⇦Chinese Physics Letters, 2024, Vol. 41, No. 5, Article code 056101 Origin of the Disparity between the Stability of Transmutated Mix-Cation and Mix-Anion Compounds Shi-Wei Ye (野仕伟)1, Song-Yuan Geng (耿松源)2, Han-Pu Liang (梁汉普)1, Xie Zhang (张燮)3*, and Su-Huai Wei (魏苏淮)1* Affiliations 1Beijing Computational Science Research Center, Beijing 100193, China 2Advanced Materials Thrust, Function Hub, Hong Kong University of Science and Technology (Guangzhou), Guangzhou 511458, China 3School of Materials Science and Engineering, Northwestern Polytechnical University, Xi'an 710072, China Received 8 March 2024; accepted manuscript online 10 April 2024; published online 23 May 2024 *Corresponding authors. Email: xie.zhang@nwpu.edu.cn; suhuaiwei@csrc.ac.cn Citation Text: Ye S W, Geng S Y, Liang H P et al. 2024 Chin. Phys. Lett. 41 056101    Abstract Transmutation is an efficient approach for material design. For example, ternary compound CuGaSe$_{2}$ in chalcopyrite structure is a promising material for novel optoelectronic and thermoelectric device applications. It can be considered as formed from the binary host compound ZnSe in zinc-blende structure by cation transmutation (i.e., replacing two Zn atoms by one Cu and one Ga). While cation-transmutated materials are common, anion-transmutated ternary materials are rare, for example, Zn$_{2}$AsBr (i.e., replacing two Se atoms by one As and one Br) is not reported. The physical origin for this puzzling disparity is unclear. In this work, we employ first-principles calculations to address this issue, and find that the distinct differences in stability between cation-transmutated (mix-cation) and anion-transmutated (mix-anion) compounds originate from their different trends of ionic radii as functions of their ionic state, i.e., for cations, the radius decreases with the increasing ionic state, whereas for anions, the radius increases with the increasing absolute ionic state. Therefore, for mix-cation compounds, the strain energy and Coulomb energy can be simultaneously optimized to make these materials stable. In contrast, for mix-anion systems, minimization of Coulomb energy will increase the strain energy, thus the system becomes unstable or less stable. Thus, the trend of decreasing strain energy and Coulomb energy is consistent in mix-cation compounds, while it is opposite in mix-anion compounds. Furthermore, the study suggests that the stability strategy for mix-anion compounds can be controlled by the ratio of ionic radii $r_{3}/r_{1}$, with a smaller ratio indicating greater stability. Our work, thus, elucidates the intrinsic stability trend of transmutated materials and provides guidelines for the design of novel ternary materials for various device applications.
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 DOI:10.1088/0256-307X/41/5/056101 © 2024 Chinese Physics Society Article Text By conducting cation or anion transmutation, many novel materials with tunable properties can be obtained from a simple material.[1-5] For example, as shown in Fig. 1, by substituting two unary group-IV Ge atoms in the original diamond structure with one group-III (group-II) element and one group-V (group-VI) element, binary zinc-blende materials such as GaAs (III–V) or ZnSe (II–VI) can be obtained. Similarly, ternary compounds can be generated from binary materials through elemental transmutation. For instance, using one group-I and one group-III atom to substitute two group-II element Zn atoms in ZnSe (with band gap of $\sim$ $2.8 $ eV), mix-cation chalcopyrite CuGaSe$_{2}$ (with reduced band gap of $\sim$ $1.7$ eV) can be obtained, which preserves the octet rule and possesses the lowest formation energy among the ordered structures and has many applications in solar cells and thermoelectrics.[1,6-13] Similar stable compounds with the chalcopyrite structure such as I–III–VI$_{2}$ (where ${\rm I} = {\rm Cu}$, Ag, or Au, ${\rm III}= {\rm Al}$, Ga, or In, and ${\rm VI} = {\rm S}$, Se, or Te) and II–IV–V$_{2}$ (where ${\rm II} = {\rm Zn}$, Cd, or Hg, ${\rm IV} ={\rm Si}$, Ge, or Sn, and ${\rm V} = {\rm P}$, As, or Sb) have received significant attention for their applications in optoelectronic devices[6-8] and thermoelectrics.[9,14-16] Although a large number of transmutated mix-cation compounds with interesting properties are known,[6-26] transmutated mix-anion compounds are rare, despite their similarity to mix-cation compounds in terms of Coulomb interaction. For example, mix-anion chalcopyrite compounds, such as II$_{2}$–V–VII (${\rm II} = {\rm Zn}$ or Cd; ${\rm V} = {\rm N}$, P, As or Sb; ${\rm VII} = {\rm F}$, Cl, Br or I), do not yet exist.[27] Currently, studies on mix-anion chalcopyrite compounds are notably scarce, suggesting significant differences from mix-cation counterparts.[28-30] Therefore, it is crucial to carefully investigate the physical mechanism behind this stability disparity. In this work, by calculating and analyzing the mixing enthalpies of the various mix-anion ternary chalcopyrite compounds and comparing to those of the mix-cation ones, the thermodynamic stabilities of ternary chalcopyrite compounds are investigated. A general trend about their stability is derived and the physical origin behind the trend is unveiled. We find that because the radius decreases with the increasing ionic state for cations, for transmutated mix-cation compounds, the strain energy and Coulomb energy can be simultaneously optimized to make these materials stable. In contrast, for mix-anion systems, because the radius increases with the increasing absolute ionic state for anions, the minimization of Coulomb energy contradicts the optimization of the strain energy, thus the system becomes unstable or less stable. These insights improve our understanding of stability of ternary and multinary compounds and greatly accelerate the future screening and design of ternary and multinary compounds for specific applications. To investigate the stability of ternary chalcopyrite compounds, the total energy and material properties of the chalcopyrites and related compounds are calculated using density functional theory (DFT) with projector-augmented wave[31] potentials as implemented in the Vienna ab initio simulation package (VASP).[32] The volume and shape of the cell, and internal atomic positions are fully relaxed using the Perdew–Burke–Ernzerhof (PBE) functional[33] until the total energy is converged to 10$^{-6}$ eV and the residual forces are below 0.01 eV$\cdot $Å$^{-1}$. All valence ${\rm d}$ electrons involved in metal elements are treated in the same footing as other valence electrons. An energy cutoff of 520 eV for the plane-wave basis and a $k$-point mesh of $5\times 5\times 4$ for sampling the Brillouin zone of the chalcopyrite primitive cell is used. The lattice parameters, bond lengths, and electronic structures are calculated using a hybrid functional as constructed by Heyd, Scuseria, and Ernzerhof (HSE) with the screening range set to 0.2 Å$^{-1}$, and the fraction of the nonlocal Fock exchange is set to be 0.25.[34] The thermodynamic stability of each chalcopyrite structure is determined by its mixing enthalpy $H_{\rm m}$, which is calculated as the total energy difference between the ternary compound and its most stable binary constituting compounds. For example, the stability of CuGaSe$_{2}$ is determined by comparing with both Cu$_{2}$Se and Ga$_{2}$Se$_{3}$, while that of Zn$_{2}$AsBr is determined by comparing with both ZnBr$_{2}$ and Zn$_{3}$As$_{2}$. Therefore, the mixing enthalpies per atom of chalcopyrite CuGaSe$_{2}$ and Zn$_{2}$AsBr are defined by \begin{eqnarray} \tag {1} &&H_{\rm m} ({\rm CuGaSe}_{2})=\frac{1}{4}\Big[ E_{\rm t}({\rm CuGaSe}_{2})-\frac{1}{2}E_{\rm t} ({\rm Cu}_{2} {\rm Se})\notag\\ &&\hphantom{H_{\rm m} ({\rm CuGaSe}_{2})=}{}-\frac{1}{2}E_{\rm t} ({\rm Ga}_{2} {\rm Se}_{3}) \Big],\\ \tag {2} &&H_{\rm m} ({\rm Zn}_{2} {\rm AsBr})=\frac{1}{4}\Big[ E_{\rm t} ({\rm Zn}_{2} {\rm AsBr})-\frac{1}{2}E_{\rm t} ({\rm ZnBr}_{2} )\notag\\ &&\hphantom{H_{\rm m} ({\rm Zn}_{2} {\rm AsBr})=}{}-\frac{1}{2}E_{\rm t} ({\rm Zn}_{3} {\rm As}_{2}) \Big], \end{eqnarray} where $E_{\rm t}$ represents the total energy per formula unit. A negative value of $H_{\rm m}$ means that decomposition into these stable binary products from ternary chalcopyrite compound requires external energy, hence the ternary chalcopyrite is thermodynamically stable. On the other hand, a positive value of $H_{\rm m}$ indicates that the chalcopyrite structure is unstable. As an example, we first investigate the prototype mix-cation and mix-anion chalcopyrite crystals CuGaSe$_{2}$ and Zn$_{2}$AsBr, respectively, which are derived from zinc-blende ZnSe by atomic transmutation with neighboring atoms in the Periodic Table, that is, for mix-cation systems, two Zn atoms are replaced by a pair of Cu and Ga atoms, and for mix-anion chalcopyrite systems, the Se atoms are replaced by a pair of As and Br atoms. The lowest energy structures of binary compounds involved in the calculation of mixing enthalpy include Zn$_{3}$As$_{2}$, ZnBr$_{2}$, Ga$_{2}$Se$_{3}$, and Cu$_{2}$Se. The ternary chalcopyrite structures are depicted in Fig. 1. The lattice parameters ($a$ and $c$), bond lengths ($r_{12}$, $r_{32}$, and $r_{22}$), the displacement parameters ($u=(1/4)+[(r_{12}^{2}-r_{32}^{2})/a^{2}]$), Bader charges ($q$), and mixing enthalpies ($H_{\rm m}$) are listed in Tables 1 and 2.
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Fig. 1. An example of generating binary and ternary compounds through elemental transmutation.
Table 1. Calculated basic physical properties of ternary compounds CuGaSe$_{2}$ and Zn$_{2}$AsBr as well as the host binary compound ZnSe, including lattice parameters $a$ and $c$ (Å), the displacement parameter $u$, bond lengths $r_{12}$, $r_{32}$, and $r_{22}$ (Å), Bader charge $q$ on each atom ($e)$, and mixing enthalpy $H_{\rm m}$ (eV/atom).
Compound $a$ $c$ $u$ $r_{12}$ $r_{32}$
CuGaSe$_{2}$ 5.64 11.09 0.2515 2.43 2.42
Zn$_{2}$AsBr 5.84 11.94 0.3057 2.75 2.38
ZnSe 5.69 – – $r_{22} = 2.47$
Compound $q_{1}$ $q_{2}$ $q_{3}$ $H_{\rm m}$
CuGaSe$_{2}$ 0.41 $-0.76$ 1.11 $-0.099$
Zn$_{2}$AsBr $-0.62$ 0.72 $-0.82$ 0.008
ZnSe – $\pm 0.72$ – –
Although we put both CuGaSe$_{2}$ (mix-cation) and Zn$_{2}$AsBr (mix-anion) compounds into the same chalcopyrite configuration, their stabilities are significantly different. As shown in Table 1, CuGaSe$_{2}$ has a negative mixing enthalpy ($-99 $ meV/atom), thus is stable again decomposition to its binary constitutes, but Zn$_{2}$AsBr has a positive mixing enthalpy (8 meV/atom), thus is unstable. The origin of this disparity in their stability is explained below. To facilitate the understanding and discussion, we consider the transmutation process in three steps to form the ternary compounds. Step 1. In this step, we fulfill the transmutation procedure by replacing the Zn atoms in mix-cation structures with Cu and Ga, or the Se atoms in mix-anion structures with As and Br. As a result, the atomic charges in the structures changed from $q_{2}$ to $q_{1}$/$q_{3}$, as shown in Table 1 and Fig. 2. The average Coulomb energy between two nearest neighbor cations and anions ($q_{1}q_{2}$ and $q_{3}q_{2}$) in the ternary compounds remains the same as that in the original ZnSe compounds ($q_{2}q_{2}$), while the different charges on the $q_{1}$ and $q_{3}$ sites leads to an extra negative Coulomb energy that is proportional to ($\delta q)^{2}$, where ($\delta q$) is the charge difference between $q_{1}$ and $q_{3}$.[35] Meanwhile, because the $q_{1}$ and $q_{3}$ ions have different sizes while the bond lengths are constrained to be the same in this step, the systems have positive strain energy. Furthermore, the mixing enthalpy is now referenced to the binary compounds with “ideal” bond lengths. Therefore, in this step, the change of mixing enthalpy $\Delta H_{1}$ depends on the bond length matching and the extra Coulomb energy. For CuGaSe$_{2}$, the difference in bond length between Cu–Se and Ga–Se bonds is small (see Table 2) because the bonds are partially covalent and partially ionic, and both are close to the Zn–Se bond length, so the increase in strain energy is limited and the extra Coulomb energy becomes dominant. This is reflected by the fact that the mixing enthalpy of CuGaSe$_{2}$ is negative in this step, as shown in Fig. 3. On the other hand, for Zn$_{2}$AsBr, as the mismatch of bond lengths between its Zn-As bonds and those in the corresponding binary compounds (Zn$_{3}$As$_{2}$) is large, i.e., the lengths of Zn–As and Zn–Se bond do not match (Table 2), there is a larger increase in strain energy. Even though the change in Coulomb energy of Zn$_{2}$AsBr is also negative, due to the significant increase in its strain energy, the mixing enthalpy is positive in this step, as shown in Fig. 3.
Table 2. Crystal system, space group, and bond lengths (Å) of the selected binary compounds related to formation of CuGaSe$_{2}$ and Zn$_{2}$AsBr.
Compound Crystal
system		 Space
group		 Average bond
length (Å)
Zn$_{3}$As$_{2}$ tetragonal $P4_2/nbc$ 2.59
ZnBr$_{2}$ tetragonal $I4_1/acd$ 2.45
Ga$_{2}$Se$_{3}$ monoclinic $C1c1$ 2.46
Cu$_{2}$Se monoclinic $P12_1/c1$ 2.47
Step 2. We adjust the volume of CuGaSe$_{2}$/Zn$_{2}$AsBr cell from its original value $V = V_{0} = V_{\rm ZnSe}$ to that of the fully relaxed chalcopyrite structural CuGaSe$_{2}$/Zn$_{2}$AsBr cell, as shown in Fig. 2. The change of enthalpy $\Delta H_{2}$ obtained in this step reflects the impact of volume change of CuGaSe$_{2}$/Zn$_{2}$AsBr cell on its mixing enthalpy. In this step, since the volume of CuGaSe$_{2}$/Zn$_{2}$AsBr has relaxed towards the value of the fully relaxed structure, $\Delta H_{2}$ is definitely negative, as shown in Fig. 3 and Table 3. For CuGaSe$_{2}$ and Zn$_{2}$AsBr, the values of $\Delta H_{2}$ are small because these structures are both transmutated from ZnSe with neighboring atoms in the Periodic Table.
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Fig. 2. Three steps of the elemental transmutation process. Step 1: element transmutation without relaxation. Step 2: volume change without relaxation. Step 3: relaxation.
cpl-41-5-056101-fig3.png
Fig. 3. The changes of mixing enthalpies during the three-step processes from ZnSe to CuGaSe$_{2}$ and to Zn$_{2}$AsBr.
Table 3. The mixing enthalpy variations in the three-step processes.
Structure $\Delta H_{1}$ (eV/atom) $\Delta H_{2}$ (eV/atom) $\Delta H_{3}$ (eV/atom)
CuGaSe$_{2}$ $-0.090$ $-0.005$ $-0.004$
Zn$_{2}$AsBr 0.155 $-0.009$ $-0.138$
Structure $H_{\rm m} = \Delta H_{1} + \Delta H_{2} + \Delta H_{3} $ (eV/atom)
CuGaSe$_{2}$ $-0.099$
Zn$_{2}$AsBr 0.008
Step 3. We fully relax the structures obtained from step 2. This process reveals the mixing enthalpy differences caused by the changes of bond lengths, denoted as $\Delta H_{3}$. In this step, the Coulomb interaction and the strain between ions occur simultaneously, attempting to find the balanced structure with the lowest energy. As shown in Fig. 3(a), for the chalcopyrite compounds, according to the crystal symmetry, the Coulombic forces applied by the two nearest trivalent ions (group-III/V ions carrying charge $q_{3}$) on the bivalent ion (group-II/VI ions carrying charge $q_{2}$) is along the $C_{2}$ principal rotation axis; the resultant force of the two nearest monovalent ions (group-I/VII ions carrying charge $q_{1}$) on the central bivalent ion is also along the $C_{2}$ axis but in opposite direction. When the bivalent ion is fixed at the tetrahedral center in step 2, i.e., the symmetric position of the zinc-blende structure, the magnitude of the Coulombic attraction between $q_{3}$ and $q_{2}$ ($F_{32}$) is greater than that ($F_{12}$) between $q_{1}$ and $q_{2}$, i.e., $| F_{32}| > |F_{12}| $, because the charge on the trivalent ion $q_{3}$ is larger than $q_{1}$ ($| q_{3}| > | q_{1}| $, see Table 1). This behavior is the same for mix-cation and mix-anion compounds because the Coulomb interaction depends on the product of the charges.
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Fig. 4. Schematic plots of the influence of Coulombic attractions and ion size on the stability of ternary chalcopyrite compounds (mix-cation AB$X_{2}$ or mix-anion C$_{2}ZY$). (a) The resultant Coulomb force $F_{32}$ ($F_{12}$) acting on the valence II ion with $q_{2}$ charge by the two nearest neighbor ions with $q_{3}$ ($q_{1}$) charges, with its magnitude determined by the charge state and the distance between the ions. (b) For a typical mix-cation ternary chalcopyrite compound, the nominal charge states of cations on both sides of the central anion are $+1$ and $+3$, where the cation with higher valence usually has a smaller ionic radius, and a shorter bond length. (c) For a typical mix-anion ternary chalcopyrite compound, the anion with higher absolute valence has a larger ionic radius and usually a longer bond length. The imbalance of the Coulombic attraction and size in the mixed-anion system will eventually force the bond lengths to be further away from the ideal values, leading to the instability.
In the case of a mix-cation structure, i.e., CuGaSe$_{2}$, the ionic radius of Cu$^{+}$ is greater than Ga$^{3+}$ because more electrons are removed from Ga, but the atomic/covalent radius of Cu is smaller than Ga. The system is actually partially covalent and partially ionic, consequently, the Cu–Se bond length is only slightly longer than the Ga–Se bond, consistent with that observed in their corresponding binary compounds, as shown in Table 2. Therefore, for CuGaSe$_{2}$, the movement of divalent ions towards their neighboring trivalent ions caused by the Coulombic attraction in step 3 also reduces the strain energy, that is, resulting in decreases in both Coulomb energy and strain energy simultaneously during this process, as shown in Fig. 4(b). Because for CuGaSe$_{2}$, the bond lengths change a little during the relaxation in step 3, $\Delta H_{3}$ (CuGaSe$_{2}$) $\sim$ $-0.004$ eV/atom has a relatively small negative value, where both the changes of Coulomb energy and of strain energy are negative. On the contrary, in the case of mix-anion structure, i.e., Zn$_{2}$AsBr, Coulomb energy and strain energy cannot be simultaneously optimized. This is because As$^{3-}$ ion is much larger than Br$^{-}$ ion since more electrons are added to As, therefore the “ideal” Zn–As bond is much longer than the Zn–Br bond, as shown in Table 2. In step 3, the Coulombic interaction brings the divalent ions closer to the trivalent ions, i.e., the bond length of Zn–As and that of Zn–Br both deviate further from the “ideal” bond lengths of Zn–As bond and Zn–Br bond, leading to an additional increase in the strain energy. This contradiction between strain and Coulombic interactions is, therefore, the origin of instability of mix-anion compounds, i.e., the stability of the Zn$_{2}$AsBr structure is worse than that of CuGaSe$_{2}$. The above analysis shows that besides the Coulombic interaction, the trend of strain energy due to the different sizes of the mixed ions is critical in determining the stability of the ternary compounds. The mix-anion and mix-cation compounds show different trends in their relationship between the size and absolute ionic charge state, so they show very different stability.
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Fig. 5. Trends of ionic radius of elements in third, fourth, and fifth periods.
For elements within the same period of the Periodic Table, cations with higher valences possess smaller ionic radii than those with lower valences because they lose more electrons (Fig. 5), whereas anions with higher absolute valences have larger ionic radii than those with lower absolute valences because more electrons are added to the anions. For isovalent elements in the same column of the Periodic Table, atomic size usually increases with the atomic number. In a ternary crystal such as the chalcopyrites, the Coulombic interaction always tends to pull the ion with $q_{2}$ charge closer to that with $q_{3}$ charge. In a mix-cation chalcopyrite, the trend of ionic sizes coincides with the Coulomb interaction that pulls $q_{2}$ ion closer to $q_{3}$ ion. Consequently, the bond lengths of the ternary compounds are close to the ideal bond lengths, i.e., the strain energy is small. Therefore, the mixing enthalpy is usually negative (stable) due to the Coulomb interaction for the mix-cation compounds. On the contrary, in a mix-anion structure, the $q_{2}$ ion also tends to move toward the $q_{3}$ ion due to the large Coulomb interaction. However, since $q_{1}$ ion is relatively smaller than $q_{3}$ ion, the shift of the $q_{2}$ ion towards the $q_{3}$ ion will cause the distance between $q_{1}$ and $q_{2}$ ions to be much longer than that in the binary compounds, and the distance between $q_{3}$ and $q_{2}$ ions to be shorter (see Tables 1 and 2). This leads to an obvious increase of the strain energy. Therefore, mix-anion ternary chalcopyrites have relatively higher strain energies, thus higher mixing enthalpy, deteriorating their thermodynamic stability. Our above analysis suggests that, for ternary chalcopyrite structures, the smaller (larger) of the trivalent (monovalent) anion atomic number, the stabler of the compounds. We validated this by calculating the bond lengths and formation enthalpies of Zn$_{2}X^{\rm VA}X^{\rm VIIA}$ ($X^{\rm VA} = {\rm P}$, As, or Sb; $X^{\rm VIIA} = {\rm Cl}$, Br, or I) compounds. The results are shown in Fig. 6, confirming our expectations.
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Fig. 6. Comparison of the bond lengths in ternary chalcopyrite crystals versus those in binary compounds, where $r_{12}$ is the bond length of monovalent ion and divalent ion, and $r_{32}$ represents that of trivalent ion and divalent ion. The mixing enthalpies ($H_{\rm m}$) indicate the stabilities of ternary structures. Stable binary phases have their bond lengths (marked by blue lines) always fall between the sum of ionic radii (yellow dashed lines) and that of covalent bond radii (green dashed lines). The more its bond length (marked by $\times$ shape) of the ternary compound exceeds this range between the dotted lines, the higher its mixing enthalpy is, and the structure becomes less thermodynamically stable.
From the reasoning mentioned above, to achieve more stable mix-anion chalcopyrite crystals, one can replace the ($-3$) valence anion with smaller anions and the ($-1$) valence anion with larger ones. To further verify this inference, a series of ternary chalcopyrite structural compounds are constructed and their mixing enthalpies are calculated, as shown in Fig. 7. More detailed results are listed in the Supplementary Materials. We find that the ratio of ionic radii ($r_{3}/r_{1}$) can be employed as a descriptor for predicting stability, where $r_{3}$ represents the ionic radii of trivalent ions; $r_1$ represents the ionic radii of monovalent ions. This guideline simplifies the stability estimation for many materials and serves as a helpful reference for future high-throughput calculations. Figure 7 shows the trend of the ratio ($r_{3}/r_{1}$) versus the mixing enthalpy. The scattered data points can be well captured by a linear fit, with an interception line dividing the diagram into two regions: (i) a stable region with ($r_{3}/r_{1}$) smaller than 1.12, and (ii) an unstable region with ($r_{3}/r_{1}$) larger than 1.12. Figure 7 confirms our conclusion that reducing the radii of cations or anions with higher absolute valences will stabilize the crystal structures. This understanding enables us to design mix-anion compounds: choosing smaller trivalent anion and larger monovalent anion will construct a more stable mix anion structure.
cpl-41-5-056101-fig7.png
Fig. 7. Mixing enthalpy of ternary chalcopyrite compounds versus the ratio of ionic radii of transmuted ions ($r_{3}/r_{1}$). The constructed ternary compounds include Zn$_{2}$N$X^{\rm VIIA}$, Zn$_{2}$As$X^{\rm VIIA}$, Cd$_{2}$As$X^{\rm VIIA}$ as mix-anion chalcopyrite structures, and Cu$X^{\rm IIIA}$Se$_{2}$, Ag$X^{\rm IIIA}$Se$_{2}$ as mix-cation chalcopyrite structures, where $X^{\rm VIIA} = {\rm Cl}$, Br, or I; $X^{\rm IIIA} ={\rm Al}$, Ga, or In. The red line is the linear fit of the scattered points.
In summary, we have revealed that the stability of ternary chalcopyrite compounds is determined by whether the strain energy and Coulomb energy can be simultaneously optimized in the structure, and explained the puzzling experimental observation that mixed-cation compounds are more stable than the mixed-anion compounds. A simple descriptor of stability is the relative ratio of ionic radii of the transmuted ions. When ions with higher absolute valences have smaller ionic radii than those of ions with lower absolute valences, the structures are more stable. Increasing the size of the high-valence ion will result in higher strain energy, which is the major cause of the instability of such crystals. Given that most cations with high valences possess smaller ionic radii than those with lower valences, mix-cation chalcopyrite crystals are more likely to be stable. On the contrary, mix-anion chalcopyrite crystals often have oversized high absolute valence anions; thus, they are more likely being unstable. A few exceptions exist, such as Zn$_{2}$AsI, Zn$_{2}$PI, Zn$_{2}$PBr, Zn$_{2}$NCl, Zn$_{2}$NBr, and Zn$_{2}$NI, where group-VII ions are larger than the group-V ions, leading to their relatively better stabilities. This work elucidates the stability mechanisms of ternary chalcopyrite crystals. The understanding constitutes the basis for designing novel ternary or multinary materials for various applications. Acknowledgments. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11991060, 12088101, 52172136, 12104035, and U2230402). We acknowledge computational resources from the Beijing Computational Science Research Center. References The prediction of semiconducting properties in inorganic compoundsSuper-Cell Structure of SemiconductorsA systematic method of deriving new semiconducting compounds by structural analogyMixed-cation perovskite solar cells in spaceUltrafast Carrier Dynamics and Terahertz Photoconductivity of Mixed-Cation and Lead Mixed-Halide Hybrid PerovskitesSolar cell efficiency tables (version 59)Increasing markets and decreasing package weight for high-specific-power photovoltaicsCd-Free Cu(In,Ga)(Se,S)2 Thin-Film Solar Cell With Record Efficiency of 23.35%Funnel-shaped electronic structure and enhanced thermoelectric performance in ultralight C x ( BN ) 1 − x biphenylene networksTunable anharmonicity versus high-performance thermoelectrics and permeation in multilayer (GaN)1–x (ZnO)xHigh‐Performance Pseudocubic Thermoelectric Materials from Non‐cubic Chalcopyrite CompoundsBand structure and stability of zinc-blende-based semiconductor polytypesTransmission electron microscopy investigation and first-principles calculation of the phase stability in epitaxial CuInS2 and CuGaSe2 filmsEnhanced Thermoelectric Properties of Cux Se (1.75≤ x ≤2.10) during Phase TransitionsUnveiling disparities and promises of Cu and Ag chalcopyrites for thermoelectricsSilver Atom Off-Centering in Diamondoid Solid Solutions Causes Crystallographic Distortion and Suppresses Lattice Thermal ConductivityComplex thermoelectric materialsTheoretical investigation of the structural stabilities, optoelectronic and thermoelectric properties of ternary alloys NaInY2 (Y = S, Se and Te) through modified Becke–Johnson exchange potentialElectronic band structure of ordered vacancy defect chalcopyrite compounds with formula II − III 2 − VI 4 Band gap energies of bulk, thin-film, and epitaxial layers of CuInSe2 and CuGaSe2Band offsets at the CdS/CuInSe2 heterojunctionElectronic structure of the ternary chalcopyrite semiconductors CuAl S 2 , CuGa S 2 , CuIn S 2 , CuAl Se 2 , CuGa Se 2 , and CuIn Se 2 Crystal phase-controlled synthesis of rod-shaped AgInTe2 nanocrystals for in vivo imaging in the near-infrared wavelength regionThermodynamic and elastic properties of AgInSe2 and AgInTe2Fabrication and properties of AgInTe2 thin filmsSynthesis and single-crystal structure determination of the zinc nitride halides Zn2NX (X=Cl, Br, I)Low-Temperature Aqueous Na-Ion Batteries: Strategies and Challenges of Electrolyte DesignGiant and composition-dependent optical band gap bowing in dilute GaSb1−xNx alloysNature of the band gap of halide perovskites ABX3 ( A = CH3 NH3 , Cs; B = Sn, Pb; X = Cl, Br, I): First-principles calculationsProjector augmented-wave methodAb-initio simulations of materials using VASP: Density-functional theory and beyondGeneralized Gradient Approximation Made SimpleErratum: “Hybrid functionals based on a screened Coulomb potential” [J. Chem. Phys. 118, 8207 (2003)]Anomalous Alloy Properties in Mixed Halide Perovskites
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