Thermoelectric Properties of Ce3Te4 under High Pressure: First-Principles Calculation
2017 Volume 58 Issue 11 Pages 1601-1605
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2017 Volume 58 Issue 11 Pages 1601-1605
Based on the first-principles calculation and Mahan-Sofo's theory, we calculated the electronic structure and the thermoelectric figure 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 figure 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 illustrates that the appropriate pressure could change the electronic structure of Ce3Te4, and improves the thermoelectric properties of Ce3Te4.
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 thermoelectric 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 conversion efficiency is relatively low. The conversion efficiency is determined by the thermoelectric figure of merit (ZT), which could be expressed as: ZT = S2σT/(κe + κl)13), where S is the seebeck coefficient, σ is the electrical conductivity, the denominator is the thermal conductivity, which is contributed by the electronic thermal conductivity κe and the phonon thermal conductivity κl14), and T is the temperature. Therefore, the thermoelectric materials with larger ZT values should have a larger seebeck coefficient and electrical conductivity, and also have a smaller thermal conductivity. However, these three parameters are related with each other. So improving the conversion efficiency of thermoelectric materials is the focus and difficulty of thermoelectric materials research.
Within the last three decades, there has been a huge development in thermoelectric materials. Rare-earth chalcogenides 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 coefficient. 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 efficient high temperature thermoelectric material.
According to the Mahan-Sofo's theory about thermoelectric materials18), the ZT values of the materials were determined by the energy of the maximum of density of states (DOS) peak around Fermi level. If the thermoelectric materials 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 significant effect on the physical and chemical properties of thermoelectric materials19). 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 find the pressure corresponding to its high thermoelectric conversion efficiency in this paper.
Ce3Te4 is the III-VI group compound, where the Ce belongs to the lanthanide and contains the f electrons. Under normal pressure, its structure is a cubic Th3P4–type structure, 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 atoms, 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.
Crystal structure of Ce3Te4. The large and small balls represent cerium and tellurium atoms, respectively.
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 gradient approximation (GGA) implemented with a Perdew-Burke-Ernzerhof (PBE) functional21,22). A 301 eV cut off energy is used for the plane-wave expansion and the spin-polarization is not considered. The lattice constants are 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 first Brillouin zone is sampled using the 10 × 10 × 10 Monkhorst-Pack k-point mesh. The high symmetry 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 optimization, the lattice constant is 9.57 Å, which is in good agreement with the results in Ref. 23).
To effectively understand the effect of pressure on thermoelectric properties of Ce3Te4 crystal, we calculate the energy band structure of Ce3Te4 crystal under different pressures 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 find that the energy level at the high symmetry points is not flat, 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 crystal. 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 gradually as the pressure increases. In addition, we also find 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.
(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 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. (b), (d), (f), (h), (j), (l) and (n) represent the energy 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.
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 characteristic. 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 noticed that the height and width of DOS peak also change with pressure.
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.
According to the Mahan-Sofo's analytic model about thermoelectric materials16,18), the electrical conductivity, thermopower, 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 measured 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 under different pressures is different. This means that the pressure 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 specific pressure could make the DOS peak of Ce3Te4 crystal close to the DOS peak of the best thermoelectric material.
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 position of DOS peak for the best thermoelectric material based on the Mahan-Sofo's theory.
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 comparing 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 structure of high efficient thermoelectric materials18). This shows that the appropriate pressure could change the electronic structure of Ce3Te4 crystal, and it is beneficial to improve the thermoelectric efficiency.
(a) The relationship between the height of DOS peak and the pressure. (b) The relationship between the full width at half maximum of DOS peak and the pressure.
In order to explore the effect of pressure on the charge density in Ce3Te4 crystal, we plot the deformation charge density of crystal under 1.1 GPa, 1.3 GPa, and 2.5 GPa pressure 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 under 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 distribution characteristic. This phenomenon, consistent with the DOS chart in the above, only occurs under some specific pressures because of the effect of the pressure on the f electronic distribution of Ce atom. And the thermoelectric properties of the materials are related to the local characteristic of the f electrons. So the specific pressure could change the thermoelectric properties of the materials.
(a), (b) and (c) are for the deformation charge density of Ce3Te4 under the pressure of 1.1 GPa, 1.3 GPa and 2.5 GPa, respectively.
In order to further explain the change of the thermoelectric properties of Ce3Te4 crystal with pressure. We refer to the Mahan-Sofo's theory to estimate the ZT values18) and use the following formulas:
| \[ ZT = k_{0}/k_{l} \] | (1) |
| \[ D(x) = e^{x}/(e^{x} + 1)^{2} \] | (2) |
| \[ k_{0} = \left( \frac{k_{B}}{e} \right)^{2} T \sigma_{0} D (b) b^{2} \] | (3) |
The relationship between the thermoelectric figure of merit and the pressure.
Based on the first-principles calculation, we study the effect 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 efficient thermoelectric materials. By comparing the maximum of peak, the full width at half maximum, and the corresponding ZT values of Ce3Te4 crystal, we find 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 thermoelectric 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 figure of merit of Ce3Te4 will not change along with the systematic shift of the bandgap obtained by DFT calculations.
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 contributed equally to this work.
