Phase-transition temperature suppression to achieve cubic GeTe and high thermoelectric performance by Bi and Mn codoping

Significance

Phase-transition behavior in thermoelectric materials is detrimental for their application in thermoelectric devices. Here we designed, and experimentally realized the high thermoelectric performance of cubic GeTe-based material by suppressing the phase transition from a cubic to a rhombohedral structure to below room temperature through a simple Bi and Mn codoping on the Ge site. Bi doping reduced the hole concentration while Mn alloying largely suppressed the phase-transition temperature and also induced modification of the valence bands. Our work provides the basis for studying phase transitions in other thermoelectric materials to optimize these materials for applications.

Abstract

Germanium telluride (GeTe)-based materials, which display intriguing functionalities, have been intensively studied from both fundamental and technological perspectives. As a thermoelectric material, though, the phase transition in GeTe from a rhombohedral structure to a cubic structure at ∼700 K is a major obstacle impeding applications for energy harvesting. In this work, we discovered that the phase-transition temperature can be suppressed to below 300 K by a simple Bi and Mn codoping, resulting in the high performance of cubic GeTe from 300 to 773 K. Bi doping on the Ge site was found to reduce the hole concentration and thus to enhance the thermoelectric properties. Mn alloying on the Ge site simultaneously increased the hole effective mass and the Seebeck coefficient through modification of the valence bands. With the Bi and Mn codoping, the lattice thermal conductivity was also largely reduced due to the strong point-defect scattering for phonons, resulting in a peak thermoelectric figure of merit (ZT) of ∼1.5 at 773 K and an average ZT of ∼1.1 from 300 to 773 K in cubic Ge_0.81Mn_0.15Bi_0.04Te. Our results open the door for further studies of this exciting material for thermoelectric and other applications.

Thermoelectric power generation (TEG), capable of directly converting heat into electricity, has reliably provided power for spacecraft explorations (1), but the low efficiency has impeded broader application. Due to the significantly improved performance realized in the last decade (2–4), TEG has drawn wide attention for energy harvesting from waste heat and natural heat that would provide an alternative approach to tackle the challenges of energy sustainability (5). The conversion efficiency of TEG is mainly determined by the material’s dimensionless thermoelectric figure of merit (ZT), ZT = [S²σ/(κ_lat + κ_ele)]T, where S, σ, κ_lat, κ_ele, and T are the Seebeck coefficient, electrical conductivity, lattice thermal conductivity, electronic thermal conductivity, and absolute temperature, respectively. Conventional methods to enhance the ZT mainly include optimizing carrier concentration and strengthening point-defect phonon scattering (6, 7), but peak ZT was limited to around unity from the 1950s to the 1990s (8). Recently proposed effective concepts or strategies, including ‘‘phonon glass electron crystal’’ to design new compounds (6), band-structure engineering to maximize the power factor (PF = S²σ) (9–13), microstructure engineering to suppress the κ_lat (14–17), and point-defect engineering to optimize performance (18–21), have led to the remarkable progress in the thermoelectric area (22–26). It should be noted that PbTe, one of the oldest and most-studied thermoelectric materials (27), plays a major role in evoking enthusiasm for current thermoelectric study since most conceptual breakthroughs have come from the recent study of the PbTe system (11, 15, 28, 29). However, the toxicity of Pb largely hinders applications for energy harvesting and therefore much scientific interest has shifted to Pb-free systems.

GeTe, one of the analogs of PbTe, has recently received intense attention from the thermoelectric community in its aim to replace traditional PbTe (30–36). GeTe undergoes a ferroelectric phase transition from the low-temperature rhombohedral structure α-GeTe (space group R3m) to cubic structure β-GeTe (space group $Fm\overline{3}m$) at the critical temperature (T_c) around 700 K (37). Due to the presence of a high concentration of Ge vacancies (38), undoped rhombohedral GeTe is a typical degenerate p-type semiconductor with intrinsically high hole concentration, which results in relatively low ZT. To overcome this shortcoming, In, Bi, or Sb doping as well as Pb alloying on the Ge site and Se alloying on the Te site have been proven to be effective in reducing the hole concentration and further enhancing ZT (30–36). However, the thermoelectric properties of all compositions previously investigated show the evident feature of phase transition in the measured temperature range. It is well known that phase-transition behavior is detrimental for applications because the sudden change in the thermal expansion coefficient would induce high internal stress between the materials and the contacts in the device that would lead to crack generation and consequently to deteriorating performance or failure under high thermal stress. Therefore, developing high-performance GeTe-based materials without the detrimental phase transition from α-GeTe to β-GeTe remains a significant challenge to be addressed. Based on the pseudobinary phase diagram of GeTe–MnTe solid solution (39), it is possible that Mn alloying on the Ge site would be an effective method to reduce the phase-transition temperature. Although Ge_1−xMn_xTe systems have been reported (40, 41), the primary focus of these studies was on the low-temperature magnetic properties of the systems.

Here we successfully achieved suppression of the phase-transition temperature from around 700 K to below 300 K, resulting in the high thermoelectric performance of cubic GeTe, by a simple Bi and Mn codoping on the Ge site using mechanical alloying and hot pressing. Bi doping reduced the hole concentration while Mn alloying induced significant valence band modification in addition to the large suppression of the phase-transition temperature. A peak ZT of ∼1.5 at 773 K and a corresponding average ZT of ∼1.1 from 300 to 773 K were achieved in cubic Ge_0.81Mn_0.15Bi_0.04Te.

Results and Discussion

The room-temperature X-ray diffraction (XRD) patterns of Ge_1−xBi_xTe samples closely match that of α-GeTe (SI Appendix, Fig. S1), confirming their room-temperature crystal structure as rhombohedral (37), but the phase-transition temperature from α-GeTe to β-GeTe decreases from 700 K (x = 0) to 585 K (x = 0.08) (SI Appendix, Fig. S2). Benefiting from the reduced hole concentration n_H upon Bi doping (Table 1), the electrical resistivity ρ shows an obvious increase to the desired value for good thermoelectric performance over the entire temperature range (Fig. 1A). As expected, Seebeck coefficient S increases upon Bi doping (Fig. 1B), in accordance with the tendency of ρ. Assuming the single parabolic band (SPB) model with acoustic phonon scattering as the dominant mechanism for carriers (6, 42), the calculated total density of states (DOS) effective mass m* continuously increases with Bi doping concentration (Table 1). Therefore, the enhancement of S could be ascribed to the combination of reduced n_H and band modification upon Bi doping. Compared with the pristine α-GeTe, Bi doping decreases PF, especially in the high-temperature range (Fig. 1C). The total thermal conductivity κ_tot shows a significant suppression upon Bi doping due to the decreased lattice thermal conductivity κ_lat, as well as the electronic thermal conductivity κ_ele. The κ_lat is obtained by subtracting κ_ele from κ_tot (Fig. 1D), where κ_ele is calculated using the Wiedemann–Franz relationship, κ_ele = LσT, in which L is the calculated Lorenz number. There is an obvious reduction of κ_lat after Bi doping, e.g., room-temperature κ_lat decreased from 2.4 W m⁻¹⋅K⁻¹ for α-GeTe to 1.0 W m⁻¹⋅K⁻¹ for α-Ge_0.92Bi_0.08Te (Fig. 1E). Bi doping on the Ge site introduces large mass fluctuations and surrounding local strain-field fluctuations due to the significant difference in the atomic mass and ionic radius between Bi and Ge atoms (43). In the low-temperature range from 300 to 523 K, α-GeTe shows the typical feature of Umklapp scattering with T^−1.2 dependence (Fig. 1E), basically consistent with the theoretical value T⁻¹. In contrast, κ_lat of α-Ge_0.92Bi_0.08Te is almost temperature independent, which may be related to the induced high degree of disorder and stronger anharmonicity by Bi doping (44, 45). The possibly incomplete subtraction of the electronic contribution may also have some effects because of the complex band structure. Due to the significantly suppressed κ_lat, Bi doping largely enhances the ZT over the whole temperature range. A peak ZT of ∼1.4 was achieved for α-Ge_0.96Bi_0.04Te, more than 50% higher than that of the pristine α-GeTe (Fig. 1F). It should also be noted that the pristine α-GeTe in our work exhibits a higher PF and ZT due to its relatively lower n_H in comparison with the previously reported samples that were synthesized by the method of melting and annealing (31, 32, 36). In general, the mechanical alloying method is able to fabricate materials with the needed chemical constituents, resulting in the lower carrier concentration in our current work. This result indicates that the combination of mechanical alloying and hot pressing is a more appropriate method to fabricate high-performance GeTe-based thermoelectric materials.

Table 1.

Electrical transport properties of α-Ge_1−xBi_xTe and α-Ge_0.96−xMn_xBi_0.04Te samples

Composition	nH, 1020 cm−3	μH, cm2 V−1⋅s−1	rH	m*, m0	μW, cm2 V−1⋅s−1
GeTe	4.2	95.3	1.0	1.6	197.9
Ge0.96Bi0.04Te	2.4	64.2	1.06	1.9	168.5
Ge0.92Bi0.08Te	1.0	51.3	1.10	2.1	159.7
Ge0.91Mn0.05Bi0.04Te	3.2	30.6	1.07	2.6	130.6
Ge0.86Mn0.1Bi0.04Te	4.1	16.9	1.08	3.9	123.0
Ge0.81Mn0.15Bi0.04Te	5.5	9.4	1.1	5.6	124.7
Ge0.76Mn0.2Bi0.04Te	10.0	4.4	1.11	9.9	136.7
Ge0.66Mn0.3Bi0.04Te	56.4	0.5	1.12	36.7	122.0

n_H is Hall carrier concentration (or hole concentration); μ_H is Hall carrier mobility (or hole mobility); r_H is Hall factor; m* is total DOS effective mass; m⁰ is the electron rest mass; and μ_W is weighted mobility.

Fig. 1.

Temperature-dependent thermoelectric properties of α-Ge_1−xBi_xTe samples (x = 0, 0.04, and 0.08). (A) ρ, (B) S, (C) PF, (D) κ_tot, (E) κ_lat, and (F) ZT.

Although Bi doping effectively reduces n_H and thus enhances ZT, the obvious phase-transition phenomenon remains. Based on the pseudobinary phase diagram of GeTe–MnTe solid solution (SI Appendix, Fig. S3) (39), Mn alloying on the Ge site is employed to possibly reduce the phase-transition temperature and obtain the cubic structure even at room temperature. XRD patterns of Ge_0.96−xMn_xBi_0.04Te samples are shown in Fig. 2A. Samples with low Mn alloying concentration (x ≤ 0.1) continue to crystallize in rhombohedral structure while samples with high Mn alloying concentration (x ≥ 0.15) crystallize in cubic structure (37, 39). In the literature, the reported critical Mn alloying composition in the pseudobinary phase diagram of GeTe–MnTe is about x = 0.18 (39). This discrepancy can be attributed to the contribution of Bi doping, which also decreases the phase-transition temperature to a certain extent. Fig. 2B shows the calculated lattice parameter and interaxial angle dependence on Mn alloying concentration. It is apparent that Mn alloying leads to an almost linear decrease of lattice parameters in solid solution. Since α-GeTe is a slightly distorted rock-salt lattice along the (111) direction (37), the interaxial angle change from non-−90° to 90° after Mn alloying is consistent with XRD measurement. Heat capacity measurements are displayed in Fig. 2C, which clearly show that Mn alloying gradually decreases the phase-transition temperature. However, it is difficult to detect the phase-transition temperature for x ≥ 0.1 by differential scanning calorimetry (DSC) measurement due to the very small or perhaps zero latent heat. Additionally, the XRD measurements of Ge_0.66Mn_0.3Bi_0.04Te sample when heating up to 473 K and cooling down to 300 K in air are performed, as shown in Fig. 2D. It is obvious that all of the obtained XRD patterns well match the cubic GeTe structure without the appearance of phase transition within the XRD detection limit. The broad peaks in the DSC measurements may be due to the high heating rate during the DSC measurements causing an incomplete phase transition.

Fig. 2.

(A) XRD patterns of Ge_0.96−xMn_xBi_0.04Te samples (x = 0, 0.05, 0.1, 0.15, 0.2, and 0.3). (B) Lattice parameter and interaxial angle dependence on Mn concentration. (C) Temperature-dependent heat capacity of Ge_0.96−xMn_xBi_0.04Te samples (x = 0, 0.01, 0.05, 0.1, 0.15, 0.2, and 0.3). (D) XRD patterns of Ge_0.66Mn_0.3Bi_0.04Te sample after heating up to 473 K and cooling down to 300 K in air.

Mn element is well known for its complex oxidation state, spanning from +2 to +7, and the most common and stable oxidation state is +2 (46). It was previously reported that Mn in GeTe–MnTe solid solution also showed the +2 that is identical to that of the host atom Ge (40, 47), but Mn alloying gradually increased the n_H of Ge_0.96−yMn_yBi_0.04Te (Table 1). Lewis et al. (38) found that the n_H of GeTe–MnTe solid solution increases with Mn concentration as a result of the increased Ge vacancies (38). The number of Ge vacancies in the GeTe system is directly related to the n_H because each Ge vacancy, acting as an acceptor center, donates one or two carriers to the valence band (38). In our first-principles calculations (addressed below) we indeed find that Mn is divalent in GeTe and that it adopts a high spin state. Mn alloying intensifies the scattering of holes, leading to the significantly decreased Hall mobility μ_H (Table 1). Thus, ρ gradually increases over the entire temperature range with increasing Mn concentration (Fig. 3A). Despite the increased n_H, S continuously increases with increasing Mn concentration (Fig. 3B), which will be addressed in detail below. It should be noted that the reduction of both ρ and S at high temperature for x ≥ 0.1 is caused by the bipolar effect, rather than the phase transition, while both the bipolar effect and phase transition contribute to those reductions for x ≤ 0.05. After Mn alloying, PF decreased somewhat due to the increased ρ (Fig. 3C). Basically, weighted mobility μ_W = μ_H(m*/m₀)^3/2, where m₀ is the free-electron mass, determines the maximum PF assuming that the carrier concentration is optimal (48). The calculated room-temperature μ_W displays the same variation trend as that of PF (Fig. 3D), both of which indicating that Mn alloying is not a valid method to enhance PF in this system.

Fig. 3.

Temperature-dependent (A) ρ, (B) S, and (C) PF of Ge_0.96−xMn_xBi_0.04Te samples (x = 0, 0.05, 0.1, 0.15, 0.2, and 0.3). (D) PF and weighted mobility μ_W dependence on Mn concentration at room temperature. The solid and dashed lines in D are included as guides for the eye.

To understand and quantify the abnormal behavior of the concurrently increased n_H and S of Ge_0.96−xMn_xBi_0.04Te with increasing Mn concentration, the corresponding m* were calculated based on the SPB model, shown in Table 1. Obviously, Mn alloying leads to the significant enhancement of m*, which is also demonstrated by the calculated Pisarenko plots displayed as dashed lines in Fig. 4A. This is consistent with the measured low μ_H of Mn alloyed samples, because heavy carriers generally diffuse with low velocities in a semiconductor. Experimental data of previously studied compositions, including Ge_1+xTe, Ge_1−xSb_xTe, and GeTe_1−xSe_x, fall on the solid black line (33), which is calculated by the modified two-band model (33), while at the same n_H, Ge_0.96−xMn_xBi_0.04Te samples exhibit a much higher S than the theoretical prediction. As a result of the high S, our PFs were observably higher than those of the previous reports (Fig. 4B).

Fig. 4.

Hall carrier-concentration-dependent (A) S and (B) PF of Ge_0.96−xMn_xBi_0.04Te and previously studied compositions, including Ge_1+xTe, Ge_1−xSb_xTe, and GeTe_1−xSe_x (33). The dashed lines in A were calculated by the SPB model with m* = 1.5, 2.5, 4, 5.5, and 10 m₀, respectively, while the red solid black line was obtained based on the modified two-valence-band model. The dashed line in B is included as a guide for the eye.

First-principles calculations, including electronic DOS and band-structure calculations, were performed to shed light on the role of Mn alloying in the significantly higher m* of rhombohedral and cubic GeTe. Fig. 5 compares the difference of the calculated DOS between pure GeTe and after Mn alloying in rhombohedral and cubic GeTe, respectively. Introducing Mn made the DOS steeper in both rhombohedral and cubic GeTe, especially near the valence band edge (e.g., from −0.05 to −0.2 eV for α-GeTe and from −0.25 to −0.3 eV for β-GeTe). This sharper DOS feature corresponds to the higher mass and is beneficial for enhancing the Seebeck coefficients, which is also consistent with the increased effective SPB m* after Mn alloying. Fig. 6 shows the calculated electronic band structures for both the pure and Mn-doped GeTe supercell with spin-orbital coupling (SOC). The primitive band structure of both the rhombohedral and cubic GeTe are essentially similar to the previously reported ones (34). For rhombohedral and cubic pristine GeTe (Fig. 6 A and C), the most beneficial feature is the multiple valence bands with relatively small band offset. Mn doping significantly increases the n_H and the corresponding Fermi level is pushed downward into the multiple valence band, resulting in the multiple valence band contribution to carrier conducting. Moreover, Mn alloying in rhombohedral GeTe realigns the bands, resulting in the contribution of the multiple band at different points (Fig. 6B). This underlines the higher DOS, which is also beneficial for achieving high S (49), as demonstrated in various systems, such as PbTe (11), SnTe (50), Mg₂Si (10, 13), etc. The calculated band structure without SOC can also support this conclusion (SI Appendix, Fig. S4). We have also shown the spin-polarized band structure in SI Appendix, Fig. S4 since the Mn alloying leads to a magnetic system (magnetic moment = 5 μB/Mn) corresponding to the high spin state of Mn²⁺, which is consistent with the previously measured electron paramagnetic resonance result (47). It should be noted that magnetic Mn²⁺ will introduce spin scattering, which is detrimental to the mobility. Thus, it will be promising and also challenging to investigate other alloying elements in the future to find nonmagnetic or weakly magnetic element ions that similarly allow carrier concentration optimization and stabilization of the cubic phase, perhaps with even higher ZT.

Fig. 5.

Comparison of the difference of the calculated DOS between pure GeTe and Mn-alloyed GeTe for (A) rhombohedral structure and (B) cubic structure. Black and red lines represent the pristine GeTe and Ge_0.875Mn_0.125Te, respectively.

Fig. 6.

The calculated electronic band structures with SOC of (A) rhombohedral structure α-GeTe, (B) α-Ge_0.875Mn_0.125Te, (C) cubic structure β-GeTe, and (D) β-Ge_0.875Mn_0.125Te. The dashed line represents the Fermi level.

The κ_tot shows a significant reduction upon Mn alloying (Fig. 7A), as a result of both the decreased κ_lat and κ_ele. Heavy Mn alloying leads to the obvious suppression of κ_lat due to the increased point-defect scattering. For example, room-temperature κ_lat decreases from 1.6 W m⁻¹⋅K⁻¹ for α-Ge_0.96Bi_0.04Te to 1.2 W m⁻¹⋅K⁻¹ for α-Ge_0.86Mn_0.1Bi_0.04Te and to 1.1 W m⁻¹⋅K⁻¹ for β-Ge_0.76Mn_0.2Bi_0.04Te (Fig. 7B). Additionally, the Debye–Callaway model, shown as the solid line in Fig. 7B (Inset), basically explains the decreasing trend of κ_lat with increasing Mn concentration (43, 51), in which the longitudinal (3,400 m/s) and transverse (1,890 m/s) sound velocities of pure GeTe are obtained from ref. 36. To confirm the origin of the reduction of κ_lat upon Mn alloying, phonon dispersion and phonon density of states (PDOS) of both α-GeTe and α-Ge_0.875Mn_0.125Te were calculated. Mn alloying in α-GeTe does not significantly alter the phonon dispersion (Fig. 7C), including acoustic modes and optical modes with low frequency. In addition, the PDOS at the low-frequency range from acoustic phonons is almost unchanged upon Mn alloying. Computational results show that Mn alloying does not significantly change the acoustic phonon properties of rhombohedral GeTe despite the induced substantial structure disorder. Furthermore, theoretical calculations of κ_lat based on the Debye–Callaway model are basically consistent with the experimental observations, which in turn indicates that Mn alloying can simply be regarded as the point-defect scattering centers. In contrast, Murphy et al. argued that soft optical mode transitions in Pb_1−xGe_xTe maximize the anharmonic acoustic–optical coupling and result in low κ_lat (52). Due to the presence of imaginary frequencies in the phonon dispersion of β-GeTe (SI Appendix, Fig. S5), it cannot provide a qualitative picture of the effect of Mn alloying on phonon transport in β-GeTe.

Fig. 7.

Temperature-dependent (A) κ_tot and (B) κ_lat of Ge_0.96−xMn_xBi_0.04Te samples. (B, Inset) Room-temperature κ_lat dependence on Mn concentration, where the solid line is calculated by the Debye–Callaway model (43, 51). (C) Phonon dispersions and (D) PDOS of α-GeTe and α-Ge_0.875Mn_0.125Te.

Due to the balance between the decreased PF and the suppressed κ_tot, the highest peak ZT at 773 K is almost unchanged after Mn alloying—they are all about 1.5 for α-Ge_0.96Bi_0.04Te, α-Ge_0.86Mn_0.1Bi_0.04Te, and β-Ge_0.81Mn_0.15Bi_0.04Te—but the low-temperature ZT is enhanced somewhat (Fig. 8A). In applications, the average ZT over the working temperature range determines the conversion efficiency of a device (53, 54). For rhombohedral GeTe-based materials, the highest average ZT from 300 to 773 K in our work is comparable with those of previous reports (Fig. 8B) (31, 33). It should be highlighted that the highest average ZT of cubic Mn-doped GeTe is higher than that of the current state-of-the-art p-type PbTe- (0.9) and SnTe- (0.4) based materials (Fig. 8B) (9, 50). Therefore, we have demonstrated the high performance of bulk cubic GeTe-based materials, for which there is no phase transition over the whole temperature range from 300 to 773 K. Additionally, Mn alloying in the GeTe system also reduces the cost of raw materials since less Ge is used. Both characteristics are beneficial for promoting the GeTe system for energy harvesting.

Fig. 8.

(A) Temperature-dependent ZT of Ge_0.96−xMn_xBi_0.04Te and (B) comparison of average ZT (from 300 to 773 K) of rhombohedral and cubic Mn-doped GeTe as well as the state-of-the-art p-type rhombohedral GeTe, cubic PbTe, and SnTe (9, 31, 33, 50).

Conclusions

In summary, we succeeded in suppressing the phase-transition temperature from ∼700 K to below ∼300 K to achieve cubic GeTe without phase transition from 300 to 773 K by a simple Bi doping and Mn alloying on the Ge site. The suppression of the phase transition to below room temperature is significant for any thermoelectric applications. Bi doping reduces the hole concentration and thus enhances ZT of the rhombohedral GeTe. Mn alloying induced significant valence band modification and increases the hole effective mass for both the rhombohedral and cubic GeTe, leading to a much higher Seebeck coefficient. The strong point-defect scattering for phonons caused by Bi and Mn largely reduces the lattice thermal conductivity, which leads to a peak ZT ∼1.5 at 773 K for cubic Ge_0.81Mn_0.15Bi_0.04Te. Our work opens the door for further studies of phase transition in other thermoelectric materials.

Experimental Section

Synthesis.

Appropriate raw materials, including Ge disks, Mn disks, Bi chunks, and Te chunks from Alfa Aesar, were weighed according to the nominal compositions Ge_1−xBi_xTe (x = 0, 0.04, and 0.08) and Ge_0.96−xMn_xBi_0.04Te (x = 0, 0.01, 0.05, 0.1, 0.15, 0.2, and 0.3), loaded into a stainless-steel jar in a glove box under argon atmosphere, and then subjected to ball milling for 5 h. The ball-milled powder was loaded into a die and hot pressed at 773 K for 2 min under a pressure of 90 MPa.

Phase and Property Characterizations.

XRD analysis was performed using a PANalytical multipurpose diffractometer with an X’celerator detector (PANalytical X'Pert Pro). Bar samples were cut from the pressed disks and used for simultaneous measurement of electrical resistivity (ρ) and Seebeck coefficient (S) on a commercial system (ULVAC ZEM-3). The thermal conductivity was calculated using κ = DC_pd, where D, C_p, and d are the thermal diffusivity, specific heat capacity, and density, respectively. The thermal diffusivity coefficient (D) was measured on a laser flash system (Netzsch LFA 457). The specific heat capacity (C_p) was measured on a DSC thermal analyzer (Netzsch DSC 404 C). The density (d) around 6.2 g cm⁻³ was determined by the Archimedes method. The room-temperature Hall coefficient R_H was measured using the Physical Properties Measurement System (Quantum Design). The Hall carrier concentration (n_H) was obtained by n_H = 1/eR_H and the Hall carrier mobility (μ_H) was calculated by σ = eμ_Hn_H, where e is the electronic charge and σ is the electrical conductivity. The uncertainty for the electrical conductivity is 3%, the Seebeck coefficient is 5%, and the thermal conductivity is 7%, so the combined uncertainty for the PF is 13% and that for ZT value is 20%. To increase the readability of the curves, error bars were not shown in the figures.

First-Principles Calculations.

The electronic band-structure calculations were performed by adopting the generalized gradient approximation of the Perdew–Burke–Ernzerhof functional for the exchange-correlation potential and the projector augmented wave method as implemented in the Vienna Ab initio Simulation Package (VASP) (55–57). The valence electrons included for Ge, Te, and Mn are 4s²4p², 5s²5p⁴, and 3p⁶4s²3d⁵, respectively. The electron wave function was expanded in a plane-wave basis set with an energy cutoff of 400 eV. The convergence of the calculations were tested with dense k-point meshes. The structures were fully relaxed until the force on each atom was less than 10⁻⁵ eV Å⁻¹ for both pure and Mn-doped GeTe. The effects of Mn doping were considered through a substitution of one Mn with one Ge atom in a 2 × 2 × 2 supercell that was built based on the original primitive cell in both cubic and rhombohedral phases. This yields a composition of Ge_0.875Mn_0.125Te. The spin polarization was included with an initial magnetic moment of 5 μB on Mn. The supercell band structures were unfolded to the primitive Brillouin zone high-symmetry path using the BandUP code (58, 59).

Phonons calculations were obtained within the harmonic approximation and using the finite displacement method based on the forces calculated via the Hellmann–Feynman theorem (60). A 2 × 2 × 2 supercell was set up for both pristine and Mn-doped rhombohedral phases, which consists of 128 atoms. The nonanalytical correction is applied by including the Born effective charges and dielectric constants calculated using the density functional perturbation theory.