Cite this Article
Wu Y Y, Zhu X L, Yang H Y, Wang Z G, Li Y H, Wang B T. First principles calculations on the thermoelectric properties of bulk Au2S with ultra-low lattice thermal conductivity. Chinese Physics B, 2020, 29(8): 087202  
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First principles calculations on the thermoelectric properties of bulk Au2S with ultra-low lattice thermal conductivity
Wu Y Y1, 2, 3, Zhu X L2, 3, Yang H Y4, Wang Z G5, Li Y H1, ‡, Wang B T2, 3, 6, §
School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
Spallation Neutron Source Science Center, Dongguan 523808, China
School of Materials Science and Engineering, Hunan University of Science and Technology, Xiangtan 411201, China
Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China
Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China

 

† Corresponding author. E-mail: liyuhong@lzu.edu.cn wangbt@ihep.ac.cn

Project supported by the National Natural Science Foundation of China (Grant Nos. 11504312, 11775102, and 11805088), the National Basic Research Program of China (Grant No. 2015CB921103), China Postdoctoral Science Foundation (Grant No. 2018M641477), Guangdong Provincial Department of Science and Technology, China (Grant No. 2018A0303100013), and the Fundamental Research Funds for the Central Universities, China (Lanzhou University, Grant No. lzujbky-2018-19).

Abstract

Sulfide nanocrystals and their composites have shown great potential in the thermoelectric (TE) field due to their extremely low thermal conductivity. Recently a solid and hollow metastable Au2S nanocrystalline has been successfully synthesized. Herein, we study the TE properties of this bulk Au2S by first-principles calculations and semiclassical Boltzmann transport theory, which provides the basis for its further experimental studies. Our results indicate that the highly twofold degeneracy of the bands appears at the Γ point in the Brillouin zone, resulting in a high Seebeck coefficient. Besides, Au2S exhibits an ultra-low lattice thermal conductivity (∼ 0.88 W⋅m−1⋅K−1 at 700 K). At 700 K, the thermoelectric figure of merit of the optimal p-type doping is close to 1.76, which is higher than 0.8 of ZrSb at 700 K and 1.4 of PtTe at 750 K. Our work clearly demonstrates the advantages of Au2S as a TE material and would greatly inspire further experimental studies and verifications.

PACS: ;72.15.Jf;;84.60.Rb;
Keyword:;first principles calculation;;thermodynamic transport properties;
1. Introduction

In order to realize human’s requirements for the sustainable development of future energy, we need to develop new and environmentally friendly renewable energy conversion technologies. Among them, temperature difference power generation is an effective energy technology. It uses the Seebeck effect of thermoelectric (TE) materials to directly convert waste heat into electrical energy without the need for mechanical moving parts and chemical reactions.[1,2] Due to its small size, lightweight, long service life, easy control, and pollution-free, TE materials have become one of the hotspots in international materials research.[3–5]

The efficiency of a TE material is mainly determined by the value ZT = S2σT/(κe + κl), where S, T, σ, κe, and κl are the Seebeck coefficient, absolute temperature, electrical conductivity, electron thermal conductivity, and lattice thermal conductivity, respectively.[6–8] Since strong connections are universal among these transport coefficients, it is difficult to optimize all the transport coefficients at the same time to obtain a designed ZT value. Currently used strategies to improve ZT include increasing the power factor (S2σ) or reducing the thermal conductivity. For example, band structure engineering has been used to improve S2σ,[9,10] and nanostructure engineering (nanostructuring, modulation doping, etc.) has been proven to effectively reduce the thermal conductivity.[11–16] Among them, nanostructuring is considered to be a promising method and can be realized by the following three ways: (i) reducing the grain size to nanoscale;[12,17–19] (ii) introducing nanocrystals into nanomaterial hosts;[19,20] (iii) introducing nanoparticles into bulk hosts.[22,23] These approaches effectively help people to ease the complex competitive relationships among those transport parameters.

Metallic sulfide nanomaterials have been widely studied in recent years as a new class of TE materials due to their advantages in modification of nanostructuring. For example, bismuth sulfide (Bi2S3),[24] copper rich sulfides (Cu2S, Cu1.98S, Cu1.97S),[25–28] silver sulfide (Ag2S),[29,30] and SnS[31] have been experimentally synthesized and proved to exhibit good TE responses. In addition, a lot of works (such as doping or synthesizing nanocomposites) have been done to improve their TE conversion efficiency. Tarachand et al.[29] investigated the effect of silver nano-inclusions on TE properties of Ag2S semiconducting nanostructures, and observed a significantly improved TE power factor for Ag2S. Furthermore, they proved that the introduction of 10 vol% of Ag2S metal nanoparticles in metallic CuS results in an increase of ZT by ∼ 37%.[20] Zheng et al.[30] prepared a series of p-type Ag-doped PbS compounds using vacuum melting combined with subsequent spark plasma sintering process, and found that the ZT value was doubled compared to that of the original PbS. All these studies show that metal sulfide nanomaterials have great potential as the ideal next-generation TE energy conversion materials.

Bulk Au2S is a cuprite type compound similar to α-Ag2S that has been experimentally synthesized.[32–34] Recently, Dalmases et al.[35] prepared solid and hollow metastable Au2S nanocrystal by sequential cation exchange reactions. Such solid and hollow Au2S nanocrystal may be applied to nanostructure engineering to obtain some nanocomposites with good TE properties. Furthermore, Au2S may have the same low intrinsic thermal conductivity as Ag2S due to their similar crystal structure and the large mass of the Au atom. Combining all these characteristics, we consider that Au2S may have a good prospect in TE applications. The TE properties of Au2S have not been reported experimentally and theoretically. In this paper, we use first-principles calculations and semi-classical Boltzmann transport theory to study the TE properties of bulk Au2S. Our results may provide forceful theoretical basis for experiments in the future and could help to encourage further experiments in TE and related fields.

2. Computational methods

The electronic properties of bulk Au2S are calculated based on density functional theory (DFT), which is implemented in the VASP.[36] Scalar relativistic projector augmented wave (PAW) pseudopotentials[37] and generalized gradient approximation of Perdew–Burke–Ernzerhof (PBE)[38] to the exchange–correlation functional are used. The s1d10 and s2p4 atomic orbitals are treated as the valence states for the Au and S atoms, respectively. The cutoff energy is set as 550 eV. 7 × 7 × 7 Monkhorst–Pack k-mesh is employed in the Brillouin-zone. The convergence criterion for electronic self-consistency is set to 10−5 eV. The forces on the atoms are smaller than 0.01 eV⋅Å−1.

The electronic transport properties are obtained by the semiclassical Boltzmann transport theory and the rigid band approach, implemented in the BoltzTraP code.[39] The constant scattering time approximation is used to calculate the constant relaxation time. This is valid if the scattering time does not vary strongly with the energy on a scale of kBT.[40] In order to obtain accurate Fourier interpolation of the Kohn–Sham eigenvalues, the k-mesh of 25 × 25 × 25 is employed in the Brillouin region. The Boltzmann transport equation as implemented in ShengBTE code[41] is used to evaluate the phonon transport properties. The VASP and Phonopy packages[42] are used to obtained Harmonic phonons. The 2 × 2 × 2 supercell and 5 × 5 × 5 k-mesh are used for calculation. The third-order force constants (3rd IFCs) are calculated using the same supercells by a finite displacement approach. The 3rd IFCs consider the interaction including the sixth nearest neighbor atoms. Here, the convergence of κ1 is carefully tested with respect to the k-grids. This system uses a 11 × 11 × 11 k-grid with good convergence.

3. Results and discussion
3.1. Crystal and band structure of bulk Au2S

Bulk Au2S belongs to the cubic crystal system, space group Pn-3m (No. 224), and its unit cell contains two formula units [4 Au and 2 S, as shown in Fig. 1(a)]. Four Au atoms are bonded around the central S atoms. The S–Au–S and Au–S–Au bond angles are 180.0° and 109.645°, respectively. Our optimized lattice parameters are a = b = c = 5.33 Å, which are similar to Suárez’s[43] calculation results (a = b = c = 5.34 Å) and slight larger than Ishikawa’s[34] experimental values (a = b = c = 5.02 Å). To our knowledge, the electronic band of bulk Au2S has not been reported in experiments. The optical band gap of colloidal Au2S nanoparticles was experimentally measured to be 1.8 ± 0.2 eV,[33] which falls within the theoretically proposed range of 1.3–2.6 eV.[44] Suárez et al.[43] reported that the PBE functional is well adapted to describe the Au2S system. Because the band gap computed at the HSE06 level is 3.0 eV, clearly overestimating the gap, we only use the PBE function to calculate the band of Au2S. As seen in Fig. 1(b), bulk Au2S is a semiconductor with a direct band gap of 1.95 eV, which is similar to Suárez’s[43] calculation (1.94 eV). Both the valence band maximum (VBM) and the conduction band minimum (CBM) are located at the Γ point. A twofold degeneracy of the valence band (VB) primarily originates from the d-orbital of Au. Such kind of degeneracy is the key for achieving high ZT,[45] and indicates excellent p-type thermoelectric performance.[46] In addition, the relatively flat degenerate state at the edge of the valence band is beneficial to obtain higher Seebeck coefficients.[47] When the sizes of the materials are reduced to the nanoscale, the quantum constraint effect will have an effect on ZT. Generally, the sharp peaks in the density of electronic states (DOS) often result in a high S2σ.[48–50] From the DOS of Au2S in Fig. 1(b), several sharp peaks are observed at the VB, which are beneficial to its electronic transport properties.

Fig. 1. (a) Crystal structure of bulk Au2S (Au: orange; S: yellow). (b) Total and partial densities of states and band structure of Au2S calculated with PBE. The Fermi level is set at zero.

To examine the stability of bulk Au2S at high temperature, we perform the ab initio molecular dynamics (AIMD) simulation at 300 K, 500 K, and 700 K with a time step of 1 fs. As shown in Fig. 2, the free energy of the system fluctuates slightly in a range of stable values during the simulation process at 300 K, 500 K, and 700 K. Besides, the illustrations show that the crystal structure of Au2S is well kept after heated for 10000 fs. As seen in Fig. 3, with the increase of temperature, the radius distribution function (RDF) of S–S, Au–Au, and Au–Au between 1–10 Å maintains a certain peak value, indicating that Au2S maintains a long-term sequence at 300–700 K. Each of these peaks has a broadening due to the strong vibrations of the particles near the lattice point at evaluated temperatures. These results clearly prove that Au2S can maintain structural stability even up to 700 K.

Fig. 2. Free energy fluctuation and the final structure of Au2S in AIMD simulations at 300 K, 500 K, and 700 K.
Fig. 3. The RDF of Au2S at 300 K, 500 K, and 700 K.
3.2. Electrical transport properties

To evaluate the TE performance of bulk Au2S, we systematically study the electronic transport coefficients (S, σ, and κe) using the Boltzmann transport theory. The electrical transport coefficients as functions of the chemical potential (μ) and temperature (T) are given by[41]

where α, β, ε, and V are the Cartesian indices, energy, and the volume of the unit cell, respectively. Σαβ(ε) is the transport distribution function and can be defined as

where N0, i, τe, and υ represent the sum of q points, the band index, the electron relaxation time, and the electron group velocity, respectively. By semi-classical Boltzmann theory with the constant relaxation approximation, we know that the electrical conductivity can be obtained from σ/τe, and τe is a constant.[53] It should be noted that the doping level of the system depends on μ within the rigid band model. The positive and negative μ imply n-type and p-type dopings, respectively.

After calculation, we find that the electrical transport properties of Au2S are isotropic, which results from its perfect lattice symmetry. The Seebeck coefficients of Au2S only along one direction as a function of μ are illustrated in Fig. 4(a). Obviously, the Seebeck coefficients decrease with the increase of temperature, which is a typical phenomenon of TE materials. At room temperature, the maximum Seebeck coefficient of Au2S for p- and n-type dopings are 2.94 mV/K and 2.68 mV/K, respectively, which are particularly higher than those of some well-researched TE materials, such as bulk Cu2S (0.3 mV/K),[28] MoSe2 (0.9 mV/K),[52] and WSe2 (0.8 mV/K).[53] The S and DOS satisfy the Mott relation[54]

where kB is the Boltzmann constant, n is the carrier density, μm is the carrier mobility. It implies that the stair-like PDOS contains several sharp peaks [seen in Fig. 1(b)], which can enhance the carrier concentration n(ε) and result in high Seebeck coefficient of Au2S.

Fig. 4. (a) Seebeck coefficients, (b) electrical conductivity, (c) electronic thermal conductivity, and (d) power factor with respect to the scattering time as functions of the chemical potential μ.

The electrical conductivity σ/τe along the one direction as a function of μ is depicted in Fig. 4(b). We find that the σ/τe of n-type doping is better than that of p-type. This is mainly because for the VBM and CBM, the curvature is different, and the group velocity of the carriers is different. As the curvature of the dispersion curve around VBM is smaller than that of CBM, the transport distribution function of n-type doping in formula (3) is larger than that of p-type, resulting in the σ/τe of n-type doping larger than that of p-type. As shown in Fig. 4(c), the κe of Au2S is defined by[55]

where L is the Lorenz number (). Similar to σ/τe, κe/τe has a weak anisotropic as well. Besides, we find that the values of κe/τe increase with the temperature and the doping, and the n-type doping is better than the p-type one. The power factor relative to the scattering time (S2σ/τe) is shown in Fig. 4(d). At room temperature (300 K), the S2σ/τe for p-type doping is higher than that of n-type. But at a higher temperature (500 K and 700 K), it has the opposite result where the S2σ/τe of n-type is higher than that of p-type, which can be attributed to its electrical transport properties.

3.3. Thermal transport properties

The phonon dispersion curves and the partial atomic phonon density of states (PhDOSs) for bulk Au2S are shown in Fig. 5. It is noted that all vibrational modes are found to be positive, indicating the dynamically stable nature at ambient pressure for our system. The Au atoms dominate the low frequency region (below ∼ 5 THz), while the high area (from ∼ 10.5 THz to 11.8 THz) is mainly contributed by the S atoms. Besides, the low-frequency optical mode crosses the acoustic branch and softens, leading to strong acoustic–optical interactions. This feature is similar to that of some well-known TE materials, such as PbSe,[56] MoSe2, and WSe2.[57] Form the PhDOS, several peaks can be observed in the optical branch region, which can give rise to the small phonon group velocity. Besides, the relative flat phonon dispersion curves also are beneficial to small phonon velocities, leading to the low lattice thermal conductivity.

Fig. 5. (a) Phonon dispersion and corresponding total and partial phonon density of states of Au2S. (b) Calculated lattice thermal conductivity with respect to temperature for Au2S.

The lattice thermal conductivity κl is defined as

where V, ci, υi, and τi are the crystal volume of the primitive cell, the model heat capacity, the phonon group velocity, and the phonon relaxation time, respectively. The ci and υi are obtained from the phonon dispersion based on the harmonic 2nd IFCs, and τi is obtained from the anharmonic 3rd IFCs. The κl of Au2S, at temperatures from 200 K to 800 K, are shown in Fig. 5(b). At room temperature, the κl of Au2S is about 1.99 W⋅m−1⋅K−1, which is smaller than that of bulk MoSe2 (2.32 W ⋅ m−1⋅K−1)[49] and MoS2 (34.50 W⋅m−1⋅K−1).[58] Furthermore, the intrinsic κl decreases upon heating. At 700 K, the lattice thermal conductivity is as low as 0.88 W⋅m−1⋅K−1. Such low κl of Au2S may originate from its mixture of optical and acoustic branches and relatively flat phonon dispersion curves.

To further understand its low thermal conductivity, we extract the phonon group velocities (ν), Grüneisen parameters (γ), phonon relaxation time (τ), and three-phonon scattering phase space (P3) with respect to frequency, as shown in Fig. 6. The phonon velocity is calculated by[59,60]

where ωλ(q) is the phonon frequency of the phonon mode λ. The Grüneisen parameter is defined as

It can be seen from Fig. 6(a) that all acoustic phonon modes (ZA, TA, and LA) have higher group velocities than the optical modes. The group velocities of all modes are very low, and the maximum group velocity is less than 2.75 km⋅s−1. On the other hand, anharmonicity determines the strength of the three-phonon scattering process. The higher anharmonicity is always related to stronger phonon–phonon interaction, lower group velocity, and lower lattice thermal conductivity. Usually, the γ is used to characterize the anharmonic scattering intensity of phonons. As seen in Fig. 6(b), the acoustic phonon has extremely high γ (about 34 for ZA), which means this system has strong anharmonicity scattering between acoustic phonons. The phonon relaxation time can be acquired by the summation of various scattering processes (umklapp scattering, boundary scattering, and defects scattering). From Fig. 6(c), the TA mode has the longest phonon relaxation time, followed the ZA and optical modes, and then the LA mode. It is worth mentioning that the phonon relaxation time of the LA branch is significantly lower than those of the other three branches, indicating that it has the smallest contribution to the total thermal conductivity.

Fig. 6. (a) Phonon group velocities, (b) Grüneisen parameters, (c) phonon relaxation time, and (d) P3 phase space with respect to frequency for Au2S.

In addition, the three-phonon scattering phase spaces are used to evaluate the phonon relaxation time of the system. The P3 can be calculated according to[61]

where Ω is a normalization factor. The large values of P3 can give rise to the large three-phonon scattering process. Figure 6(d) shows the P3 of the phonon modes relative to the phonon frequency. It can be seen that the P3 of the ZA and TA branches are larger than those of the LA branch, which is consistent with their phonon relaxation time. But it cannot be ignored that the P3 of all acoustic modes are very large, which makes them have strong phonon–phonon scattering and prevents the phonon transport. Through the above analysis, we can well understand the cause of ultra-low thermal conductivity in bulk Au2S.

3.4. Thermoelectric figure of merit

The ZT of Au2S can be calculated by combining the phonon and the electron transport coefficients. We present the ZT in Figs. 7(a) and 7(b) as a function of the chemical potential and doping concentration, respectively. Based on the deformation potential (DP) theory,[62] the calculated relaxation time is 2.1× 10−14 s. Form the formula of ZT, we know that the κe has a great influence on ZT. According to our calculated κe, we show that the ZT value of the p-type doping is larger than that of the n-type doping. Because of the low lattice thermal conductivity (∼ 0.88 W⋅m−1⋅K−1) at 700 K, the ZT can approach 1.76. This value is larger than those of many widely studied sulfide materials, such as 0.8 (CuxS at 750 K),[63] 0.33, 0.8 (ZrSb at 700 K),[64] and 1.4 (PtTe at 750 K).[65] Such high ZT values indicate that Au2S is a promising TE material. In addition, the corresponding n-type and p-type doping concentrations in Au2S to obtain the maximum ZT values at 700 K are approximately 5 × 1020 cm−3 and 2 × 1020 cm−3, respectively. Doping at such low concentrations has a slight effect on their phonon and electron structure properties. Overall speaking, Au2S is a very promising material for TE applications.

Fig. 7. The thermoelectric figure of merit as a function of (a) chemical potential and (b) carrier concentration for Au2S at 300 K, 500 K and 700 K.
4. Conclusion

In summary, we studied the TE properties of bulk Au2S using DFT and Boltzmann transport equation. Results indicate that the thermal transport property and the electrical transport of Au2S exhibit isotopic behavior. Its small ν, large γ, short τ, and large P3 significantly inhibit phonon transport and lead to low κl of 1.99 W⋅m−1⋅K−1 at room temperature. Highly degenerate VB, peaky DOS, and large Seebeck coefficients (1200 μV⋅K−1) are observed. The ZT under p-type doping can approach 1.76 at 700 K. Taken together, these results demonstrate the tremendous advantages of bulk Au2S for efficient thermal energy conversion in the 300 K to 700 K temperature range.

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