A discussion on `Domain formation and phase transitions in the wurtzite-based heterovalent ternaries: a Landau theory analysis'

Heterovalent ternary nitrides are considered one of the promising classes of materials for photovoltaics, combining attractive physical properties with low toxicity and element abundance. One of the front-runner systems under consideration is ZnSnN₂. Although it is nominally a ternary compound, no clear crystallographic evidence for cation ordering has been observed so far. An attempt to elucidate this discrepancy [Quayle (2020). Acta Cryst. A76, 410–420] was the trigger for an intensive discussion between the authors, and an agreement was reached to elaborate on some points in order to set things in perspective. Rather than using a conventional comment–answer scheme, this is published in the form of a joint discussion to celebrate constructive criticism and collegiality.

1. How it all started

Shortly after the publication of Quayle (2020), I received a message titled `On: `Domain formation and phase transitions in the wurtzite-based heterovalent ternaries: a Landau theory analysis' in which Joachim Breternitz pointed out, in the most courteous and professional manner, an error in the proposed Landau theory. What followed was an exchange concerning the nature of atomic disorder in the wurtzite-based ternaries; an exchange that has led us to a satisfying impasse. We have arrived at two distinct, seemingly plausible positions regarding ternary disorder, and we report here on our discussion in the hope that insights from the research community will lead to a resolution.

That there was an error in the Landau theory in Quayle (2020) is not in dispute. As discussed later, Quayle (2020) treats two different crystal structures, both members of the same space-group type but with different unit-cell sizes and atom locations, as one and the same, and the Landau theory built from this oversight is flawed. A straightforward fix to the analysis is achieved through the correct assignment of the two structures; however, resolving that issue introduces a disagreement between the theory and relevant experimental observations, throwing doubt on the methodology as a whole. This is the heart of the matter. Can the mismatch between the formally correct analysis and experimental observations be rectified? Or is the analysis logically sound, but not consistent with nature?

2. Introduction

Binary III–V nitrides in the wurtzite-type structure are candidate materials for photovoltaic applications (Jani et al., 2007); however, indium and gallium are rather scarce and expensive elements, and more earth-abundant alternatives are sought-after for low-cost applications. The II–IV–N₂ compounds are a promising class of materials which are formally derived from III–V materials by replacing the tri­valent cation in the binary compound by a stoichiometric mixture of divalent and tetravalent cations. With a number of cations potentially filling the divalent (e.g. Zn²⁺, Mg²⁺, Cd²⁺) and tetravalent (Si⁴⁺, Ge⁴⁺, Sn⁴⁺) positions, the chemical parameter space allows wide variations that in turn affect the bandgap and enable bandgap tuning (Martinez et al., 2017). However, in addition to mixing different tetravalent cations occupying the same sites, these materials are expected to show a second mechanism of bandgap tuning through order/disorder phenomena of the cations (Quayle et al., 2015).

ZnSnN₂ is distinguished from its sister ternary compounds. ZnSnN₂ is an earth-abundant semiconductor that has a bandgap very much in the ideal range for photovoltaics (Veal et al., 2015). Further, it has demonstrated distinct disorder properties; to date, there have been no reported observations of ordering in ZnSnN₂, even in material that emits light at room temperature (Quayle et al., 2015). Density functional theory (DFT) calculations suggest that the bandgap of ZnSnN₂ can be lowered by 0.85 eV in fully disordered mater­ial, with a randomized cation sublattice, compared with perfectly ordered material with the β-NaFeO₂-type structure (Makin et al., 2019).

The wurtzite-type structure crystallizing in space group P6₃mc possesses only one independent cation position and hence does not allow for ordering of the heterovalent ternary cations (Breternitz et al., 2019). To find a crystallographic description compatible with ordered cations, one must follow a descent into symmetry (e.g. Quayle, 2020; Breternitz & Schorr, 2021). Indeed, most of the experimentally observed ternary nitrides crystallize in the orthorhombic space group Pna2₁ in a β-NaFeO₂-type structure (β-Pna2₁), which contains two crystallographically independent cation positions (Breternitz & Schorr, 2021). The experimental observations agree with computational physics results which calculate that the β-Pna2₁ is the lowest-formation-energy phase at 0 K. The other possible ordered phases can have close formation energies however, and this is especially true in the case of ZnSnN₂. The difference in formation energy of ZnSnN₂ between the β-Pna2₁ and Pmc2₁ structures was calculated to be only 0.01 eV per formula unit (Quayle et al., 2015). The similar formation energies of the phases coupled with the lack of ordering observed experimentally raise the question of whether a mixture of phases can be present on the ZnSnN₂ lattice, and whether the phase composition can change as temperature is raised towards the melting point. Indeed, phase transitions are widely reported in the zincblende-based chalco­pyrite heterovalent ternaries with stoichiometries II–IV–V₂ and I–III–V₂ (Zunger, 1987). Another candidate atomic arrangement is an essentially randomized cation substructure, which keeps perhaps a certain degree of local bond order, but which is isotropic on the more macroscopic scale.

Quayle (2020) set out to explore a phase transition and the ordering question in ZnSnN₂ from the angle of the Landau theory. This theory is grounded on the analysis of structural similarities of different crystal structures and the identification of ordering parameters based on these similarities. The purpose of this paper is to clarify some details of the Landau theory and introduce additional considerations elucidated in the `prequel' paper by Breternitz & Schorr (2021).

3. The group–subgroup relation between β-NaFeO₂- type in space group Pna2₁ and the structure in space group Pmc2₁

A group–subgroup relationship between the hypothetical Pmc2₁ structure (in the following simply abbreviated as Pmc2₁) and a crystal structure with space group Pna2₁ is at the core of the argument by Quayle. This latter is implicitly taken as the observed β-NaFeO₂-type structure. The two phases discussed by Quayle, the hypothetical Pmc2₁ structure and the β-NaFeO₂-type structure (which we further denote as β-Pna2₁ for simplicity), are clearly both subgroups of the wurtzite type (Breternitz & Schorr, 2021). However, they do not lie on the same branch of the Bärnighausen tree (Bärnighausen, 1980), in which the group–subgroup relations are outlined graphically (Fig. 1). In fact, β-Pna2₁ is not a maximal subgroup of Pmc2₁. The group–subgroup relationship between Pmc2₁ and a structure in space group Pna2₁ that is established by Quayle (2020) through the intermediate space groups Pmn2₁, Pca2₁ or Cmc2₁ (the latter two alternatives not depicted in Fig. 1) is, nonetheless, formally valid. The transition path described does, however, end up in a unit cell that is twice as large as the one of β-Pna2₁ and is therefore not equivalent. For the sake of readability, we will further denote this hypothetical crystal structure as α-Pna2₁.

Figure 1 Group–subgroup relationship between the relevant subgroups of the wurtzite type (P63mc). The Pmc21 structure, the derived hypothetical Pna21 structure (α-Pna21) and the β-NaFeO2-type structure (β-Pna21) are highlighted for visibility. Further, the Wyckoff positions and transitions of the atomic positions are given.

We also note that all three symmetry descents lead to equivalent solutions. The statement `The Cmc2₁ phase is not compatible with the orthorhombic crystal structure' by Quayle (2020) is a common misinterpretation of klassengleiche phase transitions: the volume of the intermediate Cmc2₁ is four times that of Pmc2₁ but the latter is still a k2 subgroup, because of the introduction of centring in this symmetry transition. In return, the subsequent k2 descent from Cmc2₁ to α-Pna2₁ does not cause a volume increase, but instead is achieved through the loss of centring.

It really needs to be highlighted that the interpretation of group–subgroup relationships can be a complicated and tedious path and we would like to take this opportunity to give some guidance on the indicators that can act as a `safety net' insofar as they may point to oversights. Firstly, it is crucially important to make the distinction between space groups and space-group types, two terms that are often used synonymously. From a purely qualitative point of view, one may view the difference as follows: the 230 space-group types define all possible arrangements of the symmetry elements in three-dimensional crystal structures and their relative position to each other (e.g. the origin versus the unit-cell centre). A space group, on the other hand, can be understood as defining the concrete arrangement of the symmetry elements, or in other words, a space group also requires the unit-cell dimensions. A more formal definition may be found in sections 8.1.6 and 8.2.2 of Volume A of the International Tables for Crystallography (Hahn, 2005). If the (approximate) unit cells of the crystal structure derived from a group–subgroup relationship do not coincide with those determined experimentally, this should indeed raise serious concerns about the validity. Secondly, the atomic positions of the group–subgroup relationship derived crystal structure must be in line with those that are experimentally observed. We note that this point is even more complex in this case, as the atomic positions in the crystal structures in either Pna2₁ all lie on general positions (Wyckoff position 4a), which means that the choice of origin within the unit cell is arbitrary. In such a case, it is best either to use the relative relationships between the atomic positions, or to apply an origin shift to one of the two structures, so that they overlap.

4. How do the crystal structures compare?

The α-Pna2₁ structure derived as a hettotype of the Pmc2₁ structure has four independent crystallographic positions for the cations and four for the anions, while β-Pna2₁ has only two sites for cations and anions, respectively (Fig. 1). Given the higher number of independent crystallographic positions in α-Pna2₁, it seems possible that one could represent the β-Pna2₁ structure as α-Pna2₁. To do this, one must establish a group–subgroup relationship between the two types, so that the α-Pna2₁ structure would be a common subgroup of Pmc2₁ and β-Pna2₁. In fact, Baur & McLarnan (1982) did explicitly draw the corresponding isomorphous symmetry descent of index 2 through a doubling of the a axis in their seminal work on symmetry relationships in wurtzite materials. However, this is, unfortunately, incorrect, as no isomorphous descent of index 2 is possible in this space-group type, but only of index 3 or larger: the n-glide planes perpendicular to the a axis are situated at a = ¼ and a = ¾ in space-group type Pna2₁. If a doubled a-axis a′ is considered (as for an i2 descent), the n-glide plane lying at a′ = ¼ in this space group would have to be situated at a = ½ in the smaller unit cell, but this is not compatible with the symmetry in space-group type Pna2₁. If one triples the a axis, on the other hand, the n-glide plane at a = ¼ is compatible with the one at a = ¾. From a more visual point of view, it becomes evident that α-Pna2₁ and β-Pna2₁ cannot be linked through a direct group–subgroup relationship, since their ordering motifs are different (Fig. 2). As a matter of fact, there is no possible ordering of cations in the α-Pna2₁ structure that would fulfil the β-Pna2₁ structure, as would be a prerequisite for a group–subgroup relationship.

Figure 2 Structural representations of the β-Pna21 (a), (b) and the hypothetical α-Pna21 structure (c), (d). Views are along the crystallographic c axis (a), (c) and as a general view (b), (d). All crystallographically independent cation positions are highlighted as large spheres in different colours, while the nitro­gen atoms are very small to increase visibility of the cations. The crystal structure of the α-Pna21 structure was constructed from the wurtzite type using the TRANSTRU tool of the Bilbao Crystallographic server (Aroyo, Perez-Mato et al., 2006; Aroyo, Kirov et al., 2006; Aroyo et al., 2011).

There is another aspect of this story that deserves consideration, which comes from molecular dynamics (MD) simulations. Lany et al. (2017) reported on simulation results of possible structures of ZnSnN₂ that obey Pauling's rules, and they calculated the total energy of the α-Pna2₁ structure to be between the thermodynamically stable β-Pna2₁ structure and the Pmc2₁ structure. This result demonstrates two points: (i) the α-Pna2₁ and β-Pna2₁ structures are distinctly different, not only from a structural point of view, but also from their properties; (ii) their differences are probably not so large that the ordering of the α-Pna2₁-type structure would be completely implausible.

Fig. 3 shows the β-Pna2₁ and α-Pna2₁ structures again along with the Pmc2₁ structure from a different perspective. When viewed along the polar c axis, we see that the structures consist of different stacking motifs of the same c-axis basal plane (Quayle et al., 2015). The β-Pna2₁ structure consists of two distinct motifs (A and B), while Pmc2₁ consists only of one. In other words, the β-Pna2₁ structure is built from an ABAB stacking of the basal plane while the Pmc2₁ structure consists of an AAAA stacking. The α-Pna2₁ structure consists of both stacking motifs, AABB. When viewed in this way, it is not surprising that the calculated formation energy of the α-Pna2₁ phase is intermediate between the Pmc2₁ and β-Pna2₁ phases.

Figure 3 Comparison of the stacking motifs in the different observed and hypothetical crystal structures for ZnSnN2.

5. What are the implications for the physical model?

The symmetry relation upon which the Landau theory of Quayle (2020) is based is between the Pmc2₁ structure and the α-Pna2₁ structure; therefore, the derived order parameter describes this phase transition. Quayle (2020) however describes the transition, and order parameter, as between the Pmc2₁ structure and the β-Pna2₁ structure. What are the implications of this error?

The analysis of Quayle (2020) is, in essence, a thought experiment to explain the discrepancy between the β-Pna2₁ being reported to be the thermodynamically stable crystal structure – which also is observed for the lighter analogue ZnGeN₂ – and the X-ray diffraction (XRD) results of ZnSnN₂ adopting a wurtzite-type structure (e.g. Feldberg et al., 2012; Quayle et al., 2013; Kawamura et al., 2016).¹ The question is whether this observation is due to the fact that the different cations are truly randomly distributed, or whether they still follow certain patterns on a local level. Given the limitation of the diffraction method as averaging over space and time, both cases could result in virtually identical diffraction patterns.

In Quayle (2020), we picture the situation in which a solid-state phase transition occurs between a Pmc2₁ group phase, which is more stable at high temperatures, and a β-Pna2₁ `subgroup' phase, which is more stable at low temperatures. In a hypothetical Pmc2₁ phase, with perfect crystallinity, as the temperature is lowered towards and below the transition point, the β-Pna2₁ phase precipitates and grows to become the dominant phase. In Quayle (2020), it was proposed that the transition is from a paraelectric Pmc2₁ phase to an anti­ferroelectric β-Pna2₁ phase. Now, we see from the analysis given above that the order parameter utilized in Quayle (2020) actually describes the transition from Pmc2₁ to α-Pna2₁. For the sake of completeness, it needs to be noted that this is a thought experiment insofar as no such phase transition has been observed experimentally to date.

The misidentification of the order parameter is problematic because the α-Pna2₁ phase is not the most energetically favourable phase, according to calculations, and the α-Pna2₁ phase has never been observed experimentally in analogue ternary compounds, which have only exhibited signatures of the β-Pna2₁ phase. The analysis is formally correct however if we realize that the subgroup phase is α-Pna2₁ not β-Pna2₁. Also, it can be argued that both Pna2₁ phases are antiferroelectric, using the methodology in Quayle (2020), so that aspect of the analysis remains unchanged.

Still, for the Landau theory to truly describe nature, it must lead to a β-Pna2₁ stable phase on the low-temperature side of the transition, and the analysis in Quayle (2020) does not achieve this. The formal analysis ends at the transition from Pmc2₁ to α-Pna2₁. To find a logical path from the α-Pna2₁ phase to β-Pna2₁ we must depart from group theory arguments and note that if a ternary nitride can transition from AAAA basal plane ordering (Pmc2₁) to AABB ordering (α-Pna2₁) by way of a mechanism similar to the one described in Quayle (2020), then it is plausible that it can also transition further to the more energetically favourable ABAB ordering of β-Pna2₁.

A broader issue is raised by questioning the validity of the Landau theory. The Pmc2₁ structure is probably a very good approximation to a Pauling rule conserving wurtzite-derived structure. This may sound contradictory at first, but as outlined in Breternitz & Schorr (2021), a main feature of the hypothetical structure in Pmc2₁ is the fact that the symmetry of the space group prevents the tetrahedra around the cations from being different in size, which casts doubt on whether this system would profit from cation ordering in the first place. Given that all three structures under discussion were calculated to have very similar total energies (Lany et al., 2017), this alone may be taken as a hint that cation ordering would not be greatly energetically beneficial in this system.

Both possible explanations for the observed wurtzite-type structure – a completely random cation distribution and a partial cation ordering on the local scale – would intrinsically have to produce some cation arrangements violating Pauling's rules: in the first case these would be randomly distributed and in the second case these would reside at the interfaces between differently ordered domains.

Which of the two potential solutions, if one of them, is the one best describing reality is a question that merits further experimental studies using more local probes, such as electron diffraction and/or pair-distribution-function analyses and, potentially, spectroscopic studies, as the differences may be expressed in different optoelectronic and/or electronic properties of the observed macroscopic materials.

Footnotes

¹We note that some publications (e.g. Feldberg et al., 2012) also report monoclinic unit cells, but they are within the error of the hexagonal wurtzite type and there is hence little ground for this assumption.