Screening of II-IV-V 2 Materials for Photovoltaic Applications Based on Density Functional Theory Calculations

: The relative stability of polymorphs and their electronic structure was investigated for II-IV-V 2 materials by using ﬁrst-principles density functional theory calculations. Our calculation results show that, for Zn-, Cd-, and Be-containing compounds, nitrides favor the 2H polymorph with AB stacking sequence; however, phosphides, arsenides, and antimonides are more stable in the 3C polymorph with the ABC stacking sequence. The electronic band gap of materials was calculated by using hybrid density functional theory methods, and then materials with an ideal band gap for photovoltaic applications were chosen. The experimental synthesis of the screened materials is reported, except for CdSiSb 2 , which was found to be unstable in our calculation. The absorption coefﬁcient of the screened materials, especially ZnGeAs 2 , was high enough to make thin-ﬁlm solar cells. The higher stacking fault energy in ZnGeAs 2 than the others is consistent with the larger formation energy difference between the 2H and 3C polymorphs.


Introduction
The progress of photovoltaic technology has been largely due to the finding and optimization of new materials [1][2][3][4]. Among many approaches, cation mutation has remained a notable design principle for finding new materials [5]. Starting from diamond Si, we obtain zinc-blende CdTe (GaAs) that satisfies the octet rule by replacing every two Si atoms with Cd and Te (Ga and As). If two Cd atoms are replaced with Cu and In while Se substitutes for Te, then chalcopyrite CuInSe 2 is obtained as illustrated in Figure 1. If we further apply the operation, we can obtain kesterite Cu 2 ZnSnSe 4 . All of these materials were investigated extensively for photovoltaic applications both theoretically and experimentally [4]. II-IV-V 2 is another category of materials obtained by applying the design principle to III-V, while ZnSnN 2 is the most well-known material for photovoltaic application in this category [6].
Following the cation mutation principle, at least several tens of material can be made, theoretically, as many atoms have the same oxidation number. Such materials can be examined and screened by first-principles calculation as in previous studies [7][8][9][10]. There are many physical properties of constituent materials that determine the solar conversion efficiency [11][12][13][14]. Among them, the band gap is used as the most important metric to screen materials, as the solar conversion efficiency is fundamentally determined by the band gap [4,15]. Shaposhnikov  semiconductors [16], and concluded that ZnSiAs 2 and ZnSnAs 2 are ideal for photovoltaic applications in terms of the high absorption coefficient and the dipole matrix, not the electronic band gap. To combinatorially generate the materials, Be, Mg, Zn, Cd, Si, Ge, Sn, P, and As were used but antimonides were not considered. More recently, Pandey et al. investigated other II-IV-V 2 materials computationally and suggested several candidates for photovoltaic applications [17]. In their study, the heat of the formation of materials was calculated, using mBEEF meta-GGA functional [18], and then the electronic band gap was evaluated, using a semilocal GLLB-SC functional [19]. However, a specific exchangecorrelation functional does not perfectly predict the ground structure of all materials, and therefore it is needed to double-check the prediction using other functionals [20][21][22]. Furthermore, Be was not included in the later study, even though Be also predominantly favors a 2+ oxidation state. It is also worth noting that BeO is stable in the wurtzite phase (2H), while BeS, BeSe, and BeTe are stable in the zinc-blende structure (3C) [23]. In this article, we revisit the stability and the electronic structure of II-IV-V 2 materials to screen materials for photovoltaic applications. The relative formation energy of the 2H and the 3C polymorphs was determined by using three exchange-correlation functionals based on the majority voting. The electronic structure of the materials was calculated using a more stable structure at the hybrid density functional theory. The materials were selected based on the band gap, and the absorption coefficient of the screened materials was obtained accordingly. We also obtained the stacking fault energy for some screened materials for photovoltaic applications because the planar defects have critical roles in terms of transport properties [24,25].

Calculation Method
We performed first-principles DFT calculations using the projector-augmented wave (PAW) method [26], and implemented in the Vienna ab initio simulation package (VASP) [27]. We used three exchange-correlation functionals, such as LDA [28], SCAN+rVV10 [29], and TPSS [30], to double-check the relative stability between polymorphs. The energy cutoff for the plane waves was set to 500 eV. A 6 × 6 × 6 k-point grid was used for the primitive cells. The volume and the internal coordinates of atoms were fully relaxed until the residual forces were below 0.01 eV/Å. The electronic structure was investigated by using a hybrid density functional suggested by Heyd, Scuseria, and Ernzerhof (HSE06) [31]. For the HSE06 calculation, the energy cutoff was determined by setting the PREC tag to normal. A body-centered tetragonal unit cell of the 3C polymorph and an orthorhombic unit cell of the 2H polymorph were used throughout the study. The body-centered tetragonal and the orthorhombic cell unit cell contain 2 and 4 formula units, respectively.

Results and Discussion
ZnSnN 2 satisfies the octet rule because Zn, Sn, and N have +2, +4, −3 oxidation states, respectively. If each atom is replaced with an atom with the same oxidation state, the resulting material satisfies the octet rule as well. For instance, Zn can be replaced with Be or Cd, keeping the octet rule. Mg is another element that prefers +2 oxidation states, but we did not consider it because we restricted our focus to materials that are stable in the 2H and 3C polymorphs. MgO, MgS, and MgSe favor the rock-salt structure [23], and there is a report of a rock-salt type synthesis of MgSnN 2 [32]. Similarly, Si and Ge can substitute for Sn, while N sites can be occupied by P, As, and Sb atoms. The number of combinations satisfying the octet rule is 36.
This category of materials that consist of the tetrahedron is mostly stable in either of the 2H and the 3C polymorphs. To obtain which stacking sequence the materials favor, we compared the relative formation energy of II-IV-V 2 materials in the 2H and the 3C polymorphs. The relative formation energy of the 2H with respect to the 3C was calculated by using three different exchange-correlation functionals, as summarized in Figure 2.
The relative formation energy of 2H (eV/f.u.) Remarkably, every functional predicts that the same structure has lower formation energy. The only exception is ZnSnSb 2 , which favored the 2H polymorph a little bit in our LDA calculation. However, in the other two calculations, the ZnSnSb 2 in the 3C polymorph has lower energy than that in the 2H polymorph. Following the majority voting rule, we concluded that ZnSnSb 2 more favors the 3C structure than the 2H structure. Every nitride was calculated to be stable in the 2H polymorph; the phosphides, arsenides, and antimonides favored the 3C polymorph.
We performed hybrid DFT calculations in the 2H structure for nitride and the 3C for the others to obtain the electronic band gap, as summarized in Figure 3. We employed structures optimized by using SCAN+rVV10 functional for self-consistent field hybrid calculation. The optimized lattice constants were in good agreement with experimental values as summarized in Table 1. Some of the general chemical trends are as follows: If Zn is replaced with Cd, the band gap decreases on average, but on the other hand, the substitution of Be increases the band gap. The electronic band gap increases as Sn atoms are substituted with Ge and Si. The smaller the anion (Sb→As→P→N), the larger the band gap obtained.
B eG eN 2 B eG eP 2 B eG eA s 2 B eG eS b 2 B eS iN 2 B eS iP 2 B eS iA s 2 B eS iS b 2 B eS n N 2 B eS n P 2 B eS n A s 2 B eS n S b 2 C d G eN 2 C d G eP 2 C d G eA s 2 C d G eS b 2 C d S iN 2 C d S iP 2 C d S iA s 2 C d S iS b 2 C d S n N 2 C d S n P 2 C d S n A s 2 C d S n S b 2 Z n G eN 2 Z n G eP 2 Z n G eA s 2 Z n G eS b 2 Z n S iN 2 Z n S iP 2 Z n S iA s 2 Z n S iS b 2 Z n S n N 2 Z n S n P 2 Z n S n A s 2 Z n S n S b 2 Material The ideal materials for photovoltaic applications should have a direct band gap between 1 eV and 1.5 eV [15]. We chose materials with a direct band gap between 0.9 eV and 1.6 eV, considering a potential error in the calculation. The following five materials satisfied the criteria: CdSnP 2 , ZnSnN 2 , ZnGeAs 2 , CdSiAs 2 , and CdSiSb 2 . The experimental synthesis of CdSiAs 2 , CdSnP 2 , ZnSnN 2 , and ZnGeAs 2 were reported in the literature [33][34][35][36][37][38][39][40]. Here, we note that CdSiAs 2 , CdSnP 2 , and ZnGeAs 2 received less attention, compared to ZnSnN 2 . Here, we also note that Pandey et al. suggested ZnSiSb 2 , ZnSnN 2 , and ZnSnP 2 as small band gap materials for tandem applications. However, in our calculation, ZnSiSb 2 has an indirect band gap and ZnSnP 2 has a band gap of about 1.75 eV, and is probably suitable for a larger band gap material in photovoltaic applications. As shown in Table 1, the calculated band gaps are in good agreement with the experimental studies.
On the other hand, we could not find any experimental study of CdSiSb 2 . To check whether CdSiSb 2 can be formed in the chalcopyrite structure, we obtained the heat of formation as follows: The chemical potential of Cd, Si, and Sb was obtained from hexagonal close-packed Cd, diamond Si, and trigonal Sb, respectively. In our SCAN+rVV10 calculation, the heat of the formation was calculated to be positive (∆H f = 0.35 eV/f.u.), indicating that this material is not chemically stable. We double-checked the stability by using the LDA exchange-correlation functional [22]; the calculation result was not changed (∆H f = 0.35 eV/f.u.). This explains why there has been no experimental report of this material.
We employed the electronic band gap to screen materials; however, the absorption coefficient might be a more direct physical quantity to design solar cells. To obtain the absorption coefficient (α), we obtained the frequency-dependent dielectric function by performing hybrid DFT calculations. To reduce the computational cost, we employed a sparser k-point grid for the Fock potential by setting the reduction factor of 2 [41]. A 10 × 10 × 10 k-point grid was used for ZnSnN 2 . For the other three materials, a 14 × 14 × 14 grid was employed. Figure 4 shows the absorption coefficients of the four selected materials. CdSnP 2 , CdSiAs 2 , and ZnGeAs 2 show comparable absorption coefficients to the ordered ZnSnN 2 at wavelengths longer than 600 nm, and for the shorter wavelengths, the absorption coefficient is even higher. Among the four selected materials, ZnGeAs 2 has the highest absorption coefficient. This explains the motivation behind previous research studies to make solar cells based on these materials. Here, we point out that this class of materials is relatively new, compared to other mature technologies [6]. Further optimization should be made to improve efficiency, as solar conversion efficiency is a complex function of many parameters [12,13]. Here, we also point out that ZnSnN 2 is usually disordered and that the material properties are largely different from the ordered one considered in this study [42][43][44].  The energy difference between the 2H and the 3C polymorphs is generally small as summarized in Figure 2. This small energy difference might cause the simultaneous formation of both polymorphs in the lattice. The stacking fault is a special case in that the metastable phase (the 2H or the 3C) is sandwiched in the host material. For instance, in the zinc-blende materials, the stacking fault can be regarded as a buried wurtzite phase. This planar defect can be formed in the host material by removing or inserting a layer (e.g., · · · ABCABABC· · · in the 3C). Both computational and experimental studies show that such planar defects are harmful in terms of carrier transport and defect accumulation [45][46][47].
To examine whether the stacking disorders likely happens in the selected materials, we obtained the stacking fault energy by using the SCAN+rVV10 exchange-correlation functional. We did not calculate the defect in ZnSnN 2 because we already investigated it in our previous study. According to our recent study, the stacking fault formation energy in ZnSnN 2 is about 0.28 eV/nm 2 [48]. In CdSnP 2 , CdSiAs 2 , and ZnGeAs 2 , the stacking fault energy was 0.06 eV/nm 2 , 0.15 eV/nm 2 , and 0.47 eV/nm 2 , respectively. The higher stacking fault energy in ZnGeAs 2 is well explained by the higher relative formation energy of the 2H polymorph than the other two materials, as shown in Figure 2. The stacking fault can be formed in CdSnP 2 and CdSiAs 2 significantly, but much less formation is expected in ZnGeAs 2 . Table 1. Physical properties of the screened materials for single-junction solar cells. The lattice constants a, b, and c are those of the structure optimized by using SCAN+rVV10. The values in parentheses are experimental values. E g and E SFE stand for the electronic band gap and the stacking fault energy, respectively. The electronic band gap of ZnSnN 2 varies depending on the cation ordering [6].

Summary
We revisited II-IV-V 2 materials for photovoltaic applications based on DFT calculations. Nitrides favor AB stacking sequence, but phosphides, arsenides, and antimonides prefer ABC stacking sequence. Five materials were calculated to have an ideal band gap for a single-junction solar cell, but CdSiSb 2 was not thermodynamically stable. The absorption coefficient of CdSnP 2 , ZnSnN 2 , ZnGeAs 2 , and CdSiAs 2 was high enough to make thin-film solar cells. The stacking fault energy was low enough to be formed in CdSnP 2 and CdSiAs 2 , but much higher in ZnGeAs 2 .

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.