Introduction

Thermoelectric technology has become a hotspot in scientific society due to the capability of converting waste heat to electric energy, which aligns with the carbon neutrality initiative1,2,3. Such a process is ubiquitous and continuous with no pollution emission, providing a sustainable pathway to mitigate the global energy crisis and climate-related issues4. To obtain feasible thermoelectric devices with a decent conversion efficiency (ϕ), the key lies in finding thermoelectric materials with a high figure-of-merit, defined as:

$${ZT}={S}^{2}{\sigma }T/{\kappa }$$
(1)

where S, σ, T, and κ are the Seebeck coefficient, electrical conductivity, absolute temperature, and thermal conductivity, respectively5,6,7,8,9. Note S2σ in Eq. 1 is also termed power factor, and κ usually contains the electronic and lattice fractions, i.e., κ = \({{\kappa }}_{\rm{e}}\) + \({{\kappa }}_{{\rm{l}}}\)10,11,12. In most solid materials, their S is inversely dependent on the carrier concentration, while the σ behaves oppositely13,14, leading to an inherent ZT lower than unity. Effective strategies to increase the ZT include compromising on carrier concentration to maximise S2σ15,16,17,18, decoupling the interdependence to reach a high quality factor (B ~ S2σ/\({{\kappa }}_{{\rm{e}}}\))19,20, and decreasing the relatively independent \({{\kappa }}_{{\rm{l}}}\) by hierarchical lattice defects that scatter phonons in wide frequencies, e.g., point defects, stacking faults, grain boundaries, and nanoprecipitates21,22,23,24.

As one of the most eminent mid-temperature thermoelectric systems, GeTe favours an inherent ZT of ~0.9 due to large band asymmetry and dynamic lattice anharmonicity, which naturally secures both a large S2σ and a low \({{\kappa }}_{{\rm{l}}}\)25,26,27. As illustrated in Fig. 1a, pristine GeTe undergoes a reversible phase transformation from rhombohedral (R-) to cubic (C-) phase at 650 ± 50 K, which can be seen as a ferroelectric transition deriving from the Peierls distortion and displacement of Ge and Te sublattices driven by long-range polarisations28,29. However, the structural instability also facilitates the formation of intrinsic Ge vacancies, which act as acceptor dopants and result in excessive hole concentration (\({n}_{{\rm{h}}}\) ~ 1021 cm−3) beyond its optimal range and a degradation on the ZT30,31,32. This behaviour originates from the intrinsically low formation energy of the Ge vacancy, making GeTe thermodynamically prone to cation deficiency, off-stoichiometric compositions, and Ge precipitation. Although the high vacancies beneficial for phonon scattering, they impose a challenge in balancing thermal and electric transport properties. To overcome this, increasing the formation energy through chemical doping is an effective strategy to shift the defect formation equilibrium. Isovalent or aliovalent doping of Pb2+, Bi3+, or Sb3+ has proven effectively supress Ge vacancy formation and compensate the hole concentration33,34,35. Pb doping decrease the average chemical potential (µ) at cation sites, making Ge vacancies formation less favourable, while counter doping induces donor-like effects to optimise \({n}_{\rm {h}}\). Figure 1b compares the formation energy of a Ge vacancy as a point defect (D) in a Ge27Te27 supercell doped with different elements under both Ge-rich and Ge-poor conditions, calculated based on the density-functional theory (DFT) method and Eq. 2:

$$\Delta H({{\rm{D}}},{\mu }) \,=\, {E}_{\rm {d}}\,{-}\,{E}_{\rm {h}}\,{-}\sum {n}_{i}{\mu }_{i}+{E}_{\rm {c}}$$
(2)

where \({E}_{\rm{d}}\), \({E}_{\rm{h}}\), n, and \({E}_{\rm{c}}\) are the energy of a defective structure, the energy of a perfect structure, the number of atoms that are substituted in the system, and the correction term in the Leslie-Gillan scheme, respectively36. The subscripted “i” denotes the species of elements relevant to the formation of D. It is apparent that doping Pb enhances the ∆H for both rhombohedral and cubic phases under Ge-rich and Ge-poor conditions.

Fig. 1: Scope and background.
Fig. 1: Scope and background.
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a Schematic diagram of the reversible phase transformation between rhombohedral (R-) and cubic (C-) GeTe. b Point defect formation energy of a Ge vacancy in a Ge27Te27 supercell doped with various elements in both Ge-rich and Ge-poor conditions. c Comparison of the proposed temperature-dependent phase evolution between ref. 40. and this work, where O-, α-, and β- indicate orthorhombic, hexagonal, and cubic structure. d Comparison of ZT and average ZTavg between literature and this work.

It should be noted that doping Pb is restricted due to its well-known toxicity. Transitional metals with unfilled d-orbitals, thus a normally high µ are promising alternative dopants for GeTe-based thermoelectric materials37. Particularly, DFT calculations reveal an even higher ∆H in R-Ge26Cu1Te27 and C-Ge26Cu1Te27 over Pb-substituted structures under both conditions. Experiments can verify our theoretical analyses. For instance, Xia et al. reported a critically decreased \({n}_{{\rm{h}}}\) from 8.0 × 1020 cm−3 in GeTe to 1.3 × 1020 cm−3 by merely doping 2% CuSbTe2, along with a retained carrier mobility over 100 cm2 · V−1 · s−138. Yin et al. observed Cu ions at interstitial sites in the non-equilibrium doped Ge0.93Ti0.01Bi0.06CuxTe, which dampen \({n}_{{\rm{h}}}\) down to 1.22 × 1020 cm−339. Jin et al. alloyed GeTe with 20% CuSbSe₂, which optimises carrier concentration and triggers numerous point defects that effectively scatter phonons, resulting in a remarkable ZT value of ~ 2.610. However, interstitial Cu, albeit being an electron donor, ends up with a negative ∆H, which is harmful for suppressing Ge vacancies, especially under Ge-poor conditions. In this context, an insightful understanding of the kinetic picture in which substituting Cu and interstitial Cu could be differentiated is pivotal for GeTe-based thermoelectric materials.

In 2022, Schwarzmüller et al. investigated the temperature-dependent phase evolution in Cu2Ge16Sb2Te2040. Using X-ray absorption contrast imaging, they uncovered the endotaxial intergrowth between Cu2−xTe nanorods and Cu-doped matrix, which undergo multiple phase transformations upon heating until forming a solid solution at 855 K, as shown in Fig. 1c. Here, inspired by their work, we develop a synthesis method based on solid solution treatment to dope Cu in GeTe-based thermoelectric materials via Cu diffusion. By this method, the doped Cu fills the vacancies and substitutes Ge at cationic sites, without hopping into interstitial sites or forming a cuprous impurity phase. Interestingly, we notice an unreported phase formation and dissolution phenomenon during synthesis, which, in conjunction with Ge vacancy suppression, renders a lattice refinement in the spirit of lessening lattice defects and residual strains toward high-quality transport characteristics41,42. A desirable ZT of 2.3 at 775 K and an average ZT (ZTavg) of 1.4 at 300 − 775 K are realised in the as-synthesised Cu-doped Ge0.85Sb0.10Te, registering a relatively competitive performance in this field, as shown in Fig. 1d10,40,43,44. The experimental power density of the corresponding single-leg device achieves ~2.23 W·cm−2 when the temperature difference is 475 K.

Results and discussion

Footage of Cu ion diffusion by solid solution treatment

In this work, we first synthesise Ge0.85Sb0.10Te (GST) based on the eutectic reaction, namely weighing pure elements in nominal compositions, melting at 1225 K for 1 day, and quenching in ice water. Then, we pulverise the obtained ingots into fine powder to be mixed with Cu powder and soaked at 875 K for 3 days, followed by cooling to room temperature in a muffle furnace to synthesise Cu-doped Ge0.85Sb0.10Te (GST + xCu; with x = 0, 0.005, 0.010, 0.015, 0.020, 0.030 and 0.040 being Cu doping concentrations). The phase purities of as-synthesised samples are examined using synchrotron X-ray diffraction (XRD, see Supplementary Fig. 1), agreeing that GST is in a homogeneous R-GeTe structure. Upon doping Cu, the Ge phase separation appears, and most peaks in the full survey become sharper, indicating less lattice disorder. Figure 2a presents in situ XRD measurements of GST and GST + 0.01Cu recorded from 300 to 875 K to compare and reveal the diffusion behaviour of Cu ions. Interestingly, we find phase formation occurring from 630 to 855 K, during which the doped Cu initially forms a Cu-rich phase, then dissolves in the matrix and does not precipitate after cooling to room temperature. To support the results that we obtained from in-situ XRD, we performed an ex-situ XRD for GST + 0.04Cu with different annealing durations (0, 3, 7, and 7 days with 3 days heating and cooling cycles, Supplementary Fig. 2). There is significant Cu signal observed at the 0-day annealed sample. With the increasing annealing duration, the Cu signal disappeared and the Ge signal detected, indicating Cu diffused into the GST matrix through filling or substituting the Ge position. The Rietveld refinements are shown in Fig. 2b, and the fitting parameters are summarised in Supplementary Table 1. Doping 4 at.% Cu decreases the lattice parameter a from 4.20565(7) to 4.19909(3) Å by only 0.15%, but increases lattice parameter c from 10.4909(4) to 10.5345(2) Å by approaching 0.5%. The cation-to-anion ratio enhances from Ge0.8466Sb0.1Te to Ge0.8226Cu0.04Sb0.1Te accordingly. Pristine GST is prone to containing Ge vacancies with a reduction on the c-axis length45.

Fig. 2: X-ray investigations.
Fig. 2: X-ray investigations.
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a In situ heating XRD of Ge0.85Sb0.10Te (lower) and GST + 0.01Cu (upper) from 300 to 875 K. b Rietveld refinement of GST, GST + 0.01Cu, and GST + 0.04Cu. c Refinement of the local ranged PDF patterns of GST, GST + 0.01Cu, and GST + 0.04Cu from a 0 to 20 Å radial distance (r). d−f Refined dependencies of lattice parameter c, and the atomic displacement parameters at Ge sites with increasing rmax for GST, GST + 0.01Cu, and GST + 0.04Cu at U11 and U33. g Full XPS surveys of GST and GST + 0.01Cu. The inset shows the detailed survey of GST + 0.01Cu from 926 to 960 eV. h XANES spectra of GST and GST + 0.01Cu. The inset shows the corresponding EXAFS spectra scoping 1.5 to 3 Å.

We also investigate the change of local structures due to doping Cu in addition to the average structures using the pair distribution function (PDF) technique, which describes the distribution of distances (r) between atomic pairs of particles contained within a given volume46. The PDF patterns of GST, GST + 0.01, and GST + 0.04 collected at room temperature are fitted with a small box R-GeTe structure with lattice parameters and ADPs refined. As summarised in Fig. 2c and Supplementary Fig. 3, the worse fitting in the low-r region compared to the full-r region indicates a different short-range structure from the average structure in all three samples.

To understand the discrepancies between local and average structure, we further perform 24 refinements in a box-car manner with a fixed minimum r (rmin = 1 Å) and allowing the maximum r (rmax) to vary from 20 to 250 Å in 10 Å increments. As shown in Fig. 2d and Supplementary Figs. 4–6, the refinement residual indicator (Rwp) increases with higher rmax and saturates at the 15th refinement (rmax = 160 Å), indicating that beyond this scale, the four possible atoms at the Ge sites (i.e., vacancy, Ge, Sb, Cu) become statistically homogeneous. Moreover, the lattice parameter c (Fig. 2d) decreases dramatically with increasing rmax and saturates at the 12th refinement (rmax = 130 Å), which can be attributed to the effect of the Ge vacancy, while the lattice parameter a is almost constant. It is worth mentioning that the drop of c decreases from 0.27% in GST to 0.06% in GST + 0.04Cu, suggesting the remedy of Ge vacancy, leading to vacancy suppression. Additionally, all atomic displacement parameters (ADPs U11, U22, and U33) of Te sites increase with the increasing rmax, which results from the uncorrelated atomic motion of Te atoms (R-GST structure framework atom). A similar trend is observed in the U11 of Ge/Sb/Cu sites (Fig. 2e, f), but the U33 is significantly higher when the r range is shorter than 30 Å, suggesting the presence of locally disorder at Ge/Sb/Cu sites. In addition, the drop in U33 at the local range decreases from 42.8% in GST to 23.4% in GST + 0.04Cu, indicating the remedy of local disorder by Cu doping. The ADPs saturation occurs from the 12th refinement onwards, implying the local domain size is in average 13 nm, implying a highly dense “herringbone structure” commonly existed in GeTe-based thermoelectric materials47.

To investigate the valence state of Cu, we performed X-ray photoelectron spectroscopy (XPS) measurements. As presented in Fig. 2g and Supplementary Fig. 7, Cu signals are detectable in a binding energy window of 926–960 eV in the full surveys of GST + 0.01Cu and GST + 0.04Cu. The detailed spectra can be well fitted with Cu (I), meaning that no interstitial Cu exists in these samples under the instrument detection limit. Figure 2h and Supplementary Fig. 8 display X-ray absorption spectroscopy (XAS) results, which probe the valence state information subject to a much lower detection limit due to the high intensity and high flux of synchrotron radiation. Particularly, the X-ray absorption near edge spectroscopy (XANES) spectra show a 2 eV shift of the position of Ge_K edge from GST to GST + 0.01Cu, and the position of Cu_K edge for GST+Cu0.01 is consistent with the Cu+ sample. In addition, the extended X-ray absorption fine structure (EXAFS) spectra show different radial distances bonded to Ge, both serving as evidence of Cu substituting Ge.

Modification of crystallography and microstructures

Figure 3a, b and Supplementary Fig. 9 show the scanning electron microscopy backscattered electron (SEM-BSE) and secondary electron (SEM-SE) images to learn the microstructures. In viewing the fractographies of GST and GST + 0.04, the grains change from irregular and microscale to cubic and nanoscale with increasing x. Statistical studies based on numerous cross-sectional SEM images (Supplementary Figs. 10, 11) show that the grain size distribution is smaller and more uniform in samples with higher Cu doping levels. In the pristine sample, the grain size primarily ranges from 1.4 to 2.6 μm, whereas in the 4% Cu-doped sample, grain sizes range from 0.9 to 1.7 μm, and the quasi-layered textures disappear. We also conduct SEM analysis on polished surfaces; the observations indicate that Ge tends to precipitate for Cu dopants substituting the host positions, being consistent with the fractographies (Supplementary Fig. 12). To get down to atomistic level, scanning transmission electron microscopy (STEM) and transmission electron microscopy (TEM) are further performed on the lamella specimen of GST + 0.01Cu (see Fig. 3d, e). According to the energy-dispersive X-ray spectroscopy (EDS) mapping in Fig. 3c, the nanoprecipitates in the sample can only be identified as Ge impurity phase, and Cu distributes uniformly in the matrix, being consistent with the aforementioned results. STEM images are in [\(\bar{1}\bar{2}1\)] and [\(\bar{2}\bar{2}1\)] zone axes are displayed in Fig. 3d, e. Apparently, no Cu ions were observed at the interstitial positions in dark mode, where the intensities are inversely proportional to atomic mass. High-resolution TEM images in [101], [100], and [120] zone axes are displayed in Supplementary Fig. 13, superposed with their atomic models. The insets show the corresponding fast Fourier transfer (FFT) patterns. As can be seen, doping Cu does not change the crystallography, so it is the phase formation and dissolution and Ge vacancy suppression that triggers the lattice refinement.

Fig. 3: Microstructure analysis.
Fig. 3: Microstructure analysis.
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a, b SEM-BSE images of Ge0.85Sb0.10Te (GST) and GST + 0.04 fractographies, respectively. The insets enlarge the labelled area. c TEM-EDS mapping of GST. d, e STEM-dark field images of GST + 0.01Cu in [\(\bar{1}\bar{2}1\)] and [\(\bar{2}\bar{2}1\)] zone axes. The insets show the corresponding FFT patterns. f MD simulated atomic trajectories in the (100) plane of Ge31Cu1Te32 at 975 K for 3000 ps.

In the preceding discussion, we mentioned that GST + xCu samples are free from interstitial Cu and cuprous precipitates, as Cu substituting for Ge is thermodynamically and kinetically more stable compared with neighbouring sites. To confirm our hypothesis, ab initio molecular dynamics (MD) simulations are performed to predict the vibrational trajectories of atoms at a finite temperature. As can be seen in Fig. 3f, the liquid-like behaviour of Cu is suppressed by the immobile sublattices at 875 K. The oscillation amplitude of Cu is modestly larger than that of Ge and Te, indicating a quasi-coherent fluctuating behaviour of the constituent atoms. Note the vibration of Cu is more anisotropic than that of Ge, implying the anharmonic lattice in GST+xCu that is correlated to a low \({{\kappa }}_{{\rm{l}}}\).

Transport and thermoelectric properties

Figure 4a−c summarises the temperature-dependent thermoelectric properties of GST+xCu. In overall, the S is positive with an upward trend in both R- and C-GeTe temperature windows, indicating a p-type conductance in narrow-gap semiconductors. With increasing x, the S does not vary from the original values (by < 5 μV·K−1), being consistent with the nature of equivalent doping. In contrast, the σ decreases with increasing temperature in both R- and C-GeTe temperature windows. The sudden jump of σ at 625 K is due to the ferroelectric transition. As x increases, the σ first increases to 852 S·cm−1 at 625 K when x = 0.010, then decreases due to the influence of carrier mobility at higher doping concentrations. Increasing temperature leads to a phase transformation from an amorphous rhombohedral structure to a symmetric cubic structure, resulting in significant changes in electronic transport. Approaching to the Curie temperature of such a ferroelectric transition is accompanied with softening of lattice vibration modes, which enhances acoustic phonon scattering charge carriers and then degrades the σ48. Due to symmetrisation, the energy offset from light band to heavy band decreases, leading to band convergence and populated density of states (DOS) near the Fermi level, which further increases the σ.

Fig. 4: Electrical properties and electronic transport.
Fig. 4: Electrical properties and electronic transport.
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a−c Temperature-dependent Seebeck coefficient (S), electrical conductivity (σ), and power factor (S2σ) of GST + xCu, respectively. d Pisarenko plot of GST+xCu. e Calculated weighted mobility (\({\mu }_{{\rm{w}}}\)). f Differential electron density distribution between C-Ge32Te32 and C-Ge31Cu1Te32. g, h Unfolded band structure (left) and DOS (right) of C- and R-Ge22Sb3Cu1Te27, respectively.

As a result, the highest S2σ is obtained in GST + 0.01Cu, reaching 41.2 μW·cm−1·K−2 at 625 K. These electrical properties suggest that although doping Cu is posed to suppress Ge vacancies, its effect in decreasing hole concentration is modest. Instead, the modification of the effective mass (m*) and carrier mobility plays more important roles in enhancing S2σ49. Figure 4d shows the Pisarenko plot of S vs. Hall carrier concentration (nH) at room temperature under the single parabolic band (SPB) approximation, formulated by Eq. 3 as ref. 50:

$$S=\frac{8{{\pi }}^{2}{{k}_{\rm{B}}}^{2}}{3e{h}^{2}}{m}^{*}T{\left(\frac{{{{\pi }}}}{3{n}_{{\rm{H}}}}\right)}^{2/3}$$
(3)

where \({k}_{\rm{B}}\) and h are the Boltzmann and the Planck constants. With increasing x from 0 to 0.015, the nH of all samples remains nearly unchanged, which lies in an optimal range between 2 × 1019 and 1.1 × 1020 cm−3. Meanwhile, the m* is enlarged by 10% from 2.8 to 3.1 m0 by increasing x to 0.005, then decreased by 26% to 2.3 m0 when x = 0.015. Figure 4e calculates the temperature-dependent weighted mobility (\({\mu }_{{\rm{w}}}\)) based on Eq. 451:

$${{\mu }}_{\mathrm{w}}=\frac{{T}^{-3/2}}{{300}^{-5/2}}\left[\frac{\exp \left[\frac{\left|S\right|}{{k}_{\mathrm{B}}/e}-2\right]}{1+\exp \left[-5\left(\frac{\left|S\right|}{{k}_{\mathrm{B}}/e}-1\right)\right]}+\frac{\frac{3}{{{{{\pi }}}}^{2}}\frac{\left|S\right|}{{k}_{\rm{B}}/e}}{1+\exp \left[5\left(\frac{\left|S\right|}{{k}_{\rm{B}}/e}-1\right)\right]}\right]$$
(4)

where e is the elementary charge. As can be seen, the \({\mu }_{{\mathrm{w}}}\) increases from 145.6 cm2·V−1·s−1 in GST to 177.3 cm2·V−1·s−1 in GST + 0.01Cu.

To verify the analytical analysis, we used the van der Pauw method to measure the temperature-dependent carrier concentration and mobility, and the results are presented in Supplementary Fig. 14. Below room temperature, nH remains relatively constant for both samples. Coming to higher temperatures, an increase in nH can be observed due to thermal activation. The doped sample shows apparently lower nH owing to the addition of Cu. In contrast, our measurements reveal no significant difference in hall mobility between GST and GST + 0.01Cu. This can be attributed to two competing effects, the diffusion of Cu into the matrix fills the Ge vacancies, leading to a uniform crystal field to warrant the mobility, while simultaneously substituting Ge atoms and introducing point defects that tend to reduce mobility. The interplay between vacancy filling and cationic substitution likely results in the observed negligible change in overall Hall mobility. Besides, Cu incorporation may also induce microstructural modifications, which further influence transport properties.

The extraordinary effect of doping Cu in enhancing carrier mobility is rooted in Ge vacancy suppression and the long-range interactions conceived by the metavalent Ge–Te chemical bonds52,53. Figure 4f shows the differential electron density between C-Ge32Te32 and C-Ge31Cu1Te32, indicating that doping Cu mainly modifies the charge transfer along [001] directions. We also calculated the changes in charge density by Ge vacancy and Cu at the Ge site in GST to provide sufficient insight. In both phases (refers to Supplementary Fig. 15), the charge density is higher at the Cu substituted position. The Cu-Sb interaction is studied using electron localisation function (ELF) calculations based on a face-centre cubic supercell, showing that there is no distinct bonding features or charge localisation between Cu and Sb for GST + Cu. As shown in Supplementary Fig. 16, Cu substitution leads to electron delocalisation around the Cu atom, suggesting the Cu atom does not form strong bonds with neighbouring atoms. Figure 4g, h calculates the unfolded band structures and density-of-state (DOS) of C- and R-Ge22Sb3Cu1Te27. As can be seen, the valence band maxima (VBM) and conduction band minima (CBM) are dominated by Te_5s and Ge_4p orbitals, respectively. Comparison of the unfolded band structure of both phase Ge27Sb1Te27 (refers to Supplementary Fig. 17) reveals that Cu doping shifts C-GST from a direct semiconductor to an indirect semiconductor. This transition is further supported by a bandgap (\({E}_{\rm{g}}\)) shrinkage from 0.23 to 0.15 eV, as observed in the folded band structure (Supplementary Fig. 18). Multiple quasi-degenerated valleys also appear near VBM, which gives rise to degeneracy number (\({N}_{\rm{v}}\)), ultimately resulting in a larger m*54. Moreover, the Green’s function method is conducted to study the variation of band structure within the coherent potential approximation. As shown in Supplementary Fig. 19, Cu doping in both the cubic and rhombohedral phases results in a slight upward shift of the valence band maxima (VBM1, VBM2, and VBM3). In addition, a tiny band convergence can be observed after doping. Band convergence is associated with an enhancement in S in most cases, due to the increased DOS near the Fermi level.

Coming to thermal properties, Fig. 5a measures temperature-dependent κ, which decreases with increasing x, attaining a minimum value of 1.12 W·m−1·K−1 at 575 K in GST + 0.01Cu. Figure 5b shows \({{\kappa }}_{{\rm{l}}}\) by subtracting \({{\kappa }}_{{\rm{e}}}\) from κ, which is calculated based on the Wiedemann–Franz law. The minimum \({{\kappa }}_{{\rm{l}}}\) is 0.40 W·m−1·K−1 at 725 K in GST + 0.01Cu, lower than the amorphous limit of 0.45 W·m−1·K−155. To rationalise the decrease of \({{\kappa }}_{{\rm{l}}}\), Fig. 5c calculates the spectral lattice thermal conductivity (\({{\kappa }}_{{\rm{s}}}\)) based on phonon frequency relative to the Debye frequency (\(\omega\)/\({\omega }_{{\rm{D}}}\)). Ideally, under the relaxation approximation, the inverse of group phonon relaxation time (τ) is the addition of participant phonon scattering processes, i.e., the Matthiessen rule, formulated by Eq. 5 as:

$${{{{\tau }}}}^{-1}=\,{{{{{\tau }}}}_{{\rm{U}}}}^{-1}+{{{{{\tau }}}}_{{\rm{GB}}}}^{-1}+{{{{{\tau }}}}_{{\rm{PD}}}}^{-1}+{{\,{{{\tau }}}}_{{\rm{NP}}}}^{-1}+{{\,{{{\tau }}}}_{{\rm{SF}}}}^{-1}$$
(5)

where the subscribed U, GB, PD, NP, and SF denote phonon scattering induced by normal and Umklapp process, grain boundaries, point defects, nanoprecipitates, and stacking faults. As the integral of \({{\kappa }}_{{\rm{s}}}\) at a certain temperature refers to \({{\kappa }}_{{\rm{l}}}\), the area between two adjacent curves should equal the decreased \({{\kappa }}_{{\rm{l}}}\) by the additional phonon scattering process. Clearly, grain boundaries, point defects, and nanoprecipitates scatter low-, mid-, and high-frequency phonons. It is then clarified that hierarchical lattice defects contribute to all-scale phonon scattering and decrease \({{\kappa }}_{{\rm{l}}}\) effectively56,57.

Fig. 5: Thermal and thermoelectric properties.
Fig. 5: Thermal and thermoelectric properties.
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a, b Temperature-dependent thermal conductivity (κ) and lattice thermal conductivity (κl), respectively. c Spectral lattice thermal conductivity (κs) considering various phonon scattering processes. d Figure-of-merit (ZT) as a function of reduced chemical potential (η) and quality factor (B). e Temperature-dependent figure-of-merit (ZT). f Experimental current (I)-dependent voltage (V) and power density of GST + 0.01Cu at ΔT = 75 K, 175 K, 275 K, 375 K, and 475 K.

Recalling the enhanced \({\mu }_{{\rm{w}}}\), Fig. 5d plots ZT vs. B as a function of reduced chemical potential (η). According to the numerical modelling results, the B enhances from 1.2 in GST to 1.6 in GST + 0.01Cu, meaning a higher maximum achievable ZT when the carrier concentration is fully optimised by doping. The enhancement of B is mainly attributed to the retaining of S2σ due to vacancy suppression and lattice suppression, while the decrease of κl by hierarchical phonon scattering. Consequently, the ZT of GST + 0.01Cu achieves > 2.3 at 775 K with the error bar calculated by multiple measurements (Supplementary Fig. 20), as shown in Fig. 5e, which is 12% higher than that of GST. Envisaging practical applications during which thermoelectric materials need to work under a large temperature difference (∆T), ZTavg is even more important over ZT. Here, we calculate ZTavg based on Eq. 6:

$$Z{T}_{{\rm{avg}}}=(1/\Delta T){\int }_{{{{T}}}_{{\rm{c}}}}^{{{{T}}}_{{\rm{h}}}}{ZT}(T){{\rm{d}}}T$$
(6)

which is around 1.4 at 300 − 775 K for GST + 0.01Cu (Supplementary Fig. 21).

We then simulate the potential of applying the present samples in a single-leg device. Supplementary Fig. 22 shows that the current (I)-dependent open circuit voltage (V) increases from 6.8 to 50.7 mV with enlarging the ΔT, while output power (P) achieves a maximum value of 152.1 mW at ΔT = 475 K. Accordingly, the predicted maximum output power density reaches 3.43 W·cm−2 with 11.82% conversion efficiency (η). The rationality of our analysis is verified using finite-element analysis (FEA) simulation of a modelled device working under a non-zero ΔT, and the results and parameters used for simulation are presented in Supplementary Fig. 23 and Supplementary Tables 3−5. Experimentally, a single-leg device is fabricated. With a fixed cold side of 300 K, the V, power density, P and η as a function of I under different ΔT are shown in Fig. 5f and Supplementary Fig. 24. When ΔT = 475 K, the Pmax reaches ~ 100 mW, the maximum power output approaches 2.23 W·cm−2 with ηmax achieves 8.17% for the experimental results. The discrepancy between the experimental and theoretical curves is attributed to contact resistance, heat loss, and other effects common in practical device configurations.

In this work, we develop a new synthesis method based on solid solution treatment, which can realise doping Cu in GeTe-based thermoelectric materials through ion diffusion without forming interstitial doping or cuprous impurities. Using advanced synchrotron technologies and electron microscopies, we clarify a unique phase formation and dissolution phenomenon and Ge vacancy suppression in the as-synthesised GST + xCu samples. These factors are considered the origin of vacancy suppression and lattice refinement, which optimise electronic and thermal transport synergistically. As a result, the GST + 0.01Cu sample exhibits a high S2σ of 41.2 μW·cm−1·K−2 at 625 K and a low κ of 1.1 at 575 K that can be explained by numerical modelling and DFT calculations, leading to a desirable ZT of 2.3 at 775 K with an average value of 1.4 from 300 to 775 K, as well as consistent device performance. Our work not only provides an insight of the doping behaviour and mechanism of Cu in the GeTe system, but also develops an innovative way of using solid solution treatment to dope dynamic elements in a rigid lattice, which is transferrable to other thermoelectric systems.

Methods

Synthesis

Ge0.85Sb0.1Te + xCu (GST + xCu. x = 0, 0.005, 0.010, 0.015, 0.020, 0.030, 0.040) are synthesised using the melting-quenching-annealing method followed by solid solution treatment and spark plasma sintering (SPS-2000, SUGA Co., Ltd) technique. High-purity elements, including Ge (99.9%, Sigma-Aldrich, Australia), Sb (99.9%, Sigma-Aldrich, Australia), Te (99.9%, Sigma-Aldrich, Australia), and Cu (99.9%, Alfa-Aesar, United States), are weighed precisely based on the nominal composition. The mixture of Ge, Sb and Te are placed in a quartz tube and sealed under 10−4 Pa. The sealed tube is heated to 1225 K in 8 h and soaked for 12 h, followed by ice water quenching. The GST ingot is then ground using a mortar and pestle and mixed with the molar ratio of Cu powder listed above, which is then sealed into the quartz tube and annealed at 875 K for 3 days to allow Cu diffusion. The annealed product is then ground into powder again using a mortar and pestle and subsequently sintered into pellets by SPS at 725 K for 5 min under 60 MPa.

Characterisation

In-situ heating XRD measurements are performed using a Rigaku Smartlab diffractometer with Cu Kα radiation over the range of 20° ≤ 2θ ≤ 80° to investigate the process of Cu diffusion. Phase identification is performed using DIFFRAC.EVA v7.2 software with the ICDD PDF-5 + database. Full pattern Rietveld refinements are conducted in DIFFRAC.TOPAS v7 software. The pair distribution function (PDF) of final powder samples is measured on beamline ID31 of the European Synchrotron Radiation Facility (ESRF) in a Rapid Acquisition PDF setup. NIST SRM 660b (LaB6) is used for geometry calibration, which is carried out using the pyFAI software. PDF refinements were conducted in the xPDFsuite software. The valence states are measured using an X-ray photoelectron spectroscopy (XPS, Kratos Axis ULTRA, Kratos Analytical Limited, Japan) and analysed using CasaXPS software. The micro- and nanostructures are investigated using a field-emission scanning electron microscopy (SEM, Hitachi 7000) and a transmission electron microscopy (TEM, JEOL ARM200F NeoARM) with energy dispersive X-ray spectrometry (EDS).

Thermoelectric performance evaluation

The electrical conductivity (σ) and Seebeck coefficient (S) are measured simultaneously using a commercial system (ZEM−3, ULVAC) with rectangular bar-shaped samples in the size of ~1.5 × 1.5 × 6 mm3. The thermal conductivity (κ) is calculated by using κ = DρCp, where D is the thermal diffusivity measured by a Laser Flash thermal analyser (LFA 467, NETZSCH) under argon atmosphere, Cp is the specific heat capacity referred to the previously reported data for GST, and ρ is the density. Tests on the reproducibility and repeatability are performed. Specifically, we prepared two independent samples, and for each sample, we performed measurement in heating-cooling loops. We further used the Physical Property Measurement System (PPMS) and homemade equipment to measure temperature-dependent carrier concentration (n) and mobility (μ). The Hall coefficient (RH) is obtained by the van der Pauw method, then we calculated n and μ by n = 1/(eRH) and μ = σRH, where e is the elementary charge.

Detailed methodologies including DFT calculations, numerical modelling, and FEA simulations are referred to Supporting Information.