The nature of the high thermoelectric properties of CuInX2 (X = S, Se and Te): First-principles study

Accepted Manuscript Full Length Article The nature of the high thermoelectric properties of CuInX2 (X = S, Se and Te): First-principles study Yang Gui, Lingyun Ye, Chao Jin, Jihua Zhang, Yuanxu Wang PII: DOI: Reference:

S0169-4332(18)31887-7 https://doi.org/10.1016/j.apsusc.2018.07.013 APSUSC 39828

To appear in:

Applied Surface Science

Received Date: Revised Date: Accepted Date:

19 March 2018 20 June 2018 3 July 2018

Please cite this article as: Y. Gui, L. Ye, C. Jin, J. Zhang, Y. Wang, The nature of the high thermoelectric properties of CuInX2 (X = S, Se and Te): First-principles study, Applied Surface Science (2018), doi: https://doi.org/10.1016/ j.apsusc.2018.07.013

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The nature of the high thermoelectric properties of CuInX2 (X = S, Se and Te): First-principles study Yang Gui(1), Lingyun Ye(3), Chao Jin(1), Jihua Zhang(2), and Yuanxu Wang(1, 2)* (1) School of Physics and Electrical Engineering, Anyang Normal University, Anyang Henan,455000 China (2) Institute for Computational Materials Science, School of Physics and Electronics, Henan University, Kaifeng 475004, China. (3) Zhengzhou Chenggong University of Finance and Economics

Abstract CuInX2 (X = S, Se and Te) are members of Cu-based compounds with diamond-like structures. Their bonding characteristics, electronic structure and thermoelectric properties are studied using first-principles methods and the semiclassical Boltzmann theory. Due to the Cu-3d orbits in CuInX2, we use the mBJ+U method to obtain the accurate band gap and electronic structures. The analysis of the electronic structure for CuInX2 indicates that the combination of heavy and light bands near the Fermi level is conductive to achieve high thermoelectric performances. However, few literatures have reported the reason of the combination. Further study on AgGaTe2, CuGaTe2, and CuInX2 (X = S, Se and Te) shows that the stronger interaction between Cu-Te atoms leads to the combination of heavy and light bands in Cu-based compounds. These results provide a valuable theoretical guidance that the introduction of Cu-Te bonding in experimental synthesis is helpful to improve the thermoelectric properties.

Keywords: Cu-based compounds; Seebeck effect; thermoelectric properties; * Email: [email protected]

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1.

INTRODUCTION Thermoelectric (TE) devices, which can directly convert thermal energy to electrical energy,

have a potential to play a significant role in averting global energy crisis. The conversion efficiency of thermoelectric materials can be characterized by a non-dimensional figure of merit, ZT = S 2 σT /κ, where S is the Seebeck coeficient, T is the absolute temperature, σ is the electrical conductivity, κ is the thermal conductivity which is the sum of electronic (κe ) and lattice (κl ) thermal conductivity[1]. Cu-based compounds (Cu2SnX3, X = Se, S, Cu3SbSe4, and Cu-III-VI2, with III = Al, Ga and In, and VI = S, Se, and Te) have achieved much attention owing to their optical properties for photovoltaic application and thin film solar cell applications. Recently, some ternary of them (e.g., Cu2SnSe3, and CuInTe2) also attract eyes for their superior TE properties [2–5]. Taking CuInTe2 as an example, it has been reported as a promising TE material with a ZT of 1.18 at 850 K which is a higher value for undoped ternary chalcopyrite semiconductors [5]. CuInTe2 is a member of ternary diamond-like structure chalcopyrite semiconductors AI BIII CVI2 (AI = Cu, Ag; BIII = Al, Ga, In; CVI = S, Se, Te), which can be considered deriving from AII BVI cubic zincblende-type compounds and substituting group I and IV elements for group II cation sites. The two types of cations make these ternary chalcopyrite semiconductors AI BIII CVI2 have different structures and physical properties from their binary parent zincblende structure, especially low κ (e.g., Cu2SnSe3 and Cu2GeSe3) [4, 6–10]. It may be due to the strong phonon scattering by the ordered vacancy defects or highly distorted crystal structures, which mainly affects κl and lead to the lower κ for diamond-like materials[4, 11–15]. Therefore, the ternary diamond-like structure chalcopyrite semiconductors have relatively high ZT value. Many theoretical works have investigated the properties of CuInTe2. However, most of the studies used the exchange-correlation potential with the term of local density approximation (LDA) or 2

the generalized gradient approximation (GGA)[16-18]. Due to the existence of Cu-3d orbits in Cu-based semiconductors, LDA and GGA often wrongly predicts the band gap. The band gap of CuInTe2 and CuInS2 calculated by GGA or LDA was close to zero[16-18]. In our calculations using the GGA, the band gap of CuInS 2 , CuInSe 2 and CuInTe 2 are only 0.041eV, 0.041 eV, and 0.02 eV, respectively. These calculated results are much smaller than the experimental values. More importantly, the calculated thermoelectric properties in BoltzTraP code are strongly based on the correct electronic structure. Therefore, it is necessary to study the thermoelectric properties of CuInX 2(X=S, Se, Te). Additionaly, according to the information I have, few theoretical literature reports on the thermoelectric properties of CuInS2 and CuInSe2. This triggers us to study the electronic structures and transport properties of ternary diamond-like structure chalcopyrite semiconductors CuInX2 (X = S, Se and Te), and we deduced that CuInS2 and CuInSe2 also have promising TE properties. Previous studies on AgGaTe2 and Cu2SnX3(X=Se, S)[19] pointed out that the combination of heavy and light bands near the band edge was the key factor for their high ZT value. However, the nature of the combination of the heavy and light bands near the Fermi level is an open question. In this paper, we calculated the density of states (DOS) to analysis the possible reason which may be helpful for better understanding the Cu-based compounds and further optimize their TE performance. 2.

COMPUTATIONAL DETAIL

The structures of studied ternary chalcopyrite semiconductors AI BIII CV I2 (I = Cu, Ag; III = Ga, In; VI = S, Se, Te), were optimized with the Vienna ab initio simulation package (VASP) which based on density functional theory (DFT) [20, 21]. The exchange-correlation potential was in the form of Perdew-Burke-Ernzerhof (PBE) generalized-gradient approximation (GGA). The planewave cutoff energy was set as 400 eV, and for the Brillouin-Zone integration, a 11×11×5 Monkhorst-

3

Pack special k-points grid was used. The energy convergence criterion was chosen to be 10−6 eV. The Hellmann-Feynman force on each ion was less than 0.025 eV/ Å. The electronic structures of AI BIII CVI2 (I = Cu, Ag; III = Ga, In; VI = S, Se, Te) were calculated with the full-potential linearized augmented plane waves method [22] which implemented in the WIEN2k[23–25]. The cutoff parameter RmtKmax = 7 with Kmax is the magnitude of the largest k vector and Rmt is the smallest muffin-tin radius. Self-consistent calculations were performed with 1500 k-points in the irreducible Brillouin zone (BZ), and the total energy was converged when the energy difference was less than 0.0001 Ry. Due to the heavy atoms In and Te, Spin orbit coupling (SOC) calculations are considered in our calculation. It has been well documented that LDA or GGA cannot accurately describe the exchangecorrelation effect of strongly localized of Cu-d electrons, and fail to accurately calculate the p-d hybridization and the chemical bonding, which makes it hard to obtain reliable electronic structures for Cu-based semiconductors. Among various schemes that aim for better treatment of strongly localized d-electron systems, the parameter coulomb energy U has been widely used as a simple yet effective approach. Moreover, it will give very similar band structures and accurate band gaps comparing with the experimental data when an on-site coulomb U is considered for Cu-based semiconductors [2, 26–28]. For some Cu-based multinary semiconductors (such as Cu2 SnS3 [29] ), previous studies suggested that the band gaps were well agreement with the experimental data when the U=4 eV. To verify the effectiveness of the method, the electronic structure of Cu2 SnS3 , shown in Fig. 1, is calculated using the mBJ+U method with the plus U=4 eV on the Cu-d electrons. Comparing with the methods PBE+U and Heyd-Scuseria-Ernzerhof (HSE) hybrid function[2], it exhibits the similar band structure characteristics, especially the band edge features. In addition, the band gap of Cu2 SnS3 with the method of mBJ+U is 0.95 eV, which is well coincided with the experiment data of 0.95 eV [30]. Thus, the mBJ+U method is the right approach for Cu-based 4

compounds. However, the U is system dependent, and hence the accuracy value U used for different kinds of compounds must be debugged. In our test, we chose different U (from 0 to 6 eV) to calculate the electronic structure. Comparing the calculated results, the U=4 eV is the best choice for CuInX2. The U=4 eV have been used for other Cu-based multinary semiconductors(CuGaS2, Cu2ZnSnS4)[27]. The calculated band gaps using different exchange-correlation potential comparing with the experimental data are shown in Table 1. On the basis of the correct electronic structure with the SOC, the TE properties are obtained using the semiclassical Boltzmann theory with the relaxation time approximation which is implemented in the BoltzTraP code. In the transport calculations, we carried out the tests and found that 15000 k points in BZ was enough for the thermoelectric properties. In our study, the rigid-band approach is used to simulate thermoelectric performance as a function of the doping level. The rigid-band method simulates doping by moving Fermi level. This method has been used successfully to study the transport coefficients for many thermoelectric materials. 3.

RESULTS AND DISCUSSIONS

3.1 Formation energy The formation energy is used to judge the thermodynamic stability of the compounds. Usually, a negative formation energy indicate that the compounds is thermodynamically stable. We calculated the formation energy according to the following Eq. (1). E  E (CuInX2 )  E (solid Cu)  E (solid In)-2E (solid X)

(1)

The negative value E eV listed in Table 2 indicates that CuInX2 (X = S, Se and Te) can be experimentally synthesized. 3.2 Crystal structure CuInX2 (X = S, Se and Te) have the same tetragonal structure with space group of I-42D. The optimized lattice parameters with the PBE-GGA are satisfactory agreement with the experimental 5

values. The results are listed in Table 3. The theoretical lattice parameters of CuInX2 reproduced their experimental values within an error of less than 5%. Due to the similar crystal structures of CuInX2 (X = S, Se and Te), here, we take the crystal structure of CuInTe2 as an example and show it in Fig. 2. Each primitive cell contains eight X atoms, four Cu atoms, and four In atoms, respectively. Each kind of atoms has only one crystallographically unique atom. As seen from this figure, each Cu or In atom bond with four Te atoms, forming corner sharing tetragonal (Cu-4Te and In-4Te) network with Cu and In located in the center of tetrahedrons. The same space group and the same chalcogen elements for S, Se and Te motivate us to research the TE properties of CuInS2 and CuInSe2. 3.3 Electronic structure To address general features of the electronic structures of CuInX 2, we begin with band structures. Fig. 3 shows the band structures of CuInX2 with SOC and without SOC. We find that the some bands splitting especially at the high symmetry points. Comparatively, the bands splitting features are more obvious for CuInTe2 due to the heavier Te atoms. Meanwhile, the band gap has a small drop when the SOC is considered. However, there is little change for the bands at the Γ point. As seen from Fig. 3, the band structures of CuInX2 are similar, and they are all direct band gap semiconductors with their VBM (valence band maximum) and CBM (conduction band minimum) located in Γ point. The valence band edges of CuInX2 contain multiple bands, and one single band in the conduction band edges. Another important feature of the valence band structures is the combination of heavy and light bands near the valence bands edge. As we know[3], a heavy band is helpful for obtaining a larger effective mass which leads to a larger S, while a light band will result in a higher carrier mobility which means a higher σ. Therefore, the heavy-bands/light-bands combination in the VBM possibly brings a large power factor S 2 σ. Therefore, we concluded that CuInX2 are all promising p-type TE materials, and we paid main attention on the valence bands. 6

In Fig. 4 and Fig. 5, we show the total density of states (TDOS) and partial density of states (PDOS) of CuInX2. It is clear from Fig. 4 that the TDOS of CuInX2 (X = S, Se and Te) in valence bands (VBs) are relatively higher than that in conduction bands (CBs). According to the theory[31], the TDOS of the conduction band minimum are extremely low, which is unfavorable for electrical transport properties, indicating that p-type CuInX2 may have better TE performance. Fig. 5 is the PDOS of CuInX2 (X = S, Se and Te), which exhibits the feature of strong p-d hybridization in VBs. In the energy region of 0 eV to -5 eV, the states are mainly composed of Cu d states and X p states, and the contribution from the In states are nearly negligible. Thus, the properties of the p-type materials are mainly affected by the Cu d and X p hybridization. To understand chemical bonding character between Cu, In, and X of CuInX2, we plot the partial charge density (PCD) distribution in Fig. 6. It shows the real space distribution of the corresponding electronic states. Here, Fig. 6 (a), (b) and (c) show the energy region of 0 eV to -2 eV for CuInX2, and the distribution of the PCD in these regions appears the nearly nonoverlapping part between Cu atoms and X atoms. It means the obvious antibonding character of Cu and X. For the energy region of -3 eV to -5 eV, shown in the Fig. 6 (d), (e) and (f), the nonoverlapping part between Cu and X atoms disappear, and the distribution of the PCD exhibits strong covalent bonding character of Cu and X. Combined with the PDOS, we can get that, due to the strong p-d hybridization, the antibonding states of Cu and X atoms play an important role in the valence band edge, and they determine the transport properties of p-type CuInX2. In addition, the charge density around In atoms is little, corresponding to the very low DOS distribution shown in Fig. 4 and Fig. 5, indicating the little influence of In on the p-type electrical transport properties, and it just donates extra carriers for tuning carrier concentration for performance optimization. Therefore, the band structures close to the VBM change slightly when doping on the In-site. Thus, the Cu-X bond network in CuInX2 forms the 3D conductive network of these complex diamond-like compounds which is favorable for the hole transport. While, In atoms provide electrons to tune carrier 7

concentration, and the In site should be a good choice to optimize the thermoelectric properties in CuInX2. 3.3 Transport properties 3.3.1 Electrical conductivity The above analysis of electronic results indicates that p-type CuInX2 may have promising TE properties. Their TE properties can be enhanced by appropriately doping. For example, tuning carrier concentration has been demonstrated in the Cu 2SnSe3 system with In, Mn, or Zn on the Sn site, or Mg on the Cu site[32–34]. Thus, in current work, we simulated the doping effects and calculated the transport properties of p-type CuInX2. With the semiclassical Boltzmann theory, we only get the σ/τ. To estimate ZT value, we used semi-empirical method to calculate the relaxation time τ and get the estimated value of σ for CuInTe2. For doping dependence, there is a standard electron-phonon form, τ ∝ n−1/3, and within certain regime, there is an approximate electronphonon T dependence, σ ∝ T −1[35]. Thus, it yields ,

(1)

where C is the constant term. There are three sets of measured carrier concentration and electrical conductivity at 300 K in Ref. 5. Thus, we can calculate three constant terms with the calculated value of σ/τ at the same temperature and carrier concentration. They are C = 1.50 × 10−5, C = 1.83 × 10−5, and C = 1.85 × 10−5, and the average of constant them is C = 1.73 × 10−5. Thus, for CuInTe2, we get that . Therefore, we can obtain the estimated value of σ and κe through

(2)

k   and e  . However, there  

is no experimental σ about CuInS2 and CuInSe2. Considering the same space group and the similar crystal structure of them with CuInTe2, we used τ of CuInTe2 to roughly estimate the σ and ZT value of CuInS2 and CuInSe2. 8

The TE properties as a function of carrier concentration are shown in Fig. 8. As seen in this figure, CuInX2 have promising TE performance, and their S and σ are all high, corresponding to the discussion about the combination of heavy and light bands in their band structures. S is proportional to the DOS effective mass and is inversely to the carrier concentration. The DOS effective mass can be written as: (3) where Nv is the band degeneracy,

are the band effective mass components

along three perpendicular directions x, y, and z, respectively. The relationship between σ and carrier concentration is shown as: (4) ∝ Here, m   2 [

.

(5)

 2 E 1 ]E ( k ) E f is the band effective mass. From Fig. 8, we can see that S of CuInX2 k 2

is CuInS2> CuInSe2 >CuInTe2, while σ of CuInX2 is CuInS2 CuInSe2 > CuInTe2. According to Eqs. (3), (4), and (5), we can get that S of CuInX2 follow CuInS2 > CuInSe2 >CuInTe2 and their σ follow the inverse order. In addition, the band effective mass of the Z direction are larger than that of the X direction for the three materials. Thus, σ of the three materials along X direction has a higher value than that along Z direction. For S, the DOS effective mass is not only affected by the band effective mass, but also by the band degeneracy degree. We can see from the band structure , shown in Fig. 3, that the band degeneracy degree of the valence bands along the X direction is larger than that along the Z direction. This may be leads to the larger S along the X direction of CuInX2. 9

3.3.2 Thermal conductivity The thermal conductivity κ consists of two parts of electronic thermal conductivity (κe ) and lattice (κl ). However, only κe can be obtained in our theoretical calculations. Therefore, we calculate the κl from the Slack model[8]. The form is:

MQn 3 l  A  2T 1/ 3

(6)

Where M ,  3 , Qn ,  ,  , and T is the average atomic mass, the volume per atom, the number of atoms in the unit cell, the acoustic Debye temperature, the Grüneisen parameter, and the temperature, respectively. The parameter A is related to the Grüneisen parameter and it is A 

h 3Q N  2.43 10 8 . The Debye temperature is  D  [ n ( A )]1/3m . Here, h is 0.514 0.228 kB 4 M 1  2





the Planck’s constant, k B is the Boltzmann’s constant, N A is Avogadro’s number,  is the density,  m is obtained from the shear velocities  s and longitudinal velocities  l . The equations are as follows with B and G are the bulk modulus and shear modulus. The B and G can be calculated from the elastic constants based on the Voigt-Reuss-Hill (VRH) approximation.

s 

G

 3 4

l  ( B  G) /  1 2

1

m  [ ( 3  3 )]1/3 3 s l Therefore, we can calculate the lattice thermal conductivity and the electronic thermal conductivity. The thermal conductivity as a function of the temperature is plotted in Fig. 7. With the increase of the temperature, the κ difference between CuInX2 is getting smaller. At T>800 K, 10

the κ values are almost equal. For CuInTe2, theoretical results agree well with the experiment[5] especially at the higher temperature. 3.3.3 TE properties Fig. 8 shows the TE properties along the X direction and the Z direction at T=850 K with the κ=0.92 WmK−1. The most notable is that CuInX2 has larger S and higher σ along the X direction. Thus, it has the promising TE properties and the higher ZT value along X direction. The calculated results are consistent with the above analysis. From Fig. 8(c), the highest ZT of CuInX2 appears in the X direction and the highest ZT value for CuInS2, CuInSe2, and CuInTe2 are 1.36, 1.49 and 1.71. respectively. Thus, we can get that p-type CuInX2 are promising TE materials and its largest S and high σ occur along the X direction. In experiment for CuInTe2[5], the carrier concentration were changed with the annealing time. We choose the carrier concentration corresponding to the experiment and calculate the TE properties with the temperature. In Fig.9, we compared our calculated S, σ, and ZT value of CuInTe2 with experimental data. From Fig. 9, we can see that the calculated data is in good agreement with the measured data at T>700 K. Therefore, our calculations can be considered credible and our current work is helpful for understanding the TE performance of Cu-based ternary chalcopyrite semiconductors. 3.4 The reason of the combination of heavy and light bands The combination of heavy and light bands has a great influence on the TE performance of CuInX2 [2, 16], so it is valuable to find out the nature of the combination of heavy and light bands. From the previous work, we can see that Cu2SnS3 has no combination of heavy and light bands in the VBM or the CBM. Considering their different crystal structure, we changed the lattice constants of Cu2SnS3, and took the lattice constant value of c as a, and made β as 90 degree. That is to say, we changed the monoclinic Cu 2SnS3 to the tetragonal structure. We calculated the band structure of tetragonal Cu2 SnS3 with the same method of CuInX2, and 11

showed it in Fig. 10. From Fig. 10, we can see that the tetragonal Cu2SnS3 has higher degenerate than that of monoclinic Cu2SnS3. Therefore, the high-symmetry lattice structure support the presence of high degeneracy. Thus, the cubic and the cubic-like materials always have degenerate band edges and symmetry-related multi-valley carrier pockets [36]. CuGaTe2 and AgGaTe2 have the same tetragonal structures with CuInTe2, but they have the different distortion parameter η (=c/2a). Fig. 11 shows the band structures with SOC of CuInTe2, CuGaTe2, and AgGaTe2. From Fig. 11, we can see that the band structures of CuInTe2 and CuGaTe2 are almost identical, which indicates that once more that In site makes few contribution on the DOS of the p-type CuInX2. However, as to the band structure of AgGaTe2, the secondary band of the valence band edge falls down at , and the combination of heavy and light bands disappears. This attracts our much attention, and we may find the reason of the combination of heavy and light bands when the Cu atom is replaced by Ag atom. To find the reason, we show the PDOS of the three materials in Fig. 12. From Fig. 12 (a) and (b), we can see firstly that the distribution of the DOS of CuGaTe2 is almost the same with that of CuInTe2. The upper valence bands of CuGaTe2 are also mainly composed of Cu d states and Te p states. Comparing Fig. 12 (a) and (c), we find that the weight of Ag d states in the upper valence bands of AgGaTe2 is lower than that of Cu d states in CuInTe2 and CuGaTe2, but the weight of the DOS of Te p of AgGaTe2 in the valence band edge increases slightly. In addition, we also find that the p-d hybridization weaken when Ag substitute Cu, which results in the disappearance of the combination of heavy and light bands in the band structure (shown in Fig. 9). The bond length of Cu-Te or Ag-Te in CuInTe2, CuGaTe2 and AgGaTe2 are 2.53 Å, 2.60 Å and 2.72 Å, which also demonstrates that the p-d hybridization in AgGaTe2 is weaker than that of CuInTe2 and CuGaTe2. As is known, Cu has a higher activity than Ag. Thus, the interaction between Cu-Te atoms is stronger than that of Ag-Te atoms, which leads to the stronger p-d 12

hybridization and the combination of heavy and light bands in Cu-based compounds. Ⅲ.

CONCLUSION

The electronic structure and transport properties of Cu-based ternary chalcopyrite semiconductors have been studied with mBJ + U method based on first-principles. The electronic structure shows that the upper VBs of CuInX2 (X = S, Se and Te) have the combination of heavy and light bands. The heavy band is related to a higher S, and a light band results in a higher σ. Thus, the optimization brings about the promising TE properties of p-type materials. The TE calculations show that the S and k changes greatly at the low temperature range. Increase with temperature to more than 750 K, the value of S, k has a little change. The theoretical results is in good agreement with the experimental data especially when the T>750 K. Therefore, our current work is helpful for understanding the TE performance of Cu-based ternary chalcopyrite semiconductors. Further study show that the high-symmetry tetragonal lattice structures of CuInX2 (X = S, Se and Te) and the stronger p-d hybridization between the Cu d and X p states contribute to the high degeneration. Acknowledgments The authors thanks Dr. Zhenzhen Feng (Institute of Solid state Physics, Chinese Academy of Science) for her help in calculating the lattice thermal conductivity. This work was sponsored by the National Natural Science of China (No. 51371076, 11674083), the program for Excellent Younger teachers in universities in Henan Province of China (2015GGJS-004), the program for the Henan Postdoctoral Science Foundation.

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Figure and Table Caption Table 1 The calculated and experimental band gap (in unit of eV) of AIBIIICVI2 (I = Cu, Ag; III = Ga, In; VI = S, Se, and Te). Band gap (eV)

CuInS2

CuInSe2

CuInTe2

CuGaTe2

AgGaTe2

GGA

0.041

0.041

0.02

0.085

0.302

mBJ + U

1.16

0.88

0.97

1.20

1.14

experiments

1.53

1.04

0.96-1.06

1.20

0.90-1.3

Table 2 The calculated formation energy of CuInX2 (X=S, Se, and Te).

Compounds

Solid Cu

Solid In

2Solid X (X=S, Se Te)

Formation energy

CuInS2(-16.44 eV)

-3.72 eV

-2.51 eV

-7.59 eV

-2.62 eV

CuInSe2(-15.04 eV)

-3.72 eV

-2.51 eV

-6.97 eV

-1.84 eV

CuInTe2(-13.66 eV)

-3.72 eV

-2.51 eV

-6.27 eV

-1.16 eV

Table 3 The Lattice parameter of CuInX2 (X=S, Se, and Te).

CuInS2

CuInSe2

CuInTe2

Lattice parameter

theoretiacal

experimental

theoretiacal

experimental

theoretiacal

experimental

a=b (Ǻ)

5.57

5.53

5.86

5.78

6.29

6.184

c (Ǻ)

11.19

11.14

11.73

11.62

12.52

12.36

16

TABLE 4 The band effective mass of CuInX2 (in unit of me) along different directions. Band effective mass (me )

CuInS2

CuInSe2

CuInTe2

X direction

-0.73

-0.54

-0.36

Z direction

-1.06

-0.83

-0.57

Figure 1. (Color online) Calculated band structure of Cu2SnS3 with mBJ + U method. Figure 2. (Color online) The tetragonal structure of CuInTe 2. Here, the blue balls are Cu atoms, the red balls are In atoms, and the green balls are Te atoms. Figure 3. (Color online) Calculated band structure of Band structure of CuInX 2 (X=S, Se, and Te). Figure 4. (Color online) Calculated DOS of CuInX2(X=S, Se, and Te). The Fermi levels are set at zero. Figure. 5. (Color online) Calculated PDOS of CuInX2(X=S, Se, and Te). The Fermi levels are set at zero. Figure 6. The calculated band decomposed charge density of CuInX2 (X=S, Se, and Te). The energy windows in valence band on the Cu-X-In plane. (a), (b), and (c) show the energy region of 0 eV to -2 eV, and (d), (e), and (f) show the energy region of -3 eV to -5 eV. The unit of charge density is e/ Å3, and the contour level is from 0 to 1 with an interval of 0.005. Figure 7. (Color online) Calculated thermal conductivity of CuInX2 as a function of the temperature. Figure 8. (Color online) Calculated transport properties of CuInX2 as a function of carrier concentration from 1 × 1019 cm−3 to 1 × 1021 cm−3 at 850 K. Figure 9. The comparison between the calculated and experimental data about Seebeck coefficient, electrical conductivity, and ZT value of CuInTe2. Figure 10. Calculated band structures of tetragonal Cu2SnS3. Figure 11.Calculated band structures with SOC of CuInTe2 , CuGaTe2 and AgGaTe2. Figure 12. Calculated PDOS of CuInTe2, CuGaTe2 and AgGaTe2. The Fermi level is set at zero. 17

Figures : Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

Figure 10

Figure 11

Figure 12