Engineering Thermoelectric Performance of α‐GeTe by Ferroelectric Distortion

The rhombohedral α‐GeTe can be approximated as a slightly distorted rock‐salt structure along its [1 1 1] direction and possesses superb thermoelectric performance. However, the role of such a ferroelectric‐like structural distortion on its transport properties remains unclear. Herein, we performed a systematic study on the crystal structure and electronic band structure evolutions of Ge1‐xSnxTe alloys where the degree of ferroelectric distortion is continuously tuned. It is revealed that the band gap is maximized while multiple valence bands are converged at x = 0.6, where the ferroelectric distortion is the least but still works. Once undistorted, the band gap is considerably reduced, and the valence bands are largely separated again. Moreover, near the ferro‐to‐paraelectric phase transition Curie temperature, the lattice thermal conductivity reaches its minima because of significant lattice softening enabled by ferroelectric instability. We predict a peak ZT value of 2.6 at 673 K in α‐GeTe by use of proper dopants which are powerful in suppressing the excess hole concentrations but meanwhile exert little influence on the ferroelectric distortion.


Introduction
Middle-to-low temperature (400-700 K) waste heat utilization is of significant economic and environmental significance.[18] These strategies guarantee the highest possible ZT when the carrier concentration is fully optimized.
[21][22] At room temperature, it crystalizes in a rhombohedral structure (R3m) which can be viewed as a distorted NaCl-type structure.[25] Besides temperature, it is found that foreign dopants have a noticeable impact on the structural transformation of α-GeTe as well. [26,27]For example, experimental studies demonstrate that heavy doping of Sb, [28][29][30] Bi, [31][32][33] and Mn [34] at Ge sites can stabilize cubic structured GeTe at room temperature. [35][38] It is therefore natural to wonder about the influence of ferroelectric distortion on the thermoelectric transport behavior of α-GeTe.Unfortunately, to the best of our knowledge, very few studies really focus on this interesting and important topic. [39]n this study, a series of Ge 1-x Sn x Te samples were successfully prepared.It is demonstrated that α-GeTe and SnTe are fully miscible at any molar fraction, and the degree of ferroelectric distortion varies monotonically with compositions.We show that proper ferroelectric distortion engineering of α-GeTe contributes to optimizations of both electrical and thermal properties.Specifically, increase in ferroelectric distortion leads to an inversion of the L and Σ valence bands accompanied by a splitting of band degeneracy.Only when the ferroelectric distortion approaches (but does not equal to) zero can all valence bands be effectively converged and the band gap gets maximized.This condition is particularly favorable to achieve high PF.Moreover, near T C , ferroelectric instability is taking effect, and the energy of polar transverse optical phonons at the center of the Brillouin zone gets lower and becomes comparable to that of heatcarrying acoustic phonons.It brings about strong acoustic-optical phonon coupling and significant scattering of acoustic phonons, leading to extremely low κ lat . [40]A maximum ZT of 2.6 is predicted at 673 K for α-GeTe by finely tuning the carrier concentration without sacrificing the ferroelectric distortion too much.Our study proves that ferroelectric distortion engineering is an effective approach to DOI: 10.1002/eem2.12535 The rhombohedral α-GeTe can be approximated as a slightly distorted rocksalt structure along its [1 1 1] direction and possesses superb thermoelectric performance.However, the role of such a ferroelectric-like structural distortion on its transport properties remains unclear.Herein, we performed a systematic study on the crystal structure and electronic band structure evolutions of Ge 1-x Sn x Te alloys where the degree of ferroelectric distortion is continuously tuned.It is revealed that the band gap is maximized while multiple valence bands are converged at x = 0.6, where the ferroelectric distortion is the least but still works.Once undistorted, the band gap is considerably reduced, and the valence bands are largely separated again.Moreover, near the ferro-to-paraelectric phase transition Curie temperature, the lattice thermal conductivity reaches its minima because of significant lattice softening enabled by ferroelectric instability.We predict a peak ZT value of 2.6 at 673 K in α-GeTe by use of proper dopants which are powerful in suppressing the excess hole concentrations but meanwhile exert little influence on the ferroelectric distortion.
optimizing the thermoelectric performance of narrow gap polar semiconductors.

Results and Discussions
2.1.Phase Compositions and Microstructures of Ge 1-x Sn x Te Figure 1a depicts the powder XRD patterns of SPSed Ge 1-x Sn x Te bulk samples.With increasing x, there is a gradual evolution of crystal structure from a lower-symmetry rhombohedral one to a higher-symmetry cubic one.This can be seen from the two splitting peaks (024) and (220) in samples with x ≤ 0.6 systematically drawing closer and, finally, merging into only one peak (220) when x ≥ 0.8.Besides, the diffraction peaks regularly shift towards lower angles of 2θ with growing x, indicating the successful substitution of Ge atoms by larger-size Sn atom.A trace amount of Ge precipitates is detectable for the samples with x ≤ 0.2 at 2θ = 27.3°,which is considered normal in GeTe-rich compounds because of the inherently high concentration of cationic vacancies. [41,42]he Raman spectra of Ge 1-x Sn x Te are displayed in Figure 1b.Two prominent Raman-active modes have been identified in α-GeTe peaking at ~116 and ~161 cm −1 , respectively.The former is associated with the symmetric stretching vibrations of GeTe4 tetrahedra, while the latter arises from the vibrations of Ge atoms distinct from their distorted octahedral site. [43]With increasing x, both peaks shift to lower frequencies (namely, redshift).Under the classical vibrational model, the vibrational frequency (f) of a phonon relates to the force constant of a bond (K) and the reduced mass of the compound (M) through the following equation: [44] f Therefore, the redshift of Raman peaks indicates the successful replacement of Ge host atoms by heavier Sn atoms, which is consistent with XRD analysis.Moreover, these Raman peaks become broadened with significantly reduced intensity in the x = 0.6 sample and, finally, vanish when x = 1.0.This confirms the increased crystal structure symmetry in SnTe-alloyed α-GeTe (cubic rock-salt structure is Raman inactive). [43]ack-scattered electron (BSE) and secondary electron (SE) images together with the X-ray elemental distribution mappings for a representative sample (x = 0.4) are shown in Figure 2. Apart from the Ge precipitates, no evident phase segregations are observed at least at the micron scale, and all elements are evenly distributed within the matrix.It is in good agreement with the XRD and Raman results that GeTe and SnTe form complete solid solutions.The nominal and actual chemical compositions of all the samples are summarized in Table S1. Figure 3a,b, respectively, display the transmission electron microscope (TEM) images of samples with x = 0 and x = 0.4 viewed along [1 1 0] and [2-2 1] directions, both of which suggest the absence of any nanoscale precipitates and homogeneity of chemical compositions.47] These fishbone-like domains originate from the complex interplay between strain and electrostatic interactions during the phase transition from the high-temperature centrosymmetric phase to the lowtemperature non-centrosymmetric phase. [48]On the contrary, for the Ge 0.6 Sn 0.4 Te sample, such ferroelectric domains become very sparse.It again confirms the higher structure symmetry of GeTe after alloying with SnTe.

Structural Parameters and Ferroelectric Distortions
To gain more detailed information on the structural changes from rhombohedral α-GeTe (Figure 4a) to cubic SnTe (Figure 4b) in Ge 1- x Sn x Te solid solutions, we performed Rietveld refinement of their XRD data, Supporting Information Figure S1.The derived lattice constants a and c (Figure 4c) linearly increase with growing Sn content, which is in line with the expectation that the two compounds are fully dissolved into each other.As shown in Figure 4a, α-GeTe has two types of bonds with different lengths (d) in the rhombohedral structure, that is, a shorter one with d S = 2.83 Å and a longer one with d L = 3.17 Å, and the acute angle between the two crossing long bonds, defined as β, is 82.2°. [49]d S and d L show regular changes, that is, d S gets linearly increased and d L is almost unchanged when x is varied from 0 to 0.6, Figure 4d.In the meanwhile, β keeps going up and reaches 86.8°at x = 0.6, Figure 4e.This means that the averaging structure symmetry of Ge 1-x Sn x Te alloys becomes higher during this process.It is worth noting that, at x = 0.8, a suddenly increases and coincides with the abruptly dropping c, indicating the transformation of the structure from rhombohedral one to cubic one.Correspondingly, β approaches 90°.Further increasing x to 1 does not change β value but slightly elevates the bond length d.
[52] The degree of ferroelectric distortion in α-GeTe-based compounds can be roughly estimated by the displacement (Δε) of Ge atoms with respect to its paraelectric phase (cubic one) in the primitive cell, inset of Figure 4f.It can be clearly seen that, by alloying with SnTe, Δε of α-GeTe shows a rapid decrease trend, from 0.086 Å for x = 0 to 0.033 Å for x = 0.6 and, finally, to zero for x ≥ 0.8, where a phase transition from ferroelectric structure to the paraelectric structure is established.This means that ferroelectric distortion could be finely and continuously tuned over a wide composition range in Ge 1-x Sn x Te solid solutions, thus providing a good base for us to understand how ferroelectric distortion affects the electron and phonon transport behaviors in those technologically important thermoelectric compounds.

Ferro-to-Paraelectric Phase Transitions and Electronic Band Structures
Since SnTe alloying in α-GeTe induces great changes to the crystal structures, we suspect that it has a significant impact on the ferro-toparaelectric phase transition behavior and electronic band structures as well.Figure 5a plots the differential scanning calorimetry (DSC) curves of Ge 1-x Sn x Te samples in the temperature interval from 310 to 750 K. Pristine α-GeTe shows a sharp but asymmetric endothermal peak around 676 K upon heating, which agrees well with previous studies. [34,53,54]This temperature corresponds to the ferro-to-paraelectric phase transition temperature, namely Curie temperature (T C ).The introduction of SnTe into α-GeTe shifts the peak toward lower temperature, weakens the peak intensity, and broadens the peak width.At x ≥ 0.6, no evident endothermal peak is visible within the investigated temperature range at all.The derived T C as a function of x for Ge 1-x Sn x Te is illustrated in Figure 5b.Our data are in good agreement with the one (dotted curve) reported by Bierly et al. [55] Obviously, T C drops quickly with increasing x and falls below 300 K for the samples with x ≥ 0.6.The lowered T C of α-GeTe upon SnTe alloying is a consequence of decreased ferroelectric distortion in crystal structure.Figure 6a-f display the first-principles density functional theory (DFT) calculation results of electronic band structures for Ge 1-x Sn x Te with x = 0, 0.2, 0.4, 0.6, 0.8, and 1, respectively.Spin orbital coupling (SOC) effect [56] and experimental structural parameters from Rietveld refinements of room-temperature XRD are taken into account in all cases. [57]nTe shows a direct band gap (E g ) of ~0.2 eV with its conduction band minima (CBM) and valence band maxima (VBM) both located at L point (valley degeneracy N V = 4) of the Brillouin zone.In addition to the L valence band, there is a secondary lower-lying and heavier valence band at Σ point with a much larger N V = 12. [42]The most characterized feature in the band structure of α-GeTe is the inversion of band energy levels of L and Σ valence bands in comparison to SnTe, leading to an indirect E g of ~0.4 eV. [58]In addition, as a result of ferroelectric distortion, it splits fourfold degenerated L valence band into 3 L + 1 Z, and 12-folded Σ valence band into 6 η + 6 Σ (η is not shown here).
If we scrutinize the electronic band structures of Ge 1-x Sn x Te, it can be found that there is a regular change of band edge energies and bands splitting degree with compositions, and the results are illustrated in Figure 7, where the CBM (L point) energies are all set to zero for better visualization.Specifically, in all cases, the energies of Σ, L, and Z valence bands (L equals Z in rock-salt Ge 1-x Sn x Te, x = 0.8 and 1) first decrease and then increase with increasing x (or decreasing Δε), reaching the minima at x = 0.6 (Δε = 0.06).The difference is that the Σ valence band changes its energy much faster than the other two.Consequently, Ge 1-x Sn x Te switches its forbidden gap from an indirect one to a direct one as x goes from 0 to 1.It is worth mentioning that E g values of ferroelectrically distorted (rhombohedral) compounds are considerably larger than those of undistorted (cubic) ones, Supporting Information Figure S2.More surprisingly, in the ferroelectric phase region, E g does not intuitively increase with an increasing degree of ferroelectric distortion but has its extrema at the lowest Δε = 0.06 where the compound is near cubic.Besides, in the vicinity of Δε = 0.06, Σ, L, and Z valence bands are almost aligned in energy, giving rise to an overall high N V .This characteristic of valence band structure should be highly favorable to electrical transport properties (particularly Seebeck coefficient), on which we will elaborate below.The temperature-dependent electrical conductivities (σ) and Seebeck coefficients (S) of Ge 1-x Sn x Te solid solutions are plotted in Figure 8a,b, respectively.All samples behave like degenerate semiconductors, that is, σ decreases while S increases with increasing T. At room temperature, the value of σ drops considerably from ~7490 S cm −1 for α-GeTe to ~3130 S cm −1 for the sample Ge 0.8 Sn 0.2 Te.However, it resumes again once more SnTe is alloyed into GeTe.Table 1 summarizes the roomtemperature charge carrier concentrations (n H ) and mobilities (μ H ) for Ge 1-x Sn x Te determined by Hall measurement.All samples possess high hole concentrations in the range 10 20 -10 21 cm −3 because of the abundant cationic vacancies. [36,59]μ H , on the other hand, is much influenced by the chemical composition of samples.Specifically, a significant loss of μ H is found in the intermediate compositions with respect to the two end members α-GeTe and SnTe, and x = 0.2 sample has the lowest μ H among them.These results suggest that the variations in σ of Ge 1-x Sn x Te are mostly governed by carrier mobilities.As illustrated in the inset of Figure 8b, at 300 K, all Ge-containing samples have considerably larger values of S than bare SnTe, and S peaks at x = 0.6 and 0.8.However, at elevated temperatures (T > 500 K), α-GeTe has the highest value of S, whereas SnTe has the lowest one, and there is a monotonous decrease of S with increasing x.
To shed light on the irregular changes of electrical properties in Ge 1-x Sn x Te, we conducted temperature-variant Hall study on selected samples, and the results are illustrated in Figure 8c, where Hall coefficients at any temperature (R H,T ) were renormalized to the one at 300 K (R H,300 K ).In the temperature range of 300-673 K, R H increases in SnTe but decreases in α-GeTe with rising T. The intermediate composition Ge 0.6 Sn 0.4 Te, on the other hand, has a relatively weak dependence of R H on T. The specific variation tendencies of R H with temperatures in different Ge 1-x Sn x Te samples are actually related to their distinctive electronic band structures.In SnTe, the energy of the heavy Σ valence band is much lower than that of the L valence band (Figure 6f).Upon warming, the L valence band lowers its energy while the Σ valence band remains stationary.This leads to a net flow of holes from L to Σ valence band and an increase in R H .It is, however, not the case for α-GeTe where initially Σ valence band is above the L valence band (Figure 6a).As a consequence, there is a net flow of holes from Σ to L valence band and a decrease of R H with increasing temperature.As for the sample Ge 0.6 Sn 0.4 Te, since the three (L, Σ, and Z) valence bands are already converged, no inter-band carrier transfer is allowed, and therefore R H stays constant as temperature varies.
Although Ge 1-x Sn x Te features multiple valence bands, it is still useful to employ a single parabolic band (SPB) model to extract the effective mass (m*) of charge carriers for a rough estimation of the electronic band structure evolution with composition and temperature.If we  assume that charge carriers are mostly scattered by acoustic phonons (applicable to the vast majority of state-of-the-art thermoelectric materials, [60][61][62] including PbSe), [63] under SPB, S and n H can be expressed as a function of reduced Fermi level (η): In Equations (2-4), k B , e, and F n (η) are Boltzmann constant, elementary charge, and the n-th Fermi integrals, respectively. [60,64,65]y fitting the temperature-dependent S and n H , we are able to obtain the m* values of Ge 1- x Sn x Te alloys at different x and different T, Figure 8d and Figure S3.With increasing x, m* first increases and then decreases at almost all temperatures.Specifically, at 300 K, m* peaks at x = 0.6; while at 473 and 623 K, it peaks at x = 0.4 and 0.2, respectively.It is not difficult to understand the largest m* in x = 0.6 sample at room temperature because of its most favorable electronic band structure, where Σ, L, and Z valence bands are converged and E g is also maximized (Figures 6d and 7; Supporting Information Figure S2).However, it is not guaranteed that such a favorable electronic band structure persists in the entire temperature range because, as we know, temperature rising resembles SnTe alloying a lot in promoting the phase transition or in modifying the electronic band structure of α-GeTe.With this in mind, we are able to determine the electronic band structures of Ge 1-x Sn x Te in the x−T diagram, as schematically illustrated in Figure 9. Apparently, among the four stages that the electronic band structure of α-GeTe experiences upon SnTe alloying or subject to heating, "Stage 3" is optimal in terms of large m* and high S.With increasing temperature, "Stage 3" gradually shifts to the lower-x side, which is consistent with our experimental findings in Figure 8b that S is larger in GeTe-rich compositions, when T > 500 K and in Figure 8d that m* peaks at smaller x as T rises.
Another important conclusion that can be drawn from Figure 9 is that, if we wish to achieve high thermoelectric performance in GeTebased compounds at lower temperatures, we should weaken its ferroelectric distortion by chemical doping/alloying as much as possible; otherwise, we need to strengthen or at least not perturb too much its ferroelectric distortion.The latter, however, seems particularly challenging because chemical doping/alloying (necessary to optimize ZT) almost inevitably reduces the ferroelectric distortion of α-GeTe as a result of increased configuration entropy. [29,66,67]

Thermal Properties and Ferroelectric Instability of Ge 1-x Sn x Te
The total (κ tot ) and lattice (κ lat ) thermal conductivities of Ge 1-x Sn x Te as a function of temperature are plotted in Figure 10a,b, respectively.The alloyed samples have considerably lower values of κ tot than α-GeTe or SnTe, partly due to the reduction of κ ele (Figure S4).Moreover, GeTe-rich compositions display "V"-shaped variation in κ tot with rising temperature, and this feature is well reserved in their κ lat -T curves.It is worth noting that, with increasing x, the nadir of κ lat regularly moves toward a lower temperature range and eventually falls below room temperature when x ≥ 0.6.If we define the temperature where the nadir of κ lat appears as T nad , it would be found that the numerical values of T nad and T C are very similar at any given x (x ≤ 0.4), inset of Figure 10b.This indicates that ferroelectric instability might play a crucial role in minimizing heat conduction in GeTe-based compounds.Indeed, measurement of Raman  scattering as a function of temperature was reported for singlecrystalline α-GeTe by Steigmeier et al. [68] and a notable softening of transverse optical phonon modes was identified upon heating.This is characterized by the lowering of Raman frequency and broadening of peaks' halfwidth, which become most significant near the critical temperature of T C .Littlewood et al. [69][70][71] found that the phase transition in α-GeTe cannot be modeled by a conventional Laudau theory containing only even powers of the order parameter, implying that anharmonicity is very important in this simple binary compound.Chattopadhyay et al. [53] further proved that this anharmonicity is strongest near T C where structural refinements on α-GeTe using neutron diffraction data with harmonic thermal parameters yield the largest agreement factor.
Based on Debye-Callaway model, mass and size contrasts between host and guest atoms are considered to cause scattering of phonons in disordered alloys (e.g., Ge 1-x Sn x Te) whose lattice thermal conductivity (κ alloy lat ) can be expressed as: [40,[72][73][74][75] where κ pure lat refers to κ lat for α-GeTe (x < 0.5) and SnTe (x > 0.5), and u is given by: In Equation ( 6), Ω represents the molar volume; h is the Planck constant; and Θ D (Debye temperature), υ s (average sound velocity), and Γ tot (total scattering parameter) are calculated by Equations (7-9): Here, υ T and υ L are the transverse and longitudinal acoustic velocities (see Table 1), respectively; x is the alloying fraction; ΔM/M and Δa/a are the relative changes of average atomic mass and lattice constants, respectively; and η is determined by Poisson radio (ν p ) and Grüneisen parameter (γ):   Energy Environ.Mater.2024, 7, e12535 The ν p and γ are a function of transverse and longitudinal acoustic velocities (summarized in Table S2): Figure 10c,d compare the experimental and simulated lattice thermal conductivities of Ge 1-x Sn x Te alloys at 300 and 600 K, respectively.In both temperatures, Debye-Callaway modeling reasonably describes the experimental κ lat for most compositions.It indicates that alloying could be still effective in frustrating phonons propagation of GeTe-based compounds in addition to the aforementioned ferroelectric instability.However, in a few cases, there are non-negligible inconsistencies between experimental data and simulated results of lattice thermal conductivities, particularly at 600 K for the intermediate compositions.This could be due to two reasons: 1) the sound velocities and lattice constants of samples at 600 K are different from the ones at 300 K, which give rise to errors during simulations at high temperature; and 2) the ferroelectric instability becomes less prominent around 600 K for the samples with x = 0.2 and 0.4 (T C = 540 K and 431 K, respectively), which, however, is not taken into account in the simulations. [76]6.ZT Values of Ge 1-x Sn x Te The calculated ZT values as a function of temperature for Ge 1-x Sn x Te alloys are plotted in Figure 11a.Pristine α-GeTe with the largest ferroelectric distortion displays the highest ZT value of ~1.1 among all samples at elevated temperature (673 K).With increasing x (or decreasing ferroelectric distortion), the high-temperature thermoelectric performance seriously deteriorates, although the room-temperature ZTs of samples with x = 0.6 and 0.8 are slightly enhanced (with respect to α-GeTe) because of their optimal electronic band structures (see Stage 3 in Figure 9) as well as robust ferroelectric instability (T C = ~300 K or lower, Figure 5b).Our results unambiguously demonstrate that engineering the ferroelectric distortion is a feasible approach to regulate the electrical and thermal transport properties of polar compounds like α-GeTe.Moreover, to maximize ZT values of α-GeTe in the intermediate range, its ferroelectric distortion should be enhanced or at least not weakened when optimizing the compositions.
We utilize the SPB model [77] to predict the ZT values (673 K) of α-GeTe with varying carrier concentrations, and the results are illustrated in Figure 11b.Note that experimental κ lat of 0.76 Wm −1 K −1 and the oretically minimum κ lat of 0.45 Wm −1 K −1 are both considered: [22,60,78] the former yields a peak ZT of 1.9 at n H = 1.5 × 10 20 cm −3 , while the latter predicts an even larger peak ZT of 2.6 at n H = 9.6 × 10 19 cm −3 .b) The n H -dependent ZT values of α-GeTe predicted based on the SPB model.

Conclusions
In summary, we show that ferroelectric distortion has a significant impact on both the electronic band structures and lattice dynamics of α-GeTe by analyzing the thermoelectric transport properties of Ge 1-x Sn x Te alloys and performing first-principles theoretical calculations.It is particularly prominent near the phase transition Curie temperatures where multiple valence bands are aligned in energy due to structural modification and significant lattice softening takes place because of ferroelectric instability, both of which are favorable in terms of ZT.Our results reveal that by engineering ferroelectric distortion, it is possible to tune the Curie temperature and transport properties of α-GeTe-based compounds.We also predict a maximum ZT of 2.6 at 673 K for α-GeTe if we could explore proper chemical dopants that are effective in decreasing the hole concentrations down to 9.6 × 10 19 cm −3 and meanwhile do not weaken its ferroelectric distortion.

Experimental Section
Materials synthesis: High-purity raw elements of Ge (99.999%),Sn (99.999%), and Te (99.99999%) were weighed according to the nominal compositions of Ge 1-x Sn x Te (x = 0, 0.2, 0.4, 0.6, 0.8, and 1).They were mixed and vacuum sealed in quartz tubes prior to melting at 1273 K for 10 h.Following that, the tubes were air quenched and annealed at 823 K for 2 days.Subsequently, the annealed ingots were ground into fine powders and densified by spark plasma sintering (SPS, SPS-211LX; Fuji Electronic Industrial Co., Ltd.) at 773 K under a pressure of 50 MPa for 5 min.The relative densities exceed 95% for all samples.
Materials characterizations: The powder X-ray diffraction (XRD) patterns were collected by X-ray diffractometer (X'PertPro-PANalytical) operating at 40 kV/25 mA with a Kβ foil filter.The chemical compositions and backscattered electron (BSE) images were obtained by a Field Emission Electron Probe Microanalyzer (EPMA; JXA-8530F Plus; JEOL Co. ltd, Japan) coupled with energydispersive X-ray spectroscopy (EDS).The samples were thinned by a focused ion beam (FIB, Helios Nanolab G3 UC, FEI) for TEM observation (Talos F200s, FEI).The longitudinal and transverse sound velocities were measured using an ultrasonic instrument (ultrasonic pulser/receiver model 5072PR).Raman measurements (LabRAM HR Evolution) were performed using 523 excitation lasers in the frequency range from 50 to 200 cm −1 with 100% laser intensity.The DSC tests were carried out using the Q20 equipment (TA Instruments).
Transport properties measurements: The electrical conductivity σ and Seebeck coefficient S were simultaneously recorded using the CTA-3 (Cryoall, China) instrument under a low-pressure helium atmosphere.The total thermal conductivity κ tot was calculated by κ tot = C p Dρ, where C p is the heat capacity estimated by: C p /(k B /atom) = 3.07 + 0.00047(T−300), [63] ρ is the mass density measured by Archimedes principle, and D is the thermal diffusivity obtained from Netzsch LFA 457 instrument.The Hall coefficient (R H ) was measured using a physical property measurement system (Lake Shore 8400 Series).The carrier concentration (n H ) and carrier mobility (μ H ) were calculated from n H = 1/(eR H ) and μ H = σR H , where e is the electron charge.
Density functional theory calculations: Density functional theory (DFT) calculations were performed by using the projector-augmented wave technique (PAW) as implemented in the Vienna ab initio simulation package (VASP code). [79,80]The exchanged correlation energy was in the form of Perdew-Burke-Ernzerhof (PBE).Since Te is a heavy element, the effect of spin-orbit coupling (SOC) was included. [81]The calculations of band structures were based on the refined crystal structures of Ge 1-x Sn x Te (x = 0-1.0)at 300 K, where the cutoff energies were set to 500 eV and 100 k-points scattered between each high-symmetry point along the lines of B1-L-Γ-P-Z-B and W-L-Γ-K-L-W in the Brillouin zone. [56]

Figure 1 .
Figure 1.Room temperature a) powder XRD patterns and b) Raman spectra of Ge 1-x Sn x Te alloys.

Figure 2 .
Figure 2. a) Backscattered electron, b) secondary electron images, and corresponding elemental mappings c-e) for Ge 0.6 Sn 0.4 Te sample.

Figure 3 .
Figure 3. Transmission electron microscope images and the corresponding SAED patterns and energydispersive X-ray spectroscopy mappings for a) x = 0 and b) x = 0.4 samples.

Figure 4 .
Figure 4. Crystal structures of a) rhombohedral (an approximation of distortion from the cubic structure along the [1 1 1] direction as denoted by the pink line) and b) cubic SnTe.The room temperature structural parameters of Ge 1-x Sn x Te: c) lattice constants; d) bonds length; e) bonds angle; and f) off-centering displacement (Δε) of Ge atoms (the inset illustrates the determination of Δε).

Figure 5 .
Figure 5. a) Differential scanning calorimetry curves of Ge 1-x Sn x Te samples as a function of temperature.b) Phase transition temperature T C as a function of x.

Figure 7 .
Figure 7.The valence band edge energies of Ge 1-x Sn x Te samples as a function of x (the CBM energies at L point are all set to zero).

Figure 8 .
Figure 8.The temperature-dependent a) electrical conductivity and b) Seebeck coefficient for Ge 1-x Sn x Te samples (the inset of b) shows Seebeck coefficient at 300 K). c) Temperature-dependent Hall coefficients for Ge 1-x Sn x Te samples with x = 0, 0.4, and 1. d) The determined effective mass (m*) of Ge 1-x Sn x Te as a function of x at 300, 473, and 623 K.

Figure 9 .
Figure 9. Schematic illustration of electronic band structure evolutions in the context of x−T diagram.

Figure 10 .
Figure 10.Temperature-dependent a) total thermal conductivity and b) lattice thermal conductivity of Ge 1-x Sn x Te samples (the inset of b shows the T nad and T C as a function of x).Simulated and experimental lattice thermal conductivities for Ge 1-x Sn x Te samples as a function of x at c) 300 K and d) 600 K.

Figure 11 .
Figure 11.a) Temperature-dependent ZT values for Ge 1-x Sn x Te samples and the comparison with other reported values.[77]b) The n H -dependent ZT values of α-GeTe predicted based on the SPB model.