Regulating the Configurational Entropy to Improve the Thermoelectric Properties of (GeTe)1−x(MnZnCdTe3)x Alloys

Abstract

In thermoelectrics, entropy engineering as an emerging paradigm-shifting strategy can simultaneously enhance the crystal symmetry, increase the solubility limit of specific elements, and reduce the lattice thermal conductivity. However, the severe lattice distortion in high-entropy materials blocks the carrier transport and hence results in an extremely low carrier mobility. Herein, the design principle for selecting alloying species is introduced as an effective strategy to compensate for the deterioration of carrier mobility in GeTe-based alloys. It demonstrates that high configurational entropy via progressive MnZnCdTe₃ and Sb co-alloying can promote the rhombohedral-cubic phase transition temperature toward room temperature, which thus contributes to the enhanced density-of-states effective mass. Combined with the reduced carrier concentration via the suppressed Ge vacancies by high-entropy effect and Sb donor doping, a large Seebeck coefficient is attained. Meanwhile, the severe lattice distortions and micron-sized Zn_0.6Cd_0.4Te precipitations restrain the lattice thermal conductivity approaching to the theoretical minimum value. Finally, the maximum zT of Ge_0.82Sb_0.08Te_0.90(MnZnCdTe₃)_0.10 reaches 1.24 at 723 K via the trade-off between the degraded carrier mobility and the improved Seebeck coefficient, as well as the depressed lattice thermal conductivity. These results provide a reference for the implementation of entropy engineering in GeTe and other thermoelectric materials.

1. Introduction

On account of the rising energy costs and increasing global warming, the development of eco-friendly and sustainable energy technologies is a global challenge. Among them, thermoelectrics (TE) have been considered as one of the most compelling technologies due to the capability of direct conversion between heat and electricity, vibration-free operation, high reliability, and low environmental impact [1,2]. The energy conversion efficiency of the TE device is determined by the component materials’ dimensionless figure of merit, zT = α²σT/κ, where T, α, σ and κ denote the absolute temperature, Seebeck coefficient, electrical conductivity and thermal conductivity (including carrier component κ_e and lattice component κ_L), respectively [3]. Aiming at decoupling the adversely interdependent TE parameters {α, σ, κ} and thus the high zT, the band engineering [4,5,6] and microstructure engineering [7,8,9,10,11] are implemented to enhance the power factor PF = σα² and reduce the κ_L, simultaneously. For TE materials undergoing phase transition, implementing phase engineering can also expand the phase favorable to thermoelectric and restrain the negative phase [12,13].

The configurational entropy ΔS characterizes the disorder of the system, and it is also an important parameter for developing high-performance TE materials [13]. Different from the traditional low-entropy alloys (LEA) with ΔS < 1 R, high-entropy alloys (HEA, ΔS > 1.5 R) consist of five or more major elements with a percentage of each atom between 5% and 35% [13]. High entropy alloys can provide a good way to improve thermoelectric properties by integrating the advantages of band, microstructure and phase engineering through multi-principal-element alloying [13,14,15,16,17,18,19,20,21]. So far, entropy engineering has been successfully applied to many TE systems, such as liquid-like materials [22,23,24], IV-VI compounds (including SnTe [20], PbSe [25], GeTe [13,26], and GeSe [27,28,29]), diamond-like compounds [19], Bi₂Te₃ [30], half-Heusler alloys [31], etc.

In multicomponent materials, ΔS has a profound effect on the TE parameters. First, the materials tend to form the high-symmetric cubic phase structure as long as the ΔS is high enough; meanwhile, high ΔS can reduce the structural phase transition temperature T_c [19,20,26]. The high crystal symmetry is generally inclined to possess large band degeneracy N_V and hence the large density-of-states’ effective mass m* [32]. Moreover, the high ΔS is conducive to increasing the solubility limit of specific elements, which in turn extends the phase space for performance optimization [20]. Therefore, higher α can be expected with higher ΔS value. Furthermore, the large differences of both atomic radius and mass among various components at the same sublattice site in multicomponent materials can result in a severe lattice distortion, which reduces the phonon velocities and enhances the phonon scattering simultaneously [19,20,33,34,35]. Hence, the κ_L can be reduced. However, it is noteworthy that although the multi-principal-element alloying can lower the κ_L towards the theoretical minimum as well as improve the α [13,16,19], the severe lattice distortion can also block the carrier transport, thus remarkably deteriorating the carrier mobility μ_H and σ that are bad for the zT advance. Consequently, the successful implementation of the entropy engineering presupposes that the fall in μ_H can be offset by the enhancement of α and the reduction of κ_L [20].

There are a number of strategies to solve the dilemma of μ_H reduction in multicomponent TE materials. For TE materials with carrier mean free path approaching to the Mott-Ioffe-Regel limit, multi-principal-element alloying does not further impair μ_H, while only significantly reduces the κ_L and enhances α [17,19,20,30,31]. Unfortunately, this approach is only applicable to those TE systems with intrinsically ultralow μ_H. Second, when designing high-entropy TE materials, the selected alloying element should have a small covalent radius difference in comparison with the host atoms in order to minimize the alloy scattering potential E_al and avoid the detrimental effect of high ΔS on μ_H. Under this circumstance, only a few alloying elements are available for many TE systems, which also leads to a significant limitation in the implementation of entropy engineering. Last but not least, if the chosen alloying elements can enhance the m*, optimize the carrier concentration n_H and decline the κ_L as much as possible to compensate for the deterioration of μ_H, the figure of merit can be surely enhanced [13,20]. Apparently, this is the easiest achievable design principle for implementing entropy engineering in thermoelectrics.

GeTe is a promising group IV-VI TE material for medium-temperature power generations [36]. Owing to the presence of a large number of Ge vacancies, pristine GeTe has a very high n_H (~10²¹ cm⁻³) at room temperature, giving rise to an extremely low α (~34 μV/K) and high κ_e (~5.65 W/mK) [37]. In addition, GeTe undergoes a phase transition from the rhombohedral (R-GeTe) to cubic structure (C-GeTe) in the temperature range of 600–700 K, relying on the n_H [12,13,17,38,39,40]. The phase transition of GeTe-based materials may induce high internal stresses and damage the GeTe-based devices under frequent thermal cycles or the material/electrode interfaces, hindering their practical application [40,41]. Up to now, carrier concentration optimization, suppressing rhombohedral-cubic phase transition, introducing multiscale microstructures and promoting multivalence band convergence are the prevalent strategies to improve the TE performance of GeTe-based alloys [5,26,38,42,43,44,45,46,47].

In this work, the ΔS of GeTe-based materials is regulated and the zT value is enhanced by progressive MnZnCdTe₃ and Sb co-alloying, attesting to the efficacy of the design principle of rational selecting alloying elements in high-entropy thermoelectrics. Previous studies have demonstrated that sole Mn [38,47], Zn [44], or Cd [45,46] doping enabled the convergence of multivalence bands in GeTe and thereby increase m*. In the context of electron counting, replacing divalent Ge by isovalent Mn/Zn/Cd contributes no net n_H and therefore retains the n_H to the first order. To reduce the otherwise excessive n_H of GeTe, the introduction of trivalent Sb doping is a natural option [48]. In addition, MnZnCdTe₃ and Sb co-alloying can increase the ΔS and lower T_c towards room temperature, which is beneficial to obtain large N_V and thus increase m* [26]. This offers the possibility to retain a decent PF in GeTe-based multicomponent materials through balancing the μ_H reduction and the α enhancement in the view of increased m* and optimized n_H. Combined with the substantially diminished κ_L stemming from the severe lattice distortions and the micron-sized Zn_0.6Cd_0.4Te precipitations, the maximum zT ~1.24 at 723 K is attained in Ge_0.82Sb_0.08Te_0.90(MnZnCdTe₃)_0.10. These results deepen the understanding of the rational selection of alloying elements in high-entropy TE materials.

2. Materials and Methods

2.1. Sample Synthesis and Preparation

Appropriate amounts of high-purity (99.999%) element chunks of Ge, Te, Mn, Zn, Cd and Sb were weighed according to the nominal composition of (GeTe)_1−x(MnZnCdTe₃)_x (x = 0, 0.025, 0.050, 0.075, 0.100, and 0.125), as well as Ge_0.90−ySb_yTe_0.90(MnZnCdTe₃)_0.10 (y = 0.04, 0.06, 0.08, 0.10), sealed into quartz ampoules at 10⁻³ Pa, melted at 1373 K for 10 h and then quenched in cold water. The solidified ingots were ball milled (MSK-SFM-3, MTI Corporation) in a vacuum at 1200 rpm for 20 min to fine powders. Subsequently, the obtained fine powders were spark plasma sintered (SPS) into high-density cylinders with a diameter of 20 mm, with a thickness of 3 mm at 773 K for 5 min under the uniaxial pressure of 50 MPa.

2.2. Phase and Microstructure Characterization

The phase structures of all the GeTe-based samples were analyzed via the X-ray diffraction (XRD, CuKα, SmartLab, Rigaku^®, Tokyo, Japan). The data analysis was performed via JADE 6.0 and the lattice parameters were calculated based on the Rietveld refinement method using the Topas 3.1 software. The phase transition temperature T_c from rhombohedral to cubic was inspected with a differential scanning calorimeter (DSC TAQ2000, New Castle, USA). The microstructures were investigated by scanning electron microscopy (SEM, Hitachi SU-70, Tokyo, Japan), and the chemical composition was examined by an energy dispersive spectrometer (EDS).

2.3. Transport Property Measurements

The measurements of the Seebeck coefficient α and electrical conductivity σ were conducted by a commercial ZEM-3 (Ulvac-Riko, Chigasaki, Japan) under a protective helium atmosphere. The measurement of thermal diffusivity D was performed on a Netzsch LFA 467 HT laser flash apparatus. The specific heat C_P was estimated by the Dulong-Petit law and the sample density ρ was determined via the Archimedes method. The thermal conductivity κ was then calculated as: κ = ρC_PD. The carrier thermal conductivity κ_e was evaluated by the Wiedemann-Franz law, κ_e = LσT, where L is the Lorenz number and can be roughly expressed by $L=1.5+ \exp \left[-\frac{\left|\alpha \right|}{116}\right]$ (where L is in 10⁻⁸ WΩK⁻² and α in μV/K) [49]. The lattice thermal conductivity κ_L was calculated by subtracting κ_e from the total κ. The room temperature Hall coefficient R_H was measured on a PPMS system (Quantum Design^®) with a magnetic field scanned between ±5 T. Subsequently, the carrier concentration n_H and Hall mobility μ_H were calculated according to the equation n_H = 1/eR_H and μ_H = σR_H, respectively.

3. Results and Discussion

In fact, all the multicomponent GeTe-based alloys are inclined to form the high-symmetric cubic structures as long as the ΔS is sufficiently high [17]. Even if the entropy is not so high, boosting ΔS can still lead to the decrease in the structural transition temperature to some extent for GeTe-based alloys [13]. According to the Boltzmann theory, the configurational entropy can be defined as $\Delta S=-R{\displaystyle {\sum }_{i=1}^n{x}_i}\ln x_i$, where n is the number of alloying components in the solid solution, x_i is the mole fractions of the ith component, and R is the gas constant [14,15]. Obviously, ΔS increases from 0 R for binary GeTe to 0.84 R for (GeTe)_0.90(MnZnCdTe₃)_0.10 and then to 1.06 R with Sb co-doping in Ge_0.82Sb_0.08Te_0.90(MnZnCdTe₃)_0.10, reaching the medium-entropy region (Figure 1a). As observed from the room-temperature powder X-ray diffraction (XRD) pattern (Figure 1b), the double peaks in the 2θ ranges of both 23°–27° and 41°–45° indicate the rhombohedral structure in pristine GeTe [12,50]. As ΔS rises, the double peaks gradually converge into a single peak, demonstrating that the room-temperature crystal structure gradually changes from R-GeTe to C-GeTe [13,26]. The lattice parameters a and c are calculated based on the powder XRD diffraction patterns and the results are plotted in Figure 1c. Typically, the lattice parameter a is 4.161 Å and c is 10.658 Å for pristine GeTe. When ΔS ascends to 0.84 R, a ascends to 4.181 Å and c descends to 10.516 Å for the (GeTe)_0.90(MnZnCdTe₃)_0.10 sample. Further increasing ΔS to 1.06 R, a rises to 4.205 Å and c drops to 10.486 Å for the Ge_0.82Sb_0.08Te_0.90(MnZnCdTe₃)_0.10 sample, demonstrating the promotion of the rhombohedral-cubic phase transition with the increasing ΔS.

Differential scanning calorimetry (DSC) analysis is carried out to further detect the variation of T_c upon progressive MnZnCdTe₃ and Sb co-alloying. As depicted in Figure 1d, the T_c of pristine GeTe is 655 K, and gradually falls to 552 K for (GeTe)_0.90(MnZnCdTe₃)_0.10 with the increasing ΔS. When further boosting ΔS, the T_c of Ge_0.82Sb_0.08Te_0.90(MnZnCdTe₃)_0.10 sample declines to 320 K, which is close to the room temperature. This indicates that the increase in ΔS can observably depress the T_c and is beneficial to obtain large N_V and thus high m* [13,26], which will be discussed later.

In addition to stabilizing the high-symmetry C-GeTe phase, the high ΔS also inclines to extend the solubility limits of specific elements [20]. Typically, there are a large number of Ge precipitations as well as Ge vacancies in pristine GeTe [37,51,52]. Two weak diffraction peaks of Ge precipitation can be observed at 27.0°–27.5° and 45.0°–45.5° on the XRD pattern of binary GeTe, as marked in Figure 1b. Interestingly, the diffraction peaks of Ge secondary phases gradually diminish and eventually disappear with increasing ΔS. For the multicomponent system, the Gibbs free energy of mixing ΔG is determined by $\Delta G=\Delta H-T\Delta S$, where ΔH is the enthalpy of mixing. Obviously, if the ΔH keeps constant, a higher ΔS will give rise to a lower ΔG [20]. Under this scenario, the high ΔS may promote the Ge precipitations to be dissolved into the matrix, which in turn reduces the excessive Ge vacancies and thus mildly decreases the n_H. Furthermore, Zn_0.6Cd_0.4Te precipitations are observed when the MnZnCdTe₃ alloying content x reaches 5%, indicating the low solubility limit of MnZnCdTe₃ in GeTe. The SEM-EDS images further confirm that the dark grey regions of 0.1–2.0 μm is Zn_0.6Cd_0.4Te, which are embedded in the host matrix of GeTe-based multicomponent alloys (Figure 2). These micron-sized Zn_0.6Cd_0.4Te secondary phases are favorable for the enhancement of low-frequency phonons scattering, which is beneficial to obtain the low κ_L. It is consistent with our aim of reducing κ_L by multi-component alloying. However, the low solubility limit of MnZnCdTe₃ is not conducive to promoting band convergence and suppressing the rhombohedral-cubic phase transition, which in turn restrains the improvement of m* and PF.

In order to analyze the mechanism underlining the variation of electrical transport properties, the room temperature n_H and μ_H for all the Ge_1−x−ySb_yTe_1−x(MnZnCdTe₃)_x samples are measured. As presented in Figure 3a, the n_H falls mildly with increasing MnZnCdTe₃ alloying content. For example, the room temperature n_H decreases from 8.4 × 10²⁰ cm⁻³ for GeTe to 6.0 × 10²⁰ cm⁻³ for (GeTe)_0.90(MnZnCdTe₃)_0.10. Theoretically, single element doping with the isovalent Zn, Cd or Mn at Ge site will not have a significant effect on n_H [38,44,45,46,47]. The previous literature has demonstrated that isoelectronic Mn alloying can even promote the formation of Ge vacancy and thus boost the n_H [38,47]. In the present work, the anomalous decrease in n_H after Mn-Zn-Cd co-alloying may be ascribed to the re-dissolution of Ge precipitations into the GeTe matrix via a high entropy effect, which is in line with the XRD results. Subsequently, the donor doping of trivalent Sb further reduces the n_H substantially to 1.7 × 10²⁰ cm⁻³ for Ge_0.82Sb_0.08Te_0.90(MnZnCdTe₃)_0.10, approaching to the optimal carrier concentration of n_opt (~2 × 10²⁰ cm⁻³) for rhombohedral GeTe-based alloys [48,52,53].

Severe lattice distortion induced by multi-principal-element alloying can impede the carrier transport and hence degrade the μ_H [13,20]. As illustrated in Figure 3b, the MnZnCdTe₃ alloying dramatically deteriorates the μ_H of GeTe. Typically, the room temperature μ_H drastically reduces from 57.3 cm²V⁻¹s⁻¹ for pristine GeTe to 20.8 cm²V⁻¹s⁻¹ for (GeTe)_0.95(MnZnCdTe₃)_0.05 and further to 15.8 cm²V⁻¹s⁻¹ for (GeTe)_0.90(MnZnCdTe₃)_0.10. Moreover, Mn alloying will flatten the valence band edge and largely increase the band effective mass m_b* of GeTe, leading to a remarkable deterioration in μ_H. For instance, Zheng et al. [47] reported that the room temperature m* increased quickly from 1.44 m₀ for binary GeTe to 6.15 m₀ for Ge_0.85Mn_0.15Te, corresponding to a catastrophic reduction of room temperature μ_H from 54.17 cm²V⁻¹s⁻¹ to 4.41 cm²V⁻¹s⁻¹. Similarly, Liu et al. found that the room temperature m* for pristine GeTe was only 1.6 m₀, which was then markedly enhanced to 36.7 m₀ after 30 at% Mn alloying. Meanwhile, the room temperature μ_H diminished rapidly from 95.3 cm²V⁻¹s⁻¹ to 0.5 cm²V⁻¹s⁻¹ [38]. In view of the deterioration of μ_H approaching to the Mott-Ioffe-Regel limit upon MnZnCdTe₃ alloying, further Sb alloying does not impair the μ_H, but Sb alloying can concurrently suppress the rhombohedral-cubic phase transition, optimize the n_H and reduce the κ_L [26,31,48].

In general, the high ΔS is adverse to high σ [13,20]. Figure 4a depicts the temperature dependence of σ for all the GeTe-based samples. The σ of pristine GeTe is extremely high because of the high n_H and μ_H. After MnZnCdTe₃ alloying, the σ decreases dramatically due to the simultaneous reduction of n_H and μ_H, which originates from the dissolution of Ge precipitations, band flattening and severe lattice distortion. Typically, the room temperature σ decreases from 774 × 10³ Sm⁻¹ for binary GeTe to 152 × 10³ Sm⁻¹ for (GeTe)_0.90(MnZnCdTe₃)_0.10. In light of the μ_H approaching to the Mott-Ioffe-Regel limit, further Sb alloying reduces the n_H but has a weak impact on carrier scattering, thereby bringing out a further decreased σ. Therefore, the room temperature σ of the Ge_0.82Sb_0.08Te_0.90(MnZnCdTe₃)_0.10 sample decreases to 30 × 10³ Sm⁻¹. In addition, the σ of pristine GeTe exhibits a temperature dependence of T^−1.48, indicating that the acoustic phonon scattering (σ ~T^−1.5) is the dominant scattering mechanism in this system. The power law exponent of the σ gradually decreases with increasing ΔS because of the enhanced point defect scattering. Thus, the temperature dependence of T^−0.4 is obtained in Ge_0.82Sb_0.08Te_0.90(MnZnCdTe₃)_0.10, which is consistent with the alloy scattering mechanism (σ ~T^−0.5) [54].

The progressive alloying of MnZnCdTe₃ and Sb increases ΔS and thus suppresses T_c near room temperature, which in turn enhances the m* and hence α [13,26]. As exhibited in Figure 4b, the α boosts mildly near room temperature upon MnZnCdTe₃ alloying while increases substantially in the whole temperature range after Sb doping. For example, the room temperature α sharply increases from 32.5 μVK⁻¹ for binary GeTe to 156.4 μVK⁻¹ for Ge_0.82Sb_0.08Te_0.90(MnZnCdTe₃)_0.10.

To gain insight into the beneficial effect of band structure variation on α, the Pisarenko plots are calculated according to the single parabolic band (SPB) model [55]. As demonstrated in Figure 4c, the m* rises from 1.4 m₀ for the GeTe sample to 1.8 m₀ for the MnZnCdTe₃ alloyed sample and finally increases to 2.4 m₀ for the Ge_0.82Sb_0.08Te_0.90(MnZnCdTe₃)_0.10 sample. On the one hand, the increase in ΔS lowers the T_c of GeTe, resulting in large N_V and thus m*. On the other hand, the co-alloying of Mn, Zn and Cd is conductive to promoting the multivalence band convergence and valence band flattening, which further enhances m* [38,44,45,47]. Notably, the low solubility limit of MnZnCdTe₃ in GeTe is detrimental to band structure modulation. Besides the increase in m*, both the suppression of Ge vacancies via high entropy effect and the trivalent Sb donor doping contribute to a large reduction of n_H, giving rise to a huge enhancement of α. The increased m* and optimized n_H partially compensate for the deterioration of μ_H, thus attaining a decent power factor PF in the whole temperature region (Figure 4d) as well as a maximum PF of 1.75 × 10⁻³ Wm⁻¹K⁻² for Ge_0.82Sb_0.08Te_0.90(MnZnCdTe₃)_0.10.

The high κ (8 Wm⁻¹K⁻¹ at 300 K) limits the TE performance of p-type binary GeTe [36]. Therefore, it is necessary to reduce the otherwise too high κ in GeTe-based alloys. The severe lattice distortion induced by multi-principal-element alloying affords a natural option [15,19,20]. The temperature-dependent κ for all the GeTe-based alloys is plotted in Figure 5a. As expected, the κ of multi-principal-elements alloys are much lower than the pristine GeTe, because of the concurrent reduction of both κ_e (due to the declined σ) and κ_L. The room temperature κ substantially drops from 6.76 Wm⁻¹K⁻¹ for binary GeTe to 1.43 Wm⁻¹K⁻¹ for (GeTe)_0.90(MnZnCdTe₃)_0.10 and then to 0.98 Wm⁻¹K⁻¹ for Ge_0.82Sb_0.08Te_0.90(MnZnCdTe₃)_0.10.

The κ_L of all the samples is presented in Figure 5b. In view of the severe lattice distortions induced by large size and mass differences among various components at the same sublattice site, the multi-principal-element alloying can effectively minimize the κ_L of GeTe [13,20]. Moreover, the micron-sized Zn_0.6Cd_0.4Te secondary phases were caused by the low solubility limit of MnZnCdTe₃ in GeTe (Figure 2) that is conducive to scattering low-frequency phonons. The synergy of aforesaid two factors greatly diminish the κ_L throughout the measured temperature region. Therefore, the room-temperature κ_L declines from 3.30 Wm⁻¹K⁻¹ for binary GeTe to 0.75 Wm⁻¹K⁻¹ for (GeTe)_0.90(MnZnCdTe₃)_0.10 and further falls to 0.66 Wm⁻¹K⁻¹ for Ge_0.82Sb_0.08Te_0.90(MnZnCdTe₃)_0.10. In particular, the minimum κ_L of 0.45 Wm⁻¹K⁻¹ at 723 K is achieved in Ge_0.82Sb_0.08Te_0.90(MnZnCdTe₃)_0.10, approaching the theoretical minimum κ_L (~0.3 Wm⁻¹K⁻¹) of GeTe [17]. Moreover, the power law exponent of κ_L decreases from −1.25 for binary GeTe to −0.45 for Ge_0.82Sb_0.08Te_0.90(MnZnCdTe₃)_0.10, indicating that the dominant phonon scattering alters from acoustic phonon scattering (κ_L ~T^−1.0) to point defect scattering (κ_L ~T^−0.5) with increasing alloying species and content [20].

The trade-off between the enhanced α, the depressed κ_L and the deteriorated μ_H determines the variation of zT values with increasing ΔS (Figure 5c). Notably, the maximum zT of 1.24 at 723 K is attained in the Ge_0.82Sb_0.08Te_0.90(MnZnCdTe₃)_0.10 sample through entropy engineering, about 22% increment over the pristine GeTe. The peak zT in this study is much higher than those of the high-entropy Ge_1/3Sn_1/3Pb_1/3Te_1/3Se_1/3S_1/3 (~0.51) [18] and Ge_0.25Sn_0.25Pb_0.25Mn_0.25Te alloys (~0.92) [19] without a rational selection of alloying species. In addition, it is noteworthy that the average zT_ave value between 298 K and 723 K is 0.77, which is 113% higher than that of binary GeTe (Figure 5d). In particular, the zT_ave in this work is also much higher than those of high-entropy Ge_1/3Sn_1/3Pb_1/3Te_1/3Se_1/3S_1/3 (~0.39) and Ge_0.25Sn_0.25Pb_0.25Mn_0.25Te alloys (~0.63). These results demonstrate the validity of the screening principle for alloying species in high-entropy TE materials.

4. Conclusions

In summary, we demonstrate the power of entropy engineering on balancing the power factor and lattice thermal conductivity of GeTe in line with the rationally screening alloying species. To establish the structure-property correlation in GeTe-based multicomponent materials, entropy engineering is adopted to design the GeTe-based alloys with progressive MnZnCdTe₃ and Sb co-alloying based on the nominal composition of (Ge_1−x−ySb_yTe_1−x)(MnZnCdTe₃)_x. Multi-principal-element alloying can regulate the configurational entropy and thus improve the thermoelectric properties of the GeTe-based materials. The phase structures, microstructures and thermoelectric properties of the obtained alloys are systematically studied. The high configurational entropy reduces the rhombohedral-cubic phase transition temperature to near room temperature and Mn-Zn-Cd co-alloying promotes the multivalent bands’ convergence, which are both conducive to obtaining a high density-of-state effective mass. Meanwhile, the synergy of the decreased Ge vacancies content via the dissolution of Ge precipitations and trivalent Sb donor doping reduces the excessive carrier concentration of GeTe-based alloys. The optimized carrier concentration and the increase in the density-of-state effective mass improve the Seebeck coefficient substantially, which counteracts the decrease in electrical conductivity. Moreover, the severe lattice distortions and micron-sized Zn_0.6Cd_0.4Te second phases result in a significant reduction of lattice thermal conductivity. As a result, the maximum zT value of 1.24 at 723 K is attained in Ge_0.82Sb_0.08Te_0.90(MnZnCdTe₃)_0.10. These results not only prove the validity of rational screening principle for alloying species in high-entropy thermoelectric materials, but also trigger profound thoughts on the application of emerging entropy engineering in the field of thermoelectric materials.

Author Contributions

Conceptualization, L.H.; Data curation, Y.H., S.Z. (Shizhen Zhi) and L.H.; Formal analysis, Y.H., S.Z. (Shizhen Zhi), W.A. and L.H.; Funding acquisition, C.Z., F.L. and L.H.; Investigation, Y.H., S.Z. (Shizhen Zhi) and W.Y.; Methodology, Y.H., S.Z. (Shizhen Zhi) and W.Y.; Project administration, L.H.; Resources, C.Z., F.L., J.L. and L.H.; Software, S.Z. (Shizhen Zhi); Supervision, W.A., C.Z., F.L. and L.H.; Validation, Y.H., S.Z. (Shizhen Zhi) and L.H.; Visualization, Y.H. and L.H.; Writing—original draft, Y.H. and L.H.; Writing—review and editing, Y.H., S.Z. (Shengnan Zhang) and L.H. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data presented in this study are available on request from the corresponding author.

Conflicts of Interest

The authors declare no conflict of interest.

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Figures

Figure 1. Room temperature (a) configurational entropy ΔS, (b) powder XRD patterns, (c) lattice parameter a and c, and (d) DSC curves and the obtained phase transition temperature Tc of GeTe-based alloys.

Figure 2. The scanning electron microscopy (SEM) and the X-ray energy dispersive spectrum (EDS) elemental spot scanning of the polished Ge0.82Sb0.08Te0.90(MnZnCdTe3)0.10. The element mapping of (a) Ge, (b) Te, (c) Mn, (d) Cd, (e) Zn, (f) Sb, and (g) the X-ray energy dispersive spectrum analysis, indicating the micron-sized Zn0.6Cd0.4Te precipitations are existed.

Figure 3. The room temperature (a) carrier concentration nH, and (b) carrier mobility μH of Ge1−x−ySbyTe1−x(MnZnCdTe3)x samples.

Figure 4. Temperature dependence of (a) electrical conductivity σ, (b) Seebeck coefficient α, and (d) power factor PF for (GeTe)1−x(MnZnCdTe3)x and Ge0.82−ySbyTe0.90(MnZnCdTe3)0.10 samples. (c) Room temperature Seebeck coefficient α as a function of carrier concentration nH for Ge1−x−ySbyTe1−x(MnZnCdTe3)x samples.

Figure 5. Temperature dependences of (a) total thermal conductivity κ, (b) lattice thermal conductivity κL, and (c) zT of (GeTe)1−x(MnZnCdTe3)x and Ge0.82−ySbyTe0.90(MnZnCdTe3)0.10 samples. (d) The average zTave values between 298–723 K for our GeTe-based alloys.

References

1. Zhu, T.J.; Liu, Y.T.; Fu, C.G.; Heremans, J.P.; Snyder, J.G.; Zhao, X.B. Compromise and Synergy in High-Efficiency Thermoelectric Materials. Adv. Mater.; 2017; 29, 1605884. [DOI: https://dx.doi.org/10.1002/adma.201605884] [PubMed: https://www.ncbi.nlm.nih.gov/pubmed/28262991]

2. He, J.; Tritt, T.M. Advances in thermoelectric materials research: Looking back and moving forward. Science; 2017; 357, eaak9997. [DOI: https://dx.doi.org/10.1126/science.aak9997] [PubMed: https://www.ncbi.nlm.nih.gov/pubmed/28963228]

3. Shi, X.L.; Zou, J.; Chen, Z.G. Advanced Thermoelectric Design: From Materials and Structures to Devices. Chem. Rev.; 2020; 120, pp. 7399-7515. [DOI: https://dx.doi.org/10.1021/acs.chemrev.0c00026]

4. Tan, X.J.; Wang, H.X.; Liu, G.Q.; Noudem, J.G.; Hu, H.Y.; Xu, J.T.; Shao, H.Z.; Jiang, J. Designing band engineering for thermoelectrics starting from the periodic table of elements. Mater. Today Phys.; 2018; 7, pp. 35-44. [DOI: https://dx.doi.org/10.1016/j.mtphys.2018.10.004]

5. Pei, Y.Z.; Wang, H.; Snyder, G.J. Band Engineering of Thermoelectric Materials. Adv. Mater.; 2012; 24, pp. 6125-6135. [DOI: https://dx.doi.org/10.1002/adma.201202919] [PubMed: https://www.ncbi.nlm.nih.gov/pubmed/23074043]

6. Heremans Joseph, P.; Jovovic, V.; Toberer Eric, S.; Saramat, A.; Kurosaki, K.; Charoenphakdee, A.; Yamanaka, S.; Snyder, G.J. Enhancement of Thermoelectric Efficiency in PbTe by Distortion of the Electronic Density of States. Science; 2008; 321, pp. 554-557. [DOI: https://dx.doi.org/10.1126/science.1159725] [PubMed: https://www.ncbi.nlm.nih.gov/pubmed/18653890]

7. He, J.Q.; Kanatzidis, M.G.; Dravid, V.P. High performance bulk thermoelectrics via a panoscopic approach. Mater. Today; 2013; 16, pp. 166-176. [DOI: https://dx.doi.org/10.1016/j.mattod.2013.05.004]

8. Wang, Y.A.; Bourgès, C.; Rajamathi, R.; Nethravathi, C.; Rajamathi, M.; Mori, T. The Effect of Reactive Electric Field-Assisted Sintering of MoS₂/Bi₂Te₃ Heterostructure on the Phase Integrity of Bi₂Te₃ Matrix and the Thermoelectric Properties. Materials; 2022; 15, 53. [DOI: https://dx.doi.org/10.3390/ma15010053]

9. Chang, C.; Ibáñez, M. Enhanced Thermoelectric Performance by Surface Engineering in SnTe-PbS Nanocomposites. Materials; 2021; 14, 5416. [DOI: https://dx.doi.org/10.3390/ma14185416]

10. Zhang, Y.; Qu, W.B.; Peng, G.Y.; Zhang, C.L.; Liu, Z.Y.; Liu, J.C.; Li, S.R.; Wu, H.J.; Meng, L.J.; Gao, L.M. Seeing Structural Mechanisms of Optimized Piezoelectric and Thermoelectric Bulk Materials through Structural Defect Engineering. Materials; 2022; 15, 487. [DOI: https://dx.doi.org/10.3390/ma15020487]

11. Tang, X.; Li, Z.; Liu, W.; Zhang, Q.; Uher, C. A comprehensive review on Bi₂Te₃-based thin films: Thermoelectrics and beyond. Interdiscip. Mater.; 2022; 1, pp. 88-115. [DOI: https://dx.doi.org/10.1002/idm2.12009]

12. Chen, B.C.; Wang, K.K.; Wu, H.J. Localized crystal imperfections coupled with phase diagram engineering yield high-performance rhombohedral GeTe thermoelectrics. Mater. Today Phys.; 2022; 22, 100617. [DOI: https://dx.doi.org/10.1016/j.mtphys.2022.100617]

13. Zhi, S.Z.; Li, J.B.; Hu, L.P.; Li, J.Q.; Li, N.; Wu, H.J.; Liu, F.S.; Zhang, C.H.; Ao, W.Q.; Xie, H.P. et al. Medium Entropy-Enabled High Performance Cubic GeTe Thermoelectrics. Adv. Sci.; 2021; 8, 2100220. [DOI: https://dx.doi.org/10.1002/advs.202100220] [PubMed: https://www.ncbi.nlm.nih.gov/pubmed/34194947]

14. Shafeie, S.; Guo, S.; Hu, Q.; Fahlquist, H.; Erhart, P.; Palmqvist, A. High-entropy alloys as high-temperature thermoelectric materials. J. Appl. Phys.; 2015; 118, 184905. [DOI: https://dx.doi.org/10.1063/1.4935489]

15. Jiang, B.B.; Yu, Y.; Cui, J.; Liu, X.X.; Xie, L.; Liao, J.C.; Zhang, Q.H.; Huang, Y.; Ning, S.C.; Jia, B.H. et al. High-entropy-stabilized chalcogenides with high thermoelectric performance. Science; 2021; 371, pp. 830-834. [DOI: https://dx.doi.org/10.1126/science.abe1292]

16. Fan, Z.; Wang, H.; Wu, Y.; Liu, X.J.; Lu, Z.P. Thermoelectric high-entropy alloys with low lattice thermal conductivity. RSC Adv.; 2016; 6, pp. 52164-52170. [DOI: https://dx.doi.org/10.1039/C5RA28088E]

17. Liang, G.G.; Lyu, T.; Hu, L.P.; Qu, W.B.; Zhi, S.Z.; Li, J.B.; Zhang, Y.; He, J.; Li, J.Q.; Liu, F.S. et al. (GeTe)_(1−x)(AgSnSe₂)_(x): Strong Atomic Disorder-Induced High Thermoelectric Performance near the Ioffe-Regel Limit. ACS Appl. Mater. Interfaces; 2021; 13, pp. 47081-47089. [DOI: https://dx.doi.org/10.1021/acsami.1c14801]

18. Deng, Z.; Olvera, A.; Casamento, J.; Lopez, J.S.; Williams, L.; Lu, R.; Shi, G.; Poudeu, P.F.P.; Kioupakis, E. Semiconducting High-Entropy Chalcogenide Alloys with Ambi-ionic Entropy Stabilization and Ambipolar Doping. Chem. Mater.; 2020; 32, pp. 6070-6077. [DOI: https://dx.doi.org/10.1021/acs.chemmater.0c01555]

19. Liu, R.H.; Chen, H.Y.; Zhao, K.P.; Qin, Y.T.; Jiang, B.B.; Zhang, T.S.; Sha, G.; Shi, X.; Uher, C.; Zhang, W.Q. et al. Entropy as a Gene-Like Performance Indicator Promoting Thermoelectric Materials. Adv. Mater.; 2017; 29, 1702712. [DOI: https://dx.doi.org/10.1002/adma.201702712]

20. Hu, L.P.; Zhang, Y.; Wu, H.J.; Li, J.Q.; Li, Y.; McKenna, M.; He, J.; Liu, F.S.; Pennycook, S.J.; Zeng, X.R. Entropy Engineering of SnTe: Multi-Principal-Element Alloying Leading to Ultralow Lattice Thermal Conductivity and State-of-the-Art Thermoelectric Performance. Adv. Energy Mater.; 2018; 8, 1802116. [DOI: https://dx.doi.org/10.1002/aenm.201802116]

21. Luo, Y.B.; Hao, S.Q.; Cai, S.T.; Slade, T.J.; Luo, Z.Z.; Dravid, V.P.; Wolverton, C.; Yan, Q.Y.; Kanatzidis, M.G. High Thermoelectric Performance in the New Cubic Semiconductor AgSnSbSe₃ by High-Entropy Engineering. J. Am. Chem. Soc.; 2020; 142, pp. 15187-15198. [DOI: https://dx.doi.org/10.1021/jacs.0c07803]

22. Liu, H.L.; Shi, X.; Xu, F.F.; Zhang, L.L.; Zhang, W.Q.; Chen, L.D.; Li, Q.; Uher, C.; Day, T.; Snyder, G.J. Copper ion liquid-like thermoelectrics. Nat. Mater.; 2012; 11, pp. 422-425. [DOI: https://dx.doi.org/10.1038/nmat3273] [PubMed: https://www.ncbi.nlm.nih.gov/pubmed/22406814]

23. Zhao, K.P.; Zhu, C.X.; Qiu, P.F.; Blichfeld, A.B.; Eikeland, E.; Ren, D.D.; Iversen, B.B.; Xu, F.F.; Shi, X.; Chen, L.D. High thermoelectric performance and low thermal conductivity in Cu_2−yS_1/3Se_1/3Te_1/3 liquid-like materials with nanoscale mosaic structures. Nano Energy; 2017; 42, pp. 43-50. [DOI: https://dx.doi.org/10.1016/j.nanoen.2017.10.042]

24. Jiang, B.B.; Qiu, P.F.; Chen, H.Y.; Zhang, Q.H.; Zhao, K.P.; Ren, D.D.; Shi, X.; Chen, L.D. An argyrodite-type Ag₉GaSe₆ liquid-like material with ultralow thermal conductivity and high thermoelectric performance. Chem. Commun.; 2017; 53, pp. 11658-11661. [DOI: https://dx.doi.org/10.1039/C7CC05935C]

25. Fan, Z.; Wang, H.; Wu, Y.; Liu, X.J.; Lu, Z.P. Thermoelectric performance of PbSnTeSe high-entropy alloys. Mater. Res. Lett.; 2017; 5, pp. 187-194. [DOI: https://dx.doi.org/10.1080/21663831.2016.1244116]

26. Qiu, Y.T.; Jin, Y.; Wang, D.Y.; Guan, M.J.; He, W.K.; Peng, S.; Liu, R.H.; Gao, X.; Zhao, L.D. Realizing high thermoelectric performance in GeTe through decreasing the phase transition temperature via entropy engineering. J. Mater. Chem. A; 2019; 7, pp. 26393-26401. [DOI: https://dx.doi.org/10.1039/C9TA10963C]

27. Huang, Z.W.; Miller, S.A.; Ge, B.H.; Yan, M.T.; Anand, S.; Wu, T.M.; Nan, P.F.; Zhu, Y.H.; Zhuang, W.; Snyder, G.J. et al. High Thermoelectric Performance of New Rhombohedral Phase of GeSe stabilized through Alloying with AgSbSe₂. Angew. Chem.-Int. Ed.; 2017; 56, pp. 14113-14118. [DOI: https://dx.doi.org/10.1002/anie.201708134]

28. Lyu, T.; Li, X.; Yang, Q.X.; Cheng, J.H.; Zhang, Y.H.; Zhang, C.H.; Liu, F.S.; Li, J.Q.; Ao, W.Q.; Xie, H.P. et al. Stepwise Ge vacancy manipulation enhances the thermoelectric performance of cubic GeSe. Chem. Eng. J.; 2022; 442, 136332. [DOI: https://dx.doi.org/10.1016/j.cej.2022.136332]

29. Li, X.; Liang, Z.Y.; Li, J.B.; Cheng, F.; He, J.; Zhang, C.H.; Li, J.Q.; Liu, F.S.; Lyu, T.; Ge, B.H. et al. Crystal symmetry enables high thermoelectric performance of rhombohedral GeSe(MnCdTe₂)_x. Nano Energy; 2022; 100, 107434. [DOI: https://dx.doi.org/10.1016/j.nanoen.2022.107434]

30. Raphel, A.; Vivekanandhan, P.; Kumaran, S. High entropy phenomena induced low thermal conductivity in BiSbTe_1.5Se_1.5 thermoelectric alloy through mechanical alloying and spark plasma sintering. Mater. Lett.; 2020; 269, 127672. [DOI: https://dx.doi.org/10.1016/j.matlet.2020.127672]

31. Yan, J.L.; Liu, F.S.; Ma, G.H.; Gong, B.; Zhu, J.X.; Wang, X.; Ao, W.Q.; Zhang, C.H.; Li, Y.; Li, J.Q. Suppression of the lattice thermal conductivity in NbFeSb-based half-Heusler thermoelectric materials through high entropy effects. Scr. Mater.; 2018; 157, pp. 129-134. [DOI: https://dx.doi.org/10.1016/j.scriptamat.2018.08.008]

32. Huang, Z.W.; Zhao, L.D. Symmetry and asymmetry in thermoelectrics. J. Mater. Chem. C; 2020; 8, pp. 12054-12061. [DOI: https://dx.doi.org/10.1039/D0TC03270K]

33. Liu, Y.T.; Fu, C.G.; Xia, K.Y.; Yu, J.J.; Zhao, X.B.; Pan, H.G.; Felser, C.; Zhu, T.J. Lanthanide Contraction as a Design Factor for High-Performance Half-Heusler Thermoelectric Materials. Adv. Mater.; 2018; 30, 1800881. [DOI: https://dx.doi.org/10.1002/adma.201800881] [PubMed: https://www.ncbi.nlm.nih.gov/pubmed/29939427]

34. Zhou, Y.J.; Meng, F.C.; He, J.; Benton, A.; Hu, L.P.; Liu, F.S.; Li, J.Q.; Zhang, C.H.; Ao, W.Q.; Xie, H.P. n-Bi_2–xSb_xTe₃: A Promising Alternative to Mainstream Thermoelectric Material n-Bi₂Te_3–xSe_x near Room Temperature. ACS Appl. Mater. Interfaces; 2020; 12, pp. 31619-31627. [DOI: https://dx.doi.org/10.1021/acsami.0c07566] [PubMed: https://www.ncbi.nlm.nih.gov/pubmed/32539321]

35. Cai, J.F.; Yang, J.X.; Liu, G.Q.; Wang, H.X.; Shi, F.F.; Tan, X.J.; Ge, Z.H.; Jiang, J. Ultralow thermal conductivity and improved ZT of CuInTe₂ by high-entropy structure design. Mater. Today Phys.; 2021; 18, 100394. [DOI: https://dx.doi.org/10.1016/j.mtphys.2021.100394]

36. Zhang, X.Y.; Bu, Z.L.; Lin, S.Q.; Chen, Z.W.; Li, W.; Pei, Y.Z. GeTe Thermoelectrics. Joule; 2020; 4, pp. 986-1003. [DOI: https://dx.doi.org/10.1016/j.joule.2020.03.004]

37. Jin, Y.; Xiao, Y.; Wang, D.; Huang, Z.; Qiu, Y.; Zhao, L.-D. Realizing High Thermoelectric Performance in GeTe through Optimizing Ge Vacancies and Manipulating Ge Precipitates. ACS Appl. Energy Mater.; 2019; 2, pp. 7594-7601. [DOI: https://dx.doi.org/10.1021/acsaem.9b01585]

38. Liu, Z.H.; Sun, J.F.; Mao, J.; Zhu, H.T.; Ren, W.Y.; Zhou, J.C.; Wang, Z.M.; Singh, D.J.; Sui, J.H.; Chu, C.W. et al. Phase-transition temperature suppression to achieve cubic GeTe and high thermoelectric performance by Bi and Mn codoping. Proc. Natl. Acad. Sci. USA; 2018; 115, pp. 5332-5337. [DOI: https://dx.doi.org/10.1073/pnas.1802020115]

39. Li, J.; Zhang, X.Y.; Wang, X.; Bu, Z.L.; Zheng, L.T.; Zhou, B.Q.; Long, F.; Chen, Y.; Pei, Y.Z. High-Performance GeTe Thermoelectrics in Both Rhombohedral and Cubic Phases. J. Am. Chem. Soc.; 2018; 140, pp. 16190-16197. [DOI: https://dx.doi.org/10.1021/jacs.8b09147]

40. Liu, W.D.; Wang, D.Z.; Liu, Q.F.; Zhou, W.; Shao, Z.P.; Chen, Z.G. High-Performance GeTe-Based Thermoelectrics: From Materials to Devices. Adv. Energy Mater.; 2020; 10, 2000367. [DOI: https://dx.doi.org/10.1002/aenm.202000367]

41. Xing, T.; Song, Q.F.; Qiu, P.F.; Zhang, Q.H.; Xia, X.G.; Liao, J.C.; Liu, R.H.; Huang, H.; Yang, J.; Bai, S.Q. et al. Superior performance and high service stability for GeTe-based thermoelectric compounds. Natl. Sci. Rev.; 2019; 6, pp. 944-954. [DOI: https://dx.doi.org/10.1093/nsr/nwz052] [PubMed: https://www.ncbi.nlm.nih.gov/pubmed/34691955]

42. Li, J.; Chen, Z.W.; Zhang, X.Y.; Yu, H.L.; Wu, Z.H.; Xie, H.Q.; Chen, Y.; Pei, Y.Z. Simultaneous Optimization of Carrier Concentration and Alloy Scattering for Ultrahigh Performance GeTe Thermoelectrics. Adv. Sci.; 2017; 4, 1700341. [DOI: https://dx.doi.org/10.1002/advs.201700341] [PubMed: https://www.ncbi.nlm.nih.gov/pubmed/29270343]

43. Zhang, R.; Pei, J.; Shan, Z.H.; Zhou, W.; Wu, Y.; Han, Z.J.; Zhao, Y.H.; Li, J.F.; Ge, Z.H.; Zhang, B.P. Intrinsic vacancy suppression and band convergence to enhance thermoelectric performance of (Ge, Bi, Sb)Te crystals. Chem. Eng. J.; 2022; 429, 132275. [DOI: https://dx.doi.org/10.1016/j.cej.2021.132275]

44. Hong, M.; Wang, Y.; Feng, T.L.; Sun, Q.; Xu, S.D.; Matsumura, S.; Pantelides, S.T.; Zou, J.; Chen, Z.G. Strong Phonon-Phonon Interactions Securing Extraordinary Thermoelectric Ge_1−xSb_xTe with Zn-Alloying-Induced Band Alignment. J. Am. Chem. Soc.; 2019; 141, pp. 1742-1748. [DOI: https://dx.doi.org/10.1021/jacs.8b12624]

45. Nshimyimana, E.; Hao, S.Q.; Su, X.L.; Zhang, C.; Liu, W.; Yan, Y.G.; Uher, C.; Wolverton, C.; Kanatzidis, M.G.; Tang, X.F. Discordant nature of Cd in GeTe enhances phonon scattering and improves band convergence for high thermoelectric performance. J. Mater. Chem. A; 2020; 8, pp. 1193-1204. [DOI: https://dx.doi.org/10.1039/C9TA10436D]

46. Li, J.; Li, W.; Bu, Z.L.; Wang, X.; Gao, B.; Xiong, F.; Chen, Y.; Pei, Y.Z. Thermoelectric Transport Properties of Cd_xBi_yGe_1–x–yTe Alloys. ACS Appl. Mater. Interfaces; 2018; 10, pp. 39904-39911. [DOI: https://dx.doi.org/10.1021/acsami.8b15080]

47. Zheng, Z.; Su, X.L.; Deng, R.G.; Stoumpos, C.; Xie, H.Y.; Liu, W.; Yan, Y.G.; Hao, S.Q.; Uher, C.; Wolverton, C. et al. Rhombohedral to Cubic Conversion of GeTe via MnTe Alloying Leads to Ultralow Thermal Conductivity, Electronic Band Convergence, and High Thermoelectric Performance. J. Am. Chem. Soc.; 2018; 140, pp. 2673-2686. [DOI: https://dx.doi.org/10.1021/jacs.7b13611]

48. Li, C.; Song, H.L.; Dai, Z.B.; Zhao, Z.B.; Liu, C.Y.; Yang, H.Q.; Cui, C.Q.; Miao, L. High Thermoelectric Performance Achieved in Sb-Doped GeTe by Manipulating Carrier Concentration and Nanoscale Twin Grains. Materials; 2022; 15, 406. [DOI: https://dx.doi.org/10.3390/ma15020406]

49. Kim, H.-S.; Gibbs, Z.M.; Tang, Y.L.; Wang, H.; Snyder, G.J. Characterization of Lorenz number with Seebeck coefficient measurement. APL Mater.; 2015; 3, 041506. [DOI: https://dx.doi.org/10.1063/1.4908244]

50. Li, J.; Zhang, X.Y.; Chen, Z.W.; Lin, S.Q.; Li, W.; Shen, J.H.; Witting, I.T.; Faghaninia, A.; Chen, Y.; Jain, A. et al. Low-Symmetry Rhombohedral GeTe Thermoelectrics. Joule; 2018; 2, pp. 976-987. [DOI: https://dx.doi.org/10.1016/j.joule.2018.02.016]

51. Bayikadi, K.S.; Sankar, R.; Wu, C.T.; Xia, C.L.; Chen, Y.; Chen, L.C.; Chen, K.H.; Chou, F.C. Enhanced thermoelectric performance of GeTe through in situ microdomain and Ge-vacancy control. J. Mater. Chem. A; 2019; 7, pp. 15181-15189. [DOI: https://dx.doi.org/10.1039/C9TA03503F]

52. Li, P.G.; Ding, T.; Li, J.Q.; Zhang, C.X.; Dou, Y.B.; Li, Y.; Hu, L.P.; Liu, F.S.; Zhang, C.H. Positive Effect of Ge Vacancies on Facilitating Band Convergence and Suppressing Bipolar Transport in GeTe-Based Alloys for High Thermoelectric Performance. Adv. Funct. Mater.; 2020; 30, 1910059. [DOI: https://dx.doi.org/10.1002/adfm.201910059]

53. Dong, J.F.; Sun, F.H.; Tang, H.C.; Pei, J.; Zhuang, H.L.; Hu, H.H.; Zhang, B.P.; Pan, Y.; Li, J.F. Medium-temperature thermoelectric GeTe: Vacancy suppression and band structure engineering leading to high performance. Energy Environ. Sci.; 2019; 12, pp. 1396-1403. [DOI: https://dx.doi.org/10.1039/C9EE00317G]

54. Hu, C.L.; Xia, K.Y.; Fu, C.G.; Zhao, X.B.; Zhu, T.J. Carrier grain boundary scattering in thermoelectric materials. Energy Environ. Sci.; 2022; 15, pp. 1406-1422. [DOI: https://dx.doi.org/10.1039/D1EE03802H]

55. Zhu, J.B.; Zhang, X.M.; Guo, M.C.; Li, J.Y.; Hu, J.S.; Cai, S.T.; Cai, W.; Zhang, Y.S.; Sui, J.H. Restructured single parabolic band model for quick analysis in thermoelectricity. NPJ Comput. Mater.; 2021; 7, 116. [DOI: https://dx.doi.org/10.1038/s41524-021-00587-5]