Introduction

With the increasing energy demands and environmental concerns, seeking renewable energy solutions and developing technologies with high energy conversion efficiency are of paramount significance1. Thermoelectric (TE) technology, capable of directly converting various waste heat into electricity based on the Seebeck effect, or providing electronic cooling based on the Peltier effect, could play an pivotal role in a sustainable future2. The attributes of TE devices, including excellent reliability, scalability, no moving part or emission, and so on, make them promising for recovering low-to-intermediate temperature (500–900 K) waste heat from industry sectors or vehicle exhaust3,4. The low conversion efficiency, scarcity of constituents (e.g., Te), and thus high cost of typical TE materials, such as Bi2Te3 and PbTe, and so on, however, largely limit the large-scale applications of TE technology. With the virtues of ultralow lattice thermal conductivity (κL ~0.4 W m−1 K−1 for the pristine sample at high temperatures), high Seebeck coefficient, and good thermochemical stabilities, BiCuSeO is one of the most promising candidates for mid-temperature TE power generation applications5,6,7. High TE figures of merit (ZT) of 1.2–1.5 have been achieved in BiCuSeO by utilizing the strategies such as modulation doping8, texturing9, hierarchic structuring10, chemical bonding engineering11, and so on. As compared with typical TE semiconductors, for example, Bi2Te312 and PbTe13, however, its relatively low power factor (PF) (normally <10 μW cm−1 K−2) largely hinders the further improvement of ZT. Recently, many approaches, including carrier concentration engineering14,15, band structure engineering by modifying the band convergence16,17 or introducing sharp features in the electronic density of states (DOS)3, carrier scattering mechanism engineering18, unique band features driven by the spin–orbit coupling (SOC)19, and so on, emerge to largely improve the PF and thus ZT of TE materials.

The electronic band structure of BiCuSeO calculated by first-principles density functional theory (DFT) shows a complex constitution with multiple valleys near the valence band maximum (VBM), and the partial DOS plots further reveal that VBM is dominated by the hybridized Cu 3d and Se 4p orbitals20. These multiple conduction valleys could contribute simultaneously but weigh differently to the electrical transport at a specific Fermi energy (EF), depending on the band effective mass and their relative positions to EF. Inspired by the work done by Zhang et al.21 in the Mg3Sb2-based system, a comprehensive, systematic investigation of the correlation between band features (e.g., DOS, band convergence) and carrier transport properties is required to fundamentally understand and further optimize the carrier transport of BiCuSeO, though its large Seebeck coefficients (>150 μV K−1) and moderate PF values (~7 μW cm−1 K−2) are normally attributed to the complex band structure. In addition, the increasing band degeneracy (Nv) with down-shifting the EF, and the evident band nonparabolicity could give rise to unique electrical properties, as compared to other typical TE materials showing only single PF and/or ZT peak upon carrier concentration modification. These will bring about additional challenges in studying the band-transport correlation, but do offer opportunities to optimize the electrical transport properties.

In this work, we have successfully prepared Bi1 − xPbxCuSeO samples (x = 0–0.2) by a time- and energy-efficient self-propagating high-temperature synthesis and spark plasma sintering (SHS-SPS) technique, and systematically investigated their high-temperature TE properties and the correlation between the band structure features and electrical transport properties. With the EF shifting down, that is, increasing the carrier concentration by increasing the Pb content, two maxima in the PF and ZT values have been observed. The first one is traceable to the convergence of multiple valence bands, and the subsequent results from the conventional trade-off between the electrical conductivity (σ) and Seebeck coefficient (S). Additional PF and ZT maxima can be observed with further increase in the Pb content beyond the solubility limit (x > 0.12), primarily due to the compositing effect originated from in situ formed PbSe precipitates. Consequently, ZT peaks of 0.9, 1.1, and 1.3 at 873 K are achieved for 4, 10, and 14 at% Pb-doped BiCuSeO samples, respectively.

Results

Electronic band structure of BiCuSeO

The electronic band structure and DOS of BiCuSeO were calculated taking into account the SOC effect and are shown in Fig. 1. The modified Becke–Johnson (mBJ) potential was adopted for band-gap correction, and the calculated indirect gap is ~0.8 eV, well consistent with the experimental results22,23. The conduction band maximum locates at the Z point, while the VBM lies along the Γ–M line. Besides, the VBM shows a large dispersion, indicating its light band feature, while multiple heavy valleys along the Γ–Z–R line exist below the VBM (within several kBT, kB is the Boltzmann constant, as shown in the shaded area). The combination of multiple light and heavy bands could be beneficial for the electrical properties, as will be discussed in more details below. The complex valence band structure is also validated by the DOS plot, showing multiple sharp peaks at the valence band top (within 1.5 eV below the VBM). The first DOS peak at ~−0.2 eV, as highlighted in the shaded area, dominates the hole transport, considering the optimal carrier concentration (<1022 cm−3) for most TE semiconductors. The sharp DOS peak is mainly composed of the hybridized orbitals from Cu and Se, well consistent with the conductive functionality of (Cu2Se2)2− layers (Supplementary Fig. 2). Figure 1c shows the first Brillouin zone and Fermi surfaces plotted at different EF corresponding to the hole concentrations p of 3.0 × 1020, 5.5 × 1020, and 1.2 × 1021 cm−3. Even for lightly doped samples with EF being just below the VBM, for example, p = 3.0 × 1020 cm−3, the Fermi surface consists of multiple valleys, including dual degeneracy at the Z point, quadruplet along the Γ–M line, and eightfold degeneracy along the Z–R line. With down-shifting the EF, these valleys contribute increasingly and simultaneously to the carrier conduction. The sharp DOS peak at ~−0.2 eV is primarily due to the incorporation of multiple heavy valleys with significant band nonparabolicity (along the Γ–Z–R line). The combination of light/heavy bands and high band degeneracy Nv ~14 has been widely accepted as the origin of the high S and modest PF of BiCuSeO, considering its low μΗ and thus low σ. A systematic investigation of the complex band structure, with the aim at fully utilizing the favorable band features, however, is missing. In the following part, EF is gradually moved down facilitated by Pb doping on the Bi site, and the electronic band structure, carrier transport properties, and their correlations are systematically explored.

Fig. 1
figure 1

Electronic structure and Fermi surfaces of BiCuSeO. a Band structure of the pristine BiCuSeO. b Total electronic density of states (DOS). The shaded areas show the valence bands that dominate the hole transport. c The first Brillouin zone and Fermi surfaces at different carrier concentrations (p = 3.0 × 1020, 5.5 × 1020, and 1.2 × 1021 cm−3)

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Exploring the complex band structure

Phase compositions of the bulk samples were characterized by X-ray diffraction (XRD) (Supplementary Fig. 3a). The diffraction patterns can be well indexed to tetragonal BiCuSeO (PDF# 45-0296, Supplementary Fig. 3b) for all Pb-doped samples11, and there is no detectable impurity phase for samples with the nominal Pb content up to 12 at%. For the x > 0.12 samples, three secondary peaks appear, which can be indexed to be cubic PbSe (PDF# 06-0354)24. Meanwhile, the (211) peak of Cu2Seδ (PDF# 47-1448)25 is also detected in the x > 0.12 samples, and our previous work has confirmed that this impurity phase is mainly formed during the rapid SHS processes11. These impurity phases, as will be discussed below, show a large influence on the TE transport for the x > 0.12 samples. The calculated lattice parameters of Bi1 − xPbxCuSeO (x = 0–0.2), shown in Supplementary Fig. 3c, indicate that the solubility limit of Pb in BiCuSeO is ~12 at%, consistent with the appearance of impurity phases for x > 0.12 samples in the XRD data.

Carrier concentration p and Hall mobility μΗ values of the Pb-doped BiCuSeO at room temperature are shown in Fig. 2a and listed in Table 1. Pb doping on the Bi site successfully increases p, in agreement with the previous studies23,26,27,28. Compared with the pristine BiCuSeO, p increases by three orders of magnitude and reaches ~1.9 × 1021 cm−3 for Bi0.88Pb0.12CuSeO. The experimental p values agree well with the theoretical predication considering Pb as an acceptor for x ≤ 0.12; however, p remains unchanged when the Pb content goes beyond the solubility limit. The results are in good agreement with the XRD and lattice parameter data, indicating excellent doping efficiency of Pb. The estimated EF plotted in Fig. 2b increases gradually from the band edge to ~ 8kBT (~0.21 eV at 300 K) below for the x = 0.12 sample, which is exactly at the top of the first DOS peak. For μΗ, the pristine sample shows a value of ~10 cm2 V−1 s−1 due to its low p. The μΗ values for Pb-doped samples do not vary significantly, and are mainly in the range of 3–5 cm2 V−1 s−1, even with a large variation in EF. The large decrease of μΗ with Pb doping is primarily due to the involvement of heavy band in the electrical conduction and partially to the increased ionized impurity scattering, as confirmed by the high-temperature Hall measurements (300–650 K, Fig. 2c, d). μΗ of the pristine sample approximately follows a T−1.5 relation, characteristic of acoustic phonon scattering (Fig. 2d). The temperature exponent of μΗ for the Pb-doped samples, however, evolves gradually from −1.5 to −0.5 with the increasing Pb content, primarily due to the increased ionized impurity scattering. The inter-valley scattering or increased carrier degeneracy with down-shifting the EF and increasing the conduction valleys may also partially contribute to this29. Moreover, as shown in Fig. 2c, except for the pristine sample displaying intrinsic conduction, the doped samples show nearly temperature-independent p, typical for heavily doped semiconductors. Due to the rapid rise of DOS and Nv, we expect an unusual variation of electrical transport properties with down-shifting the EF, which will be discussed in more detail below.

Fig. 2
figure 2

Electrical transport properties of Bi1 − xPbxCuSeO (x = 0–0.2). a Carrier concentration p and Hall mobility μΗ at room temperature. b Calculated Fermi energy (EF) based on experimental p and the calculated density of states (DOS). Temperature dependences of c p and d μΗ (300–650 K). e Power factor (PF) and f dimensionless figure of merit (ZT) as a function of the nominal Pb content. Three maxima are marked as shaded areas for PF and ZT. The uncertainties of the Seebeck coefficient and the electrical conductivity measurements are ~3% and ~5%, respectively, giving rise to a PF error of ~10%. Combined with ~10% uncertainty in the thermal conductivity measurement, the measurement error of ZT is estimated to be ~20%

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Table 1 Related physical properties of Bi1 − xPbxCuSeO (x = 0–0.2)
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The PFs and ZTs are plotted as functions of Pb content at different temperatures (300, 473, 673, and 873 K), shown in Fig. 2e, f. The temperature dependence of electrical properties is shown in Supplementary Fig. 4. The maximum PF reaches ~11 μW cm−1 K−2, comparable to the best values in this material10. In particular, three PF maxima can be clearly observed with the increasing Pb content, corresponding to those of 4, 10, and 14 at%. For typical TE semiconductors with single parabolic band (SPB), there is usually one maximum at the optimal doping content and thus EF, due to the interplay between σ and S30,31. The anomalous three PF maxima should be related to the complex band structure or other extrinsic factors. Consequently, three ZT maxima can be observed for all temperatures, and at 873 K these values are ~0.9 (x = 0.04), ~1.1 (x = 0.10), and ~1.3 (x = 0.14). Clearly, the first two peaks at x = 0.04 and 0.10, within the Pb-doping limit, should be originated from the optimization of p and/or the complex electronic band structure, such as the DOS peak at ~−0.2 eV and high Nv. The third peak, as will be demonstrated below, is ascribed to the compositing effect of PbSe in Bi0.88Pb0.12CuSeO.

To clarify the origin of the multiple PF peaks, the electrical transport properties of Bi1 − xPbxCuSeO were modeled based on the Boltzmann transport theory under the relaxation-time approximation. Deformation potential coefficient, which represents the change in energy of the electronic band with elastic deformation and thus the coupling between electrons and phonons32, is an intrinsic material parameter and was set as ~25eV (normally ranges from 5 to 35eV for semiconductors33). The modeling details can be found elsewhere34, and the related parameters are listed in Supplementary Table 1. The effective masses (md*) used for modeling were derived from the DFT calculated band structure and DOS, using the equation \(\phi (\varepsilon ) = \frac{{4{\it{\pi }}(2m_d^ \ast )^{3/2}}}{{h^3}}\varepsilon ^{1/2}\), where ε is the carrier energy, \(\phi (\varepsilon )\) the DOS per unit volume, and h the Planck constant. Rigid band model was assumed while deriving the EF-dependent md*. The calculated results are plotted as blue and red lines in Fig. 3c for the pristine and 12.5 at% Pb-doped samples, respectively. The band structure and DOS of Bi0.875Pb0.125CuSeO, shown in Fig. 3a, were calculated for deriving the red line in Fig. 3c. We observe much higher m* ∝ p data, and larger DOS near the VBM (Fig. 3b) for Pb-doped samples as compared with those of the pristine BiCuSeO, where m* is the DOS effective mass. As a result, the calculated data (red line in Fig. 3c) for Bi0.875Pb0.125CuSeO are much higher than those of the pristine BiCuSeO (blue line), well consistent with the experimental data of Pb-doped samples23,26,27,28. Here the experimental m* values are calculated based on the experimental S and p, with the assumption of acoustic phonon scattering and SPB model. The details can be found in Supplementary Note 2 or elsewhere23. The estimated experimental m*, shown in Table 1, increases rapidly from ~0.8m0 (m0 is the free electron mass) for the x = 0 sample to ~7.0m0 for x = 0.14, indicating a significant band nonparabolicity. The band nonparabolicity of BiCuSeO has been widely reported, which is typical for narrow gap semiconductors35. Considering the minor contribution from Pb to the DOS of Bi0.875Pb0.125CuSeO at the top of valence band, shown in Supplementary Fig. 5, the origin of large m* for the Pb-doped samples is unclear and possibly related to the subtle change of chemical environments in the insulating (Bi2O2)2+ layer and subsequently the conductive (Cu2Se2)2− layer. In addition, the calculated m* values of various alkaline or alkaline earth-doped samples (on the Bi site)8,9,23,26,27,28,36,37,38,39,40,41, shown as the solid symbols in Fig. 3c, are apparently lower than those of Pb-doped samples, especially at high doping levels. m* values of non-Pb-doped samples are approximately consistent with the calculated line of the pristine BiCuSeO. In light of the unique role of Pb doping, the md* values of the Pb-doped samples were used for the following modeling.

Fig. 3
figure 3

Electronic structure and calculated electrical transport properties of Bi1 − xPbxCuSeO. a Electronic band structure and total density of states (DOS) of Bi0.875Pb0.125CuSeO. b Comparison of DOS at valence band top (−0.3 to 0 eV) between BiCuSeO and Bi0.875Pb0.125CuSeO. Carrier concentration dependences of c the effective mass, d Seebeck coefficient, e electrical conductivity, and f power factor for Bi1 − xPbxCuSeO. The blue and red lines in c are estimated from the DOS of the pristine and Pb-doped BiCuSeO, respectively. The red lines in d–f are calculated based on the Boltzmann transport theory and calculated effective masses md* (the red line in c) of Bi0.875Pb0.125CuSeO by assuming acoustic phonon scattering (300 K). The blue line in d is the estimated band degeneracy, and the gray line is estimated by assuming the SPB model and a constant effective mass of 2m0. The error bar of PF in f is 10%

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As shown in Fig. 3d, e, the modeled S and σ values at 300 K agree well with the experimental data, indicating the validity of our model. Seebeck coefficient estimated from the SPB model (m* = 2m0), shown as the gray line in Fig. 3d, is obviously lower than the experimental data. More importantly, our modeling well reproduces the two peaks in PF, and clarifies that the two PF maxima can be ascribed to the rapid rise in DOS due to the convergence of multiple bands and the conventional optimization of EF. With down-shifting the EF into the valence band, the involvement of multiple heavy bands, that is, the increasing Nv (Fig. 3d) in the electrical conduction largely increases DOS and thus S without sacrificing μΗ, shown in Figs 3b and 2a, resulting in the first PF maximum. Further shift of EF increases p and thus σ, while maintaining a large S due to the large band degeneracy, which gives rise to the second maximum, shown in Fig. 3f. According to our experimental data, band structure calculations, and numerical modeling, to fully utilize these beneficial band features in BiCuSeO, including the mixing of light and heavy bands and high band degeneracy, one has to down-shift the EF and touch the first DOS peak (p > 1021 cm−3). Thereby, high PF values (~10 μW cm−1 K−2) can be achieved without using other approaches such as texturing, modulation doping, chemical bonding engineering, and so on, and Pb doping on the Bi site does play a unique role in adjusting the EF and fully utilizing the electronic band structure.

Compositing effects of PbSe for the x > 0.12 samples

The first two maxima in PF and ZT can be well understood as originating from the intrinsically complex band structure; however, the appearance of third peak at x ~ 0.14 is intriguing. The x > 0.12 samples can be viewed as composites with appreciable amounts of PbSe and trace Cu2Seδ embedded in the Pb-doped BiCuSeO matrix with unchanged p and EF values, and generally the compositing could trigger unexpected effects on electron and phonon transport, such as carrier-filtering and phonon-blocking42. By scrutinizing the transport data, we can find that further increase in PF for the x > 0.12 samples predominantly results from increases in μΗ and thus σ. This μΗ increase is primarily attributed to the appearance of PbSe precipitates with high hole mobility (~1000 cm2 V−1 s−1 for lightly doped p-type PbSe at 300 K)43,44, rather than Cu2Seδ nanodots with small volume fraction and low mobility (~11.1 cm2 V−1 s−1 for the SHS-SPSed Cu2Se at 300 K)22.

To confirm our speculation, the distributions of PbSe and Cu2Seδ were investigated by a scanning electron microscope (SEM), an electron probe micro-analyzer (EPMA), and a transmission electron microscope (TEM), as shown in Fig. 4 and Supplementary Fig. 6. There is no detectable impurity phase in SEM for the x ≤ 0.12 samples (Supplementary Fig. 7a), in agreement with the XRD and Hall data. Besides, the average grain size is largely decreased for the x > 0.12 samples (Supplementary Fig. 7b), probably due to the “pinning” effect introduced by the nano-precipitates and other secondary phases45,46. Uniformly distributed Cu2Seδ nanodots with sizes of 5–10 nm (confirmed by the inter-planar spacing of 3.309 Å, corresponding to the (211) plane, PDF #47-1448)47 can be easily observed in the SHS-SPSed samples, shown in Fig. 4b, c, which is well consistent with our previous work11. The Cu2Seδ nanodots are primarily generated during the rapid and non-equilibrium SHS processes, and their amount and size both increase significantly for the x > 0.12 samples due to the non-stoichiometry, making them detectable in XRD (Supplementary Fig. 3a). In addition, PbSe precipitates with micrometer size can be readily observed in the elemental mapping (EPMA for well-polished samples), shown in Fig. 4a. Our quantitative EPMA analysis (averaged among 10 arbitrarily selected points) confirms that in Bi0.86Pb0.14CuSeO there exist a large amount of p-type PbSe precipitates embedded in the Bi1 – xPbxCuSeO (x ~ 0.12) matrix, consistent with the XRD and SEM/EDS results (Supplementary Fig. 6). In the Bi1 − xPbxCuSeO (x > 0.12) samples, in situ formed PbSe could noticeably alter the transport properties through the compositing effect. Without detailed transport properties of the PbSe precipitates, it is hard to use the Bergman composite theory to quantitatively calculate the transport properties of the composites. Additional experimental efforts need to be carried out to fully corroborate the compositing effect, which are beyond the scope of the present study.

Fig. 4
figure 4

Microstructural and compositional characterizations of Bi0.86Pb0.14CuSeO. a Elemental mapping of Pb, Bi, Se, O, and Cu. The results clearly show the PbSe precipitates (white) embedded in Bi1 – xPbxCuSeO matrix (gray). b Transmission electron microscope (TEM) image showing the homogeneously distributed nanodots. c High-resolution TEM (HRTEM) image confirming that the nanodots are Cu2Seδ

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Thermal transport properties

In addition to the complex band structure, BiCuSeO is also well known for its glass-like thermal conduction. Figure 5a shows the temperature-dependent lattice thermal conductivity. κL values for all samples decrease with the increasing temperature, approximately following a T−1 relation and thus typical of the Umklapp scattering of phonons. Moreover, Pb doping does significantly reduce κL in the whole temperature range, shown in Fig. 5b. Using the Debye–Callaway model48,49, we estimated the effect of Pb doping on the κL at 300 and 873 K by assuming the successful replacement of Pb on the Bi site, shown as the solid lines in Fig. 5b. The calculation details can be found in Supplementary Note 3 or elsewhere23,50. Here we did not consider the effect of oxygen vacancies that presumably exist in all samples with a negligible difference51, nor the influence of Cu2Seδ nanodots. This assumption is valid for the x ≤ 0.12 samples, validated by the good agreement between the model and experimental data. The model overestimates κL for x > 0.12, primarily due to the increased amounts of secondary phases, for example, Cu2Seδ nanodots, as compared with the pristine sample. Considering the small mass difference between Bi and Pb, the strong point defect scattering for phonons is mainly originated from the strain field fluctuation (ΓSF, considering the ionic radius for Bi–rBi3+ = 1.17 Å and Pb–rPb2+ = 1.29 Å), as indicated by the dominated role of ΓSF in disorder scattering parameter Γ (Supplementary Fig. 8). In addition to the strong point defect scattering for short-wavelength phonons, mesoscale grain boundaries, and nanostructures further strongly scatter the long- and middle-wavelength phonons, creating a hierarchically structural feature for scattering a wide spectrum of lattice phonons. Thereby, as shown in Fig. 5a, κL is well below the estimated minimum value (κmin ~0.59 W m−1 K−1, Cahill’s glassy limit)23, especially at high temperatures, indicating a substantial phonon scattering in the SHS-SPSed Bi1 − xPbxCuSeO. The lowest κL of ~0.13 W m−1 K−1 at 873 K can be achieved for Bi0.8Pb0.2CuSeO, which, to our best knowledge, is the lowest reported value in BiCuSeO. The ultralow κL of Bi1 − xPbxCuSeO, much lower than the estimated κmin, is presumably due to the decreased sound velocity upon doping, especially at elevated temperatures.

Fig. 5
figure 5

Thermal transport properties of Bi1 − xPbxCuSeO. a Temperature-dependent lattice thermal conductivity κL. b Pb-content-dependent κL at 300, 473, 673, and 873 K. The solid lines in b were estimated by the Debye–Callaway model for 300 and 873 K. The error bars in a, b are both 20%, as the uncertainties of the total and electronic thermal conductivity are both estimated to be ~10%

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Dimensionless figure of merit

Due to the complex band structure and high band convergence of BiCuSeO, full exploration of the favorable band features leads to high TE figure of merits, in combination with the ultralow κL originated from the hierarchical structural features. Figure 6a shows the temperature dependences of ZT for Bi1 − xPbxCuSeO, and three ZT maxima of 0.9, 1.1, and 1.3 at 873 K can be found in the x = 0.04, 0.10, and 0.14 samples, respectively. Based on first-principles calculations, numerical modeling, and experimental data, these maxima can be attributed to the convergence of multiple valence bands, the conventional optimization of the EF, as well as the effects of compositing with p-type PbSe. The anomalous multiple maxima in TE performance with shifting EF, in sharp contrast to most TE semiconductors with a single maximum, may provide a new perspective to explore the doping effects for other TE materials with complex band structures, such as chalcogenides, skutterudites, and so on, in addition to the simple doping efficiency difference based on atom size, mass, and valence shell52. Furthermore, excessive doping and the introduction of in situ formed precipitates could be a viable strategy to produce nanocomposites with compositing effects or intensified phonon scattering.

Fig. 6
figure 6

The dimensionless figure of merit ZT of Bi1 − xPbxCuSeO (x = 0–0.2). a Temperature dependence of ZT. b Comparison of the average ZTs between Bi0.86Pb0.14CuSeO and other materials reported in the literature13,14,42,53,54,55,56. The error bars in a, b are both 20%

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In addition to the high peak ZTs, the average ZT values have also been calculated as 0.7 for 300–873 K and 0.9 for 500–873 K in Bi0.86Pb0.14CuSeO. As shown in Fig. 6b, these values are comparable to those of the best mid-temperature p-type TE materials13,14,43,53,54,55,56, such as CoSb3 and PbTe. These high ZTs, combined with the good thermal and chemical stabilities (shown in Supplementary Fig. 9), as well as the time- and cost-efficient SHS-SPS technique, make Pb-doped BiCuSeO a promising material for intermediate temperature power generation applications.

Discussion

In this study, the electronic band structure and TE transport properties of BiCuSeO were systematically investigated to fully utilize the favorable band features. Pb-doped BiCuSeO samples (0–20 at%) were synthesized by an SHS-SPS technique. When shifting the Fermi energy into the valence band, convergence of multiple valence bands largely increases the density of states effective mass without scarifying the carrier mobility, and thus results in the first PF maximum at a low hole concentration of ~5.5 × 1020 cm−3. Further moving the Fermi energy deep into the valence band gives rise to the second peak in PF through balancing the electrical conductivity and Seebeck coefficient. In addition, in situ formed PbSe precipitates when the Pb content exceeds the solubility limit (x > 0.12) lead to the third PF peak due to the compositing effect. Consequently, three ZT peaks of 0.9, 1.1, and 1.3 at 873 K are observed for 4, 10, and 14 at% Pb-doped BiCuSeO samples, respectively, indicating the complex electronic structure and multiple roles of Pb in BiCuSeO. Meanwhile, the high average ZTs of 0.7 for 300–873 K and 0.9 for 500–873 K, making these materials promising for large-scale mid-temperature power generation applications. Specifically, our study demonstrates that, for material systems with complex band structures (nonparabolicity, multiple bands with varying band mass, etc.), rational tuning of the Fermi energy and dopant content within or exceeding the solubility limit may provide new opportunities to optimize the TE performance.

Methods

Material synthesis

Polycrystalline Bi1 − xPbxCuSeO (x = 0, 0.02, …, 0.2) samples were prepared by a self-propagating high-temperature synthesis and SHS-SPS technique. Stoichiometric amounts of Bi (99.99%, Aladdin), Bi2O3 (99.99%, Aladdin), PbO (99.9%, Aladdin), Cu (99.99%, Aladdin), and Se (99.99%, Aladdin) were mixed by hand grinding. The mixed powders were cold pressed into pellets and underwent the SHS processes. The details of the SHS method can be found elsewhere23. Given that a significant volatilization of Se may exist during the SHS processes, additional 5 at% Se was added. The obtained SHSed powders were then compacted into dense pellets with a diameter of ~12.5 mm and a height of ~10 mm by SPS at 923 K under a uniaxial pressure of 50 MPa for 5 min.

Structural characterization

Phase purity and crystal structure were investigated by powder XRD (Bruker D8 Advance, Germany). Chemical uniformities and compositions of the samples were characterized by an EPMA (JXA-8230, JEOL, Japan). The chemical compositions were averaged from 10 arbitrarily selected points. The microstructures of the samples with different compositions were examined by a field-emission SEM (MERLIN Compact FE-SEM, Carl Zeiss, Germany) and a high-resolution transmission electron microscope (JEOL2010, Japan).

Transport property measurement

The sintered pellets were cut into 10 × 10 × 1.5 mm3 thin square sheets and 3 × 3 × 12 mm3 bar-shaped specimens, and all transport properties were measured perpendicular to the SPS pressure direction. The rectangular bars were used for the simultaneous measurements of S and σ from room temperature to 873 K using a ZEM-3 (ULVAC, Japan). Hall coefficient measurements (300–650 K) were performed on a homemade system equipped with a 2 T electromagnet, and the magnetic field is determined by a flux meter. A four-probe configuration and pressure contact were used for the measurements. The carrier concentration (p) and Hall mobility (μΗ) were estimated from the measured Hall coefficient (RH) and electrical conductivity by the relations p = r/e|RH| (assuming the Hall factor r = 1.0) and μΗ = σ|RH|. Thermal diffusivity (D) was measured by a laser flash method (LFA-457, Netzsch, Germany) under a continuous Ar flow, shown in Supplementary Fig. 1a. The total thermal conductivity (κ) was calculated by using κ = DCpρ, where ρ is the mass density measured by the Archimedes method (Table 1) and Cp specific heat calculated by the Debye model57,58,59 (similar to the Dulong–Petit law at high temperatures, Supplementary Fig. 1b and Note 1). The uncertainties of the Seebeck coefficient and the electrical conductivity measurements are estimated to be ~3% and ~5%, respectively. The uncertainty in the thermal conductivity measurement is estimated within 10%. As a result, the uncertainty of the figure of merit ZT is accumulated to be ~20%.

Computational technique

The electronic structures were calculated using the projector augmented wave method, as implemented in the Vienna ab initio Simulation Package60,61. The mBJ exchange potential62 and the generalized gradient approximation63 were used in the calculations. To mimic the Pb doping, we constructed a 2 × 2 × 1 supercell of the primitive cell. Both supercells with or without the Pb dopants were calculated. The lattice parameters and ionic positions were fully relaxed. SOC was considered in the band structure calculations.