First-Principles Calculations of Thermoelectric Transport Properties of Quaternary and Ternary Bulk Chalcogenide Crystals

Abstract

Chalcogenide crystals have a wide range of applications, especially as thermoelectric materials for energy conversion. Thermoelectric materials can be used to generate an electric current from a temperature gradient based on the Seebeck effect and based on the Peltier effect, and they can be used in cooling applications. Using first-principles calculations and semiclassical Boltzmann theory, we have computed the Seebeck coefficient, electrical conductivity, electronic thermal conductivity, power factor, and figure of merit of 30 chalcogenide crystals. A Quantum Espresso package is used to calculate the electronic properties and locate the Fermi level. The transport properties are then calculated using the BoltzTraP code. The 30 crystals are divided into two groups. The first group has four crystals with quaternary composition (A₂BCQ₄) (A = Tl; B = Cd, Hg; C = Si, Ge, Sn; Q = S, Se, Te). The second group contains 26 crystals with the ternary composition (A’B’Q₂) (A’ = Ag, Cu, Au, Na; B’ = B, Al, Ga, In; Q = S, Se, Te). Among these 30 chalcogenide crystals, the results for 11 crystals: Tl₂CdGeSe₄, Tl₂CdSnSe₄, Tl₂HgSiSe₄, Tl₂HgSnS₄, AuBSe₂, AuBTe₂, AuAlTe₂, AuGaTe₂, AuInTe₂, AgAlSe₂, and AgAlTe₂ are revealed for the first time. In addition, temperature-dependent transport properties of pure and doped AgSbSe₂ and AgSbTe₂ crystals with dopant compositions of AgSb_0.94Cd_0.06Te₂ and AgSbTe_1.85Se_0.15 were explored. These results provide an excellent database for bulk chalcogenides crucial for a wide range of potential applications in renewable energy fields.

1. Introduction

In recent years, searching for new alternative renewable energy resources has become an essential endeavor due to severe environmental concerns and the concomitant need to reduce the use of fossil fuels [1,2,3]. This is especially true for materials that can convert energy that would otherwise go to waste into useful work. The direct conversion between heat and electricity is crucial for many applications in power generation [4,5,6,7]. In this regard, thermoelectric (TE) materials have attracted a great deal of attention because of their ability to convert the waste heat from industrial processes into usable electricity [8]. The TE effect was first noticed by Seebeck in 1821 and was coined as the Peltier effect when used to create a temperature gradient from a supplied current [9]. This method of energy generation was used in spacecraft beginning in the early 1960s [10,11]. Since then, exploring high-efficiency thermoelectric materials for clean and renewable energy production has become one of the top priorities in materials research and development. Even though many TE devices have low efficiency, they still have been widely used in many applications for energy harvesting. They are expected to have broad and diverse applications to the recovery of waste heat for useful work, as illustrated, for example, by the wristwatch design of the Seiko and Citizen companies [12,13].

The overall performance of a TE material is measured by a dimensionless figure of merit parameter ZT, given by $ZT=S^2\sigma T/\left({\kappa }_{lat}+{\kappa }_{ele}\right)$, where S, σ, T, κ_lat, and κ_ele are the Seebeck coefficient, electrical conductivity, absolute temperature, lattice thermal conductivity, and electronic thermal conductivity, respectively. σ is given by $\sigma =nem$, where n is the carrier concentration, e is the electron charge, and m is the electron mobility. The $\sigma$ and κ_ele are related by ${\kappa }_{ele}=L_0\sigma T$, where L₀ is the Lorenz number [14]. $S^2\sigma$ is called power factor (PF). For excellent TE materials, ZT should be larger than unity [15,16]. Many efforts have been dedicated to increase ZT by new innovative approaches such as using nanowires, nano-structured materials, band structure engineering, lattice anharmonicity, and two-dimensional materials technology [17,18,19,20,21,22].

Various materials are used for thermoelectricity over a wide range of temperatures, including polymeric and inorganic materials. However, inorganic semiconductors have been found to be the most efficient [23,24]. Among the inorganic materials, chalcogenide compounds have attracted a great deal of attention. They generally have high electrical conductivity and low thermal conductivity resulting in higher ZT values [25]. Chalcogenide compounds contain at least one of the chalcogen elements (S, Se, Te) and one or more electropositive, in a few cases, electronegative elements. The electropositive elements are mostly from group IB (Cu, Ag), IIB (Zn, Cd, Hg), IVA (Si, Ge, Sn), IIIA (In, Tl), IVB (Zr, Hf), IIA (Mg, Ba), and IA (Li, K, Cs) in the periodic table. In some cases, lanthanide elements such as La and Lu can also be involved. This diverse compositional space makes chalcogenide compounds a unique class of materials rarely seen in other materials such as semiconductors, large gap insulators, superconductors, silicates glasses, metallic alloys, etc. Many quaternary chalcogenide crystals, such as A^I–B^III–C^IV–X^VI system (A^I = Cu, Ag; B^III = Ga, In; C^IV = Si, Ge, Sn; X^VI = S, Se, Te), and A^I₂B^IIC^IVQ₄ system (A^I = Cu, Ag; B^II = Mg, Mn, Fe, Zn, Cd, and Hg; C^IV = Si, Ge, Sn; and Q = S, Se, Te) are very popular due to their compositional flexibility and functional turnability ideal for TE applications [26,27,28,29,30]. They are also extensively used in optical [31] and nonlinear optical devices in the visible-infrared region, photovoltaic cells [32,33,34,35,36,37,38], solar energy converters [39,40,41,42], and magnetic applications [43]. Another group of chalcogenide crystals in the form of Ag₂XYQ₄ (X = Ba, Sr; Y = Ge, Sn; Q = S, Se) are also very attractive [44].

Several experimental and theoretical pieces of research have been carried out recently to enhance the thermoelectric performance and increase the ZT of some chalcogenide crystals. C. Wang et al. [45] has succeeded recently in achieving ZT higher than one for Ag-doped crystalline CuInTe₂, where ZT was increased for Cu_1−xAg_xInTe₂ from 0.6 (at x = 0) to 1.4 (at x = 0.25). A very high enhanced figure of merit (ZT ≈ 2.6 at 573 K) was obtained in Cd-doped polycrystalline AgSbTe₂ [46]. Another study [47] showed that ZT increased to maximum value (ZT ≈ 1.35 at 600 K) for AgSbTe_2−xSe_x (at x = 0.02). S. Deng et al. [48] showed that the TE performance could be enhanced for Ga-doped CuInTe₂ crystal (CuIn_1−xGa_xTe₂), and ZT can be increased to the maximum value (ZT = 0.8 at 773 K) for the CuIn_0.8Ga_0.2Te₂ sample due to the lower thermal conductivity. Y. Zhong et al. [49] found that Ag vacancy and In dopant substitution in the AgGaTe₂ system can lower the lattice thermal conductivity (κ_lat) significantly, which can produce a higher value of ZT (ZT ≈ 1.44).

SnTe is another type of chalcogenide crystal that has been under extensive experimental and theoretical studies for a long time. Recently, A. Rifqi et al. [50] succeeded in enhancing the TE performance of Ti-doped SnTe crystal (Sn_1−xTi_xTe) and Zr-doped SnTe crystal (Sn_1−xZr_xTe). They increased ZT from 0.41 to 0.51 at 700 K for Sn_0.97Ti_0.03Te(x = 0.03), and from 0.45 to 0.55 at 700 K for Sn_0.98Zr_0.02Te (x = 0.02). New experimental techniques have been used recently to achieve higher ZT for some chalcogenides. For example, by using new mechanical alloying combined with microwave-assisted synthesis for the synthesis of single-phase cubic isocubanite CuFe₂S₃, a maximum thermoelectric figure of merit, ZT_max ≈ 0.14 at 673 K, was achieved for CuFe₂S₃ crystal [51]. Thermoelectric transport properties of another important group of chalcogenides, Tl₂PbXY₄ (X = Zr, Hf and Y = S, Se), have been reported recently by S. Azam et al. [52]. According to this study, these four crystals possess high TE performance and high ZT (ZT = 0.85, 0.71, 0.725, and 0.68 at 800 K for Tl₂PbHfS₄, Tl₂PbHfSe₄, Tl₂PbZrS₄, and Tl₂PbZrSe₄ crystals respectively).

The crystal structure, electronic structure, and the optical properties of quaternary chalcogenides Tl₂B^IIC^IVQ₄ (B^II = Cd, Hg; C^IV = Si, Ge, Sn; Q = S, Se, Te) have been reported recently [53,54,55,56]. They are semiconductors with potential TE applications but have not yet been sufficiently explored so far. Bagci et al. [57] reported the chemical-potential-dependent transport properties of the chalcopyrite crystals CuBQ₂ (Q = S, Se, Te). However, they did not include the temperature dependency of the transport properties. Although the TE properties of ADQ₂ (A = Cu, Ag; D = Ga, In; Q = Se, Te) chalcopyrite were studied extensively [58,59,60,61,62,63,64], much less research has been conducted on AuBQ₂ and AgAlQ₂ (Q = S, Se, Te). On the other hand, the electronic structure and optical properties of AuBQ₂ and AgAlQ₂ were reported recently [65,66] without the TE transport properties. The TE properties of the ternary chalcogenide crystals AgSbSe₂ and AgSbTe₂ were studied sixty years ago in the 1950s [67]. These two crystals belong to the I-V-VI₂ family, where I = Cu or Ag, V = Sb or Bi, and VI = S, Se, and Te. They were considered promising p-type TE materials because of their low thermal conductivity [68,69,70]. Both of these crystals can crystallize in a rock salt crystal structure (space group 225, $Fm\overline{3}m$) with disordered Ag and Sb atoms [71]. Another study [72] revealed that they are semiconductors with a very narrow energy band gap (E_g ≈ 0.03 eV) or that they are semi-metallic. TE performance for AgSbSe₂ and AgSbTe₂ can be significantly enhanced (e.g., ZT is greater than 1) when properly doped by Pb, Bi, Cd, Sn, Se, or Ce dopants [46,47,73,74,75,76].

Motivated by the realization of the unique properties of the above crystals, we investigate the TE transport properties of 30 chalcogenide crystals as listed in Table S1 in the Supplementary Information (SI) by using density functional theory and semiclassical Boltzmann theory. The results for these 30 crystals are presented in Section 3.1. These 30 crystals are divided into two main groups: crystals 1–4 with stoichiometry A₂BCQ₄ (A = Tl; B = Cd, Hg; C = Si, Ge, Sn; Q = S, Se, Te) (colored white in Table S1 and

Table 1. The highest values of ZT and the values of the carrier concentrations (n) at which these values of ZT occur, highest PF, and highest κ_ele/τ for the 30 chalcogenide crystals.

#	Crystal	n (e−/cm3)	Highest ZT	Highest PF (mW/cm.K2)	κele (W/m.K) at 900 K
1	Tl2CdGeSe4	1020	0.966(750 K)	0.634(800 K)	0.60
2	Tl2CdSnSe4	1019	0.947(750 K)	0.415(800 K)	0.408
3	Tl2HgSiSe4	−1020	0.915(550 K)	0.945(800 K)	0.934
4	Tl2HgSnS4	1020, −1020	0.943(900 K), 0.853(900 K)	1.49(850 K), 1.12(900 K)	0.199, 1.18
5	CuBS2	1020	0.933(900 K)	1.88(750 K)	1.77
6	CuBSe2	−1019	0.882(900 K)	2.29(900 K)	2.34
7	CuBTe2	1019	0.951(900 K)	1.57(850 K)	1.48
8	AuBSe2	1020, 1019	0.873(800 K), 0.889(650 K)	3.72(800 K), 1.57(650 K)	3.86, 1.7
9	AuBTe2	1020	0.802(900 K)	3.42(900 K)	3.84
10	AuAlTe2	1020	0.778(750 K)	2.35(750 K)	2.27
11	AuGaTe2	−1020, 1020	0.495(700 K), 0.55(400 K)	3.85(700 K), 1.58(400 K),	7.83, 5.87
12	AuInTe2	−1019, 1020	0.431(800 K), 0.536(400 K)	3.25(800 K)	7.57
13	CuAlSe2	1018, 1019, 1020	0.944(750 K), 0.935(850 K), 0.888(900 K)	0.38(750 K), 0.68(850 K), 2.15(900 K)	0.353, 0.647, 2.18
14	CuAlTe2	−1018, 1018	0.952(800 K), 0.947(850 K),	0.403(800 K), 0.577(850 K)	0.381, 0.549
15	AgAlSe2	−1019, −1018, 1018	0.960(900 K), 0.956(900 K), 0.955(900 K)	0.548(900 K), 0.748(900 K), 0.788(900 K)	0.514, 0.705, 0.743
16	AgAlTe2	1020, 1019	0.858(900 K), 0.977(250 K)	2.04(900 K)	2.14
17	CuGaS2	1021, 1020	0.607(900 K), 0.910(500 K)	5.48(900 K), 0.475(500 K)	8.13, 0.855
18	CuGaSe2	1020, −1019	0.688(600 K), 0.784(350 K)	2.26(600 K), 0.831(350 K)	3.77, 2.24
19	CuGaTe2	1020, −1018, 1019	0.739(700 K), 0.885(300 K), 0.847(400 K)	3.41(700 K), 0.352(300 K), 1.02(400 K)	4.96, 2.96, 3.04
20	AgGaS2	−1020, −1019	0.865(900 K), 0.823(900 K)	2.03(900 K), 2.81(900 K)	2.11, 3.07
21	AgGaSe2	1020	0.735(650 K)	1.86(650 K)	2.79
22	AgGaTe2	1020, 1019	0.729(800 K), 0.824(400 K)	2.86(900 K), 0.803(400 K)	3.76, 1.76
23	CuInS2	1021, 1020	0.633(900 K), 0.810(350 K)	5.03(900 K), 1.21(350 K)	7.16, 2.73
24	CuInSe2	1021	0.541(900 K)	4.67(900 K)	7.78
25	CuInTe2	1020	0.79(at 500 K)	1.85(500 K)	3.24
26	AgInS2	1020, 1018, −1018	0.824(750 K), 0.945(300 K), 0.986(250 K)	1.91(750 K), 0.151(300 K)	2.19, 0.75, 0.76
27	AgInSe2	1020	0.736(450 K)	1.31(450 K)	2.54
28	AgInTe2	1020	0.794(600 K)	1.97(600 K)	2.81
29	NaInSe2	−1019, −1018, 1018, 1019	0.956(900 K), 0.948(900 K), 0.946(900 K), 0.94(900 K)	0.308(900 K), 0.433(900 K), 0.457(900 K), 0.557(900 K)	0.29, 0.411, 0.435, 0.533
30	NaInTe2	1020	0.952(850 K)	0.335(850 K)	0.565

), crystals 5–30 with stoichiometry (A’B’Q₂) (A = Ag, Cu, Au, Na; B’ = B, Al, Ga, In; Q = S, Se, Te). The crystals 5–30 are divided into six subgroups (colored gray and white in Table S1 and Table 1). In all subsequent discuss ions, the same specific order and ID number for these crystals are maintained. The fully optimized structures are listed in Table S1 with the corresponding experimental lattice parameters. Moreover, we further investigated the crystals AgSbSe₂, AgSbTe₂, Cd-doped AgSbTe₂ (AgSb_0.94Cd_0.06Te₂), and Se-doped AgSbTe₂ (AgSbTe_1.85Se_0.15) by using the same computational methods, and their results are presented in Section 3.2. Such comprehensive studies have added valuable insight into chalcogenide crystal sciences and are to provide a comprehensive review for the industry. In the following section, we briefly describe the computational methods used, followed by the results and discussions section. We end up with a brief conclusion and our vision for the future study on how to improve their thermoelectric performance.

2. Computational Methods

In this work, two well-defined density functional theory (DFT) based methods were used for specific targeted goals, which are: (1) the Orthogonalized Linear Combination of Atomic Orbitals (OLCAO) method [77] and (2) Quantum Espresso (QE) [78]. All the calculations have been performed based on the previously studied crystals [79,80], relaxed by Vienna Ab initio Simulation Package (VASP) [81]. The Spin–Orbit Coupling (SOC) effect was not included in our DFT calculations. The OLCAO method was used to calculate the electronic structure and the interatomic bonding [77]. OLCAO is an all-electron method based on the local density approximation. It uses the atomic orbitals that are themselves expanded as Gaussian-type orbitals (GTO) on the basis of expansion of the solid-state wave function. The use of localized atomic orbitals in the basis expansion, in contrast to the plane-wave expansion, is particularly effective for both crystalline [79,82,83,84,85] and non-crystalline materials [86,87,88], especially those with complex structures typical in the biomolecular systems [89,90]. A sufficiently large number of k-points (10 × 10 × 12 for the crystals 1–4, to 16 × 16 × 8 for most crystals 5–30) were used for band structure calculations based on the size of the crystal. We used the Mulliken scheme [91] for the calculation of partial charge (PC) and interatomic bonding. The PC of an atom is defined as the charge deviation of the effective charge Q* from the charge of neutral atom (Q₀) in units of electron charge. Mathematically, ΔQ = Q₀ − Q*. Negative ΔQ implies a gain of electrons (i.e., an electronegative ion), and positive ΔQ implies a loss of electrons (i.e., electropositivity). Equations (1) and (2) show the formulae for effective charge ($Q_{\alpha }^{*}$) and bond order (BO) values, also called the overlap population, ${\rho }_{\alpha \beta }$ between any pair of atoms (α, β).

$$Q_{\alpha }^{*}={\displaystyle \sum }_i{\displaystyle \sum }_{m,occ}{\displaystyle \sum }_{j,\beta }{C}_{i\alpha }^{*m}{C}_{j\beta }^m{S}_{i\alpha ,j\beta }$$

(1)

$${\rho }_{\alpha \beta }={\displaystyle \sum }_{m, occ}{\displaystyle \sum }_{i,j}{C}_{i\alpha }^{*m}{C}_{j\beta }^m{S}_{i\alpha ,j\beta }$$

(2)

In the above equations, $S_{i\alpha .j\beta }$ are the overlap integrals between the ith orbital in the αth atom and the jth orbital in βth atom. $C_{j\beta }^m$ is the eigenvector coefficients of the mth occupied band. The BO (Equation (2)) defines the relative strength of the bond. The summation of all BO values in the crystal gives the total bond order (TBO). We obtain the total bond order density (TBOD) when TBO is normalized by the cell volume. TBOD is a single quantum mechanical metric to describe the internal cohesion of the crystal [92].

For the thermoelectric transport properties, Quantum Espresso (QE) was used to calculate the total energy, band structure, Fermi level energy, etc. The generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof (PBE) potential was used in QE [78,93]. QE uses plane waves as basis set that includes the scalar relativistic effects in the pseudo-potentials. In this study, the energy cut-off of the plane waves was set to be 40 Ryd (≈544 eV). In the QE calculations for the electronic and band structure for transport properties using BoltzTraP code, we needed denser k-point meshes for self-consistent calculations. Therefore, after careful convergence tests, a grid of 12 × 12 × 16 k-points was used for the crystals 1–4, while a grid of 18 × 18 × 9 k-points was found to be sufficient in most crystals 5–30 for the self-consistent calculations. The AgSbSe₂ and AgSbTe₂ crystals with 16 atoms in the unit cell have fully relaxed lattice parameters of a = 5.8010Å and 6.0991 Å, respectively. While a supercell with 64 atoms was created for both the Cd-doped AgSbTe2 (AgSb_0.94Cd_0.06Te₂) and Se-doped AgSbTe₂(AgSbTe_1.85Se_0.15) crystals, which have relaxed lattice parameters of a = 12.1977 Å and 12.2109 Å respectively. For the cubic supercell of AgSbTe_1.85Se_0.15, we started creating 2 × 2 × 2 supercells from the fully relaxed cubic crystal structure of AgSbTe₂ with eight atoms. In the second step, five Te atoms, located at different and totally random sites of the supercell, were replaced by five Se atoms. The sites at the six faces of the supercell were avoided. The second step was followed by fully VASP relaxation for the supercell. Another last VASP relaxation step was performed to make sure that each atom took its normal position (that minimizes the energy) in the supercell.

A special feature in QE is the ability to calculate TE transport properties using the semiclassical Boltzmann theory within the constant scattering time approximation (CSTA) as implemented in the BoltzTraP code [94]. In this theory, the Seebeck coefficient, electrical conductivity, and electronic thermal conductivity can be expressed in the following equations [94]:

$$S_{\alpha \beta }\left(T, \mu \right)= \frac{1}{eT\Omega {\sigma }_{\alpha \beta }\left(T, \mu \right)}{\displaystyle \int }{\sigma }_{\alpha \beta }\left(\epsilon \right)\left(\epsilon -\mu \right)\left[-\frac{\delta f_0\left(T, \epsilon , \mu \right)}{\delta \epsilon }\right]d\epsilon$$

(3)

$${\sigma }_{\alpha \beta }\left(T, \mu \right)= \frac{1}{\Omega }{\displaystyle \int }{\sigma }_{\alpha \beta }\left(\epsilon \right)\left[-\frac{\delta f_0\left(T, \epsilon , \mu \right)}{\delta \epsilon }\right]d\epsilon$$

(4)

$${\kappa }_{\alpha \beta }\left(T, \mu \right)= \frac{1}{e^{2}T\Omega }{\displaystyle \int }{\sigma }_{\alpha \beta }\left(\epsilon \right){\left(\epsilon -\mu \right)}^2\left[-\frac{\delta f_0\left(T, \epsilon , \mu \right)}{\delta \epsilon }\right]d\epsilon$$

(5)

Here, ${\sigma }_{\alpha \beta }$ are the transport distribution tensor elements calculated using the Fourier interpolation of the band structure. α and β are the tensor indices; Ω, μ, f₀, and T are the cell volume, chemical potential, Fermi distribution function, and absolute temperature, respectively.

The input data needed to run BoltzTraP code are the crystal structure and the band structure on a uniform grid. The BoltzTraP code solves the Boltzmann equation [95],

$$\frac{\partial }{\partial t}f+\overrightarrow{v}\frac{\partial }{\partial \overrightarrow{r}}f+\frac{eE}{ħ}\frac{\partial }{\partial \overrightarrow{k}}f={\left(\frac{\partial f}{\partial t}\right)}_{scattering}$$

(6)

where, f is the distribution function, ν is the velocity vector of the particle, and k is the wave vector.

BoltzTraP code solves the Boltzmann equation by interpolating a band structure and performing all required integrations. The BoltzTraP code has been tested over the last decade on different materials ranging from superconductors [96] to thermoelectric materials [97,98,99,100]. Good agreement with the experimental values was achieved in many cases [101,102,103]. It should be pointed out that the ZT values obtained here are slightly overestimated because the phonon contribution to the thermal conductivity was ignored due to the limitation of the BoltzTraP code. The BoltzTraP code treats the lattice thermal conductivity contributed by phonons as a constant and provides only the electronic part of the thermal conductivity. Several simplifying approximations are introduced. The chemical potential (μ) is set to the Fermi level energy (μ_F) because conductivities (thermal and electronic) depend proportionally on the relaxation time τ within the constant relaxation time framework. Those values can then simply be multiplied by the constant relaxation time (10⁻¹⁴ s, as this value is used in BoltzTraP code) to obtain the final transport properties. The Seebeck coefficient does not depend on the relaxation time within the constant relaxation time approximation. The CSTA assumes that the relaxation time τ is independent of energy. Hence the power factor is τ dependent, and ZT is τ independent. Nevertheless, this method has successfully described the transport properties of a wide range of thermoelectric materials [94,104,105,106].

3. Results and Discussion

3.1. Transport Properties of 30 Chalcogenide Crystals

The DFT calculations for the energy band gap (E_g) for some of these 30 crystals were published in our two previous works [79,80], and we have listed them in the third column of Table S2. In this work, we mostly focus on the crystals of the first group (the crystals: 1–4), and the crystals of the first (the crystals: 5–7), second (the crystals: 8–12), and third (the crystals: 13–16) subgroups of the second group (total crystals: 1–16), for which the transport properties are calculated for the first time. The calculated S and PF with varying ranges of μ for these 16 crystals at three different temperatures (300 K, 500 K, and 800 K) are shown in Figures S1 and S2 in the SI. Figures S3 and S4 show the S and PF versus the chemical potential for the remaining 14 crystals in the second main group. Figures S5–S7 show the electrical conductivity, electronic thermal conductivity, and the figure of merit versus the chemical potential for all 30 crystals. It is crucial to point out that all chemical-potential-dependent transport properties (including ZT in Figure 1) at three fixed temperatures (300 K, 500 K, and 800 K) were calculated by fixing the temperature and letting the carrier concentrations change. That means for a fixed temperature, the chemical potential is simply a function of carrier concentration, and the chemical potential µ can vary.

Table S2 shows the calculated values of S and ZT for these 30 crystals at room temperature (300 K). S can vary with the variation of the carrier concentration, which determines the value of the induced thermoelectric voltage due to the difference in temperature across the material. S is very sensitive to chemical potential, carrier concentration, and temperature. The positive value of S shows that holes have a dominant contribution to the conduction (p-type TE material), while the negative value of S shows that electrons have a dominant contribution to the conduction (n-type TE material). In insulators, S has the highest values around the Fermi energy, or $\mu -\mu F \approx 0,$ because the carrier concentrations are at their lowest values. It is evidently observed from Figures S1 and S3 that S is significantly improved in the vicinity of $\mu -\mu F=0$, which indicates that a fairly large value of S can be attained by small n-type or p-type doping. From Figures S1 and S3, we can also notice that the highest S can be found in the crystals 5-CuBS₂, 6-CuBSe₂, 29-NaInSe₂, and 30-NaInTe₂ with values: 2600, 2500, 2400, and 2100 μV/K respectively for p-type crystals. Generally, the maximum value of S is obtained at room temperature, 300 K (without fixing the carrier concentrations), and S decreases with an increase in temperature at certain values of chemical potential. Much fewer previous studies for the chemical potential transport properties of these 30 crystals were found in the literature, so we are unable to compare all our results in Table S2. S values in Table S2 may coincide with high values of electronic thermal conductivity, so it is necessary to fix the carrier concentrations when calculating S as a function of temperature.

Another important parameter that can determine the TE performance of a material is the power factor (PF). Figure S2 shows the PF as a function of the chemical potential at three different temperatures, 300 K, 500 K, and 800 K for crystals: 1–16, while the results for the remaining crystals are shown in Figure S4. As can be seen in these figures, the highest PF occurs at 800 K for all crystals, except for 29-NaInSe₂, for which the highest value of PF occurs at 500 K instead of 800 K. We mentioned previously that S has its maximum value at 300 K, but this is only true at Fermi level energy ($\mu -{\mu }_F=0$), or around Fermi level energy ($\mu -{\mu }_F\approx 0$), where σ also has its minimum values (almost zero). However, S could have its maximum values at 800 K instead of 300 K when the chemical potential μ has values that are not very close to μ_F value, and this is the main reason why PF has its maximum values at 800 K for most crystals at those values of chemical potential (μ has values that are not close to μ_F). If the maximum PF occurs when μ − μ_F < 0 (hole doping region), then the implication is -that this material works better as a p-type TE material. Conversely, when μ − μ_F > 0 (electron doping region), then the crystal works better as an n-type TE material. The material should possess a high σ to present a high TE efficiency. The variation of σ with the chemical potential at three different temperatures is depicted in Figure S5. From the graph of electrical conductivity (σ/τ) in Figure S5, most crystals that have a high σ/τ in the hole doping region also have high PF values in this region compared to the electron doping region. Conversely, most crystals, which have a high σ/τ in the electron doping region, also have a high PF in this region compared to the hole doping region. The maximum σ for some crystals is observed for n-type doping, while it is observed for p-type in other crystals. One should keep in mind that these calculations were performed without fixing the carrier concentration. In order to know more about the temperature-dependent TE properties of these crystals, it is necessary to fix the carrier’s concentration. Figure 1 displays the figure of merit ZT with varying ranges of μ for the crystals: 1–4, 8–12, and 15–16 at three different temperatures (300 K, 500 K, and 800 K). Figure S7 in the SI shows ZT versus the chemical potential for the remaining 19 crystals. The highest values of ZT as a function of μ for these 30 crystals are represented in the sixth column of Table S2. From Table S2, ZT for 1-Tl₂CdGeSe₄, 2-Tl₂HgSiSe₄, 5-CuBS₂, 6-CuBSe₂, 15-AgAlSe₂, 16-AgAlTe₂, 20-AgGaS₂, 29-NaInSe₂, and 30-NaInTe₂ is larger than unity.

The Seebeck coefficient predicted by the theory for metals and degenerate semiconductors, with a parabolic band, and the energy-independent charge carrier scattering approximation [107], depends on the carrier concentration of electrons or holes n and the effective mass m*. It can be given by [108]:

$$S= \frac{8{\pi }^2{k}_B^2}{3e{h}^2}{m}^{*}\times T{\left(\frac{\pi }{3n}\right)}^{2/3}$$

(7)

where n is the carrier concentration (electrons or holes), k_B and e are the Boltzmann constant and electronic charge, respectively. S is proportional to T, so S increases due to the increase in temperature. However, κ_ele will also increase with increasing temperature, leading to reduced ZT and TE performance. Decreasing ZT with increasing T may be a problem for the TE performance, but it can be solved sometimes by changing the carrier concentration, which in turn will modify the Fermi level energy, leading to increased ZT with the increase in temperature. Doping (i.e., the introduction of either additional hole or electron carriers) has an enormous effect on electronic transport properties. Doping will set the Fermi level (μ_F) and will directly influence the values of the transport properties.

In order to ensure that the Seebeck coefficient is large, there should only be a single type of carrier. Mixed n-type and p-type conduction will lead to both charge carriers moving to the cold end, canceling out the induced Seebeck voltages. Low carrier concentration insulators and even semiconductors may have large Seebeck coefficients; see Equation (6). However, low carrier concentration also results in low electrical conductivity. A compromise between large S and high electrical conductivity in thermoelectric materials must be struck to maximize the figure of merit ZT ($S^2\sigma T/\kappa$) where κ is the thermal conductivity.

This peak typically occurs at carrier concentrations between 10¹⁸ and 10²¹ carriers per cm³ (depending on the material system), which falls in between common metals and semiconductors—that is, concentrations found in heavily doped semiconductors. The next dataset describes the transport properties for both n-type and p-type systems at fixed doping levels ranging from 10¹⁸ to 10²¹ cm⁻³, increasing the doping by one order of magnitude at each step.

The calculated values of S, κ_ele, PF, and ZT as a function of temperature at fixed values of carrier concentration in the ranges (±10¹⁸, ±10¹⁹, ±10²⁰, ±10²¹ in e⁻/cm³) for the crystals: 1–16 are shown in Figure 2, Figure 3, Figure 4 and Figure 5, while the results for the remaining crystals are shown in Figures S9 and S10. A positive value of n indicates hole doping, while a negative value of n indicates electron doping. Figure 2 is for S, Figure 3 is for κ_ele, Figure 4 is for the PF, and Figure 5 is for ZT. The temperature range was set to be from 250 K to 900 K. Table 1 shows the highest values of ZT for these crystals, the carrier concentrations for which the highest ZT values occur, the highest PFs, and the highest κ_ele. In Figure 2, Figure 3, Figure 4 and Figure 5 and Table 1, the value of the carrier concentration (n) for each crystal was considered to be the value that can achieve the highest ZT. It is known that a narrow band gap correlates with a large carrier concentration because many charge carriers will transfer from the valence band to the conduction band. Figure 2 shows that S increases with increasing temperature at the indicated values of n and that S has positive values indicating p-type thermoelectric materials, except for the crystals 4-Tl₂HgSnS₄, 7-CuBTe₂, 11-AuGaTe₂, and 12-AuInTe₂ for which S has negative values, indicating n-type TE materials. The crystal 8-AuBSe₂ has the highest values of S and ZT among the five crystals: 8-AuBSe₂, 9-AuBTe₂, 10-AuAlTe₂, 11-AuGaTe₂, and 12-AuInTe₂. As can be seen from Table 1 and Figure 2, Figure 3, Figure 4 and Figure 5, for most of these crystals, the highest values of S and ZT occur with hole doping, except for the crystals 3-Tl₂HgSiSe₄, 6-CuBSe₂, 14-CuAlTe₂, and 20-AgGaS₂ where S and ZT can have the highest values with electron doping.

The values of the electronic thermal conductivity κ_ele/τ and PF in Figure 3 and Figure 4 must be multiplied by the scattering time factor, which was taken to be 10⁻¹⁴ s in our study, as mentioned in Section 2 (computational methods). From Table 1, the highest κ_ele occurs at 900 K for all 30 crystals. κ_ele has the largest value (κ_ele = 8.13 W/m.K) in the crystal 17-CuGaS2, while it has the smallest value (κ_ele = 0.199 W/m.K) in the crystal 4-Tl₂HgSnS₄. Hence, the values of κ_ele for all these 30 crystals are in the range from 0.199 W/m.K to 8.13 W/m.K, which indicates that the 30 crystals are very good TE materials if we know that the thermal conductivity value of the non-doped regular semiconductors such as silicon and germanium are 150 and 58 W/m.K for these two semiconductors respectively. Excellent TE materials should have a high PF (more than 3.0 mW/cm.K²). In general, if PF is greater than κ results in larger ZT and vice versa, namely 17-crystal CuGaS₂ and 29-NaInSe2, as shown in Table 1. Some crystals, such as 4-Tl₂HgSnS₂, 18-CuGaSe₂, 19-CuGaTe₂, and 29-NaInSe_2, can have a high value of ZT at both regions (hole and electron doping). From Table 1, we notice an interesting fact about crystals: 18-CuGaSe₂ and 23-CuInS₂ can have a high value of ZT (0.784 and 0.81 respectively) at a low temperature (350 K) at the following values of n: −10¹⁹ and 10²⁰ e⁻/cm³ respectively. Additionally, the crystals 19-CuGaTe₂ and 26-AgInS₂ can have high ZT (0.885 and 0.945, respectively) at 300 K at the values of n: −10¹⁸ and 10¹⁸ e⁻/cm³, respectively, and these are new and interesting findings.

The chemical potential-dependent transport properties of the crystals: 19-CuGaTe₂, 22-AgGaTe₂, 25-CuInTe₂, and 28-AgInTe₂ were calculated and can be found in the Supplementary Materials. Here, we focus only on the calculated S and ZT versus the temperature for these crystals at two wide ranges of carrier concentrations to reveal which one leads to a higher S and ZT. The ranges are: (+2 × 10¹⁹, +8 × 10¹⁹ in e⁻/cm³) and (+2 × 10¹⁸, +8 × 10¹⁸ in e⁻/cm³). Figure S11a–d shows S versus temperature for the crystals 19-CuGaTe₂ and 22-AgGaTe₂ over the two different ranges of n. S can reach its maximum value for the crystal 22-AgGaTe₂ (S ≈ 310 μV/K) and for 19-CuGaTe₂ (S ≈ 320 μV/K) at 650 K when n = 2 × 10¹⁹ for both these two crystals. In contrast, S can reach its maximum value at 450 K for 22-AgGaTe₂ (S ≈ 325 μV/K) and at 400 K for 19-CuGaTe₂ (S ≈ 400 μV/K) when n = 2 × 10¹⁸ for both crystals. Figure S11e–h shows our results for ZT at these two different ranges of n. For the crystals 19-CuGaTe₂ and 22-AgGaTe₂, in the first range, ZT hits its maximum value (ZT ≈ 0.8) at 500 K for the crystal 22-AgGaTe2 and at 450 K (ZT ≈ 0.82) for the crystal 19-CuGaTe₂ when n = 2 × 10¹⁹ for both these two crystals. While when n = 8 × 10¹⁹, ZT has its highest value at 900 K for both crystals. For the same crystals, but in the second range of carrier concentrations, ZT reaches its maximum values (ZT ≈ 0.86 for 22-AgGaTe₂, ZT ≈ 0.88 for 19-CuGaTe₂) at 350 K when n = 2 × 10¹⁸ for both crystals. For the crystals 25-CuInTe₂ and 28-AgInTe₂ at the first range of n, the situation is a bit different. ZT versus T for these two crystals is shown in Figure S12. ZT for 28-AgInTe₂ starts from its minimum value at 250 K then increases with increasing temperature to reach a maximum value (ZT ≈ 0.4) at 700 K when n = 3 × 10^19, then it starts decreasing. However, at 250 K, when n = 5 × 10¹⁹ ZT starts from its maximum value (ZT ≈ 0.84), it starts decreasing rapidly with increasing temperature. ZT for 25-CuInTe₂ starts from its minimum value at 250 K then increases with increasing temperature to reach maximum value (ZT ≈ 0.4) at 600 K when n = 4 × 10¹⁹ then starts decreasing, while ZT starts from its maximum value (ZT ≈ 0.84) at 250 K when n = 6 × 10¹⁹ then starts decreasing rapidly with increasing temperature. These are totally new findings that have not been revealed yet in any previous work.

It is not easy to find previous experimental studies on the thermoelectric properties of these 30 crystals. Nevertheless, we have identified the following available experimental data. A study by Gui etc. [109] showed that ZT values at 850 K are 1.1, 0.87, and 1.0 for CuInS₂, CuInSe₂, and CuInTe₂ crystals, respectively. Furthermore, our calculated ZT at 850 K was 0.65, 0.6, and 0.5 for these three crystals, respectively. Another four experimental studies [45,110,111,112] were conducted on the thermoelectric properties of CuInTe₂ crystal. The first two studies showed that ZT ≈ 0.6 at 800 K, which is very close to ours (ZT = 0.56 at 800 K). The third and fourth studies showed that ZT ≈ 0.35 and ZT ≈ 0.4 at 700 K, which is less than ours (ZT = 0.69 at 700 K). Another two experimental studies [113,114] were carried out on the thermoelectric properties of CuGaTe₂ crystal. Ref. [113] reported ZT ≈ 1.12 at 900 K, which is higher than our results (ZT = 0.67 at 900 K), while Ref. [114] showed that ZT ≈ 0.9 at 850 K, which is comparable to ours (ZT ≈ 0.7 at 850 K). On the other hand, Cao et al. and Yusufu et al. [58,115] experimentally found that ZT for AgGaTe₂ to be about 0.5 at 800 K and 0.4 at 850 K, less than ours (ZT = 0.729 at 800, ZT = 0.724 at 850).

3.2. Transport Properties of AgSbSe₂ and AgSbTe₂ Crystals and the Doped Ones: AgSb_0.94Cd_0.06Te₂ and AgSbTe_1.85Se_0.15

Cd-doped and Se-doped crystals (AgSb_0.94Cd_0.06Te₂ and AgSbTe_1.85Se_0.15) arise from the substitution of Cd and Se atoms in AgSbTe₂ crystal (Figure 6). The dopant compositions of AgSb_0.94Cd_0.06Te₂ and AgSbTe_1.85Se_0.15 were chosen carefully based on very recent experimental study [46]. Our calculations for the band structure of AgSbSe₂ and AgSbTe₂ crystals showed that these crystals are semi-metallic materials. Our calculations for ZT are shown in

Table 2. Our calculated ZT at 900 K and a comparison with other experimental works for the pure AgSbSe₂ and AgSbTe₂ crystals, and the doped ones: AgSb₀.₉₄Cd₀.₀₆Te₂ and AgSbTe_1.85Se₀.₁₅.

Crystal	n	ZT (Ours)	ZT
AgSbSe2	1020	0.898(750 K)	0.41(650 K) [71], 0.65(675 K) [116]
AgSbTe2	1019	0.924(750 K)	1.2(650 K) [47], 0.9(675 K) [76]
AgSb0.94Cd0.06Te2	9.0 × 1019	2.36(700 K)	2.6(573 K) [46]
AgSbTe1.85Se0.15	1020	2.39(700 K)	2.1(575 K) [21,47]

, while Figure 7 and Figure 8 show our calculations for ZT, S, σ, κ_ele, and PF as a function of temperature for the pure AgSbSe₂ and AgSbTe₂ crystals and for the doped ones (AgSb_0.94Cd_0.06Te₂ and AgSbTe_1.85Se_0.15).

As can be seen from these figures, AgSb_0.94Cd_0.06Te₂ and AgSbTe_1.85Se_0.15 crystals have less S and higher σ than the pure ones. Experimentally, S Roychowdhury et al. [46] showed that the hole concentration at 300 K of the pure AgSbTe₂ increased from $4.3\times {10}^{19} {\mathrm{cm}}^{-3}$ to $8.3\times {10}^{19} {\mathrm{cm}}^{-3}$ when doped with 6 mol% Cd(AgSb_0.94Cd_0.06Te₂), and that was the reason for reducing S values (S = ∝ 1/n) and increasing values of σ (σ ∝ n). Our study showed the same behavior for S. The doped crystals have less κ_ele than the pure ones at high temperatures, and this makes the doped crystals have higher ZT than the pure crystals. This implies that doping by Cd and Se atoms results in reduced thermal conductivity and increases the TE performance. In addition, we contend that increasing the carrier concentration (n) does not necessarily mean an increase in the values of κ_ele for the doped crystals. This can be true both experimentally and computationally since, in some cases, the doping process can create point defects, vacancies, and distortion in the bonding between atoms. This may lead to the reduction in the heat energy transferred by the charge carriers (electrons) or the phonons in the doped crystals.

Thermal conductivity has two components: lattice (phonons) and charge carriers. Increasing carrier concentration will increase thermal conductivity (if lattice does not change). However, for the dependence on temperature (our case), things may become more complicated. According to our DFT calculations, the crystals AgSbSe₂ and AgSbTe₂ are metallic materials with zero energy band gap. Wiedemann-Franz law basically relates the two conductivities of metals, i.e., thermal (κ_ele) and electrical (σ) conductivity with temperature. It states that the ratio of thermal conductivity κ_ele and electrical conductivity σ is proportional to the temperature of the specimen. G. Wiedemann and R. Franz established (based on experimental data) that the ratio $\frac{{\kappa }_{ele}}{\sigma }$ is constant at a constant temperature. L. Lorenz demonstrated that the relation $\frac{{\kappa }_{ele}}{\sigma }$ changes in direct proportion to the absolute temperature T, $\frac{{\kappa }_{ele}}{\sigma }=LT$, where T = temperature, L = 2.54 × 10⁻⁸ WΩ/K², Lorentz number (a constant). This law basically states that with an increase in temperature, the thermal conductivity of metals increases while the electrical conductivity decreases. These two properties of metals are dependent on the free electrons. An increase in temperature increases the average velocity of the free electrons leading to an increase in heat energy transfer. On the other hand, an increase in the velocity of electrons also increases the number of collisions of the free electrons with lattice ions and hence contributes to an increase in electrical resistivity or reduction in electrical conductivity (σ), and this what can be noticed in Figure 8b,c for the crystals AgSbSe₂ and AgSbTe₂. However, this law has certain limitations. The proportionality does not hold true for all ranges of temperature. It is only found valid for very high temperatures and very low temperatures.

The above argument can be valid for the metallic materials, but few experimental studies showed that AgSbSe₂ and AgSbTe₂ crystals are semiconductors with a very narrow energy band gap. When we have semiconductors, alloys, or other interesting structures, the issue is much more complex, and it is difficult to obtain a general dependence of the conductivities with the temperature. That means the relationship between the thermal and electrical conductivities in the case of semiconducting materials is very complicated and is different from one material to another. Many researchers have tried to find a common formula that correlates between the two types of conductivities, and this relationship is still under study till now.

The effective charge (Q*) on each atom of these four crystals is represented in

Table 3. Calculated average effective charge for pure AgSbSe₂ and AgSbTe₂ crystals, and the doped ones: AgSb_0.94Cd_0.06Te₂ and AgSbTe_1.85Se_0.15.

#	Crystal	Q* (in e−)
1	AgSbSe2	10.971(Ag), 4.399(Sb), 6.315(Se)
2	AgSbTe2	11.102(Ag), 4.694(Sb), 6.102(Te)
3	AgSb0.94Cd0.06Te2	11.080(Ag), 4.748(Sb), 11.243(Cd), 6.102(Te)
4	AgSbTe1.85Se0.15	11.030(Ag), 4.712(Sb), 6.120(Te), 6.266(Se)

, while Figure 9 shows the BO versus bond length (BL) for these four crystals. It is extremely helpful to investigate the interatomic bond strength between every pair of atoms in the crystal as represented by the BO values [92]. Important insights are revealed in Figure 9 and Table 3. In the AgSbSe₂ crystal, there are two bonds: Ag-Se and Sb-Se, and these two bonds are ionic bonds (metal-nonmetal bonds) where the two metal elements (Ag, Sb) lose charge to the nonmetal chalcogen element (Se). While in the AgSbTe₂ crystal, there are also two bonds: Ag-Te and Sb-Te, but the Ag-Te and Sb-Te bonds are more complicated to explain because Te is a metalloid element. Te sometimes acts similarly to a metal element and loses charge, and in different cases, it acts similarly to a nonmental element and gains charge (as our calculations show). As can be seen in Figure 9a,c, the Ag-Te bond is much stronger than the Ag-Se, Sb-Se, and Sb-Te bonds. The chalcogen elements (Se, Te) in each crystal of AgSbSe₂ and AgSbTe₂ make bonds with two atoms (two bond types): Ag and Sb. While each atom: Ag or Sb has just one bond type: Ag-Se and Sb-Se in AgSbSe₂, and Ag-Te and Sb-Te in AgSbTe₂. From the oxidation states of the Ag and Sb atoms, we know that each atom of these two atoms can bond with more than one or more different atoms in the crystal structure, and this leads to the fact that these two atoms can vibrate with higher frequency for phonons (higher thermal conductivity and less TE performance) than when they make two or three bonds with two or three other atoms in the crystal.

When the AgSbTe₂ crystal is doped with Se atoms (AgSbTe_1.85Se_0.15), we notice from Figure 9b that each atom, Ag or Sb, now has two bond types and is bonded with two atoms (Se and Te) instead of just one atom, which we could not see in the pure AgSbTe₂. Similarly, Figure 9d for the Te-doped crystal (AgSb_0.94Cd_0.06Te₂) shows that the Te atom (heavy atom) is bonded now with three atoms (Ag, Sb, and Cd) instead of just two atoms (Ag and Sb) in the pure AgSbTe₂. These three bonds in AgSb_0.94Cd_0.06Te₂ make the heavy Te atom more bound and vibrate with lower frequencies for phonons. Hence, there is less thermal conductivity and higher TE performance. We conclude that the doping process of the AgSbTe₂ crystal by Se and Cd dopants created new three bonds (Ag-Se, Sb-Se, and Cd-Te (Figure 8b–d) for atoms Ag, Sb, and Te and made them vibrate with lower frequencies for phonons (less thermal conductivity and higher TE performance) than in pure AgSbTe₂, and this may explain why AgSbTe_1.85Se_0.15 and AgSb_0.94Cd_0.06Te₂ crystals have less thermal conductivity and much higher ZT than the pure AgSbTe₂ in terms of bonding properties.

Thermal conductivity in thermoelectric materials comes from two sources: (1) electrons and holes transporting heat (κ_ele) and (2) phonons traveling through the lattice (κ_lat). Most of the electronic term (κ_ele is directly related to the electrical conductivity through the Wiedemann–Franz law:

$$\kappa ={\kappa }_{ele}+{\kappa }_{lat}$$

(8)

$${\kappa }_{ele}=L\sigma T=LnemT$$

(9)

The Lorenz factor can vary, particularly with carrier concentration. Accurate assessment of κ_ele is important, as κ_lat is often computed as the difference between κ and κ_ele (Equation (8)) using the experimental electrical conductivity. A common source of uncertainty in κ_ele occurs in low-carrier-concentration materials where the Lorenz factor can be reduced by as much as 20% from the free-electron value. Additional uncertainty in κ_ele arises from mixed conduction, which introduces a bipolar effect term into the thermal conductivity [117]. S Roychowdhury et al. [46] showed that the values of κ_lat are in the range 0.5–0.55 W/m.K and in the range 0.1–0.13 W/m.K at the temperature range 300–575 K for the two crystals AgSbTe₂ and AgSb_0.94Cd_0.06Te₂ respectively. In contrast, our calculations showed that the values of κ_ele are in the range 1.0–3.3 W/m.K and in the range 0.3–0.75 W/m.K at the temperature range 250–600 K for the two crystals AgSbTe2 and AgSb_0.94Cd_0.06Te₂ respectively. This can be used as evidence that κ_ele has a bigger effect on TE performance than κ_lat. A further calculation of κ_lat will help to explore why the doped crystals have much higher TE performance than the pure ones, which is not the focus of our study at this stage.

3.3. Correlation between Transport and Bonding Properties in 30 Chalcogenide Crystals

Identifying the underlying correlation between the TE performance (ZT) and the bonding characteristics of the 30 chalcogenide crystals is one of the main objectives of this work. In this regard, exploring the connection between the thermoelectric properties of the 30 crystals and the TBOD could be revealing. TBOD values for all 30 crystals are represented in Table S2 in the SI. TBOD describes the internal cohesion of the crystal, so that higher TBOD may imply a reduced value of thermal conductivity. In Figure S13, we plot the ZT versus TBOD for the 30 crystals. We notice that there is a general trend where ZT increases with the increase in TBOD. This correlation between TBOD and ZT could be revealed for the first time in this work. However, the crystals inside the blue circle shape (Tl₂CdGeSe₄, Tl₂CdSnSe₄, Tl₂HgSiSe₄, Tl₂HgSnS₄, NaInSe₂, and NaInTe₂) deviate from this trend. To explain this deviation, we plotted BO versus BL (Figure 10) for three crystals that deviate from the trend (Tl₂CdGeSe₄, Tl₂CdSnSe₄, and Tl₂HgSiSe₄) and for three crystals that follow the trend (CuBS₂, CuBSe₂, and CuBTe₂). From Figure 10, Tl₂CdGeSe₄, Tl₂CdSnSe₄, and Tl₂HgSiSe₄ crystals have much lower TBOD values than CuBS₂, CuBSe₂, and CuBTe₂ crystals, but they still have larger values of ZT than CuBS₂, CuBSe₂, and CuBTe₂ crystals. It is true that Tl₂CdGeSe₄, Tl₂CdSnSe₄, and Tl₂HgSiSe₄ crystals have much lower TBOD values than CuBS₂, CuBSe₂, and CuBTe₂ crystals, but at the same time, they have much smaller values of κ_ele than CuBS₂, CuBSe₂, and CuBTe₂ crystals, and this is the reason why Tl₂CdGeSe₄, Tl₂CdSnSe₄, and Tl₂HgSiSe₄ crystals still have higher values of ZT than CuBS₂, CuBSe₂, and CuBTe₂ crystal (which have larger values of κ_ele). The same explanation is valid for the other crystals (NaInSe₂ and NaInTe₂).

The crystal 19-CuGaTe₂ has higher ZT (ZT = 0.662 at 900 K) than the crystal 18-CuGaSe₂ (ZT = 0.556 at 900 K). The same trend can be seen in the crystals 27-AgInSe₂ to 28-AgInTe₂, where 28-AgInTe₂ has ZT = 0.605 at 900 K while 27-AgInSe₂ has ZT = 0.243 at the same temperature. In all these crystals, ZT decreases when moving from Te-related crystals toward Se-related crystals. In some crystals, this behavior can be related to the heavy Te atom, which results in less thermal conductivity and higher ZT. One of the important parameters that can affect the value of thermal conductivity is the bonding nature of the atoms in the crystal. Weaker bonds (lower bond order (BO)) imply higher energy phonon vibrations and, therefore, a higher value of κ and a reduced value of ZT. In contrast, stronger bonds (higher BO) imply smaller energy phonon vibrations and a reduced value of κ, and a higher value of ZT. Bonding properties calculations for these crystals have already been published in our previous work [79], and

Table 4. Bonding properties of 18-CuGaSe₂, 19-CuGaTe₂, 27-AgInSe₂, and 28-AgInTe₂ crystals.

CuGaSe2			CuGaTe2
bond	BL(Å)	BO	bond	BL(Å)	BO
Cu-Se	2.4315	0.1916	Cu-Te	2.5937	0.2286
Ga-Se	2.4746	0.2572	Ga-Te	2.6878	0.2709
AgInSe2			AgInTe2
bond	BL(Å)	BO	bond	BL(Å)	BO
Ag-Se	2.6727	0.1530	Ag-Te	2.8134	0.1814
In-Se	2.6558	0.2384	In-Te	2.8603	0.2581

represents our calculations for the bonding properties of the above crystals. As can be seen from Table 4, The crystals 19-CuGaTe₂ and 28-AgInTe₂ have bonds with higher bond order (stronger bonds) than the crystals 18-CuGaSe₂ and 27-AgInSe₂, and we believe that this is one of the main reasons that 19-CuGaTe₂ and 28-AgInTe₂ have smaller κ and higher ZT than the crystals: 18-CuGaSe₂ and 27-AgInSe₂.

4. Conclusions

A computational study was performed to investigate: 1. the thermoelectric transport properties of 30 chalcogenide crystals to provide a large set of thermoelectric transport data, which can be a good start to explore their further potentials both experimentally and theoretically, and 2. the temperature-dependent transport properties of the pure AgSbSe₂ and AgSbTe₂ and doped AgSb_0.94Cd_0.06Te₂ and AgSbTe_1.85Se_0.15 crystals. In the first part (the 30 crystals), the transport properties of the crystals: 1-Tl2CdGeSe₄, 2-Tl2CdSnSe₄, 3-Tl₂HgSiSe₄, 4-Tl₂HgSnS₄, 8-AuBSe₂, 9-AuBTe₂, 10-AuAlTe₂, 11-AuGaTe₂, 12-AuInTe₂, 15-AgAlSe₂, and 16-AgAlTe₂ are investigated for the first time. These eleven chalcogenide crystals are promising thermoelectric materials which have high Seebeck coefficients and a high figure of merit. For the crystals 19-CuGaTe₂ and 22-AgGaTe₂, we conclude that the second range of n: (+2 × 10¹⁸, +8 × 10¹⁸ in e⁻/cm³) leads to much better TE performance at low temperatures than the first range of n: (+2 × 10¹⁹, +8 × 10¹⁹ in e⁻/cm³). In the second part, it is found that the ZT value improves significantly, higher than 2.0 in p-type doped AgSb_0.94Cd_0.06Te₂ and AgSbTe_1.85Se_0.15 crystals. Our calculations show a difference in the calculated values of the Seebeck coefficient and figure of merit for some crystals compared to other previous experimental and theoretical studies. Some correlations between the calculated bonding nature of these crystals and their thermoelectric properties were revealed in this work.

Despite a large number of simplified calculations on complex thermal transport properties, some for the first time, there are obviously some serious drawbacks, such as ignoring the lattice thermal conductivity. We are encouraged by the current results and aspire to continue research in this area for more complex and interesting chalcogenide crystals. It is desirable to improve the DFT calculations with better options, such as using either hybrid potential or Becke–Johnson potential. We should also point out that there are cases [65,118] where the neglect of SOC may not result in too large a difference due to unpredictable fluctuations. Our results without SOC are within the limit of these fluctuations and will still be useful as a first step to more accurate calculations. Enhancing the thermoelectric performance of these crystals can be achieved by decreasing the values of the thermal conductivity, which is our vision for future works. This could be performed by making these crystals as 2D and 1D materials.