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Acta Crystallographica Section A: Foundations and Advances logoLink to Acta Crystallographica Section A: Foundations and Advances
. 2020 Apr 28;76(Pt 3):410–420. doi: 10.1107/S2053273320003095

Domain formation and phase transitions in the wurtzite-based heterovalent ternaries: a Landau theory analysis

Paul C Quayle a,b,c,*
PMCID: PMC7233021  PMID: 32356791

A Landau theory for the wurtzite-based heterovalent ternary semiconductor ZnSnN2 is developed and a first-order reconstructive phase transition is proposed as the cause of observed crystal structure disorder. The model implies that the phase transition is paraelectric to antiferroelectric.

Keywords: Landau theory, phase transitions, antiferroelectrics, crystalline disorder properties, heterovalent ternary semiconductors

Abstract

Characterizing the crystalline disorder properties of heterovalent ternary semiconductors continues to challenge solid-state theory. Here, a Landau theory is developed for the wurtzite-based ternary semiconductor ZnSnN2. It is shown that the symmetry properties of two nearly co-stable phases, with space groups Pmc21 and Pbn21, imply that a reconstructive phase transition is the source of crystal structure disorder via a mixture of phase domains. The site exchange defect, which consists of two adjacent antisite defects, is identified as the nucleation mechanism of the transition. A Landau potential based on the space-group symmetries of the Pmc21 and Pbn21 phases is constructed from the online databases in the ISOTROPY software suite and this potential is consistent with a system that undergoes a paraelectric to antiferroelectric phase transition. It is hypothesized that the low-temperature Pbn21 phase is antiferroelectric within the c-axis basal plane. The dipole arrangements within the Pbn21 basal plane yield a nonpolar spontaneous polarization and the electrical susceptibility derived from the Landau potential exhibits a singularity at the Néel temperature characteristic of antiferroelectric behavior. These results inform the study of disorder in the broad class of heterovalent ternary semiconductors, including those based on the zincblende structure, and open the door to the application of the ternaries in new technology spaces.

1. Introduction  

The heterovalent ternary semiconductors with stoichiometry I–III–VI2 and II–IV–V2 have more complex atomic arrangements than their binary parent semiconductors, the II–VI and III–V compounds, due to the added degree of freedom in the two-cation sublattices. Characterizing and manipulating this complexity challenges the capabilities of both theorists and experimentalists. Solid-state phase transitions are detected in many ternaries (Berger & Prochukhan, 1969 ▸; Shay & Wernick, 1975 ▸; Zunger, 1987 ▸). Mixed-phase lattices, point defects, defect complexes and line and planar defects further enrich this picture (Zhang et al., 1998 ▸; Álvarez-García et al., 2005 ▸; Oikkonen et al., 2014 ▸; Abou-Ras et al., 2016 ▸). The complexity affects and drives the performance of ternary semiconductors which enables important technologies including nonlinear optics (Petrov, 2012 ▸), thermoelectric generators (Ritz & Peterson, 2004 ▸; Cook et al., 2007 ▸; Ma et al., 2013 ▸) and thin-film photovoltaics (Jackson et al., 2011 ▸; Siebentritt, 2017 ▸). Density functional theory (DFT) (Wei et al., 1999 ▸; Lyu et al., 2019 ▸) and Monte Carlo simulations have been used to characterize this complexity (Wei et al., 1992 ▸; Ludwig et al., 2011 ▸; Ma et al., 2014 ▸; Lany et al., 2017 ▸).

Landau theory (LT) has been used to study ternaries as well (McConnell, 1978 ▸; Folmer & Franzen, 1984 ▸). LT is based on the principle that a solid-state phase transition is accompanied by a change in crystal structure symmetry. Identifying the symmetries of the phases in the transition provides information on the temperature-dependent and pressure-dependent atomic arrangements of the crystal. LT is particularly useful because it can provide information about crystalline materials that contain structure that is not periodic, making it well suited to the analysis of materials that exhibit domain structure and nanostructure (Janovec et al., 1989 ▸; Müller, 2017 ▸). First introduced in the 1930s (Landau, 1937 ▸; Landau & Lifshitz, 1959 ▸; Cowley, 1980 ▸), LT became accessible to nonspecialists in 1983 when space-group symmetry relations were consolidated in the International Tables for Crystallography (2011 ▸). Since then, complete databases have been made available online, along with a wealth of computational tools designed to enable Landau analysis.

LT played a central role in overcoming challenges in the development of chalcogen-based ternary semiconductors, which compose the commercial solar cell material Cu(In,Ga)Se2. In its early stages, the effort to synthesize device-quality CuInS2 suffered from severe cracking and nanoprecipitate formation due to a solid-state phase transition (Binsma et al., 1980 ▸; Arsene et al., 1996 ▸; Mullan et al., 1997 ▸). LT was used to characterize the phase transition and helped establish the use of non-equilibrium vapor-phase growth methods and growth temperatures significantly lower than the Curie temperature (T C) to avoid secondary-phase precipitates (Folmer & Franzen, 1984 ▸; Su et al., 2000 ▸; Abou-Ras et al., 2016 ▸).

While growing at temperatures well below T C mitigated the detrimental effects of the phase transition in chalcogen-based ternary functional materials, in other ternaries the disorder generated by the phase transition is advantageous. ZnSnP2 is a candidate photovoltaic semiconductor with an intrinsic band gap of approximately 1.7 eV and a T C of 993 K (Nakatsuka et al., 2017 ▸). Synthesis of ZnSnP2 has been done using both equilibrium solution methods at temperatures above T C and far-from-equilibrium molecular-beam epitaxy at temperatures well below T C. In both cases, tuning the cool-down rate from the growth temperature determines the degree of crystal structure disorder. Increasing the cool-down rate decreases the band gap of the material. Thus, ZnSnP2 is an adjustable-band-gap photovoltaic material (Ryan et al., 1987 ▸; Nakatsuka & Nose, 2017 ▸).

Understanding and controlling disorder in ternaries like ZnSnP2 requires precise characterization of the phases above and below the phase transition point. The identification of the two phases and of the nature of the atomic scale structure that generates lattice disorder are the main subjects of this paper. This analysis is called for because the validity of the historically predominant model of lattice disorder in ternaries has been questioned by both experiment and theory.

The predominant model holds that the high-symmetry phase above T C is characterized by an entropically random distribution of cations. This theory was first put forward by Buerger (1934 ▸) and served as the basis for the first LT analyses of chalcogen-based ternaries (McConnell, 1978 ▸; Folmer & Franzen, 1984 ▸). The random disorder model states that thermal energy randomizes the positions of the bivalent cation sublattice at the phase transition, breaks the symmetry of the low-symmetry chalcopyrite phase and yields an isotropic cation sublattice with symmetry that is equivalent to that of the sphalerite ZnS structure. Evidence for the random disorder model comes primarily from X-ray diffraction (XRD) (Shay & Wernick, 1975 ▸). Measurements of zincblende-based chalcopyrite ternaries taken near and above the Curie temperature only show peaks that are consistent with the sphalerite structure.

XRD, however, while useful for characterizing the macroscropic symmetry of the crystal, gives no information about the local environment of the atoms, since the detected signal is the average over the coherence length of the X-rays. Studies using other methods yield data that conflict with the random disorder model. For example, band-gap measurements of ZnSnP2 show decreases of 18% and 22% when comparing ordered samples with disordered (Ryan et al., 1987 ▸; St-Jean et al., 2010 ▸). DFT calculations based on the random model predict much larger decreases in band gap for the two cases of 56% and 76% (Scanlon & Walsh, 2012 ▸; Ma et al., 2014 ▸).

Ma et al. (2014 ▸) concluded that, based on DFT, a random distribution in ZnSnP2 is not possible under equilibrium conditions, a position that has been supported since then (Skachkov et al., 2016 ▸; Lany et al., 2017 ▸). The random arrangement of atoms cannot exist under equilibrium conditions because it generates too many instances in which the octet rule is violated. In the lowest-energy crystal structures of ternaries such as ZnSnP2, the lattice is populated according to the octet rule: every group V anion is bonded to two group II and two group IV cations, forming Zn2Sn2 tetrahedra. On a randomized cation sublattice, there are a statistical number of tetrahedra in which a group V anion is bonded to three group II and one group IV cations (Zn3Sn1) or vice versa, and four group II and zero group IV cations (Zn4Sn0) or vice versa. The Zn3Sn1 and Zn1Sn3 tetrahedra, and the Zn4Sn0 and Zn0Sn4 tetrahedra, especially, violate the octet rule and have high formation energies. Ma et al. (2014 ▸) predicted, based on Monte Carlo simulations, that, as the temperature of ZnSnP2 is increased, thermal energy generates an increasing probability that Zn3Sn1 and Zn1Sn3 will form; above a phase transition at ∼1100 K this probability approaches 20%. Their results also predict that, even at 20 000 K, the probability of Zn4Sn0 and Zn0Sn4 tetrahedra formation is lower than required for a randomized lattice.

Ryan et al. (1987 ▸) presented an alternative to the random disorder model. The study included a comparison of nuclear magnetic resonance (NMR) spectra taken from ordered and disordered ZnSnP2. A pair of peaks appeared in the dis­ordered samples which they assigned to the Zn3Sn1 and Zn1Sn3 tetrahedra. The data did not show any additional peaks that could be assigned to the Zn4Sn0 and Zn0Sn4 tetrahedra, however, leading them to the conclusion that those tetrahedra are not present on the lattice and that the lattice disorder is not random. In place of the random disorder model, Ryan et al. (1987 ▸) proposed that the disorder was caused by a scattering of domains embedded within the chalcopyrite structure. They hypothesized that these domains consisted of arrangements of neighboring Zn3Sn1 and Zn1Sn3 tetrahedra generated by an exchange in position of adjacent Zn and Sn atoms, and that these site-exchange defects (SEDs) create mutually compensating acceptor–donor pairs.

The model proposed by Ryan et al. (1987 ▸) has received interest lately in investigations of the crystalline material ZnSnN2. ZnSnN2 is a member of the subset of heterovalent ternaries with the wurtzite parent structure. It has emerged as a promising photovoltaic material composed of earth-abundant elements. Like ZnSnP2, a range of band gaps have been reported for ZnSnN2, spanning 1.4 to 2.0 eV (Martinez et al., 2017 ▸). Raman measurements reported by Quayle et al. (2015 ▸) show a glass-like spectrum consistent with a non­periodic lattice. All XRD measurements of ZnSnN2 to date show a macroscopically disordered spectrum consistent with wurtzite (Lyu et al., 2019 ▸). An experimental investigation of phase transitions in ZnSnN2 has not yet been carried out.

Recent studies of ZnSnN2 have incorporated both SEDs and domain mixtures into models of ZnSnN2. Lany et al. (2017 ▸) predicted, based on DFT, that the SED costs approximately 0.04 eV per pair at 0 K relative to the most stable structure. By including Zn3Sn1, Zn1Sn3, Zn4Sn0 and Zn0Sn4 tetrahedra in Monte Carlo simulations of ZnSnN2, the authors showed results like those reported by Ma et al. (2014 ▸) for ZnSnP2. At elevated temperatures between ∼1773 and 2273 K, the density of Zn3Sn1 and Zn1Sn3 tetrahedra increases to a concentration of around 10%. They reported that the density of Zn3Sn1 and Zn1Sn3 tetrahedra has only a moderate effect on the band gap, lowering it by ∼0.3 eV. Recently, Makin et al. (2019 ▸) reported that the band gaps of ZnSnN2 and MgSnN2 can be decreased via disorder tuning from 1.98 to 1.12 eV and from 3.43 to 1.87 eV, respectively.

First-principles studies agree that the most stable crystal structure of ZnSnN2 is orthorhombic Pbn21 (No. 33) (Pna21 in the standard setting) and that a second competing phase with space group Pmc21 (No. 26) is slightly less favorable (Fig. 2) (Lyu et al., 2019 ▸; Martinez et al. 2017 ▸; Lahourcade et al., 2013 ▸). [Note that the non-standard setting is used for the Pbn21 phase so its translation vectors (a and b) are aligned along the same axes as those of the Pmc21 phase, such that a_Pbn21 = 2a_Pmc21, and b_Pbn21 = b_Pmc21. CIFs and setting transformation details are provided in the supporting information.] Quayle et al. (2015 ▸) predicted that the 0 K formation energy of the Pmc21 phase is higher than that of Pbn21 by only 0.011 ± 0.003 eV, and hypothesized that the Pbn21 and Pmc21 phases form a mixed lattice that consists of Pbn21 or Pmc21 basal planes stacked along the polar c axis, similar to SiC polytypes. XRD simulations of the mixed-phase crystals show that this type of basal plane disorder ‘washes out’ Bragg reflections unique to the Pbn21 and Pmc21 crystal structures, yielding a wurtzite-like spectrum. These results provide an explanation for the observations of a wurtzite-like XRD spectrum that does not assume a randomly disordered lattice.

In this paper, we investigate an LT of ZnSnN2, assuming that a reconstructive phase transition takes place at an elevated temperature between the low-temperature Pbn21 phase and the high-temperature Pmc21 phase. The SED is proposed to be the mechanism of the transition. Skachkov et al. (2016 ▸) and Adamski et al. (2017 ▸) determined that the SED is one of the lowest formation energy defects in ZnSnN2 and its sister compound ZnGeN2. Skachkov et al. (2016 ▸) also determined that there is an energetic benefit to the clustering of SEDs in these materials; the formation energy of two neighboring SEDs is lower than two isolated SEDs. Here, we see that clusters of SEDs transform the Pmc21 crystal structure into the Pbn21 crystal structure. An analysis of the group–subgroup relation of the two phases shows that the phase relation is not direct and there are two intermediate phases that mediate the phase transition. The order parameter of the phase transition is obtained from the online databases, along with the Landau potential. A solution to the Landau potential is given, based on the analyses of similar free-energy equations in the literature.

The model presented here has broad implications for all heterovalent ternary semiconductors. Crystal structure disorder and phase transitions have been widely reported in zincblende-based ternaries, but until the work of Ma et al. (2014 ▸) it was widely accepted that the disordered phase above T C is caused by a random high-temperature phase. In the zincblende-based case, the situation is analogous to the wurtzite-based ternaries: there are exactly two atomic arrangements that satisfy the octet rule, the chalcopyrite (Inline graphic, No. 122) and CuAu (Inline graphic, No. 115) phases. High-resolution electron diffraction and electron microscopy clearly resolve a mixture of the chalcopyrite and CuAu phases in CuInS2 and CuInSe2 (Álvarez-García et al., 2005 ▸; Su et al., 2000 ▸; Su & Wei, 1999 ▸; Metzner et al., 2000 ▸; Stanbery et al., 2002 ▸). Furthermore, it is argued here that the wurtzite-based ternaries are antiferroelectric in the c plane. If confirmed experimentally, the introduction of a new class of antiferroelectric semiconductors is of both fundamental and practical interest.

The remainder of the paper is organized as follows. First, we illustrate how the Pbn21 and Pmc21 phases intermix via the clustering of SEDs. Next, we analyze the group–subgroup relations between the Pmc21, Pbn21 and intermediate phases. Following that, we use the group–subgroup relations to identify the order parameter and Landau potential of the system. The Landau potential is then solved under the assumption of strong coupling between the phases. A brief discussion of the prospect that the wurtzite-based ternaries are antiferroelectric materials is given before a summary concludes the paper.

2. Illustration of phase mixing  

Building off the model proposed by Ryan et al. (1987 ▸) for ZnSnP2, we investigate the atomic formations generated by clusters of SEDs within the ZnSnN2 lattice. In this section, we see that the Pmc21 crystal structure can transform into the Pbn21 structure via intermediate phases with space groups Pmn21 (No. 31) and Pbc21 (No. 29) (Pca21 in the standard setting) (see the supporting information).

Fig. 1 ▸ shows the primitive unit cells of the four phases of ZnSnN2. The two phases on the left are the two lowest-energy phases; they correspond to the two ways that the lattice can be populated so that each N-centered tetrahedron is type -Zn2Sn2 and the octet rule is satisfied. The two phases on the right do not satisfy the octet rule. Each tetrahedron is type -Zn3Sn1 or -Zn1Sn3.

Figure 1.

Figure 1

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The phases of ZnSnN2 involved in the transition from Pmc21 to Pbn21. Primitive unit cells are outlined in red.

Fig. 2 ▸ shows a domain of the Pbn21 phase embedded within the Pmc21 background. The unit cell at the top of Fig. 2 ▸ highlights the eight-atom Pmc21 primitive unit cell. The unit cell at the bottom highlights a 32-atom Pbn21 unit cell. There are two types of SED patterns that lead to the Pbn21 phase. One type of pattern yields an atomic arrangement consistent with the Pbc21 space group, while the second yields a Pmn21 unit cell. The Pbn21 atomic arrangement is generated when the two SED patterns coincide.

Figure 2.

Figure 2

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Illustration of the subgroup phases’ atomic arrangements embedded within the Pmc21 crystal structure. The transformation of the Pmc21 phase into the subgroup phases is induced by the SEDs, indicated by green arrows. Unit cells are outlined in red.

3. Group–subgroup relation of the ZnSnN2 phases  

A group–subgroup analysis of ZnSnN2 is based on the similarities and differences in symmetry of the Pmc21 and Pbn21 phases. The group elements of a space group are the symmetry operations – the rotations and/or translations – that transform the crystal structure back into itself. A group that results from the removal of one or more of the symmetry operations is a subgroup. The software package SYMMODES on the Bilbao Crystallographic Server (https://www.cryst.ehu.es/) gives that the Pbn21 space group is a subgroup of Pmc21 with a group–subgroup index (i) of 4 (Capillas et al., 2003 ▸; Aroyo, Kirov et al., 2006 ▸; Aroyo, Perez-Mato et al., 2006 ▸; Aroyo et al., 2011 ▸) and that the group–subgroup relation is indirect. There are three possible chains linking the two groups, Pmc21 > (Pbc21, Cmc21, Pmn21) > Pbn21 (Fig. 3 ▸). For the intermediate relations, Pmc21 > (Pbc21, Cmc21, Pmn21) and (Pbc21, Cmc21, Pmn21) > Pbn21, the subgroup indices are i = 2.

Figure 3.

Figure 3

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Diagram of the Pmc21–Pbn21 group–subgroup phase relation.

Only the Pmn21 and Pbc21 phases are valid intermediate phases. The Cmc21 phase is not compatible with the ortho­rhombic crystal structure. To demonstrate this incompatibility, we examine the subgroup indices further.

The subgroup index relates the number of symmetry operations in the space group of the group phase to those of the subgroup phase (Bilbao Crystallographic Server, Index, https://www.cryst.ehu.es/cryst/help/index.html#index; Janovec et al., 1989 ▸),

3.

where Z is the number of formula units per unit cell and |P| is the order of the point group. Each of the space groups has the same point group mm2, meaning that the second term in equation (1) can be dropped and the subgroup index is simply the ratio of the number of formula units in the two phases.

Equation (1) tells us that the unit cell of the Pbn21 structure (i = 4) included in the phase transition contains 32 atoms, since the Pmc21 structure has an eight-atom primitive unit cell, or two formula units. The unit cells of the intermediate phases (i = 2) each contain 16 atoms. Thus, each of the two transition pathways, Pmc21 to Pca21 and Pmc21 to Pbc21, is a k-type (or klassengleich) transition (Bilbao Crystallographic Server). K-type transitions take place between phases that are members of the same point group and the transitions are characterized by a doubling of the unit cell (Müller, 2017 ▸). The complete transition from Pmc21 to Pbn21 requires a quadrupling of the unit cell; the 32-atom Pbn21 unit cell is the domain containing the smallest number of atoms that can describe the two-step phase transition. It consists of two repeat units of the 16-atom Pbn21 primitive unit cell.

To see that the Cmc21 phase is not compatible with the 16-atom orthorhombic unit cells based on the Pmc21 parent structure, we tabulate all the crystal structures that are compatible with the 16-atom orthorhombic unit cells. The atomic arrangements of the 16-atom orthorhombic unit cells are subject to constraints at the boundaries to maintain periodicity. The constraints are: (i) the cations at the vertices of the orthorhombic unit cell must be the same and (ii) the cations positioned on opposite faces of the unit cells must be the same. Inspection shows that there are 35 possible unit cells that satisfy these constraints. Each of the allowed unit cells is listed in Table 3 in Appendix A , along with its space group, which was determined using the ISOTROPY software suite program FINDSYM (Stokes & Hatch, 2005 ▸; https://iso.byu.edu/iso/findsym.php). The results show that the Pmc21, Pbc21, Pmn21 and Pbn21 space groups are compatible with the atomic positions of the 16-atom orthorhombic unit cell. There are other atomic arrangements with lower-symmetry space groups that satisfy the periodicity constraints, but these arrangements violate the octet rule (Quayle et al., 2015 ▸) and are not intermediate phases, so they are not considered further here. The Cmc21 space group is not present in Table 3.

4. Determining the order parameter and Landau potential  

A solid-state phase transition is characterized by a shift in the positions of atoms on the lattice. The atomic displacements break the symmetry of the high-symmetry group phase, yielding a crystal structure with the decreased symmetry of the subgroup phase. In a single crystal, the atomic displacements are periodic and they can be collectively associated with a normal mode of the system.

The order parameter of the phase transition is often assigned to a normal-mode wavevector, although it can be insightful to assign it to a function of the wavevector. The distortion gives rise to a polarization vector in ferroelectric crystals, for example. The amplitude of the displacements from the positions in the high-symmetry phase decreases as the temperature is increased towards the transition point, and the polarization approaches zero. Accordingly, we can express the free energy of the system as a function of the order parameter and perform a Taylor expansion around T C for small values. The energy of the system, written as a function of the order parameter, is the Landau potential.

Each group–subgroup relation is associated with one or more order parameters and these order parameters were calculated for every group–subgroup possibility for all 230 space groups by Stokes & Hatch (1988 ▸). The complete listing of their results is available in the online COPL program database (Hatch & Stokes, 2002 ▸; Stokes & Hatch, 2002 ▸), and the results for our case are listed in Table 1 ▸.

Table 1. Group–subgroup data for the Pmc21 → Pbn21 transition.

K vectors Irreps and order parameters Isotropy subgroup
GM: (0, 0, 0) GM1: (a) Pmc21
X: (1/2, 0, 0) X 2: (a) Pmn21
Y: (0, 1/2, 0) Y 2: (a) Pbc21
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Since the Pbn21 phase is generated by the overlap of the atomic displacements associated with both the Pbc21 and Pmn21 phases (Fig. 2 ▸), both order parameters are required for the phase transition, and the Landau potential, which expresses the free energy of the transition, is based on the coupled order parameter. We find the Landau potential from the ISOTROPY software suite program INVARIANTS (Hatch & Stokes, 2003 ▸; https://iso.byu.edu/iso/invariants.php),

4.

where a is one order parameter and b is the other. The coefficient A is assumed to be temperature dependent,

4.

where Ta is the transition point of the a order parameter. All other constants are constant and positive. The sixth-order terms must add up to be positive so that the free energy is positive at the extremes. The sign of the coupling terms is critical in determining the nature of the system. Choosing the coupling terms to be negative allows for the description of triggered phase transitions. In the next section, it will be argued that ZnSnN2 necessarily undergoes a triggered phase transition.

5. Solution to the Landau potential  

Holakovský (1973 ▸) established that the necessary conditions for a triggered ferroelectric transition are that the lowest-order coupling term is of the form Inline graphic and the sign of the term is negative. He considered materials that transition from paraelectric to ferroelectric and have both a primary and secondary order parameter. The secondary order parameter is assigned to a polarization vector, and it is shown that a phase transition driven by the primary order parameter generates a second, ferroelectric, phase transition as a result of the coupling. The behavior of the system is determined by the strength of the coupling terms. Under sufficiently strong coupling conditions, a triggered ferroelectric transition will occur in which both phase transitions occur simultaneously. The condition for strong coupling in ZnSnN2 is determined in Appendix B .

The analysis of Holakovský (1973 ▸) built off the work of Levanyuk & Sannikov (1969 ▸), which analyzed systems with different types of coupling terms. Their work established that systems with Inline graphic coupling terms are usually associated with antiferroelectric materials. Accordingly, we look for characteristics of antiferroelectric polarization in ZnSnN2.

The crystal structure of the wurtzite-based ternaries suggests that ZnSnN2 is antiferroelectric within the c-axis basal plane. In the orthorhombic II–IV–V2 compounds, the group IV atoms each transfer an electron to the group II atoms, so that the group II atoms are negatively charged and the group IV atoms are positively charged (Ma et al., 2014 ▸; Skachkov et al., 2016 ▸). As a result, the basal plane consists of an array of dipoles. Adding up the dipoles yields a series of equal but oppositely pointing net polarization vectors (Fig. 4 ▸). In the Pmc21 basal plane, the polarization vectors cancel out completely making it paraelectric.

Figure 4.

Figure 4

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The c-plane bonding characteristics of the paraelectric Pmc21 phase (left) and antiferroelectric Pbn21 phase (right). Thin red arrows indicate the directions of the polarization vectors. Larger arrows indicate the directions of the net polarization vectors. The dashed line is along a glide mirror plane. The solid line is along a mirror plane of the two-dimensional cation sublattice plane.

Thus, we hypothesize that a solid-state phase transition in ZnSnN2 from Pmc21 to Pbn21 is paraelectric to antiferroelectric. Furthermore, the coupling of the order parameters is not merely strong, it is locked. In the same way that the secondary order parameter is assigned to the polarization vector in a triggered ferroelectric transition, both order parameters can be assigned to the two polarization vectors in the antiferroelectric Pbn21 phase. The net polarization vectors are generated by the SEDs, and the neighboring SEDs that form a Pbn21 domain within Pmc21 generate two oppositely polarized vectors. The phase transition cannot proceed further than a single SED if both order parameters are not triggered. This amounts to a simultaneous transition of both order parameters; the system is inherently strongly coupled and both order parameters are primary.

Following Levanyuk & Sannikov (1969 ▸), we analyze the system under the assumption of antiferroelectric behavior and use the polarization vectors

5.
5.

The Inline graphic and Inline graphic polarization vectors are represented in Fig. 4 ▸ and they correspond to the dipoles that sum to a finite net polarization vector, P = Inline graphic. The antiferroelectric polarization arises from the glide mirror plane that runs along the y-axis points; there is a net polarization vector Inline graphic pointing in the opposite direction to and shifted along the y axis from P, which is equivalent by symmetry.

Using equations (4) and (5), we rewrite equation (2) in terms of the polarization vectors,

5.

where α = A, β = C/2 − D/2, γ = −(3C + D), δ = −(5F + G) and ∊ = F/3 − G. In writing equation (6), we make use of the symmetry due to the mirror plane in the cationic basal plane that makes A = B, C = E, F = I and G = H.

The free energy expressed in equation (6) is similar to that described by Kittel (1950 ▸), and we can evaluate it using his methods.

In the antiferroelectric regime under no applied electric field, the spontaneous polarization is Inline graphic. Minimizing equation (6) yields

5.

At the transition point, the local minimum expressed by equation (7) is equal to the potential at Inline graphic = Inline graphic = 0,

5.

Using equations (7) and (8) we find

5.
5.

If we apply a small electric field ΔE, the net polarization is ΔP = Inline graphic, where Inline graphic. Taking the sum of Inline graphic = ΔE and Inline graphic = ΔE yields

5.

Since Inline graphic and Inline graphic, the susceptibility just below the transition temperature, now the Néel temperature (T N), is

5.

Above T N, the higher-order terms in equation (6) can be neglected and

5.

By setting Γ = 1, we can plot the susceptibility (Fig. 5 ▸) and see the singularity at T N.

Figure 5.

Figure 5

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The temperature dependence of the electrical susceptibility for a first-order antiferroelectric phase transition.

The condition for T N is found using equations (9) and (10):

5.

6. Discussion  

The temperature dependence of the electrical susceptibility plotted in Fig. 5 ▸ displays the anomaly at the Néel temperature observed in the dielectric response of antiferroelectric mater­ials such as PbZrO3 (Whatmore & Glazer, 1979 ▸; Fthenakis & Ponomareva, 2017 ▸; Shirane et al., 1951 ▸; Roberts, 1949 ▸; Liu & Dkhil, 2011 ▸; Tagantsev et al., 2013 ▸). Whether ZnSnN2 is a true antiferroelectric material depends on additional criteria, however (Rabe, 2013 ▸).

The nonpolar antiferroelectric polarization in ZnSnN2 arises from the structural distortion generated by the SED formations which transform the paraelectric Pmc21 phase. To observe the P versus E double hysteresis loop characteristic of antiferroelectrics, the material must exhibit an additional ferroelectric distortion from the paraelectric phase when subject to an external electric field (Bennett et al., 2013 ▸; Tolédano & Guennou, 2016 ▸). It is reasonable to assume that this distortion will be present. In Fig. 6 ▸ we again look at the dipole arrangement of the paraelectric Pmc21 phase. For simplicity, we only consider the two-dimensional cation sublattice plane. Given the charge states of the different cations, under the application of an external field we can expect a shift in positions of the cations due to the Coulomb force. The induced distortion will break the symmetry-based cancelation of the dipole moments, resulting in a ferroelectric polarization in the cation plane.

Figure 6.

Figure 6

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The high-temperature Pmc21 phase under (a) zero bias and (b) an applied electric field. Only the atoms in the two-dimensional cation sublattice plane are shown. Thin arrows represent polarization vectors generated by the charge states of the Zn and Sn atoms. The thicker arrows in panel (b) represent the net polarization vectors generated by the electric-field-induced atomic distortion.

7. Summary and outlook  

To summarize, based on the calculated low formation energy of the SED, and the very similar formation energies of the Pbn21 and Pmc21 crystal structures, a model for a first-order reconstructive phase transition has been developed to interpret observations of structural disorder in ZnSnN2. It has been shown that patterns of SEDs generate the Pbn21 phase within the Pmc21 phase. A Landau potential based on two primary order parameters which activate simultaneously was analyzed. A solution to the free-energy equation was derived based on the model developed by Kittel (1950 ▸) for a paraelectric to antiferroelectric transition. The electrical susceptibility expressed by equations (12) and (13), and plotted in Fig. 5 ▸, is characteristic of antiferroelectric materials.

The picture described in this work is one in which thermal energy drives the formation of SEDs, which cluster to transform the atomic arrangements on the lattice. The Pbn21 phase is more stable than the Pmc21 phase at lower temperatures. As the temperature increases, the probability of SED formation increases. A single SED serves as a nucleation site for the precipitation of a domain of the Pmc21 phase, which then increases in size as the temperature is further increased. At high temperatures, the Pmc21 phase is more stable than the Pbn21 phase and will be the dominant phase. It is likely that thermal energy will continue to generate SEDs, however, and that the high-temperature phase will be dynamic, with SED formation and annihilation varying the phase composition of the structure until the melting point.

Ternary nitrides are in the early stages of development and are little studied compared with binary nitrides or zincblende-based ternaries. A phase transition has not been investigated experimentally in ZnSnN2. A phase transition is not the only possible cause of the disorder: kinetic barriers that inhibit atom mobility during growth are suggested to contribute to disorder in ZnGeN2 (Lany et al., 2017 ▸; Blanton et al., 2017 ▸), and point defects, defect complexes and off-stoichiometry are proposed to contribute to disorder in zincblende-based ternaries generally. Cation disorder is, however, widely reported to be a major factor in heterovalent ternaries. The model presented here is an alternative to that based on entropically random disorder at high temperatures, developed to describe the order–disorder transition observed in zincblende-based ternaries.

Essential aspects of the model presented here hold in the zincblende-based ternaries in which there are also two phases that satisfy charge neutrality. As stated in the Introduction , direct evidence of chalcopyrite and CuAu phase mixing has been clearly observed in CuInSe2, and the formation of a mixture of those two phases may be unavoidable in the CIGS system (Su et al., 2000 ▸). The wurtzite-based and zincblende-based ternary systems are not completely analogous, however. The space groups Inline graphic and Inline graphic of the zincblende-based system are not group–subgroup related according to SUBGROUPGRAPH on the Bilbao Crystallographic Server (https://www.cryst.ehu.es/) (Ivantchev et al., 2000 ▸). Indeed, the wurtzite-based ternaries are exceptional in this regard. Typically, crystalline materials that undergo a reconstructive phase transition do not have group–subgroup relations between phases (Dmitriev & Toledano, 1996 ▸) and an order parameter cannot be defined. Thus, the ability to develop a Landau theory for wurtzite-based ternaries is noteworthy.

The proposal here that wurtzite-based ternaries are antiferroelectric in the c plane is based on the crystal structure of the Pbn21 phase and the form of the order parameter coupling term in the Landau potential. Validation of this model would come from observation of the double hysteresis loops in a P versus E curve which are the signature characteristics of antiferroelectric materials (Rabe, 2013 ▸). If validated, this class of materials would hold potential for new applications. The dramatic increase in electrical susceptibility near the Néel temperature has been exploited in antiferroelectrics for capacitive energy-storage technologies. The energy-storage density of antiferroelectric materials is superior to that of linear and ferroelectric dielectrics (Liu et al., 2018 ▸).

Supplementary Material

Crystal structure: contains datablock(s) global, Pca21, Pmc21, Pmn21, Pna21. DOI: 10.1107/S2053273320003095/ug5001sup1.cif

a-76-00410-sup1.cif (5.4KB, cif)

The setting transformations from Pna21 to Pbn21 and from Pca21 to Pbc21. DOI: 10.1107/S2053273320003095/ug5001sup2.pdf

a-76-00410-sup2.pdf (953.4KB, pdf)

CCDC references: 1988421, 1988422, 1988423, 1988424

Acknowledgments

The author thanks E. Blanton for helpful discussions.

Appendix A. Intermediate-phase crystal structure information

See Tables 2 ▸ and 3 ▸ for details of atomic positions and unit-cell bases.

Table 2. Atomic positions of a 16-atom orthorhombic unit cell.

Atomic position x y z
1 0 0 0
2 1/2 0 0
3 1/4 1/2 0
4 3/4 1/2 0
5 0 1/3 1/2
6 1/2 1/3 1/2
7 1/4 5/6 1/2
8 3/4 5/6 1/2
9 0 0 3/8
10 1/2 0 3/8
11 1/4 1/2 3/8
12 3/4 1/2 3/8
13 0 1/3 7/8
14 1/2 1/3 7/8
15 1/4 5/6 7/8
16 3/4 5/6 7/8
Open in a new tab

Table 3. Bases of the 35 ZnSnN2 16-atom unit cells and the resultant space groups.

Atom position  
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Space group
Zn Zn Zn Zn Sn Sn Sn Sn N N N N N N N N Cm
Zn Zn Zn Sn Zn Sn Sn Sn N N N N N N N N P1
Zn Zn Zn Sn Sn Zn Sn Sn N N N N N N N N P1
Zn Zn Zn Sn Sn Sn Zn Sn N N N N N N N N Pm
Zn Zn Zn Sn Sn Sn Sn Zn N N N N N N N N Pm
Zn Zn Sn Zn Zn Sn Sn Sn N N N N N N N N P1
Zn Zn Sn Zn Sn Zn Sn Sn N N N N N N N N P1
Zn Zn Sn Zn Sn Sn Zn Sn N N N N N N N N Pm
Zn Zn Sn Zn Sn Sn Sn Zn N N N N N N N N Pm
Zn Zn Sn Sn Zn Zn Sn Sn N N N N N N N N Pmc21 †
Zn Zn Sn Sn Zn Sn Zn Sn N N N N N N N N P1
Zn Zn Sn Sn Zn Sn Sn Zn N N N N N N N N P1
Zn Zn Sn Sn Sn Zn Sn Zn N N N N N N N N P1
Zn Zn Sn Sn Sn Zn Zn Sn N N N N N N N N P1
Zn Zn Sn Sn Sn Sn Zn Zn N N N N N N N N Pmn21 †
Zn Sn Zn Zn Zn Sn Sn Sn N N N N N N N N Pm
Zn Sn Zn Zn Sn Zn Sn Sn N N N N N N N N Pm
Zn Sn Zn Zn Sn Sn Zn Sn N N N N N N N N P1
Zn Sn Zn Zn Sn Sn Sn Zn N N N N N N N N P1
Zn Sn Zn Sn Zn Zn Sn Sn N N N N N N N N P1
Zn Sn Zn Sn Zn Sn Zn Sn N N N N N N N N Pca21 †
Zn Sn Zn Sn Zn Sn Sn Zn N N N N N N N N P21
Zn Sn Zn Sn Sn Zn Sn Zn N N N N N N N N Pbn21 †
Zn Sn Zn Sn Sn Sn Zn Zn N N N N N N N N P1
Zn Sn Zn Sn Sn Zn Zn Sn N N N N N N N N P21
Zn Sn Sn Zn Zn Zn Sn Sn N N N N N N N N P1
Zn Sn Sn Zn Zn Sn Zn Sn N N N N N N N N P21
Zn Sn Sn Zn Zn Sn Sn Zn N N N N N N N N Pca21 †
Zn Sn Sn Zn Sn Zn Sn Zn N N N N N N N N P21
Zn Sn Sn Zn Sn Zn Zn Sn N N N N N N N N Pbn21 †
Zn Sn Sn Zn Sn Sn Zn Zn N N N N N N N N P1
Zn Sn Sn Sn Zn Zn Sn Zn N N N N N N N N P1
Zn Sn Sn Sn Zn Zn Zn Sn N N N N N N N N P1
Zn Sn Sn Sn Zn Sn Zn Zn N N N N N N N N Pm
Zn Sn Sn Sn Sn Zn Zn Zn N N N N N N N N Pm
Open in a new tab
†

This phase is a maximal subgroup of Pmc21 and is involved in the Pmc21 to Pbn21 transition.

Appendix B. Strong coupling condition in wurtzite-based heterovalent ternary semiconductors

Here, we apply the methods of Holakovský (1973 ▸) to determine the strong coupling condition for wurtzite-based heterovalent ternaries.

We start from equation (2), which we restate here for convenience,

Appendix B.

We first minimize in terms of the a order parameter,

Appendix B.

which has two solutions,

Appendix B.
Appendix B.

Solution I. is valid if Inline graphic Inline graphic Inline graphic.

Solution II. is valid if Inline graphic Inline graphic Inline graphic.

At Inline graphic, Inline graphic = Inline graphic.

Since we are considering Inline graphic, we approximate,

Appendix B.

Inserting (16) and (18) into equation (15) yields the two distinct solutions which correspond to local minima in Φ,

Appendix B.
Appendix B.

where

Appendix B.
Appendix B.
Appendix B.

To interpret these equations, we consider the behavior of the order parameters as the temperature is lowered.

When T Inline graphic Ta, the displacements associated with both the a and b order parameters are inactive and a = b = 0.

At T = Ta, the displacements associated with the a order parameter are activated and the coefficients B, E and I change to B′, E′ and I′.

Just below Ta, the coupling of a to b causes an instability in b. The coefficient B′ is temperature dependent [equation (3)] and it will become negative. Once B′ is less than 0, the second transition at Tb is allowed. The nature of a second phase transition at Tb is determined by E′.

A first-order phase transition will occur at Tb if E′ Inline graphic 0. In the Ta Inline graphic T Inline graphic Tb region where b = 0 and Inline graphic = Inline graphic = 0,

Appendix B.

Once Tb is reached, E will transition to E′ and the magnitude of E′ will determine the nature of the transition. Two types of first-order phase transition can occur after E transitions to E′ at Ta:

(i) The case in which −2(BI)1/2 Inline graphic E′ Inline graphic 0 is weak coupling and Ta Inline graphic Tb.

(ii) When E′ Inline graphic −2(BI)1/2, the temperature regime Ta Inline graphic T Inline graphic Tb is forbidden.

Case (ii) is the strong coupling case in which, at Ta = Tb, the order parameters a and b become finite simultaneously.

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Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Supplementary Materials

Crystal structure: contains datablock(s) global, Pca21, Pmc21, Pmn21, Pna21. DOI: 10.1107/S2053273320003095/ug5001sup1.cif

a-76-00410-sup1.cif (5.4KB, cif)

The setting transformations from Pna21 to Pbn21 and from Pca21 to Pbc21. DOI: 10.1107/S2053273320003095/ug5001sup2.pdf

a-76-00410-sup2.pdf (953.4KB, pdf)

CCDC references: 1988421, 1988422, 1988423, 1988424

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