stacking faults spanning grain interiors could provide a pathway for As in bulk to enter grain boundaries due to low diffusion barriers 'across' (111) type wurtzite structures. Therefore, a strategy which first saturates the grain boundaries with Cl, hence removing the stacking faults, and then introducing As atoms through negatively charged ion implantation, possibly avoiding As + defects, may be more suited as this will introduce single As atoms into the grain interiors, while at the same time producing vacancies into which substitutional As can migrate.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).