Thermoelectric Properties of Ce3Te4 under High Pressure: First-Principles Calculation

Jin-Peng Li1,2, Qian-Qian Zhao1,2, Chang Liu1,2, Xiao-Chun Wang1,2,* and Yu-Jun Yang1,2

1Institute of Atomic and Molecular Physics, Jilin University, Changchun, 130012, China2Jilin Provincial Key Laboratory of Applied Atomic and Molecular Spectroscopy, Jilin University, Changchun, 130012, China

Based on the �rst-principles calculation and Mahan-Sofo’s theory, we calculated the electronic structure and the thermoelectric �gure of merit of Ce3Te4 under different pressures. The peak of DOS for Ce3Te4 shows the form of Dirac delta function around Fermi level. When the pressure is 1.1 GPa, the height of DOS peak is higher than those under other pressures, and the full width at half maximum is the narrowest. The thermoelectric �gure of merit of Ce3Te4 under 1.1 GPa, 1.3 GPa, and 2.5 GPa pressure is the highest, which is just below 14.0. This illus-trates that the appropriate pressure could change the electronic structure of Ce3Te4, and improves the thermoelectric properties of Ce3Te4. [doi:10.2320/matertrans.M2017204]

(Received July 7, 2017; Accepted September 1, 2017; Published September 29, 2017)

Keywords:　 �rst-principles calculation, thermoelectric �gure of merit, pressure regulating

1.　 Introduction

The thermoelectric materials, which could realize mutual conversion between heat and electricity, is an effective way to solve the energy problem. Currently, a number of thermo-electric materials are widely applied in the aspect of energy, military, and aerospace1–8). It has many advantages, such as no moving parts, no noise, no pollution, and safety9–12). But the scope of application is still limited because of its conver-sion ef�ciency is relatively low. The conversion ef�ciency is determined by the thermoelectric �gure of merit (ZT), which could be expressed as: ZT =  S 2σT/(κe +  κl)13), where S is the seebeck coef�cient, σ is the electrical conductivity, the de-nominator is the thermal conductivity, which is contributed by the electronic thermal conductivity κe and the phonon thermal conductivity κl

14), and T is the temperature. Therefore, the thermoelectric materials with larger ZT values should have a larger seebeck coef�cient and electrical con-ductivity, and also have a smaller thermal conductivity. However, these three parameters are related with each other. So improving the conversion ef�ciency of thermoelectric materials is the focus and dif�culty of thermoelectric materi-als research.

Within the last three decades, there has been a huge devel-opment in thermoelectric materials. Rare-earth chalco-genides have sparked interest in 1986 for the thermoelectric applications because of the thermal stability and the great contributions of d, f electrons for high Seebeck coef�cient. Recently, Andrew F. May synthesized lanthanum telluride (La3Te4) in the experiment and found that its ZT value reaches 1.1 at 1275 K15). In addition, the thermoelectric properties of cerium telluride (Ce3Te4) crystal under 0 GPa pressure with high temperature have also been calculated, whose ZT value is 13.5 at T  =  1200 K16), which is higher than that of La3Te4, although Andrew F. May found that the thermoelectric transport properties in Ce3Te4 are similar to those of La3Te4 below room temperature17). This shows that the Ce3Te4 crystal is currently the very ef�cient high tem-

perature thermoelectric material.According to the Mahan-Sofo’s theory about thermoelec-

tric materials18), the ZT values of the materials were deter-mined by the energy of the maximum of density of states (DOS) peak around Fermi level. If the thermoelectric mate-rials are desired to have higher ZT values, it is necessary that the DOS exhibits a peak similar to the Dirac delta function around Fermi level. Therefore, the thermoelectric properties of crystal are closely related to the electronic structure. And the pressure and temperature have a signi�cant effect on the physical and chemical properties of thermoelectric materi-als19). However, the change of entropy caused by the cell volume change is very small, so the calculation is easier by changing the pressure than by changing the temperature. Therefore, we explore the high temperature thermoelectric properties of Ce3Te4 crystal under different pressures and �nd the pressure corresponding to its high thermoelectric conversion ef�ciency in this paper.

2.　 Calculation Method

Ce3Te4 is the III-VI group compound, where the Ce be-longs to the lanthanide and contains the f electrons. Under normal pressure, its structure is a cubic Th3P4–type struc-ture, and the space group is I-34d. The lattice constant is a =  b =  c =  9.57 Å and the unit cell volume is 875.37 Å3. 12 Ce atoms and 16 Te atoms are contained in the unit cell. However, in our calculation, we choose the smallest unit cell as the initial structure, containing 6 Ce atoms and 8 Te at-oms, as shown in Fig. 1. In the calculation process, we treat the 4f, 5d, 6s electrons of Ce atom and the 5s, 5p electrons of Te atom as the valence electrons.

In this paper, the energy band structure, DOS, and charge density of Ce3Te4 crystal were calculated using the Vienna ab initio Simulation Package (VASP)20), which is based on the density function theory (DFT) and the generalized gradi-ent approximation (GGA) implemented with a Perdew-Burke-Ernzerhof (PBE) functional21,22). A 301 eV cut off en-ergy is used for the plane-wave expansion and the spin-polarization is not considered. The lattice constants are * Corresponding author, E-mail: wangxiaochun@jlu.edu.cn

Materials Transactions, Vol. 58, No. 11 (2017) pp. 1601 to 1605 ©2017 The Japan Institute of Metals and Materials

fully relaxed until the energy changes are less than 1.0 ×  10−4 eV and the interatomic forces are less than 0.02 eV Å−1. The �rst Brillouin zone is sampled using the 10 ×  10 ×  10 Monkhorst-Pack k-point mesh. The high sym-metry points of energy band structure are set to Γ (0,0,0), Η (1/2,−1/2,1/2), Ρ (1/4,1/4,1/4), Ν (0,0,1/2). After optimi-zation, the lattice constant is 9.57 Å, which is in good agree-ment with the results in Ref. 23).

3.　 Results and Discussion

To effectively understand the effect of pressure on ther-moelectric properties of Ce3Te4 crystal, we calculate the en-ergy band structure of Ce3Te4 crystal under different pres-sures as shown in Fig. 2. The order of high symmetry points of Ce3Te4 crystal in reciprocal space is set to Γ-Η-Ν-Γ-Ρ, and the Fermi level is set to 0 eV. We �nd that the energy level at the high symmetry points is not �at, which shows that the Ce3Te4 crystal is an anisotropic crystal. It is worth noting that the conduction band has an obvious overlap with the valence band around Fermi level under all pressures, which shows that the Ce3Te4 crystal is a semi-metallic crys-tal. The lowest point of the overlapping bands appeared at the Γ point, and the highest point of the energy band below the bandgap around Fermi level appeared in the Γ-H line, which shows that the Ce3Te4 crystal is an indirect bandgap crystal. And the bandgap width also tends to narrow gradu-ally as the pressure increases. In addition, we also �nd that there is a region with very high DOS above the Fermi level, and its broadening also changes with pressure. Therefore, the change of pressure could affect the electronic structure of Ce3Te4 crystal, especially the energy band structure in the high density distribution region.

Figure 3 presents the variation of DOS with the energy under different pressures. It is worth noting that the DOS around Fermi level under all pressures shows the form of Dirac delta function. The partial DOS (PDOS) shows the f electrons of the Ce atom mainly contribute to the DOS around Fermi level, while the s, p, d electrons of the Ce atom are relatively weak. This is because the f electrons of the Ce atom have a very obvious localized distribution char-acteristic. We can also see that the p electrons of the Te atom mainly contribute to the DOS within the energy range from −5 eV to −1 eV. However, the electrons of the Te atom have a low contribution to the DOS around Fermi level, which is negligible compared to the peak provided by the f electrons of the Ce atom. Therefore, we can determine that the DOS

of Ce3Te4 crystal around Fermi level is mainly provided by the f electrons of the Ce atom. In addition, it would also be

Fig. 2　(a), (c), (e), (g), (i), (k) and (m) represent the energy band structure of Ce3Te4 within the energy range from −6 eV to 2 eV under the pres-sure of 1.0 GPa, 1.1 GPa, 1.3 GPa, 1.5 GPa, 2.0 GPa, 2.5 GPa and 3.0 GPa, respectively. (b), (d), (f), (h), (j), (l) and (n) represent the en-ergy band structure of Ce3Te4 within the energy range from −0.5 eV to 0.5 eV under the pressure of 1.0 GPa,1.1 GPa, 1.3 GPa, 1.5 GPa, 2.0 GPa, 2.5 GPa and 3.0 GPa, respectively.

Fig. 1　Crystal structure of Ce3Te4. The large and small balls represent ce-rium and tellurium atoms, respectively.

1602 J.-P. Li, Q.-Q. Zhao, C. Liu, X.-C. Wang and Y.-J. Yang

noticed that the height and width of DOS peak also change with pressure.

According to the Mahan-Sofo’s analytic model about ther-moelectric materials16,18), the electrical conductivity, ther-mopower, and electronic thermal conductivity are related to the common transport distribution function s(x), where x is the dimensionless electron energy scaled by kBT and mea-sured from Fermi level, and the ZT value is optimized when the DOS peak locates at x0~±2.4 kBT above the Fermi level. If the thermoelectric material is operated at T =  1200 K, the optimal DOS peak would locate at 0.25 eV. The relationship between the energy of the DOS peak around Fermi level for

Ce3Te4 crystal and the pressure is calculated as shown in Fig. 4, which shows that the DOS peak of Ce3Te4 crystal un-der different pressures is different. This means that the pres-sure could change the electronic structure of Ce3Te4 crystal. Moreover, we see that the DOS peak under 1.1 GPa, 1.3 GPa, and 2.5 GPa pressure is closest to 0.25 eV, which means that the speci�c pressure could make the DOS peak of Ce3Te4 crystal close to the DOS peak of the best thermo-electric material.

As shown in Fig. 5, when the pressure is 1.1 GPa, the height of DOS peak has a maximum value of 500 states/(eV.Unit cell) and a small full width of 0.23 eV at half maximum within the pressure range from 0 GPa to 3.0 GPa. By com-paring the results under 0 GPa pressure, we can see that the DOS peak of under non-zero pressure is higher and its full width at half maximum is smaller than that under 0 GPa pressure. These changes are consistent with the requirement of Mahan-Sofo’s theoretical model for the electronic struc-ture of high ef�cient thermoelectric materials18). This shows that the appropriate pressure could change the electronic structure of Ce3Te4 crystal, and it is bene�cial to improve the thermoelectric ef�ciency.

In order to explore the effect of pressure on the charge density in Ce3Te4 crystal, we plot the deformation charge

Fig. 3　The DOS and the PDOS of Ce3Te4 under the pressure of 1.0 GPa, 1.1 GPa, 1.3 GPa, 1.5 GPa, 2.0 GPa, 2.5 GPa and 3.0 GPa.

Fig. 4　The relationship between the energy at the DOS peak around Fermi level of Ce3Te4 and the pressure. The line at 0.25 eV is to show the posi-tion of DOS peak for the best thermoelectric material based on the Mahan-Sofo’s theory.

Fig. 5　(a) The relationship between the height of DOS peak and the pres-sure. (b) The relationship between the full width at half maximum of DOS peak and the pressure.

1603Thermoelectric Properties of Ce3Te4 under High Pressure: First-Principles Calculation

density of crystal under 1.1 GPa, 1.3 GPa, and 2.5 GPa pres-sure as shown in Fig. 6. Here, the three pressure values are chosen based on the pressure of the three extreme points of curves in Fig. 5(a). The deformation charge density is the difference between the total charge density of the system and the charge density before the atomic interaction. Figure 6 shows that Ce atom exhibits obvious anisotropy un-der these pressures. The difference is that the charge density around the Ce atom under 1.1 GPa and 1.3 GPa pressure is higher than that under 2.5 GPa pressure. It shows that the electrons around the Ce atom have a very obvious local dis-tribution characteristic. This phenomenon, consistent with the DOS chart in the above, only occurs under some speci�c pressures because of the effect of the pressure on the f elec-tronic distribution of Ce atom. And the thermoelectric prop-erties of the materials are related to the local characteristic of the f electrons. So the speci�c pressure could change the thermoelectric properties of the materials.

In order to further explain the change of the thermoelec-tric properties of Ce3Te4 crystal with pressure. We refer to the Mahan-Sofo’s theory to estimate the ZT values18) and use the following formulas:

ZT = k0/kl (1)

D(x) = ex/(ex + 1)2 (2)

k0 =kB

e

2

Tσ0D(b)b2 (3)

where k0 is the electronic thermal conductivity, kl is the lat-tice thermal conductivity, T is the temperature, b is the posi-tion of DOS peak around Fermi level, e is the electronic charge, σ0 =  e2/(ħa0) with ħ being the reduced Plank’s con-stant and a0 being the Bohr’s radius. By setting some idealis-tic parameters, such as temperature T =  1200 K, lattice ther-mal conductivity kl  =  1.0 W/m·K, mean-free-path l  =  a  = 

0.3 nm, and b =  ±2.4 kBT, the curves of the thermoelectric �gure of merit of Ce3Te4 crystal under different pressures are calculated as shown in Fig. 7. The highest ZT values of Ce3Te4 crystal are obtained under 1.1 GPa, 1.3 GPa and 2.5 GPa pressure, which are just below 14.0. This means that the appropriate pressure could improve the thermoelec-tric properties of Ce3Te4 crystal. In addition, we also see that the actual ZT values of Ce3Te4 crystal are lower than the es-timated values because some ideal conditions are used in the estimation, such as setting the lattice thermal conductivity, and ignoring the effect of the low s, p, d electronic distribu-tion on the ZT values. However, it does not affect the trend of the ZT curves with pressure.

4.　 Conclusions

Based on the �rst-principles calculation, we study the ef-fect of pressure on the thermoelectric properties of Ce3Te4 crystal. Under different pressures, there is a high and narrow DOS peak around Fermi level for Ce3Te4 crystal, whose form looks like the Dirac delta function. The calculation of PDOS indicates that this peak is provided by the f electrons of the Ce atom. This phenomenon meets the requirements of Mahan-Sofo’s theory for the electronic structure of high ef�-cient thermoelectric materials. By comparing the maximum of peak, the full width at half maximum, and the corre-sponding ZT values of Ce3Te4 crystal, we �nd that the Dirac delta function form peak of Ce3Te4 crystal has a higher height and a narrower width when the applied pressure is 1.1 GPa, 1.3 GPa, and 2.5 GPa. And the ZT values under 1.1 GPa, 1.3 GPa, and 2.5 GPa pressure are just below 14.0, which is higher than that under other pressures. It indicates that the appropriate pressure could improve the thermoelec-tric properties of thermoelectric materials, which provides a valuable way to improve the thermoelectric properties of materials. Furthermore, in order to compare with the results in our previous work, we use DFT to calculate the bandgap. The mechanism of the pressure effect on the thermoelectric �gure of merit of Ce3Te4 will not change along with the sys-tematic shift of the bandgap obtained by DFT calculations.

Fig. 6　(a), (b) and (c) are for the deformation charge density of Ce3Te4 un-der the pressure of 1.1 GPa, 1.3 GPa and 2.5 GPa, respectively.

Fig. 7　The relationship between the thermoelectric �gure of merit and the pressure.

1604 J.-P. Li, Q.-Q. Zhao, C. Liu, X.-C. Wang and Y.-J. Yang

Acknowledgments

This work was supported by the Natural Science Foundation of Jilin Province of China under Grant No. 20170101154JC. Jin-Peng Li and Qian-Qian Zhao contrib-uted equally to this work.

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1605Thermoelectric Properties of Ce3Te4 under High Pressure: First-Principles Calculation