Potential thermoelectric material Tl₃XS₄ (X = V, Nb, Ta) with ultralow lattice thermal conductivity

Abstract

The demand for sustainable energy solutions has driven intensive research into advanced thermoelectric (TE) materials, to harness waste heat for efficient power generation. Recently, several studies have revealed that Tl₃VS₄ possesses an ultralow lattice thermal conductivity, despite its simple body-centered cubic lattice structure. This paper focuses on the TE properties of Tl₃XS₄ (X = V, Nb, Ta) compounds through a systematic exploration utilizing first-principles calculations and semiclassical Boltzmann transport theory. The results of the AIMD simulation and phonon calculation reveal the excellent dynamic stability and thermal stability of Tl₃XS₄ at 300, 500, and 700 K. Moreover, we find that the Tl₃XS₄ compounds present ultralow lattice thermal conductivity (<0.5 W m⁻¹ K⁻¹ at 300 K), and the nanostructure strategy is effective. Based on the outstanding Seebeck coefficient and ultralow lattice thermal conductivity, the optimal ZT values of Tl₃VS₄, Tl₃NbS₄, and Tl₃TaS₄ at 300 K are determined to be 1.17 (p-type), 0.84 (n-type), and 0.65 (p-type), respectively. Additionally, at each considered temperature, the maximum ZT values of p-type (n-type) Tl₃XS₄ follow the order: Tl₃VS₄ > Tl₃NbS₄ > Tl₃TaS₄ (Tl₃NbS₄ > Tl₃VS₄ > Tl₃TaS₄). Our results demonstrate that Tl₃XS₄ (X = V, Nb, Ta) compounds are promising thermoelectric materials. This exhaustive research enhanced our nuanced comprehension of the electronic, dynamic, and thermoelectric attributes of Tl₃XS₄ (X = V, Nb, Ta), thereby offering valuable insights into the TE field.

---

1. Introduction

The exploration of advanced thermoelectric (TE) materials has become paramount in the pursuit of sustainable energy technologies, as these materials offer the unique capability to convert waste heat into valuable electrical power. Thermoelectric materials are characterized by TE coefficients that play a crucial role in determining their efficiency. The thermoelectric power factor (PF), defined as PF = S²σ, where S is the Seebeck coefficient and σ is the electrical conductivity,¹ represents a crucial metric. High power factors contribute to efficient TE energy conversion.² Simultaneously, the dimensionless figure of merit (ZT) encapsulates the overall efficiency of TE materials and is given by ZT = S²σT/(κ_e + κ_l),^3,4 where κ_e and κ_l are the electronic and lattice thermal conductivities,⁵ respectively. Achieving a high ZT value is paramount for the practical application of thermoelectric materials in energy conversion devices.

Traditionally, thermoelectric research has been guided by empirical models, such as the pioneering work of Slack,⁶ emphasizing the exploration of materials with large unit cells and complex structures to achieve ultralow lattice thermal conductivities. Several materials, predominantly exhibiting complex structures, have been suggested for utilization in TE applications, such as skutterudites,^7–9 half-Heuslers,^10–13 and clathrates.^14–16 However, Tl₃VSe₄ challenges this paradigm by showcasing low lattice thermal conductivities despite its simple body-centered cubic lattice structure.¹⁷ This enigma prompts a reevaluation of existing models and calls for a deeper exploration of the underlying mechanisms governing thermoelectric behavior.

In 2018, Mukhopadhyay¹⁷ introduced a nuanced perspective to clarify the ultralow lattice thermal conductivity in Tl₃VSe₄. His work delves into the intricate interplay of heat transport mechanisms, involving lattice phonons and localized oscillators, thereby providing valuable insights into the TE phenomena exhibited by these compounds.^18–20 The application of multichannel transport theory further enriches our understanding, highlighting the need for a comprehensive approach to decipher the TE behavior of the Tl₃XSe₄ family of compounds. In addition, theoretical studies suggested that Tl₃TaSe₄ and Tl₃VS₄ have ultralow thermal conductivity (0.1–0.4 W m⁻¹ K⁻¹ at room temperature).²¹ The high σ and ultralow κ_l contribute to the high ZT value of 0.85 and 0.84 at 300 K for Tl₃TaSe₄ and Tl₃VS₄, respectively. However, the relaxation time has been set at a constant of τ = 9 × 10⁻¹⁴ s in their calculations of PF and ZT. Recently, Wang et al. revealed that Tl₃XSe₄ (X = V, Nb, Ta) is a potential TE material with excellent performance, including large Seebeck coefficient, high electrical conductivity, high power factor, and low thermal conductivity.²² In conclusion, these compounds of the Tl₃XSe₄^17–20,22 family typically exhibit ultralow κ_l and excellent TE properties and are expected to be candidates for a new generation of TE applications.

The simplicity of Tl₃XY₄ (X = V, Nb, Ta; Y = S, Se) structures, marked by high lattice symmetry and minimal disorder, positions these compounds as promising alternatives for scalable and efficient TE applications. However, the intricacies of their TE behavior necessitate a comprehensive investigation. Notice that the Tl₃XS₄-compounds (X = V, Nb, Ta) with a cubic body-centered lattice have been prepared and synthesized successfully,^23,24 which boosts the investigation of TE properties. However, it is important to highlight that there is a lack of systematic research addressing the electronic, phonon, and thermoelectric properties of Tl₃XS₄ (X = V, Nb, Ta) in the existing literature. This study employs first-principles calculations and semiclassical Boltzmann transport theory to understand the electronic structure, phonon dispersion, thermal conductivity, and thermoelectric transmission characteristics of Tl₃XS₄ (X = V, Nb, Ta). By unraveling the underlying mechanisms, we aim to contribute to the evolving field of thermoelectric materials and shed light on the unique properties of Tl₃XS₄ compounds.

2. Computational details

Density functional theory (DFT) calculations of Tl₃XS₄ (X = V, Nb, Ta) were comprehensively performed using the Vienna Ab initio Simulation Package (VASP).²⁵ All calculations utilized a plane-wave basis set and the projector-augmented wave (PAW) method,^26,27 employing a 500 eV plane-wave cutoff energy. The exchange–correlation functional was treated within the generalized gradient approximation (GGA), utilizing the Perdew–Burke–Ernzerhof (PBE) parameterization.²⁸ A convergence criterion of energy and force were set to 10⁻⁶ eV and 10⁻⁴ eV Å⁻¹ for computational efficiency and accuracy, respectively. To capture the dynamic behavior of Tl₃XS₄ at elevated temperatures, AIMD²⁹ simulations were conducted using VASP within the NVT ensemble based on the machine-learned force fields (MLFF)^30–32 feature. The Nosé–Hoover thermostat maintain temperatures of 300, 500, and 700 K.

The Ab initio scattering and transport (AMSET) package³³ was employed to calculate the electron transport properties, including the Seebeck coefficient S, electrical conductivity σ, carrier scattering rates, and carrier mobilities. The AMSET code employs the momentum relaxation time approximation (MRTA) to solve the Boltzmann transport equation. It provides reliable estimates of electron relaxation times by incorporating multiple scattering mechanisms, including acoustic deformation potential scattering (ADP), polar optical phonon scattering (POP), ionized impurity scattering (IMP), and piezoelectric scattering (PIE).³⁴ These estimates are in excellent agreement with experimental measurements and electron–phonon couplings computed through the Wannier function.³³ The lattice thermal conductivities are obtained utilizing the ShengBTE code,³⁵ in which a dense q-point mesh of 20 × 20 × 20 and the nearest neighbors of 8 (cutoff value) were adopted. Interatomic force constants (IFCs) including the second-order and third-order were extracted from DFT calculations using VASP, and a 2 × 2 × 2 supercell was employed for accurate force constant determination. The Dynaphopy code³⁶ and Phonopy code³⁷ were utilized to gain insights into anharmonic effects and calculate phonon dispersions at the finite temperatures of 300, 500, and 700 K.

3. Results and discussion

3.1. Crystal structures and electronic structures

The Tl₃XS₄ (X = V, Nb, Ta) compounds exhibit a simple body-centered cubic phase with I4̄3m (no. 217) space group. Fig. 1 demonstrate that the XS₄⁻ tetrahedra hold both the body center and vertices in the cubic lattice, while Tl atoms locate at the center of each edge and face of the cube. The relaxed lattice constants of Tl₃XS₄ (X = V, Nb, Ta) are given in Table 1, and the information of Tl₃XSe₄ (X = V, Nb, Ta) also been listed.

Fig. 1 The top views and side views of the crystal structure for Tl₃VS₄ (a), Tl₃NbS₄ (b), and Tl₃TaS₄ (c).

Table 1 Calculated lattice constants and band gap for Tl₃XS₄ (X = V, Nb, Ta). The results of Tl₃XSe₄ (X = V, Nb, Ta) are used for comparison

	a = b = c (Å)	E g (eV)
Tl3VS4	7.67	2.88 (HSE06) 2.06 (PBE)
Tl3NbS4	7.81	3.24 (HSE06) 2.61 (PBE)
Tl3TaS4	7.84	3.31 (HSE06) 2.62 (PBE)
Tl3VSe4^22	7.89	2.33 (HSE06) 1.60 (PBE)
Tl3NbSe4^22	8.03	3.01 (HSE06) 2.18 (PBE)
Tl3TaSe4^22	8.05	2.89 (HSE06) 2.26 (PBE)

---

Using the PBE functional and HSE06 hybrid functional, we calculate the band structures of Tl₃XS₄ (X = V, Nb, Ta) (Fig. 2(a), (c) and (e)). The band gaps of Tl₃VS₄, Tl₃NbS₄, and Tl₃TaS₄ based on the PBE (HSE06) method are 2.06 (2.88), 2.61 (3.24), and 2.62 (3.31), respectively (the details are listed in Table 1). Additionally, the density of states (DOS) based on the HSE06 method revealed that p orbitals of S atoms largely contribute to the total DOS in the valence band maximum (VBM) with some contributions from s states of Tl atoms and V/Nb/Ta/-d orbitals, while the conduction band minimum (CBM) is predominately contributed by d orbitals of V/Nb/Ta atoms with some contributions from p states of Tl atoms. This is consistent with previously reported theoretical and experimental research findings.^21,38,39

Fig. 2 Band structure and DOS, phonon structures and phonon DOS for Tl₃VS₄ (a) and (b), Tl₃NbS₄ (c) and (d), and Tl₃TaS₄ (e) and (f).

3.2. Stability and phonon structure

As is commonly understood, the establishment of a stable structure forms the foundation for the subsequent exploration of diverse material properties. To evaluate the dynamic stability of Tl₃XS₄ (X = V, Nb, Ta) and explore its phonon transport properties, we conduct calculations for the phonon band and DOS, as illustrated in Fig. 2(b), (d) and (f). Notably, imaginary frequency modes have not been observed in the phonon band of Tl₃XS₄ at 300, 500, and 700 K, indicating the extraordinary dynamical stabilities of Tl₃XS₄. The phonon DOS provides insights into the respective contributions of various atoms to the overall phonon spectrum. In the case of Tl₃XS₄, the predominant influence on the acoustic modes arises from the heavy Tl atoms. This observation can be attributed to the contribution of the atom to the phonon band, which is expressed as the ratio of atomic mass to unit cell mass.⁴⁰ Besides, the optical modes primarily originate from the vibrations of lighter atoms S, with some additional contributions stemming from the vibrations of V/Nb/Ta. These findings are consistent with previously reported results.²¹ Therefore, for Tl₃XS₄ compounds, the contributions to lattice thermal conductivity from acoustic and optical modes predominantly arise from Tl and S atoms, respectively.

Furthermore, utilizing AIMD simulations, we obtain the free energy of Tl₃XS₄ (X = V, Nb, Ta) at temperatures of 300, 500, and 700 K (Fig. 3). A sufficiently long time of 20 ps was employed to guarantee the dependability of the simulation. These findings indicate the exceptional thermal stability of these materials across the three specified temperatures. After long enough simulation, the free energy of Tl₃XS₄ (X = V, Nb, Ta) remains stable, indicating the outstanding thermal stability of compounds at the specified temperatures.

Fig. 3 Free energy as a function of simulation time in AIMD calculations for Tl₃XS₄ (X = V, Nb, Ta) at 300, 500, and 700 K.

3.3. Electrical transport properties

The absolute Seebeck coefficient of both p-type and n-type Tl₃VS₄ are presented in Fig. 4(a), (c) and (e). Generally, the Seebeck coefficient is defined as:⁴¹where e is the electron charge. It is observed that there is a minimal difference in the Seebeck coefficients of Tl₃XS₄ compounds under n-type and p-type doping. Specifically, for Tl₃VS₄ and Tl₃TaS₄, the Seebeck coefficient under p-type doping is slightly higher compared to n-type doping. However, Tl₃NbS₄ exhibits an opposite trend in its Seebeck coefficient compared to Tl₃VS₄ and Tl₃TaS₄, showing higher values under n-type doping. Furthermore, regardless of the doping type, the Seebeck coefficient of Tl₃XS₄ compounds increases with temperature and decreases with the increase in carrier concentration. These observations align with the trends predicted by eqn (1). It is noteworthy that Tl₃XS₄ compounds exhibit high Seebeck coefficients. Specifically, at 300 K, the Seebeck coefficients within the doping concentration range of 10¹⁸–10²¹ cm⁻³ are approximately 100–850 μV K⁻¹, consistent with previous reports.²¹ Such outstanding Seebeck coefficients are comparable to excellent TE materials such as LaCuOSe/BiCuOSe.⁴²

Fig. 4 The absolute value of Seebeck coefficient |S| and electrical conductivity σ for Tl₃VS₄ (a) and (b), Tl₃NbS₄ (c) and (d), and Tl₃TaS₄ (e) and (f) as a function of carrier concentration.

The results of the electrical conductivities with carrier concentrations are shown in Fig. 4(b), (d) and (f). The electrical conductivity of the material is closely related to the carrier concentration (n), as defined by:⁴³

σ = nμe (2)

in which μ is the carrier mobility. According to eqn (2), the σ increases with carrier concentration, which is consistent with our calculation results including p- and n-type Tl₃XS₄ (X = V, Nb, Ta). Moreover, there is a notable reduction in electrical conductivity as temperature increases, attributable to the increased carrier scattering and reduced carrier mobility at high temperatures. Based on the calculation results of AMSET, the carrier relaxation times τ^tot and mobilities μ^tot of Tl₃XS₄ (X = V, Nb, Ta) under ADP, IMP, and POP scattering mechanisms were calculated using the following equations:⁴⁴where , τⁱ, and μⁱ (i = ADP, IMP, POP) are the carrier scattering rates, relaxation times, and carrier mobilities under the ADP, IMP, and POP scattering mechanisms, respectively. As shown in Fig. 5, the Tl₃XS₄ (X = V, Nb, Ta) system exhibits shorter carrier relaxation times (on the order of ∼10⁻¹⁵ s) and smaller mobilities, which can be attributed to the strong carrier scattering. It is noteworthy that the carrier relaxation times of most thermoelectric semiconductor materials are typically on the order of ∼10⁻¹⁵ s.^42,45,46 For both p-type and n-type Tl₃XS₄ (X = V, Nb, Ta), the relaxation time decreases (Fig. 5(a), (c) and (e)) and the mobility decreases (Fig. 5(b), (d) and (f)) with the temperature from 300 K to 700 K, which is due to the enhanced carrier scattering with the increase of temperature. Additionally, as the carrier concentration increases, both relaxation time and mobility show a decreasing trend, which is consistent with their linear relationship μ = eτ/m*. Since the effect of carrier concentration on mobility is not significant, the electrical conductivity exhibits a linear increasing trend with carrier concentration (as shown in Fig. 4(b), (d), (f)), which is consistent with eqn (2).

Fig. 5 The carrier relaxation time (τ) and carrier mobility (μ) for Tl₃VS₄ (a) and (b), Tl₃NbS₄ (c) and (d), and Tl₃TaS₄ (e) and (f) as a function of carrier concentration.

3.4. Thermal transport properties

Electronic thermal conductivity (κ_e) exhibits a significant dependence on the Lorenz number (L), which is defined by the Wiedemann–Franz law as:⁴⁷

κ_e = LσT (5)

For most metals, L converges to degenerate limit (D-limit) of ∼2.44 × 10⁻⁸ WΩ K⁻².⁴⁷ Although the Lorenz number for some heavily doped semiconductor thermoelectric materials is very close to the degenerate limit, its error can still reach up to ∼40%. This discrepancy arises because L is typically closely related to the scattering factor (λ), Seebeck coefficient (S), and chemical potential (η), according to the single parabolic band (SPB) model:⁴⁷

where F_j(η) represents the Fermi integral. Furthermore, Snyder et al.⁴⁷ proposed a satisfactory approximate equation for L (referred to as Snyder's model in this paper):where L is measured in 10⁻⁸ WΩ K⁻², and S is expressed in μV K⁻¹. Under the assumption of a single parabolic band and acoustic phonon scattering, Snyder's model achieves an accuracy within 5%. Based on the eqn (6)–(8), we obtained the L values corresponding the S of p-type and n-type Tl₃XS₄ (X = V, Nb, Ta) at different temperatures, as shown in Fig. 6(a), (c), (e). To verify the accuracy of the results, we further calculated the values of L by using Snyder's model and compared with the calculation results of SPB model, as shown in Fig. S1–S3 (ESI†). The L values calculated by the two methods show close agreement, and the trend of L as a function of S is consistent with previous reports,⁴⁷ confirming the reliability of our calculations. Since κ_e is directly proportional to σ, its behavior closely follows that of σ under varying carrier concentrations or temperatures (as depicted in Fig. 6(b), (d), (f)).

Fig. 6 The absolute Seebeck coefficient |S| dependent Lorenz number calculated by the SPB model of Tl₃VS₄ (a), Tl₃NbS₄ (c), and Tl₃TaS₄ (e). The electronic thermal conductivity κ_e for Tl₃VS₄ (b), Tl₃NbS₄ (d), and Tl₃TaS₄ (f) as a function of carrier concentration.

It is commonly acknowledged that the thermal conductivity in semiconductors is predominantly dominated by phonon part (κ_l).⁴⁸ The lattice thermal conductivity of Tl₃XS₄ (X = V, Nb, Ta) at 300, 500, and 700 K is presented in Fig. 7(a). It is noteworthy that the entire compounds of the Tl₃XS₄ family exhibit exceptionally low lattice thermal conductivity. Specifically, the κ_l of the Tl₃XS₄ compounds is consistently below 0.5 W m⁻¹ K⁻¹ at room temperature. Furthermore, with increasing temperature, the scattering between phonons intensifies, leading to a decrease in κ_l as temperature rises. Moreover, we conducted calculations for the cumulative lattice thermal conductivity under different phonon mean free paths (MFP). The results indicate that, for Tl₃XS₄ compounds with a bulk structure, diminishing the material dimensions through a nanostructure strategy can effectively reduce its lattice thermal conductivity. For instance, at 300 K, when the size of Tl₃VS₄ samples is shortened to 3.34 nm, the κ_l can be halved, as plotted by the dashed line in Fig. 7(b). Besides, the cumulative κ_l of Tl₃VS₄, Tl₃NbS₄, and Tl₃TaS₄ are shown in the inset of Fig. 7(b), (c), and (d), respectively. Notably, at 300 K, nearly 90% of κ_l for Tl₃XS₄ originates from the acoustic modes and low-frequency optical modes (<2.5 THz). This means that the lattice thermal conductivity of Tl₃XS₄ is primarily contributed by acoustic phonons.

Fig. 7 (a) The lattice thermal conductivity κ_lvs. temperature of Tl₃XS₄ (X = V, Nb, Ta). (b)–(d) Cumulative lattice thermal conductivity as a function of phonon mean free path at 300, 500, and 700 K, inset of (b), (c) and (d) is cumulative lattice thermal conductivity as a function of frequency. The phonon lifetime (e) and phonon group velocity (f) vs. frequency at 300 K.

The κ_l is calculated as:

where V and N_q are the cell volume and the number of sampled q points, respectively, C_λ is the heat capacity, v_λ is the group velocity, τ_λ is the phonon lifetime. To investigate the underlying factors of the low κ_l in Tl₃XS₄, we calculated the phonon lifetime and group velocity at 300 K, as depicted in Fig. 7(e) and (f). This analysis was pursued as κ_l is predominantly influenced by the phonon lifetime and group velocity. Due to the significant band gap in the phonon dispersion of Tl₃XS₄, phonon lifetimes and phonon group velocities are predominantly distributed in the ranges of 0–6 THz and beyond 11 THz. This distribution aligns well with the frequency positions corresponding to the phonon spectrum. Since the κ_l is mainly decided by the acoustic modes of the phonon band, we focus on the phonon lifetimes and phonon group velocities at the range of low frequencies. It is worth noting that the phonon lifetimes in the acoustic modes and low-frequency optical modes region follow the order: Tl₃TaS₄ > Tl₃VS₄ > Tl₃NbS₄ (as shown in Fig. 7(e)). From Fig. 7(f), the phonon group velocities also exhibit a similar trend, specifically, the group velocity of Tl₃TaS₄ is the largest and that of Tl₃NbS₄ is the smallest. These results indicate that the κ_l of the Tl₃XS₄ (X = V, Nb, Ta) follows the order: Tl₃NbS₄ < Tl₃VS₄ < Tl₃TaS₄.

3.5. Power factor and figure of merit

Based on PF = S²σ, the power factors of Tl₃XS₄ are given in Fig. 8(a), (c), (e). It is evident that the PF of Tl₃XS₄ significantly depends on the carrier type. Specifically, the p-type power factors of Tl₃VS₄ are higher than that of n-type, while Tl₃NbS₄ exhibits an opposite trend. This phenomenon is agreement with the dependence of the Seebeck coefficient on doping type, because the power factor is primarily determined by the Seebeck coefficient. Furthermore, the power factors exhibit a characteristic of initially increasing and then decreasing with the carrier concentration for both p- and n-type doping, reaching its maximum at a certain doping concentration. This is due to the increase of σ and the decrease of S with carrier concentration. Notably, at 300 K, the highest PF values of p-type (n-type) Tl₃XS₄ follow the order: Tl₃VS₄ > Tl₃TaS₄ > Tl₃NbS₄ (Tl₃NbS₄ > Tl₃VS₄ > Tl₃TaS₄). Specifically, at 300 K, the maximum PF values of p-type (n-type) Tl₃VS₄, Tl₃NbS₄, and Tl₃TaS₄ are 1.82 (1.13), 0.94 (1.62), and 0.95 (0.90) mW m⁻¹ K⁻², respectively.

Fig. 8 The power factor PF and figure of merit ZT for Tl₃VS₄ (a) and (b), Tl₃NbS₄ (c) and (d), and Tl₃TaS₄ (e) and (f) under different carrier concentration at 300, 500, and 700 K.

Finally, the figure of merit with carrier concentrations of Tl₃XS₄ at 300, 500, and 700 K are presented in Fig. 8(b), (d) and (f). At all temperatures, the maximum ZT values of p-type Tl₃XS₄ follow the order: Tl₃VS₄ > Tl₃NbS₄ > Tl₃TaS₄, and the values of n-type follow the relationship: Tl₃NbS₄ > Tl₃VS₄ > Tl₃TaS₄. For example, at 300 K, the optimal ZT values of p-type (n-type) Tl₃VS₄, Tl₃NbS₄, and Tl₃TaS₄ are 1.17 (0.82), 0.74 (0.84), and 0.65 (0.44), respectively. The optimal ZT value of Tl₃XS₄ (X = V, Nb, Ta) at 300 K and corresponding parameters are listed in Table 2. Significantly, the peak ZT values are achieved at the carrier concentration of about 10²⁰ cm⁻³, meeting the criteria for the optimal doping concentration in heavily doped semiconductors.^49,50 The maximum ZT value for Tl₃VS₄, Tl₃NbS₄, and Tl₃TaS₄ is obtained at 700 K, for which the values are 2.79 (p-type), 2.85 (n-type), and 2.39 (n-type), respectively. Such outstanding ZT values can be on par with other excellent TE materials, such as LaCuOSe and BiCuOSe.⁴² Specifically, the optimal ZT values of p-type (n-type) LaCuOSe and BiCuOSe are reported to be about 0.32 (1.46) and 0.75 (1.20) at 900 K,⁴² respectively. In a word, these remarkable ZT values suggest that Tl₃XS₄ (X = V, Nb, Ta) compounds with a simple body-centered cubic structure are promising TE materials.

Table 2 Optimal ZT values and corresponding parameter conditions for Tl₃XS₄ (X = V, Nb, Ta) at 300 K

	Carrier type	n (10^20 cm^−3)	|S| (μV K^−1)	σ (10^4 S m^−1)	PF (mW m^−1 K^−2)	ZT
Tl3VS4	Hole	4.23	254	2.32	1.49	1.17
	Electron	2.33	239	1.72	0.98	0.82
Tl3NbS4	Hole	1.28	222	1.69	0.83	0.74
	Electron	10.4	178	4.08	1.29	0.84
Tl3TaS4	Hole	2.33	212	1.93	0.87	0.65
	Electron	2.33	183	1.73	0.58	0.44

---

4. Conclusions

Collectively, utilizing the first-principles calculations, we investigated the electronic and thermal transport properties of Tl₃XS₄ (X = V, Nb, Ta) compounds with the simple body-centered cubic structure. First, the results of the AIMD simulation and phonon calculation reveal excellent dynamic stability and thermal stability of Tl₃XS₄ at 300, 500, and 700 K. Furthermore, the thermoelectric coefficients including the Seebeck coefficient S, electrical conductivity σ, power factor PF, and lattice thermal conductivity κ_l are further investigated at the considered temperatures. Our results demonstrate that the Tl₃XS₄ family exhibits ultralow κ_l (< 0.5 W m⁻¹ K⁻¹ at 300 K). The κ_l is primarily contributed by acoustic phonons, and the nanostructure strategy can effectively reduce κ_l. The investigations of phonon lifetimes and group velocities explain the lower κ_l of Tl₃NbS₄ compared to Tl₃VS₄ and Tl₃TaS₄. Finally, the combination of outstanding S and ultralow κ_l contributes to the remarkable ZT value. The optimal ZT values at 300 K for Tl₃VS₄, Tl₃NbS₄, and Tl₃TaS₄ are determined to be 1.17 (p-type), 0.84 (n-type), and 0.65 (p-type), respectively. For each considered temperature, the maximum ZT values of p-type (n-type) Tl₃XS₄ follow the order: Tl₃VS₄ > Tl₃NbS₄ > Tl₃TaS₄ (Tl₃NbS₄ > Tl₃VS₄ > Tl₃TaS₄). In a word, our results demonstrate that Tl₃XS₄ (X = V, Nb, Ta) compounds are promising thermoelectric materials, and the outstanding TE properties are comparable to excellent TE materials (i.e., LaCuOSe and BiCuOSe).

Data availability

Data available on request from the authors.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 12365007, 12164050 and 12264022), the Special Basic Cooperative Research Programs of Yunnan Provincial Undergraduate Universities’ Association (Grant No. 202301BA070001-121, 202301BA070001-122, and 202301BA070001-094), and the Science Research Foundation of the Department of Education in Yunnan Province, China (Grant No. 2024J1056).

References

1. L. Xie, C. Ming, Q. Song, C. Wang, J. Liao, L. Wang, C. Zhu, F. Xu, Y.-Y. Sun, S. Bai and L. Chen, Sci. Adv., 2023, 9, eadg7919.
2. X.-L. Shi, J. Zou and Z.-G. Chen, Chem. Rev., 2020, 120, 7399–7515.
3. D. Yang, Q. Gui, W. Yao, G. Wang and X. Zhou, Inorg. Chem., 2021, 60, 12331–12338.
4. J. Dong, D. Zhang, J. Liu, Y. Jiang, X. Y. Tan, N. Jia, J. Cao, A. Suwardi, Q. Zhu, J. Xu, J.-F. Li and Q. Yan, Inorg. Chem., 2023, 62, 17905–17912.
5. L.-D. Zhao, V. P. Dravid and M. G. Kanatzidis, Energy Environ. Sci., 2014, 7, 251–268.
6. G. A. Slack, J. Phys. Chem. Solids, 1973, 34, 321–335.
7. R. Wei, G. Huiyuan, Z. Zihao and Z. Lixia, Phys. Rev. Lett., 2017, 118, 245901.
8. B. C. Sales, D. Mandrus and R. K. Williams, Science, 1996, 272, 1325–1328.
9. G. Rogl, A. Grytsiv, P. Rogl, N. Peranio, E. Bauer, M. Zehetbauer and O. Eibl, Acta Mater., 2014, 63, 30–43.
10. J. Carrete, W. Li, N. Mingo, S. Wang and S. Curtarolo, Phys. Rev. X, 2014, 4, 011019.
11. J. R. Sootsman, D. Y. Chung and M. G. Kanatzidis, Angew. Chem., Int. Ed., 2009, 48, 8616–8639.
12. H. Zhu, R. He, J. Mao, Q. Zhu, C. Li, J. Sun, W. Ren, Y. Wang, Z. Liu, Z. Tang, A. Sotnikov, Z. Wang, D. Broido, D. J. Singh, G. Chen, K. Nielsch and Z. Ren, Nat. Commun., 2018, 9, 1–9.
13. H. Zhu, J. Mao, Y. Li, J. Sun, Y. Wang, Q. Zhu, G. Li, Q. Song, J. Zhou, Y. Fu, R. He, T. Tong, Z. Liu, W. Ren, L. You, Z. Wang, J. Luo, A. Sotnikov, J. Bao, K. Nielsch, G. Chen, D. J. Singh and Z. Ren, Nat. Commun., 2019, 10, 1–8.
14. T. Takabatake, K. Suekuni, T. Nakayama and E. Kaneshita, Rev. Mod. Phys., 2014, 86, 669–716.
15. G. S. Nolas, J. L. Cohn, G. A. Slack and S. B. Schujman, Appl. Phys. Lett., 1998, 73, 178–180.
16. Y. Saiga, K. Suekuni, S. K. Deng, T. Yamamoto, Y. Kono, N. Ohya and T. Takabatake, J. Alloys Compd., 2010, 507, 1–5.
17. S. Mukhopadhyay, D. S. Parker, B. C. Sales, A. A. Puretzky, M. A. McGuire and L. Lindsay, Science, 2018, 360, 1455–1458.
18. Z. Zeng, C. Zhang, Y. Xia, Z. Fan, C. Wolverton and Y. Chen, Phys. Rev. B, 2021, 103, 224307.
19. Y. Xia, K. Pal, J. He, V. Ozoliņš and C. Wolverton, Phys. Rev. Lett., 2020, 124, 065901.
20. A. Jain, Phys. Rev. B, 2020, 102, 201201.
21. S. Mukhopadhyay and T. L. Reinecke, J. Phys. Chem. Lett., 2019, 10, 4117–4122.
22. B. Li, C. Zhang, Z. Sun, T. Han, X. Zhang, J. Du, J. Wang, X. Xiao and N. Wang, Phys. Chem. Chem. Phys., 2022, 24, 24447–24456.
23. C. Crevecoeur, Acta Cryst., 1964, 17, 757.
24. K. Wacker and P. Buck, Cryst. Res. Technol., 1985, 20, 1025–1030.
25. G. Kresse and J. Furthmüller, Phys. Rev. B: Condens. Matter Mater. Phys., 1996, 54, 11169–11186.
26. G. Kresse and D. Joubert, Phys. Rev. B:Condens. Matter Mater. Phys., 1999, 59, 1758.
27. P. E. Blöchl, Phys. Rev. B:Condens. Matter Mater. Phys., 1994, 50, 17953.
28. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1997, 77, 3865.
29. S. Nosé, J. Chem. Phys., 1984, 81, 511.
30. R. Jinnouchi, J. Lahnsteiner, F. Karsai, G. Kresse and M. Bokdam, Phys. Rev. Lett., 2019, 122, 225701.
31. R. Jinnouchi, F. Karsai and G. Kresse, Phys. Rev. B, 2019, 100, 014105.
32. R. Jinnouchi, F. Karsai, C. Verdi, R. Asahi and G. Kresse, J. Chem. Phys., 2020, 152, 234102.
33. A. M. Ganose, J. Park, A. Faghaninia, R. Woods-Robinson, K. A. Persson and A. Jain, Nat. Commun., 2021, 12, 2222.
34. W. Dou, K. B. Spooner, S. R. Kavanagh, M. Zhou and D. O. Scanlon, J. Am. Chem. Soc., 2024, 146, 17679–17690.
35. W. Li, J. Carrete, N. A. Katcho and N. Mingo, Comput. Phys. Commun., 2014, 185, 1747–1758.
36. A. Carreras, A. Togo and I. Tanaka, Comput. Phys. Commun., 2017, 221, 221–234.
37. A. Togo and I. Tanaka, Scr. Mater., 2015, 108, 1–5.
38. T. V. Vu, A. A. Lavrentyev, B. V. Gabrelian, K. F. Kalmykova, V. V. Sidorkin, D. M. Hoat, H. D. Tong, D. P. Tran, O. Y. Khyzhun and D. D. Vo, Opt. Mater., 2020, 99, 109601.
39. A. A. Lavrentyev, A. N. Gusatinskii, I. Y. Nikiforov, A. P. Popov and V. V. Ilyasov, J. Phys.: Condens. Matter, 1993, 5, 1447–1454.
40. E. Osei-Agyemang, C. E. Adu and G. Balasubramanian, npj Comput. Mater., 2019, 5, 116.
41. C. Lee, J.-H. Shim and M.-H. Whangbo, Inorg. Chem., 2018, 57, 11895–11900.
42. N. Wang, M. Li, H. Xiao, Z. Gao, Z. Liu, X. Zu, S. Li and L. Qiao, npj Comput. Mater., 2021, 7, 18.
43. G. Tan, L.-D. Zhao and M. G. Kanatzidis, Chem. Rev., 2016, 116, 12123–12149.
44. X. Song, Y. Zhao, J. Ni, S. Meng and Z. Dai, Mater. Today Phys., 2023, 31, 100990.
45. N. Wang, M. Li, H. Xiao, X. Zu and L. Qiao, Phys. Rev. Appl., 2020, 13, 024038.
46. X. Song, G. Wang, L. Zhou, H. Yang, X. Li, H. Yang, Y. Shen, G. Xu, Y. Luo and N. Wang, ACS Appl. Energy Mater., 2023, 6, 9756–9763.
47. H.-S. Kim, Z. M. Gibbs, Y. Tang, H. Wang and G. J. Snyder, APL Mater., 2015, 3, 041506.
48. J. Li, W. Hu and J. Yang, J. Am. Chem. Soc., 2022, 144, 4448–4456.
49. G. Tan, L.-D. Zhao and M. G. Kanatzidis, Chem. Rev., 2016, 116, 12123–12149.
50. M. A. Haque, S. Kee, D. R. Villalva, W.-L. Ong and D. Baran, Adv. Sci., 2020, 7, 1903389.

---

Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4cp04000g

---