In the co-substitution approach,^[9] a new compound is derived from the parent compound by replacing one type of elements by pairs of elements from other groups in the Periodic Table while keeping the number of valence electrons per atom unchanged (e.g., replacing Zn by Cu and In). The mechanism for stabilizing a compound in co-substitution is analogous to the co-doping^[54–56] physics involving pairs of donor and acceptor defects. The electrons transferring from donor to acceptor levels, and the donor-acceptor electrostatic attraction along with the atomic relaxations upon donor–acceptor pairing, are the main factors that stabilize the co-doping system. Co-substitution can be viewed as a 100% co-doping. In addition to the above energy saving factors, the ordering energy gained by forming ordered pattern in 100% co-doping that reduces strain energy, further stabilizes the co-substitution system. Co-substitution has been widely used in experimental and theoretical studies, leading to the discovery of many new stable compounds.^[11,57–63] Several mutated types of co-substitutions were also used in materials design, e.g., co-substitution with ordered vacancies^{[10,18,64–67]} and co-substitution with filled interstitials.^{[17,19,68–74]}

The substitution along with the formation of ordered vacancies in diamond-like structures was discussed in 1960 as a super-cell structure of semiconductors^[10] with 1 where n_i is the number of atoms of the ith kind (vacancies are considered as a kind of atoms) and v_i is its number of valence electrons. In this type of co-substitution one of the substituents was assigned to have zero valence electron and is represented by a vacancy. The stability of ordered vacancy compounds is well understood from defect physics, as clearly demonstrated in Refs. [64] and [65] for CuInSe₂, showing that the cluster of two Cu vacancies plus one In on Cu defect can have a large energy gain (3.45 eV) of clustering, much larger than the formation energies of individual defects. Therefore, a series of Cu–In–Se compounds, Cu_(n−3m)In_(n+m)Se_2n (m = 1, 2, 3, . . .; n = 3, 4, 5, . . .), can form with different amounts of such defect clusters in CuInSe₂, such as CuIn₃Se₅, CuIn₅Se₈, Cu₂In₄Se₇, etc. It is not surprising from the chemical formulae point of view for the existence of these compounds, since their formulae are all combinations of Cu₂Se and In₂Se₃. However, it is surprising and interesting that they all have CuInSe₂-like structures with ordered defect clusters. It is also calibrated theoretically that, by forming ordered patterns, the ordered defective compounds gain an ordering energy of about 0.8 eV per cluster in the Cu–In–Se system.^[64]

According to Eq. (1), the vacancy also has four valence electrons assigned to it in the diamond-like structures, which contribute to the formation of lone pair electrons of the four nearby anions pointing towards the vacancy as illustrated in the ordered vacancy compound CdIn₂Se₄^[66] that can be derived from CdSe by forming defect cluster of one Cd vacancy plus two In on Cd defects per four CdSe formulae. Other ordered vacancy compounds such as Ga₂Se₃^[67] (one Cd vacancy plus two Ga on Cd defects in 3 CdSe) and Sc₂S₃^[18] (one Ca vacancy plus two Sc on Ca defects in 3 CaS) were also studied previously.

There are plenty of crystal structures in nature that can be derived from other structures by filling some of their interstitial sites.^[68–74] Filling the interstitial sites in semiconductors usually leads to metallic phases, as the interstitial defects are usually donors. If combining acceptor-like substitutions (e.g., Sn on Sb) with filled interstitials to compensate the variation of valence electrons, one can design semiconductors stabilized by electron count even in complicated structures, such as the LaCo₄Sb₉Sn₃ filled Skutterudite semiconductor derived from the CoSb₃ Skutterudite semiconductor by filling La with three valence electrons on the interstitial site and substituting three Sb by three Sn atoms losing three valence electrons.^[69]

The co-substitution approach along with filled interstitials has been extensively applied to diamond-like structures, leading to the discovery of many semiconducting compounds.^[71–74] The diamond structure is one of most loosely packed three-dimensionally bonded structures in nature. In the unit cell of diamond structure, there are four fcc sub-lattices (see Fig. 1) and two of them are filled by atoms with the other two being empty; filling one of the empty fcc sub-lattice leads to the filled tetrahedral structure^[17] also called half-Heusler structure (filling the two empty fcc sub-lattice leads to Heusler structure). This type of filled interstitial semiconductors was first discovered by Nowotny et al.^[72] and Juza et al.^[71] It is found that the filled interstitial could help to move some electronic states away from the band edges leading to direct band gaps, such as for indirect gap GaP, filling Li at the interstitial site and substituting Ga by Zn for valence electron compensation leading to a direct-gap semiconductor LiZnP.^[17] Theoretical calculations^[19] show that the filled interstitial structure has both strongly ionic bonds between anion and the interstitial (such as P–Li) and strongly covalent bonds between anion and the substituted cation (such P–Zn). Analogous to ordered vacancy compounds, filled interstitial compounds can appear in different stoichiometry’s depending on the concentration of interstitial-substitution defect complexes. For example, there are ABX-type (e.g., LiZnP),^[17] A₃BX₂-type (e.g., Li₃AlN₂),^[74] and A₃X-type (e.g., Li₃Sb)^[73] filled interstitial structures derived from diamond-like structure with eight valence electrons per anion. Note that the A₃X-type filled interstitial structures have a Heusler-like structure.^[73]

The half-Heusler compounds (e.g., LiZnSb) can be derived from the Heusler-like A₃X-type compounds (e.g., Li₃Sb) by co-substitution with ordered vacancies (e.g., substituting one Li by Zn and removing another Li), which can also be viewed as filled interstitial structures derived from zinc blende compounds as discussed above. In the next section, we will discuss the focusing study of new half-Heusler compounds.

Prediction of the lowest-energy structure of a multinary compound can be solved successfully based on, e.g., the global space group optimization (GSGO) method^[75] that starts from randomly selected lattice vectors and random atomic positions within a cell as input for a sequence of ab initio calculations of total energy of locally relaxed trial structures to search for a global minimum via an genetic-algorithm selection. The GSGO method is successfully applied in predicting the lowest-energy structures of ABX compounds (e.g., NaBaAs^[76]). Analogous methods including simulated annealing,^[77] evolutionary algorithm,^[78,79] random search,^[80] and particle-swarm optimization,^[81] are also successful for a number of focused cases of structure prediction.^[78–81] However, these coordination-based optimization methods^[75,78–81] typically require significant computational resources to solve the lowest-energy structure of a single compound and is not tractable for high-throughput predictions of new materials.

Alternatively, one can use prior knowledge of the crystal structures of the existing compounds by constructing a tensor T(c_i, e_j)^[82,83] for the known structures of given compositions (c_i) and sets of elemental constituents (e_j). T(c_i, e_j) = “null” if for the composition c_i, e.g., ABX, and the set of elemental constituents e_j, e.g., NaZnN, the compound has not been reported yet. One can then construct a probability density P(T(c_i, e_j)) based on the tensor T(c_i, e_j), considering (i) compounds in the same composition c_i (e.g., NaZnN derived from LiZnN, NaZnP, NaZnAs, or NaZnSb, each derivation suggesting a structure), and (ii) compounds that the composition c_i can be derived from using e.g., the substitution approaches discussed in Section 2 (e.g., LiMgSb derived from GaSb based on co-substitution with filled interstitials). The probability density P(T(c_i, e_j)) can be used to suggest the possible crystal structures of unknown compounds. This methodology has been successfully applied in predicting the lowest-energy structures of a few hundred of new ABX compounds,^[76,84] finding dozens of cubic half-Heusler structures as well as hundreds of non-cubic structures.

The next step after structural determination for predicting the new compound is the evaluation of its stability relative to its competing phases by comparing their total energies or formation enthalpies. However, the standard approximations to density functional theory (DFT),^[85,86] namely, the local density approximation (LDA)^[87,88] or the generalized gradient approximation (GGA)^[89] have to some extent numeric errors for predicting compounds’ formation enthalpies, especially when the compounds and their competing phases belong to different chemical classes (e.g., metal or non-metal). In Ref. [90], a computationally inexpensive theoretical approach based on GGA+U calculations with fitted elemental-phase reference energies (FEREs) that can be used for accurate predictions of the ΔH_f values of binary and multinary solid compounds involving chemical bonding between metals and nonmetals (such as pnictides, chalcogenides, halides). A set of FERE values^[90] were obtained using an extensive set of 252 binary compounds with measured heat of formations, showing that the 252 formation enthalpies can be reproduced with the mean absolute error of 0.054 eV/atom after fitting instead of 0.250 eV/atom before fitting (see Fig. 3). Therefore, by providing accurate ΔH_f, the FERE approach^[90] can be applied for accurate predictions of the compound thermodynamic stability with respect to decomposition into the competing phases.

Fig. 3.

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	Fig. 3. (color online) Histogram showing absolute errors of the GGA (upper part) and of the FERE approach (lower part) in reproducing measured enthalpies of formation for 45 binary pnictides and chalcogenides containing 3d transition metals. Dashed lines represent the mean absolute error of the two methods corresponding to the full set of 252 binary compounds (Figure adapted from Ref. [90]).

In Refs. [76] and [84] on the prediction of new ABX half-Heusler compounds, the thermodynamic stability analysis is performed in the elemental chemical potentials space using the information of compounds’ formation enthalpies (ΔH_f). A new compound (e.g., ABX) is considered stable under the thermodynamic equilibrium conditions if there are values of the elemental chemical potentials (e.g., Δμ_I, I = A, B, X) that do not favor the formation of elemental (thus Δμ_I < 0, I = A, B, X), binary (e.g., A_lB_m thus lΔμA + m Δ μ_B < ΔH_f(A_lB_m)), ternary (e.g., A_lB_mX_n thus lΔμA + mΔμ_B + nΔμX < ΔH_f(A_lB_mXn)) competing phases, but favor the formation of the target compound under thermodynamic equilibrium condition (e.g., for ABX, Δμ_A + ΔμB + Δμ_X = ΔH_f(ABX)). The area in the elemental chemical potentials space that includes all the ΔμI values favoring the target compound but not competing phases can then be identified as the stability region. Figure 4 shows the calculated stability regions (in yellow)^[91] of a few recently predicted ABX compounds.^[84]

Fig. 4.

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	Fig. 4. (color online) The allowed chemical potential regions (yellow) in the chemical potential space of three cubic semiconducting ABX compounds, indicating also some of the phases competing with the stability of the ABX structure (Figure adapted from Ref. [91]).

Following the above two steps of materials prediction (structure determination and thermodynamic stability analysis), 235 new ABX compounds with eight valence electrons (e.g., LiMgSb)^[76] and 54 new ABX with 18 valence electrons (e.g., TaIrGe, see Fig. 5)^[84] were predicted to be thermodynamically stable in their predicted lowest-energy structures. The compounds that already existed in ICSD^[92] were in indicated by check marks in Fig. 5 and analogous figures in Refs. [76] and [84], The chemical trends concerning the appearance of cubic half-Heusler structure versus non-cubic structures as lowest-energy structures for each new or previously known ABX compounds were analyzed based on chemical bonding and elemental identity, suggesting that the half-Heusler structure is preferred mainly for compounds with covalent bonding features.^[84] Dozens of the predicted ABX compounds^[76,84] were realized in experiments,^{[32,35,84,93,94]} confirming the theoretically predicted crystal structures and materials stability. Interesting physical properties and functionalities have been found in the new ABX compounds as discussed below.

Fig. 5.

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Fig. 5. (color online) ABX compounds with 18-valence electrons. Elements A are in green, B in blue and X in red. Check mark, previously reported compounds; plus sign, unreported and predicted here to be stable; minus sign, unreported and predicted here to be unstable. Green background indicates the predicted p-type transparent conductor; red background indicates the predicted topological semimetals.[84]

3.3.1. Topological semimetals

A number of ABX half-Heusler materials were found to be topological semimetals^[26–29] analogous to HgTe including ScPtBi (with 18 valence electrons) that was a hypothetical half-Heusler structure considered in Ref. [26]. It is confirmed in Ref. [84] that ScPtBi is indeed thermodynamically stable with a cubic half-Heusler structure. Furthermore, two new topological semimetals (HfIrAs and HfIrBi) were predicted in Ref. [84] (see Fig. 6). The interesting trend in the HfIr(As, Sb, Bi) group of compounds with HfIrAs and HfIrBi being topological semimetal while HfIrSb in the middle being normal insulator^[84] is related to the relatively high position of the Sb-6s atomic level compared to the As-5s level and Bi-7s level.

Fig. 6.

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	Fig. 6. (color online) Topological band characters of predicted ABX semiconductors and semimetals (Figure adapted from Ref. [84]).

3.3.3. Thermoelectric properties

The existing 18-electron ABX compounds and their alloys are widely used in thermoelectric devices.^[21–25]

Many of the predicted ABX compounds could be candidate thermoelectric materials, e.g., high thermopower (−153.9 mV·K⁻¹) and power factor (5.2 μW·cm⁻¹·K⁻²) was found for the predicted and experimentally confirmed compound ZrNiPb,^[84] whose neighboring compound ZrNiSn has already been widely studied for thermoelectric applications^[21–25] (Figure 7 shows the calculated defect properties of ZrNiSn^[91]).

Fig. 7.

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Fig. 7. (color online) Formation energy of the dominant defects versus parametric Fermi level in ZrNiSn from hybrid functional calculations. Main panel: positively charged electron–donor defects shown in red and negatively charged electron–acceptor defects shown in blue; their charge states denoted in numbers and their charge transition levels shown in open circles. The Fermi level, VBM and CBM variables are denoted by EF, EV, and EC, respectively. The inset shows the allowed chemical stability region, whereas the red circle shows the specific chemical condition used for the current formation energy versus EF calculations. The small vertical arrows indicate the position of the equilibrium Fermi level at growth condition of 850 °C (Figure adapted from Ref. [91]).

3.3.5. Flexible three-dimensional inorganic semiconductors

In the MgSrSi-type crystal structure (Pnma), there are one-dimensional (1D) Mg–Si ribbons embedded in the three-dimensional (3D) matrix of Sr.

Figure 8 shows the predicted MgSrSi-type structure of RbCuTe with a semiconducting band gap suitable for PV absorbers,^[76] illustrating clearly the 1D Cu–Te ribbons in the 3D Rb matrix, which was confirmed in subsequent synthesis and structure characterization.^[35] RbCuTe was then found to be a flexible 3D inorganic semiconductor that can tolerant up to 30% strain with impunity due to the concerted rotation of the 1D Cu–Te ribbons in the inter-dimensional hybrid structure (IDHS). Theoretical design of this type of IDHSs in 3D inorganic compounds could suggest novel materials and new techniques for the flexible electronics technology.

Fig. 8.

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	Fig. 8. (color online) Crystal structure of RbCuTe[35] in the MgSrSi-type structure. Green sphere: Rb. Red sphere: Cu. Blue sphere: Te.