# Investigation of PbSnTeSe High-Entropy Thermoelectric Alloy: A DFT Approach

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}σ) than p-type PbSnTeSe because of larger electrical conductivity for n-type doping. Despite high electrical thermal conductivities, the calculated ZT are satisfactory. The maximum ZT (about 1.1) is found at 500 K for n-type doping. These results confirm that PbSnTeSe HEA is a promising thermoelectric material.

## 1. Introduction

^{2}σT/κ [5] is used to evaluate the thermoelectric conversion efficiency in which S, σ, T, and κ are the Seebeck coefficient, electrical conductivity, temperature, and thermal conductivity (electronic and lattice), respectively. However, due to the strong coupling between these parameters, it is not easy to improve the thermoelectric efficiency [6]. Therefore, in order to improve the thermoelectric efficiency, various routes have been explored, e.g., using band engineering to improve power factor (S

^{2}σ) [7,8], and reducing the dimensionality of the material to reduce the lattice thermal conductivity [9,10,11]. In addition to improving the thermoelectric efficiency of existing materials with these strategies, finding new thermoelectric materials is also an important approach [12,13,14].

_{mix}), defined as ∆S

_{mix}= −R∑

_{i}c

_{i}lnc

_{i}, where c

_{i}and R are the compositional ratio and the gas constant, respectively [15]. It is generally admitted that the high entropy of mixing favors the formation of solid solutions and reduces the number of phases [15]. Because of the lattice distortion effects [43], which reduce phonon velocity and enhance the scattering of phonons, high-entropy alloys generally have low lattice thermal conductivity [44,45,46]. As high-entropy sulfides, Cu

_{5}Sn

_{1.2}MgGeZnS

_{9}has been reported with a ZT value of 0.58 at 773 K [47]. The high-entropy metal chalcogenide (Ag,Pb,Bi)(S,Se,Te) alloy with a NaCl-type structure has been investigated and it was found that this compound is a n-type semiconductor with very low κ

_{L}and good power factor resulting in a figure of merit of 0.54 at 723 K [48]. Recently, an n-type PbSe-based high-entropy material formed by entropy-driven structural stabilization was studied for its thermoelectric properties. The ZT value was found to reach 1.8 at 900 K, which corresponds to a material exhibiting good thermoelectric properties [49]. Apart from HEA, other, more conventional, types of compounds have been reported bearing low thermal conductivity, such as Zintl phases (e.g., Ba

_{2}ZnSb

_{2}[50]), argyrodites [51], sulfide-containing films [52], rare-earth molybdates [53], perovskites (e.g., [54]), and defective metal chalcogenide thin films [55], to cite a few. Typically, the thermal conductivity of these compounds lies below 1 W/(m K). The reasons, or the conjunction of reasons, for this low thermal conductivity have been identified by means of the combined density-functional theory and Boltzmann transport theory approaches. The investigations have evidenced large Grüneisen parameters, low-lying optical and acoustic phonon frequencies, short phonon lifetime, and the presence of defects. We shall mention that, to date, several papers report on the prediction of the thermal conductivity of materials using a combined approached based on ab initio molecular dynamics used to train a machine-learned force field subsequently used with the Green–Kubo formalism to derive the heat flux and thermal conductivity [56,57,58]. These approaches are similar to the one used in this work, which is based on the work from Verdi et al. [59]. It has been shown that this machine-learned approach is very effective and yields property data close to those obtained from the Boltzmann transport theory (see, e.g., Ref. [60]).

## 2. Computational Details

## 3. Results and Discussion

#### 3.1. Structural and Electronic Properties of PbSnTeSe High-Entropy Alloy

#### 3.2. Seebeck Coefficient, Electrical Conductivity, and Power Factor of PbSnTeSe

_{B}, m*

_{d}, T and n are the Planck constant, Boltzmann constant, density of states effective mass and carrier concentration, respectively. According to this expression, apart from the effect of temperature, the Seebeck coefficient is governed by the ratio m*

_{d}n

^{−2/3}. Assuming the simple evolution of the density of states (DOS) for 3D materials as the square-root of the state energies (see, e.g., Figure 39.1 in Ref. [78]), at low doping level the curvature radius increases drastically and hence the DOS mass, and overall, the Seebeck coefficient increases sharply. As the doping level increases further, the DOS mass becomes roughly constant, and the n

^{−2/3}term starts dominating in the Mott formula, leading to a decrease of S. For both low and highly doped compound (n ~ 10

^{17}cm

^{−3}and n ≥ 10

^{21}cm

^{−3}), the Seebeck coefficient is improved with the increase in the temperature, which can also be understood by this formula. In the meantime, the maximum S values are reduced with the temperature increase. The peak values of S for PbSnTeSe are all in the range of 160–190 μV/K with little difference between n-type and p-type.

^{19}cm

^{−3}) $\sigma $ decreases with the increase in temperature, which can be understood from the Drude–Sommerfeld formula [79,80,81]:

^{2}) and 7–8 mW/(m K

^{2}), respectively. In each case, the optimal PF values for n-type doping are larger than those for p-type doping because of the higher electrical conductivity, indicating that n-type doping is more efficient than the p-type at improving the TE performance of PbSnTeSe.

#### 3.3. Electronic Thermal Conductivity of PbSnTeSe

_{0}that takes the value 2.44 × 10

^{8}WΩK

^{−1}. Whereas the deviation of the L/L

_{0}ratio from one is still an open question for nanoscale materials [84], it seems that the deviation from the Wiedemann–Franz law occurs mainly at low temperature (well below 300 K) where lattice vibrations increase the L/L

_{0}ratio above 1. However, this tendency can also be counteracted by electronic corrections, leading finally to a small change of the L/L

_{0}ratio (between 0.8 and 1.2, at most). For bulk compounds, the same effects can occur. Therefore, we are confident that the conclusions presented hereafter should not change drastically with L. Figure 6 shows the calculated ${k}_{e}$ at 300 K, 500 K, and 700 K as a function of the carrier concentration. Due to the linear correlation between ${k}_{e}$ and $\sigma $, the impact of carrier concentration and temperature on the electronic thermal conductivity is the same to that on the electrical conductivity. For both n-type and p-type doping, ${k}_{e}$ increases with the increase in carrier concentration, and increases (decreases, resp.) with temperature for low (high, resp.) carrier concentrations. In each case, the n-type PbSnTeSe has a larger ${k}_{e}$ than the p-type.

#### 3.4. Machine-Learned Force-Field Potential

#### 3.5. Lattice Thermal Conductivity of PbSnTeSe

^{−1}m

^{−1}, which is a very low value of the lattice thermal conductivity, favorable for thermoelectric materials. This is due to the lattice distortion effect of high-entropy alloys that can reduce phonon velocity and enhance phonon scattering, resulting in low thermal conductivity. Additionally, one can observe that when the temperature rises, the thermal conductivity of the lattice decreases, which is also due to an increase in phonon scattering at higher temperatures.

#### 3.6. Figure of Merit of PbSnTeSe

#### 3.7. Comparison with Available Data on PbSnTeSe and Other HEA

^{19}e/cm

^{−3}with the experimental results from Fan et al. The calculated total thermal conductivity (2.3 W/(mK)) is higher than that obtained experimentally, which should degrade the theoretical ZT value, but this is not what we observe. By contrast the calculated power factor S

^{2}σ is much higher (almost eight times as high). As our Seebeck coefficient is the same as the experimental one, the reason for the difference is to be found in the electrical conductivity. Indeed, σ

_{theo}amounts to 15 × 10

^{4}S/m. This large value can be explained by two factors. First, the calculated gap that includes the spin–orbit interaction is smaller by a factor of three than the experimental result, and second, we are modeling a pure, defect-free compound. In real compounds, electrons are scattered by impurities, defects, and grains boundaries, which are not accounted for in our model.

^{4}S/m) as that of PbSnTeSe. Assuming further that the thermal conductivity is similar for both compounds, one can infer that these compounds perform equally. From the experimental side [86], GeSnPbSSeTe shows a similar Seebeck coefficient, but the electrical conductivity is much lower, being of the order of 600 S/m.

_{0.25}Pb

_{0.25}Mn

_{0.25}Ge

_{0.25}Te was investigated both experimentally and theoretically by Wang et al. [87]. A ZT value of 1.0 was found at 700 K, probably due to a drastic decrease in the thermal conductivity down to 0.76 W/(mK) by entropy engineering, compared to SnTe (4 W/(mK)). Compared to PbSnTeSe, the Seebeck coefficient of Sn

_{0.25}Pb

_{0.25}Mn

_{0.25}Ge

_{0.25}Te is twice as small (~100 μV/K), but the power factor is notably higher (14 × 10

^{−4}W/(mK

^{2})). It was indeed observed that the electrical conductivity increases with alloying with more elements. This result shows that, PbSnTeSe could be an interesting candidate for thermoelectric application as a high-entropy alloy materials, but there is probably room for improvement, in particular on the electrical conductivity by further alloying with other elements.

## 4. Conclusions

^{−1}m

^{−1}. It has been found that the PF values for n-type doping are always larger than those for p-type doping because of the higher electrical conductivity. The n-type PbSnTeSe exhibits better thermoelectric properties than the p-type. The maximum ZT (≈1.1) is found at 500 K for n-type doping. These results confirm that the PbSnTeSe HEA is a promising thermoelectric (TE) material.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Calculated band structures of PbSnTeSe without spin–orbit coupling (SOC) (

**a**) and with SOC (

**b**).

**Figure 3.**Seebeck coefficient (S) for n-type (

**a**) and p-type (

**b**) PbSnTeSe at 300 K, 500 K and 700 K as a function of carrier concentration.

**Figure 4.**Electrical conductivity (σ) for n-type (

**a**) and p-type (

**b**) PbSnTeSe at 300 K, 500 K and 700 K as a function of carrier concentration.

**Figure 5.**Power factor (PF) for n-type (

**a**) and p-type (

**b**) PbSnTeSe at 300 K, 500 K and 700 K as a function of carrier concentration.

**Figure 6.**Electronic part of the thermal conductivity (κ

_{e}) for n-type (

**a**) and p-type (

**b**) PbSnTeSe at 300 K, 500 K, and 700 K as a function of carrier concentration.

**Figure 7.**Estimated Bayesian error in the forces (

**a**), energies (

**b**), and stress tensors (

**c**) predicted by the MLFF for the training dataset.

**Figure 8.**Lattice thermal properties at 300 K from GK theory. (

**a**) Heat-flux autocorrelation function (HFACF) normalized by its zero-time value and (

**b**) lattice thermal conductivity as a function of correlation time at T = 300 K.

**Figure 10.**Figure of merit (ZT) for n-type (

**a**) and p-type (

**b**) PbSnTeSe at 300 K, 500 K, and 700 K as a function of carrier concentration.

**Table 1.**Elastic constant C, deformation potential E

_{1}, effective mass m* and relaxation times τ at 300 K, 500 K and 700 K of PbSnTeSe.

Carrier Type | C | E_{1} | m* | τ (fs) | τ (fs) | τ (fs) |
---|---|---|---|---|---|---|

eV/Å^{3} | (eV) | (m_{e}) | 300 K | 500 K | 700 K | |

Hole | 0.195 | 12.945 | 0.520 | 17.3 | 8.06 | 4.87 |

Electron | 0.195 | 6.434 | 0.303 | 158 | 73.4 | 44.3 |

**Table 2.**Root-means-square errors in the energies, forces, and stress tensors predicted by the MLFF for the training dataset.

Energy (meV/Atom) | Force (eV/Å) | Stress (kB) |
---|---|---|

1.045 | 0.057 | 0.244 |

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**MDPI and ACS Style**

Xia, M.; Record, M.-C.; Boulet, P. Investigation of PbSnTeSe High-Entropy Thermoelectric Alloy: A DFT Approach. *Materials* **2023**, *16*, 235.
https://doi.org/10.3390/ma16010235

**AMA Style**

Xia M, Record M-C, Boulet P. Investigation of PbSnTeSe High-Entropy Thermoelectric Alloy: A DFT Approach. *Materials*. 2023; 16(1):235.
https://doi.org/10.3390/ma16010235

**Chicago/Turabian Style**

Xia, Ming, Marie-Christine Record, and Pascal Boulet. 2023. "Investigation of PbSnTeSe High-Entropy Thermoelectric Alloy: A DFT Approach" *Materials* 16, no. 1: 235.
https://doi.org/10.3390/ma16010235